Subgroups — C₃₁×#3519 #16843452

Order 3968.

1252 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#553	C₂	1	2	yes	maximal, complemented
4	C₃₁×#553	C₂	1	2	yes	maximal, complemented
5	C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#546	C₂	1	4	yes	maximal, complemented
7	C₃₁×#546	C₂	1	2	yes	maximal, complemented
8	D₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄²×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₃₁×#540	C₂	1	2	yes	maximal, complemented
11	C₃₁×#538	C₂	1	4	yes	maximal, complemented
12	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#525	C₂	1	2	yes	maximal, complemented
15	C₂×C₃₁×#121	C₂	1	2	yes	maximal, complemented
16	C₃₁×#518	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
19	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
22	C₂²×D₄×C₃₁	C₂²	1	2	yes	complemented
23	C₂²×D₄×C₃₁	C₂²	1	4	yes	complemented
24	C₂³×C₄×C₃₁	C₂²	1	2	yes	complemented
25	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
26	C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#127	C₂²	1	2	yes	complemented
28	C₃₁×#126	C₂²	1	2	yes	complemented
29	C₃₁×#126	C₂²	1	4	yes	complemented
30	C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#125	C₂²	1	2	yes	complemented
32	C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#124	C₂²	1	2	yes	complemented
34	C₃₁×#124	C₂²	1	4	yes	complemented
35	C₃₁×#123	C₂²	1	4	yes	complemented
36	C₃₁×#123	C₂²	1	2	yes	complemented
37	C₃₁×#123	C₂²	1	4	yes	complemented
38	C₃₁×#123	C₂²	1	4	yes	complemented
39	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
41	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
42	C₃₁×#121	C₂²	1	1	yes	characteristic
43	C₃₁×#121	C₂²	1	2	yes	complemented
44	C₃₁×#121	C₂²	1	4	yes	complemented
45	C₃₁×#121	C₂²	1	4	yes	complemented
46	C₃₁×#121	C₂²	1	2	yes	complemented
47	C₃₁×#121	C₂²	1	2	yes	complemented
48	C₃₁×#121	C₂²	1	2	yes	complemented
49	C₃₁×#121	C₂²	1	2	yes	complemented
50	C₃₁×#121	C₂²	1	4	yes	complemented
51	C₃₁×#121	C₂²	1	4	yes	complemented
52	C₃₁×#121	C₂²	1	2	yes	complemented
53	C₃₁×#121	C₂²	1	2	yes	complemented
54	C₃₁×#121	C₂²	1	4	yes	complemented
55	C₃₁×#120	C₂²	1	4	yes	complemented
56	C₃₁×#120	C₂²	1	4	yes	complemented
57	C₃₁×#120	C₂²	1	4	yes	complemented
58	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
59	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
60	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
61	C₄×D₄×C₃₁	C₂²	1	2	yes	—
62	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
63	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
64	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
65	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
66	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
67	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
68	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
69	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
70	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
71	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
72	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
73	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
74	C₃₁×#117	C₂²	1	2	yes	complemented
75	C₃₁×#117	C₂²	1	2	yes	complemented
76	C₃₁×#117	C₂²	1	2	yes	complemented
77	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
78	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
79	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
80	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
81	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
82	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
83	C₂⁴×C₃₁	C₂³	1	2	yes	—
84	C₂⁴×C₃₁	C₂³	1	4	yes	—
85	#41×C₃₁	—	4	4	no	—
86	#41×C₃₁	—	4	4	no	—
87	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
89	C₂×D₄×C₃₁	—	2	4	no	—
90	C₂×D₄×C₃₁	—	2	8	no	—
91	C₂×D₄×C₃₁	—	4	4	no	—
92	C₂×D₄×C₃₁	—	4	8	no	—
93	C₂×D₄×C₃₁	C₂³	1	2	yes	—
94	C₂×D₄×C₃₁	C₂³	1	4	yes	—
95	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₃₁	C₂³	1	2	yes	—
97	C₂×D₄×C₃₁	C₂³	1	4	yes	—
98	C₂×D₄×C₃₁	C₂³	1	2	yes	—
99	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₃₁	C₂³	1	2	yes	—
101	C₂×D₄×C₃₁	C₂³	1	4	yes	—
102	C₂×D₄×C₃₁	C₂³	1	2	yes	—
103	C₂×D₄×C₃₁	C₂³	1	4	yes	—
104	C₂×D₄×C₃₁	C₂³	1	4	yes	—
105	C₂×D₄×C₃₁	C₂³	1	4	yes	—
106	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	—	4	4	no	—
108	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
116	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
118	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
119	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
120	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
121	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
122	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
123	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
124	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
125	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
126	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
132	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
134	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
135	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
137	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
138	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
139	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
140	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
141	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
142	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
143	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
144	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
145	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
146	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
147	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
148	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
149	C₄²×C₃₁	C₂³	1	1	yes	characteristic
150	C₄²×C₃₁	C₂³	1	1	yes	characteristic
151	C₄²×C₃₁	C₂³	1	1	yes	characteristic
152	C₄²×C₃₁	C₂³	1	1	yes	characteristic
153	C₄²×C₃₁	C₂³	1	4	yes	—
154	C₄²×C₃₁	C₂³	1	2	yes	—
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	—	2	4	no	—
157	C₂³×C₃₁	—	2	8	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	4	no	—
160	C₂³×C₃₁	—	4	4	no	—
161	C₂³×C₃₁	C₂⁴	1	2	yes	—
162	C₂³×C₃₁	C₂⁴	1	2	yes	—
163	C₂³×C₃₁	C₂⁴	1	4	yes	—
164	C₂³×C₃₁	C₂⁴	1	2	yes	—
165	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
166	Q₈×C₃₁	—	2	2	no	—
167	Q₈×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	2	4	no	—
169	D₄×C₃₁	—	2	8	no	—
170	D₄×C₃₁	—	2	4	no	—
171	D₄×C₃₁	—	2	8	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	2	4	no	—
174	D₄×C₃₁	—	2	2	no	—
175	D₄×C₃₁	—	2	4	no	—
176	D₄×C₃₁	—	4	4	no	—
177	D₄×C₃₁	—	4	4	no	—
178	D₄×C₃₁	—	4	4	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	4	4	no	—
181	D₄×C₃₁	#41	1	4	yes	complemented
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	4	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	4	no	—
187	C₂×C₄×C₃₁	—	2	4	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	4	no	—
192	C₂×C₄×C₃₁	—	4	4	no	—
193	C₂×C₄×C₃₁	—	4	4	no	—
194	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
196	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
199	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
201	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
202	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
203	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
204	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
206	#3519	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂²×C₃₁	—	2	1	no	—
208	C₂²×C₃₁	—	2	1	no	—
209	C₂²×C₃₁	—	2	2	no	—
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	2	2	no	—
213	C₂²×C₃₁	—	2	4	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	4	no	—
216	C₂²×C₃₁	—	2	4	no	—
217	C₂²×C₃₁	—	2	2	no	—
218	C₂²×C₃₁	—	2	2	no	—
219	C₂²×C₃₁	—	2	2	no	—
220	C₂²×C₃₁	—	4	8	no	—
221	C₂²×C₃₁	—	8	8	no	—
222	C₂²×C₃₁	C₂×#41	1	4	yes	—
223	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
224	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
225	C₄×C₃₁	—	2	1	no	—
226	C₄×C₃₁	—	2	2	no	—
227	C₄×C₃₁	—	2	1	no	—
228	C₄×C₃₁	—	2	1	no	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	2	1	no	—
232	C₄×C₃₁	—	2	2	no	—
233	C₄×C₃₁	—	2	4	no	—
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	C₂×#41	1	2	yes	—
236	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
237	#555	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#554	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#553	C₂×C₃₁	1	2	yes	complemented
240	#553	C₂×C₃₁	1	2	yes	complemented
241	#550	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#546	C₂×C₃₁	1	4	yes	complemented
243	#546	C₂×C₃₁	1	2	yes	complemented
244	D₄²	C₂×C₃₁	1	1	yes	characteristic, complemented
245	D₄²	C₂×C₃₁	1	2	yes	complemented
246	#540	C₂×C₃₁	1	2	yes	complemented
247	#538	C₂×C₃₁	1	4	yes	complemented
248	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#525	C₂×C₃₁	1	2	yes	complemented
251	C₂×#121	C₂×C₃₁	1	2	yes	complemented
252	#518	C₂×C₃₁	1	2	yes	complemented
253	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂×C₃₁	—	2	2	no	—
255	C₂×C₃₁	—	2	4	no	—
256	C₂×C₃₁	—	4	2	no	—
257	C₂×C₃₁	—	4	4	no	—
258	C₂×C₃₁	—	4	2	no	—
259	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
260	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
261	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
262	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
263	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
264	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
265	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
266	C₂²×D₄	C₂²×C₃₁	1	2	yes	complemented
267	C₂²×D₄	C₂²×C₃₁	1	4	yes	complemented
268	C₂³×C₄	C₂²×C₃₁	1	2	yes	complemented
269	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
270	#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
271	#127	C₂²×C₃₁	1	2	yes	complemented
272	#126	C₂²×C₃₁	1	2	yes	complemented
273	#126	C₂²×C₃₁	1	4	yes	complemented
274	#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
275	#125	C₂²×C₃₁	1	2	yes	complemented
276	#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
277	#124	C₂²×C₃₁	1	2	yes	complemented
278	#124	C₂²×C₃₁	1	4	yes	complemented
279	#123	C₂²×C₃₁	1	4	yes	complemented
280	#123	C₂²×C₃₁	1	2	yes	complemented
281	#123	C₂²×C₃₁	1	4	yes	complemented
282	#123	C₂²×C₃₁	1	4	yes	complemented
283	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
284	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
285	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
286	#121	C₂²×C₃₁	1	1	yes	characteristic
287	#121	C₂²×C₃₁	1	2	yes	complemented
288	#121	C₂²×C₃₁	1	4	yes	complemented
289	#121	C₂²×C₃₁	1	4	yes	complemented
290	#121	C₂²×C₃₁	1	2	yes	complemented
291	#121	C₂²×C₃₁	1	2	yes	complemented
292	#121	C₂²×C₃₁	1	2	yes	complemented
293	#121	C₂²×C₃₁	1	2	yes	complemented
294	#121	C₂²×C₃₁	1	4	yes	complemented
295	#121	C₂²×C₃₁	1	4	yes	complemented
296	#121	C₂²×C₃₁	1	2	yes	complemented
297	#121	C₂²×C₃₁	1	2	yes	complemented
298	#121	C₂²×C₃₁	1	4	yes	complemented
299	#120	C₂²×C₃₁	1	4	yes	complemented
300	#120	C₂²×C₃₁	1	4	yes	complemented
301	#120	C₂²×C₃₁	1	4	yes	complemented
302	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
303	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
304	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
305	C₄×D₄	C₂²×C₃₁	1	2	yes	—
306	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
307	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
309	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
310	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
311	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
312	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
313	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
314	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
315	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
316	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
317	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
318	#117	C₂²×C₃₁	1	2	yes	complemented
319	#117	C₂²×C₃₁	1	2	yes	complemented
320	#117	C₂²×C₃₁	1	2	yes	complemented
321	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
322	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
323	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
324	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
325	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
326	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
327	C₃₁	#3519	1	1	yes	characteristic, direct product factor, Sylow
328	C₂⁴	C₂³×C₃₁	1	2	yes	—
329	C₂⁴	C₂³×C₃₁	1	4	yes	—
330	#41	—	4	4	no	—
331	#41	—	4	4	no	—
332	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	8	no	—
336	C₂×D₄	—	4	4	no	—
337	C₂×D₄	—	4	8	no	—
338	C₂×D₄	C₂³×C₃₁	1	2	yes	—
339	C₂×D₄	C₂³×C₃₁	1	4	yes	—
340	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×D₄	C₂³×C₃₁	1	2	yes	—
342	C₂×D₄	C₂³×C₃₁	1	4	yes	—
343	C₂×D₄	C₂³×C₃₁	1	2	yes	—
344	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₂×D₄	C₂³×C₃₁	1	2	yes	—
346	C₂×D₄	C₂³×C₃₁	1	4	yes	—
347	C₂×D₄	C₂³×C₃₁	1	2	yes	—
348	C₂×D₄	C₂³×C₃₁	1	4	yes	—
349	C₂×D₄	C₂³×C₃₁	1	4	yes	—
350	C₂×D₄	C₂³×C₃₁	1	4	yes	—
351	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	—	4	4	no	—
353	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
357	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
358	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
359	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
360	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
361	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
362	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
363	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
364	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
365	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
366	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
368	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
370	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
371	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
372	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
374	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
375	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
376	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
377	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
378	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
379	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
380	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
381	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
382	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
383	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
384	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
385	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
386	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
387	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
388	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
389	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
390	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
391	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
392	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
393	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
394	C₄²	C₂³×C₃₁	1	1	yes	characteristic
395	C₄²	C₂³×C₃₁	1	1	yes	characteristic
396	C₄²	C₂³×C₃₁	1	1	yes	characteristic
397	C₄²	C₂³×C₃₁	1	1	yes	characteristic
398	C₄²	C₂³×C₃₁	1	4	yes	—
399	C₄²	C₂³×C₃₁	1	2	yes	—
400	C₂³	—	2	4	no	—
401	C₂³	—	2	4	no	—
402	C₂³	—	2	8	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	4	4	no	—
406	C₂³	C₂⁴×C₃₁	1	2	yes	—
407	C₂³	C₂⁴×C₃₁	1	2	yes	—
408	C₂³	C₂⁴×C₃₁	1	4	yes	—
409	C₂³	C₂⁴×C₃₁	1	2	yes	—
410	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
411	Q₈	—	2	2	no	—
412	Q₈	—	4	2	no	—
413	D₄	—	2	4	no	—
414	D₄	—	2	8	no	—
415	D₄	—	2	4	no	—
416	D₄	—	2	8	no	—
417	D₄	—	2	2	no	—
418	D₄	—	2	4	no	—
419	D₄	—	2	2	no	—
420	D₄	—	2	4	no	—
421	D₄	—	4	4	no	—
422	D₄	—	4	4	no	—
423	D₄	—	4	4	no	—
424	D₄	—	4	2	no	—
425	D₄	—	4	4	no	—
426	D₄	#41×C₃₁	1	4	yes	complemented
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	4	no	—
437	C₂×C₄	—	4	4	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
441	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
442	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
444	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
445	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
446	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
447	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
448	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
449	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
450	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
451	C₂²	—	2	1	no	—
452	C₂²	—	2	1	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	2	2	no	—
455	C₂²	—	2	2	no	—
456	C₂²	—	2	2	no	—
457	C₂²	—	2	4	no	—
458	C₂²	—	2	2	no	—
459	C₂²	—	2	4	no	—
460	C₂²	—	2	4	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	2	no	—
464	C₂²	—	4	8	no	—
465	C₂²	—	8	8	no	—
466	C₂²	C₂×#41×C₃₁	1	4	yes	—
467	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
468	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
469	C₄	—	2	1	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	1	no	—
472	C₄	—	2	1	no	—
473	C₄	—	2	2	no	—
474	C₄	—	2	2	no	—
475	C₄	—	2	1	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	4	no	—
478	C₄	—	2	2	no	—
479	C₄	C₂×#41×C₃₁	1	2	yes	—
480	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
481	C₂	—	2	2	no	—
482	C₂	—	2	4	no	—
483	C₂	—	4	2	no	—
484	C₂	—	4	2	no	—
485	C₂	—	4	4	no	—
486	C₂	C₂²×#41×C₃₁	1	1	yes	characteristic
487	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
488	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#555	C₂	semidirect
2. C₃₁×#554	C₂	semidirect
3. C₃₁×#553	C₂	semidirect
4. C₃₁×#553	C₂	semidirect
5. C₃₁×#550	C₂	semidirect
6. C₃₁×#546	C₂	semidirect
7. C₃₁×#546	C₂	semidirect
8. D₄²×C₃₁	C₂	semidirect
9. D₄²×C₃₁	C₂	semidirect
10. C₃₁×#540	C₂	semidirect
11. C₃₁×#538	C₂	semidirect
12. C₃₁×#532	C₂	semidirect
13. C₃₁×#529	C₂	semidirect
14. C₃₁×#525	C₂	semidirect
15. C₂×C₃₁×#121	C₂	semidirect
16. C₃₁×#518	C₂	semidirect
17. C₂×C₄×D₄×C₃₁	C₂	semidirect
18. C₂×#41×C₃₁	C₂²	semidirect
19. C₂×#41×C₃₁	C₂²	semidirect
22. C₂²×D₄×C₃₁	C₂²	semidirect
23. C₂²×D₄×C₃₁	C₂²	semidirect
24. C₂³×C₄×C₃₁	C₂²	semidirect
25. C₃₁×C₄⋊Q₈	C₂²	semidirect
26. C₃₁×#127	C₂²	semidirect
27. C₃₁×#127	C₂²	semidirect
28. C₃₁×#126	C₂²	semidirect
29. C₃₁×#126	C₂²	semidirect
30. C₃₁×#125	C₂²	semidirect
31. C₃₁×#125	C₂²	semidirect
32. C₃₁×#124	C₂²	semidirect
33. C₃₁×#124	C₂²	semidirect
34. C₃₁×#124	C₂²	semidirect
35. C₃₁×#123	C₂²	semidirect
36. C₃₁×#123	C₂²	semidirect
37. C₃₁×#123	C₂²	semidirect
38. C₃₁×#123	C₂²	semidirect
39. C₃₁×C₂²⋊Q₈	C₂²	semidirect
40. C₃₁×C₂²⋊Q₈	C₂²	semidirect
41. C₃₁×C₂²⋊Q₈	C₂²	semidirect
43. C₃₁×#121	C₂²	semidirect
44. C₃₁×#121	C₂²	semidirect
45. C₃₁×#121	C₂²	semidirect
46. C₃₁×#121	C₂²	semidirect
47. C₃₁×#121	C₂²	semidirect
48. C₃₁×#121	C₂²	semidirect
49. C₃₁×#121	C₂²	semidirect
50. C₃₁×#121	C₂²	semidirect
51. C₃₁×#121	C₂²	semidirect
52. C₃₁×#121	C₂²	semidirect
53. C₃₁×#121	C₂²	semidirect
54. C₃₁×#121	C₂²	semidirect
55. C₃₁×#120	C₂²	semidirect
56. C₃₁×#120	C₂²	semidirect
57. C₃₁×#120	C₂²	semidirect
58. C₄×Q₈×C₃₁	C₂²	semidirect
62. C₄×D₄×C₃₁	C₂²	semidirect
63. C₄×D₄×C₃₁	C₂²	semidirect
64. C₄×D₄×C₃₁	C₂²	semidirect
65. C₄×D₄×C₃₁	C₂²	semidirect
66. C₄×D₄×C₃₁	C₂²	semidirect
67. C₄×D₄×C₃₁	C₂²	semidirect
68. C₄×D₄×C₃₁	C₂²	semidirect
69. C₄×D₄×C₃₁	C₂²	semidirect
70. C₄×D₄×C₃₁	C₂²	semidirect
71. C₄×D₄×C₃₁	C₂²	semidirect
72. C₄×D₄×C₃₁	C₂²	semidirect
73. C₄×D₄×C₃₁	C₂²	semidirect
74. C₃₁×#117	C₂²	semidirect
75. C₃₁×#117	C₂²	semidirect
76. C₃₁×#117	C₂²	semidirect
77. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
78. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
79. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
80. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
81. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
82. C₂×C₄²×C₃₁	C₂²	semidirect
181. D₄×C₃₁	#41	semidirect
206. #3519	C₃₁	direct (N × H)
237. #555	C₂×C₃₁	semidirect
238. #554	C₂×C₃₁	semidirect
239. #553	C₂×C₃₁	semidirect
240. #553	C₂×C₃₁	semidirect
241. #550	C₂×C₃₁	semidirect
242. #546	C₂×C₃₁	semidirect
243. #546	C₂×C₃₁	semidirect
244. D₄²	C₂×C₃₁	semidirect
245. D₄²	C₂×C₃₁	semidirect
246. #540	C₂×C₃₁	semidirect
247. #538	C₂×C₃₁	semidirect
248. #532	C₂×C₃₁	semidirect
249. #529	C₂×C₃₁	semidirect
250. #525	C₂×C₃₁	semidirect
251. C₂×#121	C₂×C₃₁	semidirect
252. #518	C₂×C₃₁	semidirect
253. C₂×C₄×D₄	C₂×C₃₁	semidirect
262. C₂×#41	C₂²×C₃₁	semidirect
263. C₂×#41	C₂²×C₃₁	semidirect
266. C₂²×D₄	C₂²×C₃₁	semidirect
267. C₂²×D₄	C₂²×C₃₁	semidirect
268. C₂³×C₄	C₂²×C₃₁	semidirect
269. C₄⋊Q₈	C₂²×C₃₁	semidirect
270. #127	C₂²×C₃₁	semidirect
271. #127	C₂²×C₃₁	semidirect
272. #126	C₂²×C₃₁	semidirect
273. #126	C₂²×C₃₁	semidirect
274. #125	C₂²×C₃₁	semidirect
275. #125	C₂²×C₃₁	semidirect
276. #124	C₂²×C₃₁	semidirect
277. #124	C₂²×C₃₁	semidirect
278. #124	C₂²×C₃₁	semidirect
279. #123	C₂²×C₃₁	semidirect
280. #123	C₂²×C₃₁	semidirect
281. #123	C₂²×C₃₁	semidirect
282. #123	C₂²×C₃₁	semidirect
283. C₂²⋊Q₈	C₂²×C₃₁	semidirect
284. C₂²⋊Q₈	C₂²×C₃₁	semidirect
285. C₂²⋊Q₈	C₂²×C₃₁	semidirect
287. #121	C₂²×C₃₁	semidirect
288. #121	C₂²×C₃₁	semidirect
289. #121	C₂²×C₃₁	semidirect
290. #121	C₂²×C₃₁	semidirect
291. #121	C₂²×C₃₁	semidirect
292. #121	C₂²×C₃₁	semidirect
293. #121	C₂²×C₃₁	semidirect
294. #121	C₂²×C₃₁	semidirect
295. #121	C₂²×C₃₁	semidirect
296. #121	C₂²×C₃₁	semidirect
297. #121	C₂²×C₃₁	semidirect
298. #121	C₂²×C₃₁	semidirect
299. #120	C₂²×C₃₁	semidirect
300. #120	C₂²×C₃₁	semidirect
301. #120	C₂²×C₃₁	semidirect
302. C₄×Q₈	C₂²×C₃₁	semidirect
306. C₄×D₄	C₂²×C₃₁	semidirect
307. C₄×D₄	C₂²×C₃₁	semidirect
308. C₄×D₄	C₂²×C₃₁	semidirect
309. C₄×D₄	C₂²×C₃₁	semidirect
310. C₄×D₄	C₂²×C₃₁	semidirect
311. C₄×D₄	C₂²×C₃₁	semidirect
312. C₄×D₄	C₂²×C₃₁	semidirect
313. C₄×D₄	C₂²×C₃₁	semidirect
314. C₄×D₄	C₂²×C₃₁	semidirect
315. C₄×D₄	C₂²×C₃₁	semidirect
316. C₄×D₄	C₂²×C₃₁	semidirect
317. C₄×D₄	C₂²×C₃₁	semidirect
318. #117	C₂²×C₃₁	semidirect
319. #117	C₂²×C₃₁	semidirect
320. #117	C₂²×C₃₁	semidirect
321. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
322. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
323. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
324. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
325. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
326. C₂×C₄²	C₂²×C₃₁	semidirect
327. C₃₁	#3519	direct (N × H)
426. D₄	#41×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#555	2
C₃₁×#554	2
C₃₁×#553	2
C₃₁×#550	2
C₃₁×#546	2
D₄²×C₃₁	2
C₃₁×#540	2
C₃₁×#538	2
C₃₁×#532	2
C₃₁×#529	2
C₃₁×#525	2
C₂×C₃₁×#121	2
C₃₁×#518	2
C₂×C₄×D₄×C₃₁	2
#3519	31

Subgroups — C31×#3519 #16843452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3519 #16843452