Subgroups — C₃₁×#61817 #21518576

Order 7936.

2194 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×#3497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₈×C₃₁	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3358	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3226	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3100	C₂	1	6	yes	maximal, complemented
7	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
9	C₂³×Q₈×C₃₁	—	2	1	no	—
10	C₃₁×#550	C₂²	1	4	yes	complemented
11	D₄×Q₈×C₃₁	—	2	4	no	—
12	D₄×Q₈×C₃₁	C₂²	1	8	yes	—
13	D₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
14	C₃₁×#548	—	2	4	no	—
15	C₃₁×#537	—	2	3	no	—
16	C₃₁×#532	C₂²	1	6	yes	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	—	2	3	no	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
21	C₃₁×#500	C₂²	1	12	yes	complemented
22	C₃₁×#486	C₂²	1	12	yes	complemented
23	C₃₁×#480	C₂²	1	12	yes	—
24	C₃₁×#474	C₂²	1	12	yes	—
25	C₃₁×#474	C₂²	1	24	yes	complemented
26	C₃₁×#459	C₂²	1	12	yes	—
27	C₃₁×#447	C₂²	1	4	yes	—
28	C₈×Q₈×C₃₁	C₂²	1	8	yes	complemented
29	C₄×D₈×C₃₁	C₂²	1	24	yes	—
30	C₃₁×#431	C₂²	1	6	yes	—
31	C₃₁×2₋¹⁺⁴	—	4	8	no	—
32	C₂×#41×C₃₁	—	2	4	no	—
33	C₂×#41×C₃₁	—	2	6	no	—
34	C₂²×Q₈×C₃₁	—	2	4	no	—
35	C₂²×Q₈×C₃₁	—	2	8	no	—
36	C₂²×Q₈×C₃₁	—	2	3	no	—
37	C₂²×Q₈×C₃₁	—	4	8	no	—
38	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	—	2	3	no	—
41	C₂×D₈×C₃₁	C₂³	1	8	yes	—
42	C₃₁×C₄⋊Q₈	—	2	6	no	—
43	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
44	C₃₁×C₄⋊Q₈	C₂³	1	12	yes	—
45	C₃₁×#127	C₂³	1	6	yes	—
46	C₃₁×#124	—	2	6	no	—
47	C₃₁×C₂²⋊Q₈	—	2	24	no	—
48	C₃₁×C₂²⋊Q₈	—	2	12	no	—
49	C₃₁×#121	—	2	6	no	—
50	C₃₁×#121	—	2	12	no	—
51	C₃₁×#121	C₂³	1	4	yes	—
52	C₄×Q₈×C₃₁	D₄	1	4	yes	—
53	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
54	C₄×D₄×C₃₁	—	2	6	no	—
55	C₄×D₄×C₃₁	—	2	6	no	—
56	C₄×D₄×C₃₁	C₂³	1	12	yes	—
57	C₄×D₄×C₃₁	C₂³	1	12	yes	—
58	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
59	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
62	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
63	C₃₁×#107	C₂³	1	24	yes	—
64	C₃₁×C₄⋊C₈	C₂³	1	12	yes	—
65	C₃₁×#102	C₂³	1	8	yes	—
66	C₃₁×#102	C₂³	1	24	yes	—
67	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
68	C₄×C₈×C₃₁	C₂³	1	12	yes	—
69	C₂⁴×C₃₁	—	2	1	no	—
70	#41×C₃₁	—	4	16	no	—
71	#41×C₃₁	—	4	24	no	—
72	C₂×Q₈×C₃₁	—	2	6	no	—
73	C₂×Q₈×C₃₁	—	2	12	no	—
74	C₂×Q₈×C₃₁	—	4	32	no	—
75	C₂×Q₈×C₃₁	—	4	16	no	—
76	C₂×Q₈×C₃₁	—	4	12	no	—
77	C₂×Q₈×C₃₁	D₈	1	8	yes	complemented
78	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
79	C₂×D₄×C₃₁	—	2	4	no	—
80	C₂×D₄×C₃₁	—	2	4	no	—
81	C₂×D₄×C₃₁	—	2	1	no	—
82	C₂×D₄×C₃₁	—	2	6	no	—
83	C₂×D₄×C₃₁	—	2	6	no	—
84	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
85	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
86	C₂²×C₄×C₃₁	—	2	3	no	—
87	C₂²×C₄×C₃₁	—	2	12	no	—
88	C₂²×C₄×C₃₁	—	2	12	no	—
89	C₂²×C₄×C₃₁	—	4	12	no	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
91	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
92	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
93	D₈×C₃₁	—	2	16	no	—
94	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
95	C₄⋊C₄×C₃₁	—	2	12	no	—
96	C₄⋊C₄×C₃₁	—	2	3	no	—
97	C₄⋊C₄×C₃₁	—	2	6	no	—
98	C₄⋊C₄×C₃₁	C₂×D₄	1	12	yes	—
99	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
100	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
101	C₂²⋊C₄×C₃₁	—	2	12	no	—
102	C₂²⋊C₄×C₃₁	—	2	4	no	—
103	C₂²⋊C₄×C₃₁	—	2	12	no	—
104	C₄²×C₃₁	C₂×D₄	1	6	yes	—
105	C₄²×C₃₁	C₂⁴	1	6	yes	—
106	#61817	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	2	4	no	—
108	C₂³×C₃₁	—	2	2	no	—
109	C₂³×C₃₁	—	2	4	no	—
110	C₂³×C₃₁	—	4	2	no	—
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₃₁	—	4	24	no	—
114	Q₈×C₃₁	—	4	12	no	—
115	Q₈×C₃₁	C₂×D₈	1	16	yes	—
116	D₄×C₃₁	—	2	4	no	—
117	D₄×C₃₁	—	2	2	no	—
118	D₄×C₃₁	—	2	8	no	—
119	D₄×C₃₁	—	4	8	no	—
120	D₄×C₃₁	—	4	12	no	—
121	D₄×C₃₁	—	8	6	no	—
122	D₄×C₃₁	2₋¹⁺⁴	1	4	yes	complemented
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	4	no	—
125	C₂×C₄×C₃₁	—	2	6	no	—
126	C₂×C₄×C₃₁	—	4	3	no	—
127	C₂×C₄×C₃₁	—	4	6	no	—
128	C₂×C₄×C₃₁	—	4	24	no	—
129	C₂×C₄×C₃₁	—	4	24	no	—
130	C₂×C₄×C₃₁	C₂×D₈	1	12	yes	—
131	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
133	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
134	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
135	C₈×C₃₁	—	2	8	no	—
136	#3497	C₂×C₃₁	1	1	yes	characteristic, complemented
137	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
138	Q₈×D₈	C₂×C₃₁	1	8	yes	complemented
139	#3358	C₂×C₃₁	1	8	yes	complemented
140	#3226	C₂×C₃₁	1	6	yes	complemented
141	#3100	C₂×C₃₁	1	6	yes	complemented
142	#2982	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	2	1	no	—
145	C₂²×C₃₁	—	2	4	no	—
146	C₂²×C₃₁	—	4	8	no	—
147	C₂²×C₃₁	—	4	2	no	—
148	C₂²×C₃₁	—	4	2	no	—
149	C₂²×C₃₁	—	4	4	no	—
150	C₂²×C₃₁	—	4	4	no	—
151	C₂²×C₃₁	—	4	4	no	—
152	C₂²×C₃₁	C₂²×D₈	1	2	yes	—
153	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
154	C₄×C₃₁	—	2	6	no	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	4	4	no	—
157	C₄×C₃₁	—	4	6	no	—
158	C₄×C₃₁	C₂²×D₈	1	12	yes	—
159	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
160	C₂×2₋¹⁺⁴	—	2	1	no	—
161	C₂³×Q₈	—	2	1	no	—
162	#550	C₂²×C₃₁	1	4	yes	complemented
163	D₄×Q₈	—	2	4	no	—
164	D₄×Q₈	C₂²×C₃₁	1	8	yes	—
165	D₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
166	#548	—	2	4	no	—
167	#537	—	2	3	no	—
168	#532	C₂²×C₃₁	1	6	yes	—
169	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
170	C₂×C₂²⋊Q₈	—	2	3	no	—
171	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
172	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
173	#500	C₂²×C₃₁	1	12	yes	complemented
174	#486	C₂²×C₃₁	1	12	yes	complemented
175	#480	C₂²×C₃₁	1	12	yes	—
176	#474	C₂²×C₃₁	1	12	yes	—
177	#474	C₂²×C₃₁	1	24	yes	complemented
178	#459	C₂²×C₃₁	1	12	yes	—
179	#447	C₂²×C₃₁	1	4	yes	—
180	C₈×Q₈	C₂²×C₃₁	1	8	yes	complemented
181	C₄×D₈	C₂²×C₃₁	1	24	yes	—
182	#431	C₂²×C₃₁	1	6	yes	—
183	C₂×C₃₁	—	2	2	no	—
184	C₂×C₃₁	—	4	4	no	—
185	C₂×C₃₁	—	8	4	no	—
186	C₂×C₃₁	#3486	1	1	yes	characteristic
187	C₂×C₃₁	C₂³×D₈	1	1	yes	characteristic
188	C₂×C₃₁	#3586	1	1	yes	characteristic
189	2₋¹⁺⁴	—	4	8	no	—
190	C₂×#41	—	2	4	no	—
191	C₂×#41	—	2	6	no	—
192	C₂²×Q₈	—	2	4	no	—
193	C₂²×Q₈	—	2	8	no	—
194	C₂²×Q₈	—	2	3	no	—
195	C₂²×Q₈	—	4	8	no	—
196	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
197	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
198	C₂³×C₄	—	2	3	no	—
199	C₂×D₈	C₂³×C₃₁	1	8	yes	—
200	C₄⋊Q₈	—	2	6	no	—
201	C₄⋊Q₈	C₂³×C₃₁	1	12	yes	—
202	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
203	#127	C₂³×C₃₁	1	6	yes	—
204	#124	—	2	6	no	—
205	C₂²⋊Q₈	—	2	24	no	—
206	C₂²⋊Q₈	—	2	12	no	—
207	#121	—	2	6	no	—
208	#121	—	2	12	no	—
209	#121	C₂³×C₃₁	1	4	yes	—
210	C₄×Q₈	D₄×C₃₁	1	4	yes	—
211	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
212	C₄×D₄	—	2	6	no	—
213	C₄×D₄	—	2	6	no	—
214	C₄×D₄	C₂³×C₃₁	1	12	yes	—
215	C₄×D₄	C₂³×C₃₁	1	12	yes	—
216	C₂×C₄⋊C₄	—	2	3	no	—
217	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
218	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
219	C₂×C₂²⋊C₄	—	2	3	no	—
220	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
221	#107	C₂³×C₃₁	1	24	yes	—
222	C₄⋊C₈	C₂³×C₃₁	1	12	yes	—
223	#102	C₂³×C₃₁	1	8	yes	—
224	#102	C₂³×C₃₁	1	24	yes	—
225	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
226	C₄×C₈	C₂³×C₃₁	1	12	yes	—
227	C₃₁	#61817	1	1	yes	characteristic, direct product factor, Sylow
228	C₂⁴	—	2	1	no	—
229	#41	—	4	16	no	—
230	#41	—	4	24	no	—
231	C₂×Q₈	—	2	6	no	—
232	C₂×Q₈	—	2	12	no	—
233	C₂×Q₈	—	4	32	no	—
234	C₂×Q₈	—	4	16	no	—
235	C₂×Q₈	—	4	12	no	—
236	C₂×Q₈	D₈×C₃₁	1	8	yes	complemented
237	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	6	no	—
242	C₂×D₄	—	2	6	no	—
243	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
244	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
245	C₂²×C₄	—	2	3	no	—
246	C₂²×C₄	—	2	12	no	—
247	C₂²×C₄	—	2	12	no	—
248	C₂²×C₄	—	4	12	no	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
251	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
252	D₈	—	2	16	no	—
253	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
254	C₄⋊C₄	—	2	12	no	—
255	C₄⋊C₄	—	2	3	no	—
256	C₄⋊C₄	—	2	6	no	—
257	C₄⋊C₄	C₂×D₄×C₃₁	1	12	yes	—
258	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
259	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
260	C₂²⋊C₄	—	2	12	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	12	no	—
263	C₄²	C₂×D₄×C₃₁	1	6	yes	—
264	C₄²	C₂⁴×C₃₁	1	6	yes	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
271	Q₈	—	4	24	no	—
272	Q₈	—	4	12	no	—
273	Q₈	C₂×D₈×C₃₁	1	16	yes	—
274	D₄	—	2	4	no	—
275	D₄	—	2	2	no	—
276	D₄	—	2	8	no	—
277	D₄	—	4	8	no	—
278	D₄	—	4	12	no	—
279	D₄	—	8	6	no	—
280	D₄	C₃₁×2₋¹⁺⁴	1	4	yes	complemented
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	6	no	—
284	C₂×C₄	—	4	3	no	—
285	C₂×C₄	—	4	6	no	—
286	C₂×C₄	—	4	24	no	—
287	C₂×C₄	—	4	24	no	—
288	C₂×C₄	C₂×D₈×C₃₁	1	12	yes	—
289	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
291	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
292	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
293	C₈	—	2	8	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	C₂²×D₈×C₃₁	1	2	yes	—
304	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	6	no	—
307	C₄	—	4	4	no	—
308	C₄	—	4	6	no	—
309	C₄	C₂²×D₈×C₃₁	1	12	yes	—
310	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
311	C₂	—	2	2	no	—
312	C₂	—	4	4	no	—
313	C₂	—	8	4	no	—
314	C₂	C₃₁×#3486	1	1	yes	characteristic
315	C₂	C₂³×D₈×C₃₁	1	1	yes	characteristic
316	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3497	C₂	semidirect
2. C₂×D₄×Q₈×C₃₁	C₂	semidirect
3. Q₈×D₈×C₃₁	C₂	semidirect
4. C₃₁×#3358	C₂	semidirect
5. C₃₁×#3226	C₂	semidirect
6. C₃₁×#3100	C₂	semidirect
7. C₃₁×#2982	C₂	semidirect
10. C₃₁×#550	C₂²	semidirect
13. D₄×Q₈×C₃₁	C₂²	semidirect
21. C₃₁×#500	C₂²	semidirect
22. C₃₁×#486	C₂²	semidirect
25. C₃₁×#474	C₂²	semidirect
28. C₈×Q₈×C₃₁	C₂²	semidirect
77. C₂×Q₈×C₃₁	D₈	semidirect
106. #61817	C₃₁	direct (N × H)
122. D₄×C₃₁	2₋¹⁺⁴	semidirect
136. #3497	C₂×C₃₁	semidirect
137. C₂×D₄×Q₈	C₂×C₃₁	semidirect
138. Q₈×D₈	C₂×C₃₁	semidirect
139. #3358	C₂×C₃₁	semidirect
140. #3226	C₂×C₃₁	semidirect
141. #3100	C₂×C₃₁	semidirect
142. #2982	C₂×C₃₁	semidirect
162. #550	C₂²×C₃₁	semidirect
165. D₄×Q₈	C₂²×C₃₁	semidirect
173. #500	C₂²×C₃₁	semidirect
174. #486	C₂²×C₃₁	semidirect
177. #474	C₂²×C₃₁	semidirect
180. C₈×Q₈	C₂²×C₃₁	semidirect
227. C₃₁	#61817	direct (N × H)
236. C₂×Q₈	D₈×C₃₁	semidirect
280. D₄	C₃₁×2₋¹⁺⁴	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3497	2
C₂×D₄×Q₈×C₃₁	2
Q₈×D₈×C₃₁	2
C₃₁×#3358	2
C₃₁×#3226	2
C₃₁×#3100	2
C₃₁×#2982	2
#61817	31

Subgroups — C31×#61817 #21518576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61817 #21518576