Subgroups — C₃₁×#33769 #21490528

Order 7936.

576 conjugacy classes of subgroups.

The subgroup lattice has 334 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×#2236	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2232	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2220	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2215	C₂	1	1	yes	maximal, characteristic, 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×#366	C₂²	1	1	yes	characteristic
27	C₃₁×#366	C₂²	1	1	yes	characteristic
28	C₃₁×#366	C₂²	1	2	yes	—
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	1	yes	characteristic
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
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	1	no	—
112	C₁₆×C₃₁	—	2	1	no	—
113	C₁₆×C₃₁	—	2	2	no	—
114	#33769	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	#2236	C₂×C₃₁	1	2	yes	complemented
146	#2232	C₂×C₃₁	1	2	yes	complemented
147	#2220	C₂×C₃₁	1	2	yes	complemented
148	#2216	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#2215	C₂×C₃₁	1	1	yes	characteristic, 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	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	—	2	2	no	—
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	8	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	8	2	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	#534	1	1	yes	characteristic
169	C₄×C₃₁	#575	1	1	yes	characteristic
170	C₂×#135	C₂²×C₃₁	1	2	yes	—
171	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
172	#537	—	2	2	no	—
173	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
175	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
177	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
178	#468	—	2	2	no	—
179	#468	—	2	2	no	—
180	#467	C₂²×C₃₁	1	2	yes	—
181	#467	C₂²×C₃₁	1	2	yes	—
182	#466	C₂²×C₃₁	1	2	yes	—
183	#466	C₂²×C₃₁	1	2	yes	—
184	C₂×#107	C₂²×C₃₁	1	2	yes	—
185	#417	—	2	2	no	—
186	#366	C₂²×C₃₁	1	1	yes	characteristic
187	#366	C₂²×C₃₁	1	1	yes	characteristic
188	#366	C₂²×C₃₁	1	2	yes	—
189	#358	C₂²×C₃₁	1	2	yes	—
190	#358	C₂²×C₃₁	1	2	yes	—
191	#357	C₂²×C₃₁	1	2	yes	—
192	#357	C₂²×C₃₁	1	2	yes	—
193	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
194	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
195	C₂×C₃₁	—	4	2	no	—
196	C₂×C₃₁	—	4	1	no	—
197	C₂×C₃₁	—	16	2	no	—
198	C₂×C₃₁	#3186	1	1	yes	characteristic
199	C₂×C₃₁	#3415	1	1	yes	characteristic
200	C₂×C₃₁	#3417	1	1	yes	characteristic
201	C₂×#41	—	2	2	no	—
202	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
203	#135	—	4	2	no	—
204	#135	—	4	2	no	—
205	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×D₈	C₂³×C₃₁	1	2	yes	—
208	#131	—	4	1	no	—
209	#131	—	4	1	no	—
210	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
211	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
213	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	#121	—	4	2	no	—
220	#121	—	4	2	no	—
221	#121	—	4	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	Q₃₂	—	4	1	no	—
224	SD₃₂	—	4	2	no	—
225	D₁₆	—	4	1	no	—
226	M₃₂	—	4	1	no	—
227	C₂×C₁₆	—	4	1	no	—
228	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
230	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
231	#107	—	2	2	no	—
232	#107	C₂³×C₃₁	1	2	yes	—
233	#107	C₂³×C₃₁	1	2	yes	—
234	#103	—	2	2	no	—
235	#103	—	2	2	no	—
236	#102	—	2	2	no	—
237	#102	—	2	2	no	—
238	C₃₁	#33769	1	1	yes	characteristic, direct product factor, Sylow
239	#41	—	4	1	no	—
240	#41	—	4	1	no	—
241	#41	—	4	4	no	—
242	#41	—	4	4	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
250	C₂×D₄	D₈×C₃₁	1	2	yes	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	2	no	—
257	SD₁₆	—	4	4	no	—
258	D₈	—	2	2	no	—
259	D₈	—	2	2	no	—
260	M₁₆	—	4	2	no	—
261	M₁₆	—	4	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
268	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₁₆	—	2	1	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	—	2	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
283	C₂³	C₂×D₈×C₃₁	1	2	yes	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	4	no	—
292	D₄	—	16	2	no	—
293	D₄	—	16	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	2	no	—
308	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
309	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	C₃₁×#534	1	1	yes	characteristic
326	C₄	C₃₁×#575	1	1	yes	characteristic
327	C₂	—	4	2	no	—
328	C₂	—	4	1	no	—
329	C₂	—	16	2	no	—
330	C₂	C₃₁×#3186	1	1	yes	characteristic
331	C₂	C₃₁×#3415	1	1	yes	characteristic
332	C₂	C₃₁×#3417	1	1	yes	characteristic

	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×#2236	C₂	semidirect
3. C₃₁×#2232	C₂	semidirect
4. C₃₁×#2220	C₂	semidirect
5. C₃₁×#2216	C₂	semidirect
6. C₃₁×#2215	C₂	semidirect
7. C₃₁×#2189	C₂	semidirect
8. C₃₁×#2186	C₂	semidirect
9. C₃₁×#2113	C₂	semidirect
114. #33769	C₃₁	direct (N × H)
144. #3061	C₂×C₃₁	semidirect
145. #2236	C₂×C₃₁	semidirect
146. #2232	C₂×C₃₁	semidirect
147. #2220	C₂×C₃₁	semidirect
148. #2216	C₂×C₃₁	semidirect
149. #2215	C₂×C₃₁	semidirect
150. #2189	C₂×C₃₁	semidirect
151. #2186	C₂×C₃₁	semidirect
152. #2113	C₂×C₃₁	semidirect
238. C₃₁	#33769	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×#2236	2
C₃₁×#2232	2
C₃₁×#2220	2
C₃₁×#2216	2
C₃₁×#2215	2
C₃₁×#2189	2
C₃₁×#2186	2
C₃₁×#2113	2
#33769	31

Subgroups — C31×#33769 #21490528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33769 #21490528