Subgroups — C₃₁×#19994 #21476753

Order 7936.

652 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3341	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3305	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1677	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1631	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1624	C₂	1	2	yes	maximal, complemented
10	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#555	C₂²	1	2	yes	—
13	C₃₁×#550	C₂²	1	2	yes	—
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#493	—	2	1	no	—
16	C₃₁×#493	C₂²	1	1	yes	characteristic
17	C₃₁×#492	C₂²	1	2	yes	—
18	C₃₁×#486	C₂²	1	2	yes	—
19	C₃₁×#482	—	2	2	no	—
20	C₃₁×#482	—	2	2	no	—
21	C₃₁×#477	C₂²	1	2	yes	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#466	—	2	1	no	—
24	C₃₁×#466	—	2	2	no	—
25	C₃₁×#459	—	2	2	no	—
26	C₃₁×#459	C₂²	1	2	yes	—
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#449	—	2	2	no	—
29	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₃₁	—	2	1	no	—
32	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	2	yes	—
35	C₃₁×#427	—	2	1	no	—
36	C₃₁×#416	—	2	2	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#334	C₂²	1	1	yes	characteristic
39	C₃₁×#333	C₂²	1	2	yes	—
40	C₃₁×#331	C₂²	1	2	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
43	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂×SD₁₆×C₃₁	—	4	2	no	—
47	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
48	C₂×D₈×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	1	no	—
51	C₂×D₈×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	—	2	1	no	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₃₁×#127	—	2	2	no	—
57	C₃₁×#127	C₂³	1	2	yes	—
58	C₃₁×#124	—	4	2	no	—
59	C₃₁×#121	—	2	2	no	—
60	C₃₁×#121	—	2	2	no	—
61	C₃₁×#121	—	2	2	no	—
62	C₃₁×#121	—	4	2	no	—
63	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	C₂³	1	2	yes	—
66	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#117	—	2	2	no	—
68	C₃₁×#117	—	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×#106	—	2	2	no	—
74	C₃₁×#106	D₄	1	1	yes	characteristic
75	C₃₁×C₄⋊C₈	—	2	1	no	—
76	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
77	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₃₁×#103	—	2	2	no	—
80	C₃₁×#103	D₄	1	1	yes	characteristic
81	C₃₁×#102	—	2	2	no	—
82	C₃₁×#102	—	2	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×C₂²⋊C₈	—	2	2	no	—
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₄×C₈×C₃₁	—	2	1	no	—
91	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	#41×C₃₁	—	4	4	no	—
95	#41×C₃₁	—	8	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂×D₄×C₃₁	—	4	2	no	—
103	C₂×D₄×C₃₁	—	4	2	no	—
104	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	4	2	no	—
109	SD₁₆×C₃₁	—	2	2	no	—
110	SD₁₆×C₃₁	—	8	4	no	—
111	D₈×C₃₁	—	2	2	no	—
112	D₈×C₃₁	—	4	2	no	—
113	D₈×C₃₁	—	8	1	no	—
114	D₈×C₃₁	—	8	2	no	—
115	D₈×C₃₁	—	8	2	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	4	2	no	—
121	C₂×C₈×C₃₁	—	8	2	no	—
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²⋊C₄×C₃₁	—	2	2	no	—
133	C₂²⋊C₄×C₃₁	—	2	1	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	4	2	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₄²×C₃₁	—	2	1	no	—
138	C₄²×C₃₁	—	4	2	no	—
139	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
140	#19994	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	1	no	—
142	C₂³×C₃₁	—	2	2	no	—
143	C₂³×C₃₁	—	4	2	no	—
144	Q₈×C₃₁	—	2	1	no	—
145	Q₈×C₃₁	—	8	2	no	—
146	Q₈×C₃₁	#135	1	2	yes	—
147	D₄×C₃₁	—	2	2	no	—
148	D₄×C₃₁	—	2	1	no	—
149	D₄×C₃₁	—	2	1	no	—
150	D₄×C₃₁	—	2	2	no	—
151	D₄×C₃₁	—	4	2	no	—
152	D₄×C₃₁	—	4	2	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	2	no	—
155	D₄×C₃₁	—	8	4	no	—
156	D₄×C₃₁	—	8	2	no	—
157	D₄×C₃₁	—	16	2	no	—
158	D₄×C₃₁	—	16	2	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	8	4	no	—
169	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
172	C₈×C₃₁	—	2	2	no	—
173	C₈×C₃₁	—	2	1	no	—
174	C₈×C₃₁	—	2	1	no	—
175	C₈×C₃₁	—	2	1	no	—
176	C₈×C₃₁	—	4	1	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	—	4	1	no	—
179	C₈×C₃₁	#135	1	2	yes	—
180	#3391	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#3341	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#3305	C₂×C₃₁	1	2	yes	complemented
183	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1677	C₂×C₃₁	1	2	yes	complemented
185	#1676	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#1631	C₂×C₃₁	1	2	yes	complemented
188	#1624	C₂×C₃₁	1	2	yes	complemented
189	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
191	C₂²×C₃₁	—	4	2	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	—	4	2	no	—
196	C₂²×C₃₁	—	4	1	no	—
197	C₂²×C₃₁	—	8	2	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	D₄²	1	1	yes	characteristic
201	C₄×C₃₁	—	2	2	no	—
202	C₄×C₃₁	—	2	1	no	—
203	C₄×C₃₁	—	4	2	no	—
204	C₄×C₃₁	—	4	1	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	8	2	no	—
207	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
208	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
209	C₄×C₃₁	#576	1	1	yes	characteristic
210	C₄×C₃₁	#577	1	1	yes	characteristic
211	#555	C₂²×C₃₁	1	2	yes	—
212	#550	C₂²×C₃₁	1	2	yes	—
213	#500	C₂²×C₃₁	1	1	yes	characteristic
214	#493	—	2	1	no	—
215	#493	C₂²×C₃₁	1	1	yes	characteristic
216	#492	C₂²×C₃₁	1	2	yes	—
217	#486	C₂²×C₃₁	1	2	yes	—
218	#482	—	2	2	no	—
219	#482	—	2	2	no	—
220	#477	C₂²×C₃₁	1	2	yes	—
221	#476	C₂²×C₃₁	1	2	yes	—
222	#466	—	2	1	no	—
223	#466	—	2	2	no	—
224	#459	—	2	2	no	—
225	#459	C₂²×C₃₁	1	2	yes	—
226	#459	C₂²×C₃₁	1	2	yes	—
227	#449	—	2	2	no	—
228	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
229	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
230	C₄×D₈	—	2	1	no	—
231	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
233	C₈×D₄	C₂²×C₃₁	1	2	yes	—
234	#427	—	2	1	no	—
235	#416	—	2	2	no	—
236	#335	C₂²×C₃₁	1	1	yes	characteristic
237	#334	C₂²×C₃₁	1	1	yes	characteristic
238	#333	C₂²×C₃₁	1	2	yes	—
239	#331	C₂²×C₃₁	1	2	yes	—
240	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
241	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
242	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₈²	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₃₁	—	8	2	no	—
245	C₂×C₃₁	—	8	1	no	—
246	C₂×C₃₁	—	16	2	no	—
247	C₂×C₃₁	#3280	1	1	yes	characteristic
248	C₂×C₃₁	#3284	1	1	yes	characteristic
249	C₂×C₃₁	#3292	1	1	yes	characteristic
250	C₂×#41	—	2	2	no	—
251	C₂×SD₁₆	—	4	2	no	—
252	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
253	C₂×D₈	—	2	2	no	—
254	C₂×D₈	—	2	2	no	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₈	—	2	1	no	—
259	C₂²×C₈	—	2	2	no	—
260	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
261	#127	—	2	2	no	—
262	#127	C₂³×C₃₁	1	2	yes	—
263	#124	—	4	2	no	—
264	#121	—	2	2	no	—
265	#121	—	2	2	no	—
266	#121	—	2	2	no	—
267	#121	—	4	2	no	—
268	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₄×D₄	—	2	2	no	—
270	C₄×D₄	C₂³×C₃₁	1	2	yes	—
271	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
272	#117	—	2	2	no	—
273	#117	—	2	2	no	—
274	#107	—	2	1	no	—
275	#107	—	2	1	no	—
276	#107	—	2	2	no	—
277	#107	D₄×C₃₁	1	1	yes	characteristic
278	#106	—	2	2	no	—
279	#106	D₄×C₃₁	1	1	yes	characteristic
280	C₄⋊C₈	—	2	1	no	—
281	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
282	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
283	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
284	#103	—	2	2	no	—
285	#103	D₄×C₃₁	1	1	yes	characteristic
286	#102	—	2	2	no	—
287	#102	—	2	2	no	—
288	#102	—	2	2	no	—
289	#102	—	2	2	no	—
290	#102	—	2	2	no	—
291	#102	D₄×C₃₁	1	2	yes	—
292	#102	D₄×C₃₁	1	1	yes	characteristic
293	C₂²⋊C₈	—	2	2	no	—
294	C₂²⋊C₈	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₃₁	#19994	1	1	yes	characteristic, direct product factor, Sylow
300	#41	—	4	4	no	—
301	#41	—	8	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	—	4	2	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
311	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	4	2	no	—
315	SD₁₆	—	2	2	no	—
316	SD₁₆	—	8	4	no	—
317	D₈	—	2	2	no	—
318	D₈	—	4	2	no	—
319	D₈	—	8	1	no	—
320	D₈	—	8	2	no	—
321	D₈	—	8	2	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	4	2	no	—
327	C₂×C₈	—	8	2	no	—
328	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²⋊C₄	—	2	2	no	—
339	C₂²⋊C₄	—	2	1	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	4	2	no	—
345	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
346	C₂³	—	2	1	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	4	2	no	—
349	Q₈	—	2	1	no	—
350	Q₈	—	8	2	no	—
351	Q₈	C₃₁×#135	1	2	yes	—
352	D₄	—	2	2	no	—
353	D₄	—	2	1	no	—
354	D₄	—	2	1	no	—
355	D₄	—	2	2	no	—
356	D₄	—	4	2	no	—
357	D₄	—	4	2	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	8	4	no	—
361	D₄	—	8	2	no	—
362	D₄	—	16	2	no	—
363	D₄	—	16	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	8	4	no	—
374	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
376	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
377	C₈	—	2	2	no	—
378	C₈	—	2	1	no	—
379	C₈	—	2	1	no	—
380	C₈	—	2	1	no	—
381	C₈	—	4	1	no	—
382	C₈	—	4	2	no	—
383	C₈	—	4	1	no	—
384	C₈	C₃₁×#135	1	2	yes	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	8	2	no	—
392	C₂²	—	8	2	no	—
393	C₂²	—	8	2	no	—
394	C₂²	D₄²×C₃₁	1	1	yes	characteristic
395	C₄	—	2	2	no	—
396	C₄	—	2	1	no	—
397	C₄	—	4	2	no	—
398	C₄	—	4	1	no	—
399	C₄	—	4	1	no	—
400	C₄	—	8	2	no	—
401	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
402	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
403	C₄	C₃₁×#576	1	1	yes	characteristic
404	C₄	C₃₁×#577	1	1	yes	characteristic
405	C₂	—	8	2	no	—
406	C₂	—	8	1	no	—
407	C₂	—	16	2	no	—
408	C₂	C₃₁×#3280	1	1	yes	characteristic
409	C₂	C₃₁×#3284	1	1	yes	characteristic
410	C₂	C₃₁×#3292	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×#3391	C₂	semidirect
2. C₃₁×#3341	C₂	semidirect
3. C₃₁×#3305	C₂	semidirect
4. C₃₁×#1706	C₂	semidirect
5. C₃₁×#1677	C₂	semidirect
6. C₃₁×#1676	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1631	C₂	semidirect
9. C₃₁×#1624	C₂	semidirect
10. C₈×SD₁₆×C₃₁	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
140. #19994	C₃₁	direct (N × H)
180. #3391	C₂×C₃₁	semidirect
181. #3341	C₂×C₃₁	semidirect
182. #3305	C₂×C₃₁	semidirect
183. #1706	C₂×C₃₁	semidirect
184. #1677	C₂×C₃₁	semidirect
185. #1676	C₂×C₃₁	semidirect
186. #1667	C₂×C₃₁	semidirect
187. #1631	C₂×C₃₁	semidirect
188. #1624	C₂×C₃₁	semidirect
189. C₈×SD₁₆	C₂×C₃₁	semidirect
190. C₈×D₈	C₂×C₃₁	semidirect
299. C₃₁	#19994	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×#3391	2
C₃₁×#3341	2
C₃₁×#3305	2
C₃₁×#1706	2
C₃₁×#1677	2
C₃₁×#1676	2
C₃₁×#1667	2
C₃₁×#1631	2
C₃₁×#1624	2
C₈×SD₁₆×C₃₁	2
C₈×D₈×C₃₁	2
#19994	31

Subgroups — C31×#19994 #21476753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19994 #21476753