Subgroups — C₃₁×#33770 #21490529

Order 7936.

576 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3061	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2235	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2233	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2219	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2218	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2217	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2189	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2186	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#135	C₂²	1	2	yes	—
11	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
12	C₃₁×#537	—	2	2	no	—
13	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#468	—	2	2	no	—
19	C₃₁×#468	—	2	2	no	—
20	C₃₁×#467	C₂²	1	2	yes	—
21	C₃₁×#467	C₂²	1	2	yes	—
22	C₃₁×#466	C₂²	1	2	yes	—
23	C₃₁×#466	C₂²	1	2	yes	—
24	C₂×C₃₁×#107	C₂²	1	2	yes	—
25	C₃₁×#417	—	2	2	no	—
26	C₃₁×#367	C₂²	1	2	yes	—
27	C₃₁×#367	C₂²	1	1	yes	characteristic
28	C₃₁×#367	C₂²	1	1	yes	characteristic
29	C₃₁×#358	C₂²	1	2	yes	—
30	C₃₁×#358	C₂²	1	2	yes	—
31	C₃₁×#357	C₂²	1	2	yes	—
32	C₃₁×#357	C₂²	1	2	yes	—
33	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₂×#41×C₃₁	—	2	2	no	—
36	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×#135	—	4	2	no	—
38	C₃₁×#135	—	4	2	no	—
39	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₃₁	—	2	2	no	—
41	C₂×D₈×C₃₁	C₂³	1	2	yes	—
42	C₃₁×#131	—	4	1	no	—
43	C₃₁×#131	—	4	1	no	—
44	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
45	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₂²⋊Q₈	—	2	2	no	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	4	2	no	—
51	C₃₁×#121	—	2	2	no	—
52	C₃₁×#121	—	2	2	no	—
53	C₃₁×#121	—	4	2	no	—
54	C₃₁×#121	—	4	2	no	—
55	C₃₁×#121	—	4	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₃₁×Q₃₂	—	4	1	no	—
58	C₃₁×SD₃₂	—	4	2	no	—
59	C₃₁×D₁₆	—	4	1	no	—
60	C₃₁×M₃₂	—	4	1	no	—
61	C₂×C₁₆×C₃₁	—	4	1	no	—
62	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#107	—	2	2	no	—
66	C₃₁×#107	C₂³	1	2	yes	—
67	C₃₁×#107	C₂³	1	2	yes	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×#102	—	2	2	no	—
71	C₃₁×#102	—	2	2	no	—
72	#41×C₃₁	—	4	1	no	—
73	#41×C₃₁	—	4	1	no	—
74	#41×C₃₁	—	4	4	no	—
75	#41×C₃₁	—	4	4	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂×D₄×C₃₁	—	4	2	no	—
81	C₂×D₄×C₃₁	—	4	2	no	—
82	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
83	C₂×D₄×C₃₁	D₈	1	2	yes	—
84	C₂²×C₄×C₃₁	—	4	2	no	—
85	C₂²×C₄×C₃₁	—	4	2	no	—
86	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₃₁	—	2	2	no	—
89	Q₁₆×C₃₁	—	2	2	no	—
90	SD₁₆×C₃₁	—	4	4	no	—
91	D₈×C₃₁	—	2	2	no	—
92	D₈×C₃₁	—	2	2	no	—
93	M₁₆×C₃₁	—	4	2	no	—
94	M₁₆×C₃₁	—	4	1	no	—
95	C₂×C₈×C₃₁	—	2	1	no	—
96	C₂×C₈×C₃₁	—	2	1	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	4	2	no	—
105	C₄⋊C₄×C₃₁	—	4	2	no	—
106	C₄⋊C₄×C₃₁	—	4	2	no	—
107	C₂²⋊C₄×C₃₁	—	4	2	no	—
108	C₂²⋊C₄×C₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	4	2	no	—
111	C₁₆×C₃₁	—	2	2	no	—
112	C₁₆×C₃₁	—	2	1	no	—
113	C₁₆×C₃₁	—	2	1	no	—
114	#33770	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	4	2	no	—
116	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
117	C₂³×C₃₁	C₂×D₈	1	2	yes	—
118	Q₈×C₃₁	—	4	1	no	—
119	Q₈×C₃₁	—	4	2	no	—
120	Q₈×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	4	2	no	—
122	D₄×C₃₁	—	4	1	no	—
123	D₄×C₃₁	—	4	2	no	—
124	D₄×C₃₁	—	4	2	no	—
125	D₄×C₃₁	—	8	4	no	—
126	D₄×C₃₁	—	16	2	no	—
127	D₄×C₃₁	—	16	2	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	8	4	no	—
136	C₂×C₄×C₃₁	—	16	2	no	—
137	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₃₁	—	2	1	no	—
141	C₈×C₃₁	—	2	2	no	—
142	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
143	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
144	#3061	C₂×C₃₁	1	2	yes	complemented
145	#2235	C₂×C₃₁	1	2	yes	complemented
146	#2233	C₂×C₃₁	1	2	yes	complemented
147	#2219	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#2218	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#2217	C₂×C₃₁	1	2	yes	complemented
150	#2189	C₂×C₃₁	1	2	yes	complemented
151	#2186	C₂×C₃₁	1	2	yes	complemented
152	#2113	C₂×C₃₁	1	1	yes	characteristic, complemented
153	C₂²×C₃₁	—	2	2	no	—
154	C₂²×C₃₁	—	2	2	no	—
155	C₂²×C₃₁	—	2	2	no	—
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	—	2	1	no	—
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	8	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	—	8	2	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	#534	1	1	yes	characteristic
168	C₄×C₃₁	#575	1	1	yes	characteristic
169	C₂×#135	C₂²×C₃₁	1	2	yes	—
170	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
171	#537	—	2	2	no	—
172	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
174	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
176	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
177	#468	—	2	2	no	—
178	#468	—	2	2	no	—
179	#467	C₂²×C₃₁	1	2	yes	—
180	#467	C₂²×C₃₁	1	2	yes	—
181	#466	C₂²×C₃₁	1	2	yes	—
182	#466	C₂²×C₃₁	1	2	yes	—
183	C₂×#107	C₂²×C₃₁	1	2	yes	—
184	#417	—	2	2	no	—
185	#367	C₂²×C₃₁	1	2	yes	—
186	#367	C₂²×C₃₁	1	1	yes	characteristic
187	#367	C₂²×C₃₁	1	1	yes	characteristic
188	#358	C₂²×C₃₁	1	2	yes	—
189	#358	C₂²×C₃₁	1	2	yes	—
190	#357	C₂²×C₃₁	1	2	yes	—
191	#357	C₂²×C₃₁	1	2	yes	—
192	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
193	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
194	C₂×C₃₁	—	4	2	no	—
195	C₂×C₃₁	—	4	1	no	—
196	C₂×C₃₁	—	16	2	no	—
197	C₂×C₃₁	#3186	1	1	yes	characteristic
198	C₂×C₃₁	#3416	1	2	yes	—
199	C₂×#41	—	2	2	no	—
200	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
201	#135	—	4	2	no	—
202	#135	—	4	2	no	—
203	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×D₈	C₂³×C₃₁	1	2	yes	—
206	#131	—	4	1	no	—
207	#131	—	4	1	no	—
208	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
209	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
211	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	4	2	no	—
215	#121	—	2	2	no	—
216	#121	—	2	2	no	—
217	#121	—	4	2	no	—
218	#121	—	4	2	no	—
219	#121	—	4	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	Q₃₂	—	4	1	no	—
222	SD₃₂	—	4	2	no	—
223	D₁₆	—	4	1	no	—
224	M₃₂	—	4	1	no	—
225	C₂×C₁₆	—	4	1	no	—
226	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
227	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
228	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
229	#107	—	2	2	no	—
230	#107	C₂³×C₃₁	1	2	yes	—
231	#107	C₂³×C₃₁	1	2	yes	—
232	#103	—	2	2	no	—
233	#103	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	—	2	2	no	—
236	C₃₁	#33770	1	1	yes	characteristic, direct product factor, Sylow
237	#41	—	4	1	no	—
238	#41	—	4	1	no	—
239	#41	—	4	4	no	—
240	#41	—	4	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
248	C₂×D₄	D₈×C₃₁	1	2	yes	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	SD₁₆	—	4	4	no	—
256	D₈	—	2	2	no	—
257	D₈	—	2	2	no	—
258	M₁₆	—	4	2	no	—
259	M₁₆	—	4	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
266	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₁₆	—	2	2	no	—
277	C₁₆	—	2	1	no	—
278	C₁₆	—	2	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
281	C₂³	C₂×D₈×C₃₁	1	2	yes	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	1	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	8	4	no	—
290	D₄	—	16	2	no	—
291	D₄	—	16	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	16	2	no	—
301	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
303	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	2	no	—
306	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
307	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₃₁×#534	1	1	yes	characteristic
323	C₄	C₃₁×#575	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	4	1	no	—
326	C₂	—	16	2	no	—
327	C₂	C₃₁×#3186	1	1	yes	characteristic
328	C₂	C₃₁×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3061	C₂	semidirect
2. C₃₁×#2235	C₂	semidirect
3. C₃₁×#2233	C₂	semidirect
4. C₃₁×#2219	C₂	semidirect
5. C₃₁×#2218	C₂	semidirect
6. C₃₁×#2217	C₂	semidirect
7. C₃₁×#2189	C₂	semidirect
8. C₃₁×#2186	C₂	semidirect
9. C₃₁×#2113	C₂	semidirect
114. #33770	C₃₁	direct (N × H)
144. #3061	C₂×C₃₁	semidirect
145. #2235	C₂×C₃₁	semidirect
146. #2233	C₂×C₃₁	semidirect
147. #2219	C₂×C₃₁	semidirect
148. #2218	C₂×C₃₁	semidirect
149. #2217	C₂×C₃₁	semidirect
150. #2189	C₂×C₃₁	semidirect
151. #2186	C₂×C₃₁	semidirect
152. #2113	C₂×C₃₁	semidirect
236. C₃₁	#33770	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×#3061	2
C₃₁×#2235	2
C₃₁×#2233	2
C₃₁×#2219	2
C₃₁×#2218	2
C₃₁×#2217	2
C₃₁×#2189	2
C₃₁×#2186	2
C₃₁×#2113	2
#33770	31

Subgroups — C31×#33770 #21490529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33770 #21490529