Subgroups — C₃₁×#3540 #16843473

Order 3968.

1058 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×H(3,𝔽₄)	C₂	1	2	yes	maximal, complemented
2	C₃₁×#560	C₂	1	2	yes	maximal, complemented
3	C₃₁×#559	C₂	1	2	yes	maximal, complemented
4	C₃₁×#556	C₂	1	2	yes	maximal, complemented
5	C₃₁×#555	C₂	1	2	yes	maximal, complemented
6	C₃₁×#551	C₂	1	2	yes	maximal, complemented
7	C₃₁×#548	C₂	1	2	yes	maximal, complemented
8	C₃₁×#546	C₂	1	2	yes	maximal, complemented
9	C₃₁×#542	C₂	1	2	yes	maximal, complemented
10	C₃₁×#542	C₂	1	4	yes	maximal, complemented
11	C₃₁×#538	C₂	1	4	yes	maximal, complemented
12	C₃₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#535	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#525	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#525	C₂	1	2	yes	maximal, complemented
16	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×#41×C₃₁	C₂²	1	2	yes	—
18	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
20	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
21	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
22	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
23	C₃₁×#127	C₂²	1	2	yes	—
24	C₃₁×#126	C₂²	1	4	yes	complemented
25	C₃₁×#126	C₂²	1	4	yes	complemented
26	C₃₁×#126	C₂²	1	4	yes	complemented
27	C₃₁×#124	C₂²	1	2	yes	—
28	C₃₁×#124	C₂²	1	4	yes	—
29	C₃₁×#124	C₂²	1	2	yes	complemented
30	C₃₁×#124	C₂²	1	4	yes	complemented
31	C₃₁×#124	C₂²	1	2	yes	complemented
32	C₃₁×#124	C₂²	1	4	yes	complemented
33	C₃₁×#124	C₂²	1	4	yes	complemented
34	C₃₁×#124	C₂²	1	4	yes	complemented
35	C₃₁×#123	C₂²	1	1	yes	characteristic
36	C₃₁×#123	C₂²	1	4	yes	complemented
37	C₃₁×#123	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#123	C₂²	1	2	yes	complemented
39	C₃₁×#123	C₂²	1	2	yes	complemented
40	C₃₁×#123	C₂²	1	4	yes	complemented
41	C₃₁×#123	C₂²	1	4	yes	complemented
42	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
43	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
44	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
45	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
46	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
47	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
48	C₃₁×#121	C₂²	1	2	yes	—
49	C₃₁×#121	C₂²	1	2	yes	—
50	C₃₁×#121	C₂²	1	2	yes	—
51	C₃₁×#121	C₂²	1	4	yes	—
52	C₃₁×#121	C₂²	1	2	yes	complemented
53	C₃₁×#121	C₂²	1	4	yes	complemented
54	C₃₁×#121	C₂²	1	4	yes	complemented
55	C₃₁×#120	C₂²	1	1	yes	characteristic
56	C₃₁×#120	C₂²	1	2	yes	—
57	C₃₁×#120	C₂²	1	4	yes	—
58	C₃₁×#120	C₂²	1	1	yes	characteristic, complemented
59	C₃₁×#120	C₂²	1	4	yes	complemented
60	C₃₁×#120	C₂²	1	2	yes	complemented
61	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
62	C₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
63	C₄×D₄×C₃₁	C₂²	1	2	yes	—
64	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
65	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
66	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
67	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
68	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
69	C₃₁×#117	C₂²	1	1	yes	characteristic, complemented
70	C₃₁×#117	C₂²	1	2	yes	complemented
71	C₃₁×#117	C₂²	1	1	yes	characteristic, complemented
72	C₃₁×#117	C₂²	1	1	yes	characteristic, complemented
73	C₃₁×#117	C₂²	1	4	yes	complemented
74	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
75	C₂⁴×C₃₁	C₂³	1	2	yes	—
76	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
77	#41×C₃₁	—	4	2	no	—
78	#41×C₃₁	—	4	4	no	—
79	C₂×Q₈×C₃₁	—	4	2	no	—
80	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
81	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
82	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
83	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×D₄×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	C₂³	1	2	yes	—
86	C₂×D₄×C₃₁	C₂³	1	2	yes	—
87	C₂×D₄×C₃₁	C₂³	1	4	yes	—
88	C₂×D₄×C₃₁	C₂³	1	4	yes	—
89	C₂×D₄×C₃₁	C₂³	1	2	yes	—
90	C₂×D₄×C₃₁	C₂³	1	4	yes	—
91	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₃₁	C₂³	1	4	yes	—
93	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₃₁	C₂³	1	2	yes	—
96	C₂²×C₄×C₃₁	—	4	2	no	—
97	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
99	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
102	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
103	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
106	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
111	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₄⋊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	4	yes	—
120	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
123	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
124	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
125	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
129	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
131	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
132	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
133	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₄²×C₃₁	C₂³	1	2	yes	—
135	C₄²×C₃₁	C₂³	1	4	yes	—
136	C₄²×C₃₁	C₂³	1	2	yes	—
137	C₄²×C₃₁	C₂³	1	2	yes	—
138	C₄²×C₃₁	C₂³	1	4	yes	—
139	C₂³×C₃₁	—	4	1	no	—
140	C₂³×C₃₁	—	4	2	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	—	4	4	no	—
143	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
144	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
145	C₂³×C₃₁	C₂⁴	1	2	yes	—
146	C₂³×C₃₁	C₂⁴	1	4	yes	—
147	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
148	Q₈×C₃₁	—	2	4	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	4	4	no	—
151	Q₈×C₃₁	—	4	1	no	—
152	D₄×C₃₁	—	2	4	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	2	no	—
155	D₄×C₃₁	—	4	4	no	—
156	D₄×C₃₁	—	4	4	no	—
157	D₄×C₃₁	—	4	4	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	1	no	—
160	D₄×C₃₁	—	4	1	no	—
161	D₄×C₃₁	—	4	1	no	—
162	D₄×C₃₁	—	4	4	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	4	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	4	no	—
172	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
173	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
174	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
175	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
179	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
180	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	#3540	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂²×C₃₁	—	2	1	no	—
184	C₂²×C₃₁	—	2	2	no	—
185	C₂²×C₃₁	—	2	1	no	—
186	C₂²×C₃₁	—	2	1	no	—
187	C₂²×C₃₁	—	2	4	no	—
188	C₂²×C₃₁	—	2	4	no	—
189	C₂²×C₃₁	—	2	2	no	—
190	C₂²×C₃₁	—	2	2	no	—
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	2	2	no	—
193	C₂²×C₃₁	—	2	4	no	—
194	C₂²×C₃₁	—	2	2	no	—
195	C₂²×C₃₁	—	4	4	no	—
196	C₂²×C₃₁	—	8	4	no	—
197	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	2	1	no	—
200	C₄×C₃₁	—	2	1	no	—
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	2	4	no	—
203	C₄×C₃₁	—	2	2	no	—
204	C₄×C₃₁	—	2	4	no	—
205	C₄×C₃₁	—	2	4	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
208	H(3,𝔽₄)	C₂×C₃₁	1	2	yes	complemented
209	#560	C₂×C₃₁	1	2	yes	complemented
210	#559	C₂×C₃₁	1	2	yes	complemented
211	#556	C₂×C₃₁	1	2	yes	complemented
212	#555	C₂×C₃₁	1	2	yes	complemented
213	#551	C₂×C₃₁	1	2	yes	complemented
214	#548	C₂×C₃₁	1	2	yes	complemented
215	#546	C₂×C₃₁	1	2	yes	complemented
216	#542	C₂×C₃₁	1	2	yes	complemented
217	#542	C₂×C₃₁	1	4	yes	complemented
218	#538	C₂×C₃₁	1	4	yes	complemented
219	#536	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#535	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#525	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#525	C₂×C₃₁	1	2	yes	complemented
223	C₂×C₃₁	—	4	1	no	—
224	C₂×C₃₁	—	4	2	no	—
225	C₂×C₃₁	—	4	1	no	—
226	C₂×C₃₁	—	4	4	no	—
227	C₂×C₃₁	—	4	1	no	—
228	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₃₁	#585	1	2	yes	—
230	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×#41	C₂²×C₃₁	1	2	yes	—
232	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
233	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
234	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
235	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
236	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
237	#127	C₂²×C₃₁	1	2	yes	—
238	#126	C₂²×C₃₁	1	4	yes	complemented
239	#126	C₂²×C₃₁	1	4	yes	complemented
240	#126	C₂²×C₃₁	1	4	yes	complemented
241	#124	C₂²×C₃₁	1	2	yes	—
242	#124	C₂²×C₃₁	1	4	yes	—
243	#124	C₂²×C₃₁	1	2	yes	complemented
244	#124	C₂²×C₃₁	1	4	yes	complemented
245	#124	C₂²×C₃₁	1	2	yes	complemented
246	#124	C₂²×C₃₁	1	4	yes	complemented
247	#124	C₂²×C₃₁	1	4	yes	complemented
248	#124	C₂²×C₃₁	1	4	yes	complemented
249	#123	C₂²×C₃₁	1	1	yes	characteristic
250	#123	C₂²×C₃₁	1	4	yes	complemented
251	#123	C₂²×C₃₁	1	1	yes	characteristic, complemented
252	#123	C₂²×C₃₁	1	2	yes	complemented
253	#123	C₂²×C₃₁	1	2	yes	complemented
254	#123	C₂²×C₃₁	1	4	yes	complemented
255	#123	C₂²×C₃₁	1	4	yes	complemented
256	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
257	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
258	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
259	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
260	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
261	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
262	#121	C₂²×C₃₁	1	2	yes	—
263	#121	C₂²×C₃₁	1	2	yes	—
264	#121	C₂²×C₃₁	1	2	yes	—
265	#121	C₂²×C₃₁	1	4	yes	—
266	#121	C₂²×C₃₁	1	2	yes	complemented
267	#121	C₂²×C₃₁	1	4	yes	complemented
268	#121	C₂²×C₃₁	1	4	yes	complemented
269	#120	C₂²×C₃₁	1	1	yes	characteristic
270	#120	C₂²×C₃₁	1	2	yes	—
271	#120	C₂²×C₃₁	1	4	yes	—
272	#120	C₂²×C₃₁	1	1	yes	characteristic, complemented
273	#120	C₂²×C₃₁	1	4	yes	complemented
274	#120	C₂²×C₃₁	1	2	yes	complemented
275	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
276	C₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
277	C₄×D₄	C₂²×C₃₁	1	2	yes	—
278	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
279	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
280	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
281	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
282	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
283	#117	C₂²×C₃₁	1	1	yes	characteristic, complemented
284	#117	C₂²×C₃₁	1	2	yes	complemented
285	#117	C₂²×C₃₁	1	1	yes	characteristic, complemented
286	#117	C₂²×C₃₁	1	1	yes	characteristic, complemented
287	#117	C₂²×C₃₁	1	4	yes	complemented
288	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
289	C₃₁	#3540	1	1	yes	characteristic, direct product factor, Sylow
290	C₂⁴	C₂³×C₃₁	1	2	yes	—
291	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
292	#41	—	4	2	no	—
293	#41	—	4	4	no	—
294	C₂×Q₈	—	4	2	no	—
295	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
296	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
297	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
298	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×D₄	—	4	2	no	—
300	C₂×D₄	C₂³×C₃₁	1	2	yes	—
301	C₂×D₄	C₂³×C₃₁	1	2	yes	—
302	C₂×D₄	C₂³×C₃₁	1	4	yes	—
303	C₂×D₄	C₂³×C₃₁	1	4	yes	—
304	C₂×D₄	C₂³×C₃₁	1	2	yes	—
305	C₂×D₄	C₂³×C₃₁	1	4	yes	—
306	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×D₄	C₂³×C₃₁	1	4	yes	—
308	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂×D₄	C₂³×C₃₁	1	2	yes	—
311	C₂²×C₄	—	4	2	no	—
312	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
314	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
317	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
318	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
320	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
321	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
322	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
323	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
324	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
325	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
326	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
327	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
328	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
329	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
330	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
331	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
332	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
333	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
334	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
335	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
336	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
337	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
339	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
341	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
342	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
343	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
344	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
346	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
347	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
348	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
349	C₄²	C₂³×C₃₁	1	2	yes	—
350	C₄²	C₂³×C₃₁	1	4	yes	—
351	C₄²	C₂³×C₃₁	1	2	yes	—
352	C₄²	C₂³×C₃₁	1	2	yes	—
353	C₄²	C₂³×C₃₁	1	4	yes	—
354	C₂³	—	4	1	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	4	no	—
358	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
359	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
360	C₂³	C₂⁴×C₃₁	1	2	yes	—
361	C₂³	C₂⁴×C₃₁	1	4	yes	—
362	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
363	Q₈	—	2	4	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	4	no	—
366	Q₈	—	4	1	no	—
367	D₄	—	2	4	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	4	no	—
371	D₄	—	4	4	no	—
372	D₄	—	4	4	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	1	no	—
375	D₄	—	4	1	no	—
376	D₄	—	4	1	no	—
377	D₄	—	4	4	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	4	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	4	no	—
387	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
388	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
389	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
390	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
392	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
393	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
394	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
395	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
397	C₂²	—	2	1	no	—
398	C₂²	—	2	2	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	2	1	no	—
401	C₂²	—	2	4	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	4	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	8	4	no	—
411	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
412	C₄	—	2	2	no	—
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	2	1	no	—
416	C₄	—	2	4	no	—
417	C₄	—	2	2	no	—
418	C₄	—	2	4	no	—
419	C₄	—	2	4	no	—
420	C₄	—	2	2	no	—
421	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
422	C₂	—	4	2	no	—
423	C₂	—	4	1	no	—
424	C₂	—	4	1	no	—
425	C₂	—	4	4	no	—
426	C₂	—	4	1	no	—
427	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
428	C₂	C₃₁×#585	1	2	yes	—

	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×H(3,𝔽₄)	C₂	semidirect
2. C₃₁×#560	C₂	semidirect
3. C₃₁×#559	C₂	semidirect
4. C₃₁×#556	C₂	semidirect
5. C₃₁×#555	C₂	semidirect
6. C₃₁×#551	C₂	semidirect
7. C₃₁×#548	C₂	semidirect
8. C₃₁×#546	C₂	semidirect
9. C₃₁×#542	C₂	semidirect
10. C₃₁×#542	C₂	semidirect
11. C₃₁×#538	C₂	semidirect
12. C₃₁×#536	C₂	semidirect
13. C₃₁×#535	C₂	semidirect
14. C₃₁×#525	C₂	semidirect
15. C₃₁×#525	C₂	semidirect
18. C₂²×Q₈×C₃₁	C₂²	semidirect
19. C₂²×D₄×C₃₁	C₂²	semidirect
20. C₂³×C₄×C₃₁	C₂²	semidirect
21. C₃₁×C₄⋊Q₈	C₂²	semidirect
22. C₃₁×C₄⋊Q₈	C₂²	semidirect
24. C₃₁×#126	C₂²	semidirect
25. C₃₁×#126	C₂²	semidirect
26. C₃₁×#126	C₂²	semidirect
29. C₃₁×#124	C₂²	semidirect
30. C₃₁×#124	C₂²	semidirect
31. C₃₁×#124	C₂²	semidirect
32. C₃₁×#124	C₂²	semidirect
33. C₃₁×#124	C₂²	semidirect
34. C₃₁×#124	C₂²	semidirect
36. C₃₁×#123	C₂²	semidirect
37. C₃₁×#123	C₂²	semidirect
38. C₃₁×#123	C₂²	semidirect
39. C₃₁×#123	C₂²	semidirect
40. C₃₁×#123	C₂²	semidirect
41. C₃₁×#123	C₂²	semidirect
42. C₃₁×C₂²⋊Q₈	C₂²	semidirect
43. C₃₁×C₂²⋊Q₈	C₂²	semidirect
44. C₃₁×C₂²⋊Q₈	C₂²	semidirect
45. C₃₁×C₂²⋊Q₈	C₂²	semidirect
46. C₃₁×C₂²⋊Q₈	C₂²	semidirect
47. C₃₁×C₂²⋊Q₈	C₂²	semidirect
52. C₃₁×#121	C₂²	semidirect
53. C₃₁×#121	C₂²	semidirect
54. C₃₁×#121	C₂²	semidirect
58. C₃₁×#120	C₂²	semidirect
59. C₃₁×#120	C₂²	semidirect
60. C₃₁×#120	C₂²	semidirect
61. C₄×Q₈×C₃₁	C₂²	semidirect
62. C₄×Q₈×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. C₃₁×#117	C₂²	semidirect
70. C₃₁×#117	C₂²	semidirect
71. C₃₁×#117	C₂²	semidirect
72. C₃₁×#117	C₂²	semidirect
73. C₃₁×#117	C₂²	semidirect
74. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
182. #3540	C₃₁	direct (N × H)
208. H(3,𝔽₄)	C₂×C₃₁	semidirect
209. #560	C₂×C₃₁	semidirect
210. #559	C₂×C₃₁	semidirect
211. #556	C₂×C₃₁	semidirect
212. #555	C₂×C₃₁	semidirect
213. #551	C₂×C₃₁	semidirect
214. #548	C₂×C₃₁	semidirect
215. #546	C₂×C₃₁	semidirect
216. #542	C₂×C₃₁	semidirect
217. #542	C₂×C₃₁	semidirect
218. #538	C₂×C₃₁	semidirect
219. #536	C₂×C₃₁	semidirect
220. #535	C₂×C₃₁	semidirect
221. #525	C₂×C₃₁	semidirect
222. #525	C₂×C₃₁	semidirect
232. C₂²×Q₈	C₂²×C₃₁	semidirect
233. C₂²×D₄	C₂²×C₃₁	semidirect
234. C₂³×C₄	C₂²×C₃₁	semidirect
235. C₄⋊Q₈	C₂²×C₃₁	semidirect
236. C₄⋊Q₈	C₂²×C₃₁	semidirect
238. #126	C₂²×C₃₁	semidirect
239. #126	C₂²×C₃₁	semidirect
240. #126	C₂²×C₃₁	semidirect
243. #124	C₂²×C₃₁	semidirect
244. #124	C₂²×C₃₁	semidirect
245. #124	C₂²×C₃₁	semidirect
246. #124	C₂²×C₃₁	semidirect
247. #124	C₂²×C₃₁	semidirect
248. #124	C₂²×C₃₁	semidirect
250. #123	C₂²×C₃₁	semidirect
251. #123	C₂²×C₃₁	semidirect
252. #123	C₂²×C₃₁	semidirect
253. #123	C₂²×C₃₁	semidirect
254. #123	C₂²×C₃₁	semidirect
255. #123	C₂²×C₃₁	semidirect
256. C₂²⋊Q₈	C₂²×C₃₁	semidirect
257. C₂²⋊Q₈	C₂²×C₃₁	semidirect
258. C₂²⋊Q₈	C₂²×C₃₁	semidirect
259. C₂²⋊Q₈	C₂²×C₃₁	semidirect
260. C₂²⋊Q₈	C₂²×C₃₁	semidirect
261. C₂²⋊Q₈	C₂²×C₃₁	semidirect
266. #121	C₂²×C₃₁	semidirect
267. #121	C₂²×C₃₁	semidirect
268. #121	C₂²×C₃₁	semidirect
272. #120	C₂²×C₃₁	semidirect
273. #120	C₂²×C₃₁	semidirect
274. #120	C₂²×C₃₁	semidirect
275. C₄×Q₈	C₂²×C₃₁	semidirect
276. C₄×Q₈	C₂²×C₃₁	semidirect
278. C₄×D₄	C₂²×C₃₁	semidirect
279. C₄×D₄	C₂²×C₃₁	semidirect
280. C₄×D₄	C₂²×C₃₁	semidirect
281. C₄×D₄	C₂²×C₃₁	semidirect
282. C₄×D₄	C₂²×C₃₁	semidirect
283. #117	C₂²×C₃₁	semidirect
284. #117	C₂²×C₃₁	semidirect
285. #117	C₂²×C₃₁	semidirect
286. #117	C₂²×C₃₁	semidirect
287. #117	C₂²×C₃₁	semidirect
288. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
289. C₃₁	#3540	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×H(3,𝔽₄)	2
C₃₁×#560	2
C₃₁×#559	2
C₃₁×#556	2
C₃₁×#555	2
C₃₁×#551	2
C₃₁×#548	2
C₃₁×#546	2
C₃₁×#542	2
C₃₁×#538	2
C₃₁×#536	2
C₃₁×#535	2
C₃₁×#525	2
#3540	31

Subgroups — C31×#3540 #16843473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3540 #16843473