Subgroups — C₃₁×#19954 #21476713

Order 7936.

618 conjugacy classes of subgroups.

The subgroup lattice has 344 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×#3303	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1691	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1683	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1647	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1634	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#555	C₂²	1	2	yes	—
11	C₃₁×#547	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#498	C₂²	1	2	yes	—
14	C₃₁×#494	C₂²	1	1	yes	characteristic
15	C₃₁×#487	C₂²	1	2	yes	—
16	C₃₁×#485	—	2	2	no	—
17	C₃₁×#482	—	2	2	no	—
18	C₃₁×#481	—	2	2	no	—
19	C₃₁×#476	C₂²	1	2	yes	—
20	C₃₁×#476	C₂²	1	2	yes	—
21	C₃₁×#474	C₂²	1	2	yes	—
22	C₃₁×#469	—	2	2	no	—
23	C₃₁×#465	—	2	2	no	—
24	C₃₁×#461	C₂²	1	2	yes	—
25	C₃₁×#449	—	2	2	no	—
26	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#436	C₂²	1	2	yes	—
29	C₈×D₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#428	—	2	2	no	—
31	C₃₁×#416	—	2	2	no	—
32	C₃₁×#335	C₂²	1	1	yes	characteristic
33	C₃₁×#335	C₂²	1	1	yes	characteristic
34	C₃₁×#333	C₂²	1	2	yes	—
35	C₃₁×#332	C₂²	1	2	yes	—
36	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
37	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₃₁	—	4	1	no	—
41	C₂×SD₁₆×C₃₁	—	2	2	no	—
42	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	—	2	2	no	—
46	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₃₁×#124	—	4	2	no	—
49	C₃₁×#123	—	4	2	no	—
50	C₃₁×C₂²⋊Q₈	—	2	2	no	—
51	C₃₁×#121	—	2	2	no	—
52	C₃₁×#121	—	2	2	no	—
53	C₃₁×#121	—	2	2	no	—
54	C₄×D₄×C₃₁	C₂³	1	2	yes	—
55	C₄×D₄×C₃₁	C₂³	1	2	yes	—
56	C₃₁×#117	—	2	2	no	—
57	C₃₁×#117	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₃₁×#107	—	2	2	no	—
60	C₃₁×#107	—	4	1	no	—
61	C₃₁×#107	D₄	1	1	yes	characteristic
62	C₃₁×#107	D₄	1	1	yes	characteristic
63	C₃₁×#106	—	2	2	no	—
64	C₃₁×#106	—	2	2	no	—
65	C₃₁×#106	—	2	2	no	—
66	C₃₁×#106	—	2	2	no	—
67	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
68	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#103	—	2	2	no	—
72	C₃₁×#102	—	2	2	no	—
73	C₃₁×#102	—	2	2	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	D₄	1	2	yes	—
76	C₃₁×#102	D₄	1	2	yes	—
77	C₃₁×C₂²⋊C₈	—	2	2	no	—
78	C₃₁×C₂²⋊C₈	—	2	2	no	—
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	#41×C₃₁	—	4	4	no	—
83	#41×C₃₁	—	8	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	4	2	no	—
87	C₂×D₄×C₃₁	—	4	2	no	—
88	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	4	2	no	—
93	Q₁₆×C₃₁	—	8	2	no	—
94	SD₁₆×C₃₁	—	8	2	no	—
95	D₈×C₃₁	—	2	2	no	—
96	D₈×C₃₁	—	4	1	no	—
97	M₁₆×C₃₁	—	4	4	no	—
98	C₂×C₈×C₃₁	—	2	1	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	8	2	no	—
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₃₁	—	2	2	no	—
114	C₂²⋊C₄×C₃₁	—	2	2	no	—
115	C₂²⋊C₄×C₃₁	—	4	2	no	—
116	C₂²⋊C₄×C₃₁	—	4	2	no	—
117	C₂²⋊C₄×C₃₁	—	4	2	no	—
118	C₄²×C₃₁	—	4	2	no	—
119	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
120	#19954	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	2	2	no	—
122	C₂³×C₃₁	—	2	2	no	—
123	Q₈×C₃₁	—	4	2	no	—
124	Q₈×C₃₁	—	4	2	no	—
125	Q₈×C₃₁	—	8	2	no	—
126	D₄×C₃₁	—	2	2	no	—
127	D₄×C₃₁	—	2	2	no	—
128	D₄×C₃₁	—	2	2	no	—
129	D₄×C₃₁	—	2	2	no	—
130	D₄×C₃₁	—	8	4	no	—
131	D₄×C₃₁	—	16	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	8	4	no	—
141	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₃₁	—	2	2	no	—
145	C₈×C₃₁	—	2	1	no	—
146	C₈×C₃₁	—	2	1	no	—
147	C₈×C₃₁	—	2	1	no	—
148	C₈×C₃₁	—	4	2	no	—
149	C₈×C₃₁	—	4	2	no	—
150	C₈×C₃₁	#135	1	2	yes	—
151	#3354	C₂×C₃₁	1	2	yes	complemented
152	#3303	C₂×C₃₁	1	2	yes	complemented
153	#1707	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#1691	C₂×C₃₁	1	2	yes	complemented
155	#1683	C₂×C₃₁	1	2	yes	complemented
156	#1647	C₂×C₃₁	1	2	yes	complemented
157	#1634	C₂×C₃₁	1	2	yes	complemented
158	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
159	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
160	C₂²×C₃₁	—	4	2	no	—
161	C₂²×C₃₁	—	4	2	no	—
162	C₂²×C₃₁	—	4	2	no	—
163	C₂²×C₃₁	—	4	2	no	—
164	C₂²×C₃₁	—	4	2	no	—
165	C₂²×C₃₁	—	4	2	no	—
166	C₂²×C₃₁	D₄²	1	1	yes	characteristic
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	4	2	no	—
169	C₄×C₃₁	—	4	2	no	—
170	C₄×C₃₁	—	8	2	no	—
171	C₄×C₃₁	—	8	2	no	—
172	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
173	C₄×C₃₁	#575	1	1	yes	characteristic
174	C₄×C₃₁	#576	1	1	yes	characteristic
175	C₄×C₃₁	#578	1	1	yes	characteristic
176	#555	C₂²×C₃₁	1	2	yes	—
177	#547	C₂²×C₃₁	1	2	yes	—
178	#500	C₂²×C₃₁	1	1	yes	characteristic
179	#498	C₂²×C₃₁	1	2	yes	—
180	#494	C₂²×C₃₁	1	1	yes	characteristic
181	#487	C₂²×C₃₁	1	2	yes	—
182	#485	—	2	2	no	—
183	#482	—	2	2	no	—
184	#481	—	2	2	no	—
185	#476	C₂²×C₃₁	1	2	yes	—
186	#476	C₂²×C₃₁	1	2	yes	—
187	#474	C₂²×C₃₁	1	2	yes	—
188	#469	—	2	2	no	—
189	#465	—	2	2	no	—
190	#461	C₂²×C₃₁	1	2	yes	—
191	#449	—	2	2	no	—
192	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
193	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
194	#436	C₂²×C₃₁	1	2	yes	—
195	C₈×D₄	C₂²×C₃₁	1	2	yes	—
196	#428	—	2	2	no	—
197	#416	—	2	2	no	—
198	#335	C₂²×C₃₁	1	1	yes	characteristic
199	#335	C₂²×C₃₁	1	1	yes	characteristic
200	#333	C₂²×C₃₁	1	2	yes	—
201	#332	C₂²×C₃₁	1	2	yes	—
202	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
203	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
204	C₈²	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×C₃₁	—	8	2	no	—
206	C₂×C₃₁	—	8	2	no	—
207	C₂×C₃₁	#3273	1	1	yes	characteristic
208	C₂×C₃₁	#3283	1	1	yes	characteristic
209	C₂×C₃₁	#3291	1	1	yes	characteristic
210	C₂×#41	—	2	2	no	—
211	C₂×Q₁₆	—	4	1	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
214	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
215	C₂×M₁₆	—	2	2	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
218	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
219	#124	—	4	2	no	—
220	#123	—	4	2	no	—
221	C₂²⋊Q₈	—	2	2	no	—
222	#121	—	2	2	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	C₄×D₄	C₂³×C₃₁	1	2	yes	—
226	C₄×D₄	C₂³×C₃₁	1	2	yes	—
227	#117	—	2	2	no	—
228	#117	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	#107	—	2	2	no	—
231	#107	—	4	1	no	—
232	#107	D₄×C₃₁	1	1	yes	characteristic
233	#107	D₄×C₃₁	1	1	yes	characteristic
234	#106	—	2	2	no	—
235	#106	—	2	2	no	—
236	#106	—	2	2	no	—
237	#106	—	2	2	no	—
238	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
239	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
240	#103	—	2	2	no	—
241	#103	—	2	2	no	—
242	#103	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	D₄×C₃₁	1	2	yes	—
247	#102	D₄×C₃₁	1	2	yes	—
248	C₂²⋊C₈	—	2	2	no	—
249	C₂²⋊C₈	—	2	2	no	—
250	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
251	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
253	C₃₁	#19954	1	1	yes	characteristic, direct product factor, Sylow
254	#41	—	4	4	no	—
255	#41	—	8	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
261	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	Q₁₆	—	8	2	no	—
266	SD₁₆	—	8	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	4	1	no	—
269	M₁₆	—	4	4	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	8	2	no	—
274	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
284	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
292	C₂³	—	2	2	no	—
293	C₂³	—	2	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	2	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	16	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
314	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	4	2	no	—
320	C₈	—	4	2	no	—
321	C₈	C₃₁×#135	1	2	yes	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	D₄²×C₃₁	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	2	no	—
334	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
335	C₄	C₃₁×#575	1	1	yes	characteristic
336	C₄	C₃₁×#576	1	1	yes	characteristic
337	C₄	C₃₁×#578	1	1	yes	characteristic
338	C₂	—	8	2	no	—
339	C₂	—	8	2	no	—
340	C₂	C₃₁×#3273	1	1	yes	characteristic
341	C₂	C₃₁×#3283	1	1	yes	characteristic
342	C₂	C₃₁×#3291	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×#3303	C₂	semidirect
3. C₃₁×#1707	C₂	semidirect
4. C₃₁×#1691	C₂	semidirect
5. C₃₁×#1683	C₂	semidirect
6. C₃₁×#1647	C₂	semidirect
7. C₃₁×#1634	C₂	semidirect
8. C₃₁×#1589	C₂	semidirect
9. C₈×D₈×C₃₁	C₂	semidirect
120. #19954	C₃₁	direct (N × H)
151. #3354	C₂×C₃₁	semidirect
152. #3303	C₂×C₃₁	semidirect
153. #1707	C₂×C₃₁	semidirect
154. #1691	C₂×C₃₁	semidirect
155. #1683	C₂×C₃₁	semidirect
156. #1647	C₂×C₃₁	semidirect
157. #1634	C₂×C₃₁	semidirect
158. #1589	C₂×C₃₁	semidirect
159. C₈×D₈	C₂×C₃₁	semidirect
253. C₃₁	#19954	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×#3354	2
C₃₁×#3303	2
C₃₁×#1707	2
C₃₁×#1691	2
C₃₁×#1683	2
C₃₁×#1647	2
C₃₁×#1634	2
C₃₁×#1589	2
C₈×D₈×C₃₁	2
#19954	31

Subgroups — C31×#19954 #21476713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19954 #21476713