Subgroups — C₃₁×#3547 #16843480

Order 3968.

1022 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#562	C₂	1	4	yes	maximal, complemented
2	C₃₁×#555	C₂	1	2	yes	maximal, complemented
3	C₃₁×#555	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#553	C₂	1	2	yes	maximal, complemented
6	C₃₁×#552	C₂	1	2	yes	maximal, complemented
7	D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#548	C₂	1	2	yes	maximal, complemented
9	C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#544	C₂	1	2	yes	maximal, complemented
11	C₃₁×#542	C₂	1	4	yes	maximal, complemented
12	C₃₁×#540	C₂	1	2	yes	maximal, complemented
13	C₃₁×#537	C₂	1	2	yes	maximal, complemented
14	C₃₁×#536	C₂	1	2	yes	maximal, complemented
15	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₃₁	C₂²	1	2	yes	—
19	C₂×#41×C₃₁	C₂²	1	2	yes	—
20	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
21	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
25	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₃₁×#127	C₂²	1	1	yes	characteristic
27	C₃₁×#126	C₂²	1	2	yes	—
28	C₃₁×#126	C₂²	1	4	yes	—
29	C₃₁×#125	C₂²	1	1	yes	characteristic
30	C₃₁×#125	C₂²	1	2	yes	—
31	C₃₁×#125	C₂²	1	4	yes	—
32	C₃₁×#125	C₂²	1	2	yes	—
33	C₃₁×#124	C₂²	1	1	yes	characteristic
34	C₃₁×#124	C₂²	1	2	yes	—
35	C₃₁×#124	C₂²	1	4	yes	—
36	C₃₁×#124	C₂²	1	4	yes	—
37	C₃₁×#124	C₂²	1	4	yes	—
38	C₃₁×#123	C₂²	1	2	yes	—
39	C₃₁×#123	C₂²	1	4	yes	—
40	C₃₁×#123	C₂²	1	4	yes	—
41	C₃₁×#123	C₂²	1	4	yes	—
42	C₃₁×#123	C₂²	1	4	yes	—
43	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
44	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
45	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
46	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
47	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
48	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
49	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
50	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
51	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
52	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
53	C₃₁×#121	C₂²	1	1	yes	characteristic
54	C₃₁×#121	C₂²	1	1	yes	characteristic
55	C₃₁×#121	C₂²	1	1	yes	characteristic
56	C₃₁×#121	C₂²	1	2	yes	—
57	C₃₁×#121	C₂²	1	4	yes	—
58	C₃₁×#121	C₂²	1	2	yes	—
59	C₃₁×#121	C₂²	1	4	yes	—
60	C₃₁×#121	C₂²	1	4	yes	—
61	C₃₁×#121	C₂²	1	2	yes	—
62	C₃₁×#121	C₂²	1	4	yes	—
63	C₃₁×#121	C₂²	1	2	yes	—
64	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
66	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
67	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
68	C₄×D₄×C₃₁	C₂²	1	2	yes	—
69	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
70	C₄×D₄×C₃₁	C₂²	1	2	yes	—
71	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
72	C₄×D₄×C₃₁	C₂²	1	2	yes	—
73	C₄×D₄×C₃₁	C₂²	1	2	yes	—
74	C₄×D₄×C₃₁	C₂²	1	4	yes	—
75	C₃₁×#117	C₂²	1	2	yes	—
76	C₃₁×#117	C₂²	1	2	yes	—
77	C₃₁×#117	C₂²	1	2	yes	—
78	C₃₁×#117	C₂²	1	4	yes	—
79	C₃₁×#117	C₂²	1	2	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
82	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
83	#41×C₃₁	—	4	4	no	—
84	#41×C₃₁	—	4	4	no	—
85	C₂×Q₈×C₃₁	—	4	4	no	—
86	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
88	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
89	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
90	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
91	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₃₁	C₂³	1	2	yes	—
95	C₂×D₄×C₃₁	C₂³	1	4	yes	—
96	C₂×D₄×C₃₁	C₂³	1	2	yes	—
97	C₂×D₄×C₃₁	C₂³	1	2	yes	—
98	C₂×D₄×C₃₁	C₂³	1	4	yes	—
99	C₂×D₄×C₃₁	C₂³	1	2	yes	—
100	C₂×D₄×C₃₁	C₂³	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
108	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
109	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
110	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
116	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
118	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
119	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
123	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
126	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
127	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
132	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂²⋊C₄×C₃₁	C₂³	1	4	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	2	yes	—
140	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
141	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
142	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
143	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
144	C₄²×C₃₁	C₂³	1	1	yes	characteristic
145	C₄²×C₃₁	C₂³	1	1	yes	characteristic
146	C₄²×C₃₁	C₂³	1	1	yes	characteristic
147	C₄²×C₃₁	C₂³	1	1	yes	characteristic
148	C₄²×C₃₁	C₂³	1	4	yes	—
149	C₄²×C₃₁	C₂³	1	4	yes	—
150	C₄²×C₃₁	C₂³	1	2	yes	—
151	C₂³×C₃₁	C₂⁴	1	2	yes	—
152	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
153	C₂³×C₃₁	C₂⁴	1	2	yes	—
154	C₂³×C₃₁	C₂⁴	1	2	yes	—
155	Q₈×C₃₁	—	2	2	no	—
156	Q₈×C₃₁	—	2	4	no	—
157	Q₈×C₃₁	—	4	4	no	—
158	Q₈×C₃₁	—	4	2	no	—
159	Q₈×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	2	2	no	—
161	D₄×C₃₁	—	2	2	no	—
162	D₄×C₃₁	—	2	2	no	—
163	D₄×C₃₁	—	4	2	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	4	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	4	4	no	—
170	C₂×C₄×C₃₁	—	2	4	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	4	4	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	4	no	—
178	C₂×C₄×C₃₁	—	4	4	no	—
179	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
183	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
187	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
188	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
189	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
190	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
192	#3547	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	—	2	2	no	—
196	C₂²×C₃₁	—	2	2	no	—
197	C₂²×C₃₁	—	2	1	no	—
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	2	2	no	—
200	C₂²×C₃₁	—	2	2	no	—
201	C₂²×C₃₁	—	2	2	no	—
202	C₂²×C₃₁	—	2	1	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	4	no	—
210	C₄×C₃₁	—	2	2	no	—
211	C₄×C₃₁	—	2	1	no	—
212	C₄×C₃₁	—	2	2	no	—
213	C₄×C₃₁	—	2	2	no	—
214	C₄×C₃₁	—	2	1	no	—
215	C₄×C₃₁	—	2	4	no	—
216	C₄×C₃₁	—	2	1	no	—
217	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
218	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
219	#562	C₂×C₃₁	1	4	yes	complemented
220	#555	C₂×C₃₁	1	2	yes	complemented
221	#555	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#554	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#553	C₂×C₃₁	1	2	yes	complemented
224	#552	C₂×C₃₁	1	2	yes	complemented
225	D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#548	C₂×C₃₁	1	2	yes	complemented
227	#547	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#544	C₂×C₃₁	1	2	yes	complemented
229	#542	C₂×C₃₁	1	4	yes	complemented
230	#540	C₂×C₃₁	1	2	yes	complemented
231	#537	C₂×C₃₁	1	2	yes	complemented
232	#536	C₂×C₃₁	1	2	yes	complemented
233	#533	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂×C₃₁	—	4	2	no	—
237	C₂×C₃₁	—	4	1	no	—
238	C₂×C₃₁	—	4	2	no	—
239	C₂×C₃₁	—	4	2	no	—
240	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
242	C₂×C₃₁	#585	1	1	yes	characteristic
243	C₂×#41	C₂²×C₃₁	1	2	yes	—
244	C₂×#41	C₂²×C₃₁	1	2	yes	—
245	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
246	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
247	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
249	C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
250	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
251	#127	C₂²×C₃₁	1	1	yes	characteristic
252	#126	C₂²×C₃₁	1	2	yes	—
253	#126	C₂²×C₃₁	1	4	yes	—
254	#125	C₂²×C₃₁	1	1	yes	characteristic
255	#125	C₂²×C₃₁	1	2	yes	—
256	#125	C₂²×C₃₁	1	4	yes	—
257	#125	C₂²×C₃₁	1	2	yes	—
258	#124	C₂²×C₃₁	1	1	yes	characteristic
259	#124	C₂²×C₃₁	1	2	yes	—
260	#124	C₂²×C₃₁	1	4	yes	—
261	#124	C₂²×C₃₁	1	4	yes	—
262	#124	C₂²×C₃₁	1	4	yes	—
263	#123	C₂²×C₃₁	1	2	yes	—
264	#123	C₂²×C₃₁	1	4	yes	—
265	#123	C₂²×C₃₁	1	4	yes	—
266	#123	C₂²×C₃₁	1	4	yes	—
267	#123	C₂²×C₃₁	1	4	yes	—
268	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
270	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
271	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
272	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
273	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
274	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
275	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
276	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
277	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
278	#121	C₂²×C₃₁	1	1	yes	characteristic
279	#121	C₂²×C₃₁	1	1	yes	characteristic
280	#121	C₂²×C₃₁	1	1	yes	characteristic
281	#121	C₂²×C₃₁	1	2	yes	—
282	#121	C₂²×C₃₁	1	4	yes	—
283	#121	C₂²×C₃₁	1	2	yes	—
284	#121	C₂²×C₃₁	1	4	yes	—
285	#121	C₂²×C₃₁	1	4	yes	—
286	#121	C₂²×C₃₁	1	2	yes	—
287	#121	C₂²×C₃₁	1	4	yes	—
288	#121	C₂²×C₃₁	1	2	yes	—
289	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
291	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
292	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
293	C₄×D₄	C₂²×C₃₁	1	2	yes	—
294	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
295	C₄×D₄	C₂²×C₃₁	1	2	yes	—
296	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
297	C₄×D₄	C₂²×C₃₁	1	2	yes	—
298	C₄×D₄	C₂²×C₃₁	1	2	yes	—
299	C₄×D₄	C₂²×C₃₁	1	4	yes	—
300	#117	C₂²×C₃₁	1	2	yes	—
301	#117	C₂²×C₃₁	1	2	yes	—
302	#117	C₂²×C₃₁	1	2	yes	—
303	#117	C₂²×C₃₁	1	4	yes	—
304	#117	C₂²×C₃₁	1	2	yes	—
305	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
306	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
307	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
308	C₃₁	#3547	1	1	yes	characteristic, direct product factor, Sylow
309	#41	—	4	4	no	—
310	#41	—	4	4	no	—
311	C₂×Q₈	—	4	4	no	—
312	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
314	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
315	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
316	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
317	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
318	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
320	C₂×D₄	C₂³×C₃₁	1	2	yes	—
321	C₂×D₄	C₂³×C₃₁	1	4	yes	—
322	C₂×D₄	C₂³×C₃₁	1	2	yes	—
323	C₂×D₄	C₂³×C₃₁	1	2	yes	—
324	C₂×D₄	C₂³×C₃₁	1	4	yes	—
325	C₂×D₄	C₂³×C₃₁	1	2	yes	—
326	C₂×D₄	C₂³×C₃₁	1	2	yes	—
327	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
330	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
332	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
334	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
335	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
336	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
337	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
339	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
340	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
341	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
342	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
343	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
346	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
347	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
348	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
349	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
350	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
351	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
352	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
354	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
355	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
356	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
357	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
358	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
359	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
360	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
361	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
362	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
363	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
364	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
365	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
366	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
367	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
368	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
369	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
370	C₄²	C₂³×C₃₁	1	1	yes	characteristic
371	C₄²	C₂³×C₃₁	1	1	yes	characteristic
372	C₄²	C₂³×C₃₁	1	1	yes	characteristic
373	C₄²	C₂³×C₃₁	1	1	yes	characteristic
374	C₄²	C₂³×C₃₁	1	4	yes	—
375	C₄²	C₂³×C₃₁	1	4	yes	—
376	C₄²	C₂³×C₃₁	1	2	yes	—
377	C₂³	C₂⁴×C₃₁	1	2	yes	—
378	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
379	C₂³	C₂⁴×C₃₁	1	2	yes	—
380	C₂³	C₂⁴×C₃₁	1	2	yes	—
381	Q₈	—	2	2	no	—
382	Q₈	—	2	4	no	—
383	Q₈	—	4	4	no	—
384	Q₈	—	4	2	no	—
385	Q₈	—	4	2	no	—
386	D₄	—	2	2	no	—
387	D₄	—	2	2	no	—
388	D₄	—	2	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	4	2	no	—
391	D₄	—	4	4	no	—
392	D₄	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	4	4	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
407	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
409	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
410	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
411	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
413	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
414	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
415	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
416	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
418	C₂²	—	2	2	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	1	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	2	2	no	—
430	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
431	C₄	—	2	2	no	—
432	C₄	—	2	1	no	—
433	C₄	—	2	2	no	—
434	C₄	—	2	4	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	4	no	—
441	C₄	—	2	1	no	—
442	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
443	C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
444	C₂	—	4	2	no	—
445	C₂	—	4	1	no	—
446	C₂	—	4	2	no	—
447	C₂	—	4	2	no	—
448	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
449	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
450	C₂	C₃₁×#585	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×#562	C₂	semidirect
2. C₃₁×#555	C₂	semidirect
3. C₃₁×#555	C₂	semidirect
4. C₃₁×#554	C₂	semidirect
5. C₃₁×#553	C₂	semidirect
6. C₃₁×#552	C₂	semidirect
7. D₄×Q₈×C₃₁	C₂	semidirect
8. C₃₁×#548	C₂	semidirect
9. C₃₁×#547	C₂	semidirect
10. C₃₁×#544	C₂	semidirect
11. C₃₁×#542	C₂	semidirect
12. C₃₁×#540	C₂	semidirect
13. C₃₁×#537	C₂	semidirect
14. C₃₁×#536	C₂	semidirect
15. C₃₁×#533	C₂	semidirect
16. C₃₁×#532	C₂	semidirect
17. C₃₁×#529	C₂	semidirect
192. #3547	C₃₁	direct (N × H)
219. #562	C₂×C₃₁	semidirect
220. #555	C₂×C₃₁	semidirect
221. #555	C₂×C₃₁	semidirect
222. #554	C₂×C₃₁	semidirect
223. #553	C₂×C₃₁	semidirect
224. #552	C₂×C₃₁	semidirect
225. D₄×Q₈	C₂×C₃₁	semidirect
226. #548	C₂×C₃₁	semidirect
227. #547	C₂×C₃₁	semidirect
228. #544	C₂×C₃₁	semidirect
229. #542	C₂×C₃₁	semidirect
230. #540	C₂×C₃₁	semidirect
231. #537	C₂×C₃₁	semidirect
232. #536	C₂×C₃₁	semidirect
233. #533	C₂×C₃₁	semidirect
234. #532	C₂×C₃₁	semidirect
235. #529	C₂×C₃₁	semidirect
308. C₃₁	#3547	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#562	2
C₃₁×#555	2
C₃₁×#554	2
C₃₁×#553	2
C₃₁×#552	2
D₄×Q₈×C₃₁	2
C₃₁×#548	2
C₃₁×#547	2
C₃₁×#544	2
C₃₁×#542	2
C₃₁×#540	2
C₃₁×#537	2
C₃₁×#536	2
C₃₁×#533	2
C₃₁×#532	2
C₃₁×#529	2
#3547	31

Subgroups — C31×#3547 #16843480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3547 #16843480