Subgroups — C₃₁×#19953 #21476712

Order 7936.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3354	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3341	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1691	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1646	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1635	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₃₁×#555	C₂²	1	2	yes	—
14	C₃₁×#547	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#498	C₂²	1	2	yes	—
17	C₃₁×#494	C₂²	1	1	yes	characteristic
18	C₃₁×#487	C₂²	1	2	yes	—
19	C₃₁×#482	—	2	2	no	—
20	C₃₁×#482	—	2	2	no	—
21	C₃₁×#481	—	2	2	no	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#476	C₂²	1	2	yes	—
24	C₃₁×#474	C₂²	1	2	yes	—
25	C₃₁×#469	—	2	2	no	—
26	C₃₁×#467	—	2	1	no	—
27	C₃₁×#466	—	2	1	no	—
28	C₃₁×#461	C₂²	1	2	yes	—
29	C₃₁×#452	—	2	2	no	—
30	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#436	C₂²	1	2	yes	—
33	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#428	—	2	2	no	—
36	C₃₁×#427	—	2	1	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#335	C₂²	1	1	yes	characteristic
39	C₃₁×#333	C₂²	1	2	yes	—
40	C₃₁×#332	C₂²	1	2	yes	—
41	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₃₁×#135	—	4	4	no	—
47	C₂×Q₁₆×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	—	2	1	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
51	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	—	2	1	no	—
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₃₁×#124	—	4	2	no	—
58	C₃₁×#123	—	4	2	no	—
59	C₃₁×C₂²⋊Q₈	—	2	2	no	—
60	C₃₁×#121	—	2	2	no	—
61	C₃₁×#121	—	2	2	no	—
62	C₃₁×#121	—	2	2	no	—
63	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₄×D₄×C₃₁	C₂³	1	2	yes	—
66	C₃₁×#117	—	2	2	no	—
67	C₃₁×#117	—	2	2	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₃₁×#107	—	2	1	no	—
70	C₃₁×#107	—	2	1	no	—
71	C₃₁×#107	—	2	2	no	—
72	C₃₁×#107	D₄	1	1	yes	characteristic
73	C₃₁×#107	D₄	1	1	yes	characteristic
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×#106	—	2	2	no	—
76	C₃₁×#106	—	2	2	no	—
77	C₃₁×#106	—	2	2	no	—
78	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	—	4	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	D₄	1	2	yes	—
87	C₃₁×#102	D₄	1	1	yes	characteristic
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	#41×C₃₁	—	4	4	no	—
96	#41×C₃₁	—	8	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	4	2	no	—
100	C₂×D₄×C₃₁	—	4	2	no	—
101	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	4	2	no	—
108	Q₁₆×C₃₁	—	4	2	no	—
109	Q₁₆×C₃₁	—	8	1	no	—
110	SD₁₆×C₃₁	—	4	4	no	—
111	D₈×C₃₁	—	2	2	no	—
112	D₈×C₃₁	—	2	2	no	—
113	M₁₆×C₃₁	—	4	4	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	4	2	no	—
119	C₂×C₈×C₃₁	—	4	2	no	—
120	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²⋊C₄×C₃₁	—	2	1	no	—
133	C₂²⋊C₄×C₃₁	—	2	2	no	—
134	C₂²⋊C₄×C₃₁	—	2	1	no	—
135	C₂²⋊C₄×C₃₁	—	4	2	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₂²⋊C₄×C₃₁	—	4	2	no	—
138	C₄²×C₃₁	—	4	2	no	—
139	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
140	#19953	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	1	no	—
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	2	no	—
144	Q₈×C₃₁	—	4	2	no	—
145	Q₈×C₃₁	—	4	2	no	—
146	Q₈×C₃₁	—	8	2	no	—
147	D₄×C₃₁	—	2	1	no	—
148	D₄×C₃₁	—	2	1	no	—
149	D₄×C₃₁	—	2	2	no	—
150	D₄×C₃₁	—	2	2	no	—
151	D₄×C₃₁	—	2	1	no	—
152	D₄×C₃₁	—	2	1	no	—
153	D₄×C₃₁	—	8	4	no	—
154	D₄×C₃₁	—	16	2	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	1	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	1	no	—
162	C₂×C₄×C₃₁	—	4	1	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	8	4	no	—
167	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
170	C₈×C₃₁	—	2	1	no	—
171	C₈×C₃₁	—	2	1	no	—
172	C₈×C₃₁	—	4	1	no	—
173	C₈×C₃₁	—	4	2	no	—
174	C₈×C₃₁	—	4	1	no	—
175	C₈×C₃₁	—	4	1	no	—
176	C₈×C₃₁	#135	1	2	yes	complemented
177	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
178	#3354	C₂×C₃₁	1	2	yes	complemented
179	#3341	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#3283	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#1707	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1691	C₂×C₃₁	1	2	yes	complemented
183	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1646	C₂×C₃₁	1	2	yes	complemented
186	#1635	C₂×C₃₁	1	2	yes	complemented
187	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
188	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₂²×C₃₁	—	4	1	no	—
190	C₂²×C₃₁	—	4	1	no	—
191	C₂²×C₃₁	—	4	2	no	—
192	C₂²×C₃₁	—	4	2	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	—	4	1	no	—
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	1	no	—
198	C₂²×C₃₁	D₄²	1	1	yes	characteristic
199	C₄×C₃₁	—	2	1	no	—
200	C₄×C₃₁	—	4	1	no	—
201	C₄×C₃₁	—	4	1	no	—
202	C₄×C₃₁	—	4	2	no	—
203	C₄×C₃₁	—	8	2	no	—
204	C₄×C₃₁	—	8	2	no	—
205	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
206	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₄×C₃₁	#577	1	1	yes	characteristic
208	C₄×C₃₁	#578	1	1	yes	characteristic
209	C₂×#135	—	2	1	no	—
210	#555	C₂²×C₃₁	1	2	yes	—
211	#547	C₂²×C₃₁	1	2	yes	—
212	#500	C₂²×C₃₁	1	1	yes	characteristic
213	#498	C₂²×C₃₁	1	2	yes	—
214	#494	C₂²×C₃₁	1	1	yes	characteristic
215	#487	C₂²×C₃₁	1	2	yes	—
216	#482	—	2	2	no	—
217	#482	—	2	2	no	—
218	#481	—	2	2	no	—
219	#476	C₂²×C₃₁	1	2	yes	—
220	#476	C₂²×C₃₁	1	2	yes	—
221	#474	C₂²×C₃₁	1	2	yes	—
222	#469	—	2	2	no	—
223	#467	—	2	1	no	—
224	#466	—	2	1	no	—
225	#461	C₂²×C₃₁	1	2	yes	—
226	#452	—	2	2	no	—
227	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
229	#436	C₂²×C₃₁	1	2	yes	—
230	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
231	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
232	#428	—	2	2	no	—
233	#427	—	2	1	no	—
234	#335	C₂²×C₃₁	1	1	yes	characteristic
235	#335	C₂²×C₃₁	1	1	yes	characteristic
236	#333	C₂²×C₃₁	1	2	yes	—
237	#332	C₂²×C₃₁	1	2	yes	—
238	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
239	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
240	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
241	C₈²	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₃₁	—	8	1	no	—
243	C₂×C₃₁	—	8	1	no	—
244	C₂×C₃₁	—	8	2	no	—
245	C₂×C₃₁	#3272	1	1	yes	characteristic
246	C₂×C₃₁	#3283	1	1	yes	characteristic
247	C₂×C₃₁	#3290	1	1	yes	characteristic
248	C₂×#41	—	2	2	no	—
249	#135	—	4	4	no	—
250	C₂×Q₁₆	—	2	1	no	—
251	C₂×Q₁₆	—	2	1	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
254	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
255	C₂×M₁₆	—	2	2	no	—
256	C₂²×C₈	—	2	1	no	—
257	C₂²×C₈	—	2	1	no	—
258	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
259	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
260	#124	—	4	2	no	—
261	#123	—	4	2	no	—
262	C₂²⋊Q₈	—	2	2	no	—
263	#121	—	2	2	no	—
264	#121	—	2	2	no	—
265	#121	—	2	2	no	—
266	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
267	C₄×D₄	C₂³×C₃₁	1	2	yes	—
268	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
269	#117	—	2	2	no	—
270	#117	—	2	2	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	#107	—	2	1	no	—
273	#107	—	2	1	no	—
274	#107	—	2	2	no	—
275	#107	D₄×C₃₁	1	1	yes	characteristic
276	#107	D₄×C₃₁	1	1	yes	characteristic
277	#106	—	2	2	no	—
278	#106	—	2	2	no	—
279	#106	—	2	2	no	—
280	#106	—	2	2	no	—
281	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
282	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
283	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
284	#103	—	2	2	no	—
285	#103	—	4	2	no	—
286	#102	—	2	2	no	—
287	#102	—	2	2	no	—
288	#102	—	2	2	no	—
289	#102	D₄×C₃₁	1	2	yes	—
290	#102	D₄×C₃₁	1	1	yes	characteristic
291	#102	D₄×C₃₁	1	1	yes	characteristic
292	C₂²⋊C₈	—	2	1	no	—
293	C₂²⋊C₈	—	2	1	no	—
294	C₂²⋊C₈	—	2	2	no	—
295	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₃₁	#19953	1	1	yes	characteristic, direct product factor, Sylow
299	#41	—	4	4	no	—
300	#41	—	8	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
306	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
307	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	4	2	no	—
312	Q₁₆	—	4	2	no	—
313	Q₁₆	—	8	1	no	—
314	SD₁₆	—	4	4	no	—
315	D₈	—	2	2	no	—
316	D₈	—	2	2	no	—
317	M₁₆	—	4	4	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	4	2	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
334	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂²⋊C₄	—	2	1	no	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	—	2	1	no	—
339	C₂²⋊C₄	—	4	2	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₄²	—	4	2	no	—
343	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	2	2	no	—
347	Q₈	—	4	2	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	8	2	no	—
350	D₄	—	2	1	no	—
351	D₄	—	2	1	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	2	no	—
354	D₄	—	2	1	no	—
355	D₄	—	2	1	no	—
356	D₄	—	8	4	no	—
357	D₄	—	16	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	8	4	no	—
370	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
371	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
373	C₈	—	2	1	no	—
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	4	1	no	—
378	C₈	—	4	1	no	—
379	C₈	C₃₁×#135	1	2	yes	complemented
380	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
381	C₂²	—	4	1	no	—
382	C₂²	—	4	1	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	1	no	—
390	C₂²	D₄²×C₃₁	1	1	yes	characteristic
391	C₄	—	2	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	4	2	no	—
395	C₄	—	8	2	no	—
396	C₄	—	8	2	no	—
397	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
398	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
399	C₄	C₃₁×#577	1	1	yes	characteristic
400	C₄	C₃₁×#578	1	1	yes	characteristic
401	C₂	—	8	1	no	—
402	C₂	—	8	1	no	—
403	C₂	—	8	2	no	—
404	C₂	C₃₁×#3272	1	1	yes	characteristic
405	C₂	C₃₁×#3283	1	1	yes	characteristic
406	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3354	C₂	semidirect
2. C₃₁×#3341	C₂	semidirect
3. C₃₁×#3283	C₂	semidirect
4. C₃₁×#1707	C₂	semidirect
5. C₃₁×#1691	C₂	semidirect
6. C₃₁×#1675	C₂	semidirect
7. C₃₁×#1669	C₂	semidirect
8. C₃₁×#1646	C₂	semidirect
9. C₃₁×#1635	C₂	semidirect
10. C₃₁×#1589	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
140. #19953	C₃₁	direct (N × H)
176. C₈×C₃₁	#135	semidirect
178. #3354	C₂×C₃₁	semidirect
179. #3341	C₂×C₃₁	semidirect
180. #3283	C₂×C₃₁	semidirect
181. #1707	C₂×C₃₁	semidirect
182. #1691	C₂×C₃₁	semidirect
183. #1675	C₂×C₃₁	semidirect
184. #1669	C₂×C₃₁	semidirect
185. #1646	C₂×C₃₁	semidirect
186. #1635	C₂×C₃₁	semidirect
187. #1589	C₂×C₃₁	semidirect
188. C₈×D₈	C₂×C₃₁	semidirect
298. C₃₁	#19953	direct (N × H)
379. C₈	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3354	2
C₃₁×#3341	2
C₃₁×#3283	2
C₃₁×#1707	2
C₃₁×#1691	2
C₃₁×#1675	2
C₃₁×#1669	2
C₃₁×#1646	2
C₃₁×#1635	2
C₃₁×#1589	2
C₈×D₈×C₃₁	2
#19953	31

Subgroups — C31×#19953 #21476712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19953 #21476712