Subgroups — C₃₁×#21259 #21478018

Order 7936.

1566 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#419	C₂	1	4	yes	maximal
4	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1981	C₂	1	8	yes	maximal, complemented
6	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₃₁×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₃₁×C₄⋊Q₈	C₄	1	8	yes	complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
13	C₂×C₃₁×#117	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#423	—	2	4	no	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
18	C₃₁×#419	—	2	16	no	—
19	C₂×C₃₁×#103	C₂²	1	16	yes	—
20	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
21	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
22	C₃₁×#376	—	2	1	no	—
23	C₄³×C₃₁	—	2	1	no	—
24	C₂²×Q₈×C₃₁	—	2	4	no	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂²×Q₈×C₃₁	—	2	8	no	—
27	C₂²×Q₈×C₃₁	—	2	8	no	—
28	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	—
29	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
30	C₂³×C₄×C₃₁	—	2	2	no	—
31	C₂³×C₄×C₃₁	D₄	1	2	yes	—
32	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₃₁	—	2	8	no	—
34	C₂×M₁₆×C₃₁	—	2	4	no	—
35	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
36	C₃₁×C₄⋊Q₈	—	2	8	no	—
37	C₃₁×C₄⋊Q₈	—	2	8	no	—
38	C₃₁×C₄⋊Q₈	—	2	8	no	—
39	C₃₁×C₄⋊Q₈	C₂×C₄	1	16	yes	—
40	C₃₁×#117	—	2	4	no	—
41	C₃₁×#117	D₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
46	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄²×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	—	2	4	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	D₄	1	4	yes	—
53	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
54	C₃₁×C₄⋊C₈	—	2	8	no	—
55	C₃₁×#103	—	2	32	no	—
56	C₃₁×#95	—	2	4	no	—
57	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₃₁	—	2	8	no	—
59	C₂×Q₈×C₃₁	—	2	4	no	—
60	C₂×Q₈×C₃₁	—	2	32	no	—
61	C₂×Q₈×C₃₁	—	2	8	no	—
62	C₂×Q₈×C₃₁	—	2	8	no	—
63	C₂×Q₈×C₃₁	—	4	8	no	—
64	C₂×Q₈×C₃₁	—	4	8	no	—
65	C₂×Q₈×C₃₁	—	4	8	no	—
66	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₃₁	—	2	8	no	—
68	C₂²×C₄×C₃₁	—	2	8	no	—
69	C₂²×C₄×C₃₁	—	2	4	no	—
70	C₂²×C₄×C₃₁	—	2	4	no	—
71	C₂²×C₄×C₃₁	—	4	8	no	—
72	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
73	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₃₁	#41	1	4	yes	—
80	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₃₁	—	4	8	no	—
82	C₂×C₈×C₃₁	—	2	8	no	—
83	C₂×C₈×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	2	16	no	—
85	C₄⋊C₄×C₃₁	—	2	16	no	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
88	C₂²⋊C₄×C₃₁	—	2	8	no	—
89	C₄²×C₃₁	—	2	8	no	—
90	C₄²×C₃₁	—	4	8	no	—
91	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
92	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
93	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
94	C₄²×C₃₁	C₂×D₄	1	4	yes	—
95	#21259	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₃₁	—	2	4	no	—
97	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂³×C₃₁	#121	1	4	yes	—
99	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	Q₈×C₃₁	—	2	16	no	—
101	Q₈×C₃₁	—	4	4	no	—
102	Q₈×C₃₁	—	4	4	no	—
103	Q₈×C₃₁	—	4	16	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	16	no	—
106	C₂×C₄×C₃₁	—	2	8	no	—
107	C₂×C₄×C₃₁	—	2	8	no	—
108	C₂×C₄×C₃₁	—	4	8	no	—
109	C₂×C₄×C₃₁	—	4	8	no	—
110	C₂×C₄×C₃₁	—	4	16	no	—
111	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
115	C₂×C₄×C₃₁	#121	1	4	yes	—
116	C₂×C₄×C₃₁	#124	1	4	yes	—
117	C₂×C₄×C₃₁	#127	1	4	yes	—
118	C₂×C₄×C₃₁	#137	1	8	yes	—
119	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
123	C₈×C₃₁	—	4	4	no	—
124	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
125	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
126	C₂×#419	C₂×C₃₁	1	4	yes	—
127	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
128	#1981	C₂×C₃₁	1	8	yes	complemented
129	C₂²×C₃₁	—	2	4	no	—
130	C₂²×C₃₁	—	2	8	no	—
131	C₂²×C₃₁	#390	1	4	yes	—
132	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃₁	C₂×#121	1	2	yes	—
134	C₂²×C₃₁	C₂×#137	1	4	yes	—
135	C₄×C₃₁	—	2	4	no	—
136	C₄×C₃₁	—	4	8	no	—
137	C₄×C₃₁	—	4	8	no	—
138	C₄×C₃₁	#390	1	4	yes	—
139	C₄×C₃₁	#419	1	8	yes	complemented
140	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
141	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
142	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
143	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
144	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
145	C₂×C₄⋊Q₈	—	2	1	no	—
146	C₂×C₄⋊Q₈	—	2	1	no	—
147	C₂×C₄⋊Q₈	—	2	4	no	—
148	C₂×C₄⋊Q₈	C₄×C₃₁	1	8	yes	complemented
149	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
150	C₂×#117	C₂²×C₃₁	1	2	yes	—
151	C₂²×C₄⋊C₄	—	2	2	no	—
152	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
153	#423	—	2	4	no	—
154	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
155	#419	—	2	16	no	—
156	C₂×#103	C₂²×C₃₁	1	16	yes	—
157	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
158	C₄×C₂²⋊C₄	—	2	2	no	—
159	#376	—	2	1	no	—
160	C₄³	—	2	1	no	—
161	C₂×C₃₁	—	2	4	no	—
162	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
163	C₂×C₃₁	C₂×#419	1	2	yes	—
164	C₂×C₃₁	#3039	1	4	yes	—
165	C₂²×Q₈	—	2	4	no	—
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×Q₈	—	2	8	no	—
168	C₂²×Q₈	—	2	8	no	—
169	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	—
170	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
171	C₂³×C₄	—	2	2	no	—
172	C₂³×C₄	D₄×C₃₁	1	2	yes	—
173	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
174	C₂×M₁₆	—	2	8	no	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	—	2	8	no	—
179	C₄⋊Q₈	—	2	8	no	—
180	C₄⋊Q₈	C₂×C₄×C₃₁	1	16	yes	—
181	#117	—	2	4	no	—
182	#117	D₄×C₃₁	1	8	yes	—
183	C₂×C₄⋊C₄	—	2	16	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	4	no	—
190	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
191	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₃₁	1	4	yes	—
194	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
195	C₄⋊C₈	—	2	8	no	—
196	#103	—	2	32	no	—
197	#95	—	2	4	no	—
198	C₃₁	#21259	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₂×Q₈	—	2	8	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	32	no	—
203	C₂×Q₈	—	2	8	no	—
204	C₂×Q₈	—	2	8	no	—
205	C₂×Q₈	—	4	8	no	—
206	C₂×Q₈	—	4	8	no	—
207	C₂×Q₈	—	4	8	no	—
208	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	4	8	no	—
214	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
215	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
221	C₂²×C₄	#41×C₃₁	1	4	yes	—
222	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
223	M₁₆	—	4	8	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₄⋊C₄	—	2	16	no	—
227	C₄⋊C₄	—	2	16	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
230	C₂²⋊C₄	—	2	8	no	—
231	C₄²	—	2	8	no	—
232	C₄²	—	4	8	no	—
233	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
234	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
235	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
236	C₄²	C₂×D₄×C₃₁	1	4	yes	—
237	C₂³	—	2	4	no	—
238	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
239	C₂³	C₃₁×#121	1	4	yes	—
240	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
241	Q₈	—	2	16	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	4	16	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	4	16	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
255	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
256	C₂×C₄	C₃₁×#121	1	4	yes	—
257	C₂×C₄	C₃₁×#124	1	4	yes	—
258	C₂×C₄	C₃₁×#127	1	4	yes	—
259	C₂×C₄	C₃₁×#137	1	8	yes	—
260	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
262	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	8	no	—
267	C₂²	C₃₁×#390	1	4	yes	—
268	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
269	C₂²	C₂×C₃₁×#121	1	2	yes	—
270	C₂²	C₂×C₃₁×#137	1	4	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	4	8	no	—
273	C₄	—	4	8	no	—
274	C₄	C₃₁×#390	1	4	yes	—
275	C₄	C₃₁×#419	1	8	yes	complemented
276	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
277	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
278	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
279	C₂	—	2	4	no	—
280	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
281	C₂	C₂×C₃₁×#419	1	2	yes	—
282	C₂	C₃₁×#3039	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃₁×#1981	C₂	semidirect
11. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
95. #21259	C₃₁	direct (N × H)
128. #1981	C₂×C₃₁	semidirect
139. C₄×C₃₁	#419	semidirect
148. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
198. C₃₁	#21259	direct (N × H)
275. C₄	C₃₁×#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₄⋊Q₈	2
C₂×C₃₁×#423	2
C₂×C₃₁×#419	2
C₄×C₃₁×#117	2
C₃₁×#1981	2
#21259	31

Subgroups — C31×#21259 #21478018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21259 #21478018