Subgroups — C₃₁×#3194 #16843127

Order 3968.

466 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#525	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#483	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#482	C₂	1	2	yes	maximal, complemented
5	C₃₁×#480	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#469	C₂	1	2	yes	maximal, complemented
7	C₃₁×#468	C₂	1	2	yes	maximal, complemented
8	C₃₁×#450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#449	C₂	1	2	yes	maximal, complemented
10	C₃₁×#447	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#407	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₃₁	—	2	1	no	—
14	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
17	C₂×D₈×C₃₁	C₂²	1	2	yes	complemented
18	C₂×M₁₆×C₃₁	C₂²	1	2	yes	complemented
19	C₃₁×#123	—	2	1	no	—
20	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#121	—	2	1	no	—
23	C₃₁×#121	—	2	1	no	—
24	C₃₁×#121	C₂²	1	1	yes	characteristic
25	C₃₁×#121	C₂²	1	2	yes	—
26	C₃₁×#121	C₂²	1	2	yes	—
27	C₃₁×#121	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#120	—	2	1	no	—
29	C₄×D₄×C₃₁	—	2	1	no	—
30	C₄×D₄×C₃₁	—	2	1	no	—
31	C₃₁×#117	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
34	C₃₁×#107	C₂²	1	2	yes	complemented
35	C₃₁×#106	C₂²	1	2	yes	complemented
36	C₃₁×#103	C₂²	1	2	yes	complemented
37	C₃₁×#102	C₂²	1	2	yes	complemented
38	C₃₁×#102	C₂²	1	2	yes	complemented
39	C₃₁×#102	C₂²	1	2	yes	complemented
40	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
43	C₂⁴×C₃₁	—	2	1	no	—
44	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
45	#41×C₃₁	—	4	2	no	—
46	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	—	2	1	no	—
49	C₂×D₄×C₃₁	—	2	1	no	—
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	—	2	1	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	4	4	no	—
55	C₂×D₄×C₃₁	C₂³	1	2	yes	—
56	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	1	no	—
58	C₂²×C₄×C₃₁	—	2	1	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	SD₁₆×C₃₁	—	4	2	no	—
67	D₈×C₃₁	—	4	2	no	—
68	M₁₆×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	C₂³	1	2	yes	—
70	C₂×C₈×C₃₁	C₂³	1	2	yes	—
71	C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂²⋊C₄×C₃₁	—	2	1	no	—
79	C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂²⋊C₄×C₃₁	—	2	1	no	—
83	C₂²⋊C₄×C₃₁	—	2	1	no	—
84	C₄²×C₃₁	—	2	1	no	—
85	C₂³×C₃₁	—	2	2	no	—
86	C₂³×C₃₁	—	2	1	no	—
87	C₂³×C₃₁	—	2	2	no	—
88	C₂³×C₃₁	—	4	1	no	—
89	C₂³×C₃₁	—	4	1	no	—
90	C₂³×C₃₁	—	4	2	no	—
91	C₂³×C₃₁	—	4	2	no	—
92	C₂³×C₃₁	—	4	2	no	—
93	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
96	Q₈×C₃₁	—	2	2	no	—
97	D₄×C₃₁	—	2	2	no	—
98	D₄×C₃₁	—	2	2	no	—
99	D₄×C₃₁	—	2	2	no	—
100	D₄×C₃₁	—	4	2	no	—
101	D₄×C₃₁	—	4	2	no	—
102	D₄×C₃₁	—	4	4	no	—
103	D₄×C₃₁	—	4	4	no	—
104	D₄×C₃₁	—	4	2	no	—
105	D₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	4	1	no	—
114	C₂×C₄×C₃₁	—	4	1	no	—
115	C₂×C₄×C₃₁	—	4	1	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
122	C₈×C₃₁	—	2	2	no	—
123	C₈×C₃₁	—	2	2	no	—
124	#3194	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₃₁	—	2	1	no	—
126	C₂²×C₃₁	—	2	1	no	—
127	C₂²×C₃₁	—	2	1	no	—
128	C₂²×C₃₁	—	2	1	no	—
129	C₂²×C₃₁	—	2	1	no	—
130	C₂²×C₃₁	—	2	1	no	—
131	C₂²×C₃₁	—	2	1	no	—
132	C₂²×C₃₁	—	2	1	no	—
133	C₂²×C₃₁	—	4	4	no	—
134	C₂²×C₃₁	—	4	1	no	—
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	—	4	1	no	—
138	C₂²×C₃₁	—	4	1	no	—
139	C₂²×C₃₁	—	4	2	no	—
140	C₂²×C₃₁	—	8	4	no	—
141	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
142	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	2	1	no	—
146	C₄×C₃₁	—	4	1	no	—
147	C₄×C₃₁	—	4	2	no	—
148	C₄×C₃₁	—	4	1	no	—
149	C₄×C₃₁	—	4	1	no	—
150	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
151	#525	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂×#121	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#483	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#482	C₂×C₃₁	1	2	yes	complemented
155	#480	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#469	C₂×C₃₁	1	2	yes	complemented
157	#468	C₂×C₃₁	1	2	yes	complemented
158	#450	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#449	C₂×C₃₁	1	2	yes	complemented
160	#447	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#407	C₂×C₃₁	1	1	yes	characteristic, complemented
162	C₂×C₃₁	—	2	2	no	—
163	C₂×C₃₁	—	4	1	no	—
164	C₂×C₃₁	—	4	1	no	—
165	C₂×C₃₁	—	8	1	no	—
166	C₂×C₃₁	—	8	2	no	—
167	C₂×C₃₁	#534	1	1	yes	characteristic
168	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₃₁	#575	1	1	yes	characteristic
170	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic, complemented
171	C₂²×D₄	—	2	1	no	—
172	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
173	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
175	C₂×D₈	C₂²×C₃₁	1	2	yes	complemented
176	C₂×M₁₆	C₂²×C₃₁	1	2	yes	complemented
177	#123	—	2	1	no	—
178	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
179	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
180	#121	—	2	1	no	—
181	#121	—	2	1	no	—
182	#121	C₂²×C₃₁	1	2	yes	—
183	#121	C₂²×C₃₁	1	2	yes	—
184	#121	C₂²×C₃₁	1	1	yes	characteristic
185	#121	C₂²×C₃₁	1	1	yes	characteristic, complemented
186	#120	—	2	1	no	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	—	2	1	no	—
189	#117	C₂²×C₃₁	1	1	yes	characteristic, complemented
190	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	#107	C₂²×C₃₁	1	2	yes	complemented
193	#106	C₂²×C₃₁	1	2	yes	complemented
194	#103	C₂²×C₃₁	1	2	yes	complemented
195	#102	C₂²×C₃₁	1	2	yes	complemented
196	#102	C₂²×C₃₁	1	2	yes	complemented
197	#102	C₂²×C₃₁	1	2	yes	complemented
198	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
199	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
200	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
201	C₃₁	#3194	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	—	2	1	no	—
203	C₂⁴	D₄×C₃₁	1	1	yes	characteristic
204	#41	—	4	2	no	—
205	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	C₂³×C₃₁	1	2	yes	—
215	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
225	SD₁₆	—	4	2	no	—
226	D₈	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	C₂×C₈	C₂³×C₃₁	1	2	yes	—
229	C₂×C₈	C₂³×C₃₁	1	2	yes	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
234	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
235	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
254	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	2	2	no	—
258	D₄	—	2	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	4	2	no	—
264	D₄	—	4	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	2	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	8	4	no	—
299	C₂²	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
300	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	4	1	no	—
311	C₂	—	4	1	no	—
312	C₂	—	8	2	no	—
313	C₂	—	8	1	no	—
314	C₂	C₃₁×#534	1	1	yes	characteristic
315	C₂	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
316	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#525	C₂	semidirect
2. C₂×C₃₁×#121	C₂	semidirect
3. C₃₁×#483	C₂	semidirect
4. C₃₁×#482	C₂	semidirect
5. C₃₁×#480	C₂	semidirect
6. C₃₁×#469	C₂	semidirect
7. C₃₁×#468	C₂	semidirect
8. C₃₁×#450	C₂	semidirect
9. C₃₁×#449	C₂	semidirect
10. C₃₁×#447	C₂	semidirect
11. C₃₁×#407	C₂	semidirect
12. C₂×#41×C₃₁	C₂²	semidirect
16. C₂×SD₁₆×C₃₁	C₂²	semidirect
17. C₂×D₈×C₃₁	C₂²	semidirect
18. C₂×M₁₆×C₃₁	C₂²	semidirect
21. C₃₁×C₂²⋊Q₈	C₂²	semidirect
27. C₃₁×#121	C₂²	semidirect
31. C₃₁×#117	C₂²	semidirect
34. C₃₁×#107	C₂²	semidirect
35. C₃₁×#106	C₂²	semidirect
36. C₃₁×#103	C₂²	semidirect
37. C₃₁×#102	C₂²	semidirect
38. C₃₁×#102	C₂²	semidirect
39. C₃₁×#102	C₂²	semidirect
42. C₃₁×C₂²⋊C₈	C₂²	semidirect
124. #3194	C₃₁	direct (N × H)
151. #525	C₂×C₃₁	semidirect
152. C₂×#121	C₂×C₃₁	semidirect
153. #483	C₂×C₃₁	semidirect
154. #482	C₂×C₃₁	semidirect
155. #480	C₂×C₃₁	semidirect
156. #469	C₂×C₃₁	semidirect
157. #468	C₂×C₃₁	semidirect
158. #450	C₂×C₃₁	semidirect
159. #449	C₂×C₃₁	semidirect
160. #447	C₂×C₃₁	semidirect
161. #407	C₂×C₃₁	semidirect
170. C₂×#41	C₂²×C₃₁	semidirect
174. C₂×SD₁₆	C₂²×C₃₁	semidirect
175. C₂×D₈	C₂²×C₃₁	semidirect
176. C₂×M₁₆	C₂²×C₃₁	semidirect
179. C₂²⋊Q₈	C₂²×C₃₁	semidirect
185. #121	C₂²×C₃₁	semidirect
189. #117	C₂²×C₃₁	semidirect
192. #107	C₂²×C₃₁	semidirect
193. #106	C₂²×C₃₁	semidirect
194. #103	C₂²×C₃₁	semidirect
195. #102	C₂²×C₃₁	semidirect
196. #102	C₂²×C₃₁	semidirect
197. #102	C₂²×C₃₁	semidirect
200. C₂²⋊C₈	C₂²×C₃₁	semidirect
201. C₃₁	#3194	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×#525	2
C₂×C₃₁×#121	2
C₃₁×#483	2
C₃₁×#482	2
C₃₁×#480	2
C₃₁×#469	2
C₃₁×#468	2
C₃₁×#450	2
C₃₁×#449	2
C₃₁×#447	2
C₃₁×#407	2
#3194	31

Subgroups — C31×#3194 #16843127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3194 #16843127