Subgroups — C₃₁×#22538 #21479297

Order 7936.

970 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₃₁	C₂	1	2	yes	maximal
3	C₃₁×#1966	C₂	1	6	yes	maximal
4	C₃₁×#1801	C₂	1	3	yes	maximal
5	C₄×C₃₁×#101	C₂	1	3	yes	maximal
6	C₃₁×#557	—	2	2	no	—
7	C₃₁×#557	—	2	6	no	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
9	C₂×C₃₁×#125	C₄	1	6	yes	—
10	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	—
11	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#446	—	2	6	no	—
14	C₈×Q₈×C₃₁	—	2	2	no	—
15	C₃₁×#423	C₂²	1	6	yes	—
16	C₂×C₃₁×#101	C₂²	1	6	yes	—
17	C₂×C₃₁×#101	C₂²	1	6	yes	—
18	C₄×M₁₆×C₃₁	C₂²	1	6	yes	—
19	C₃₁×#343	C₂²	1	3	yes	—
20	C₃₁×#343	C₂²	1	1	yes	characteristic
21	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
23	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
24	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
25	C₃₁×C₄⋊Q₈	—	2	6	no	—
26	C₃₁×C₄⋊Q₈	—	2	6	no	—
27	C₃₁×#125	—	2	12	no	—
28	C₃₁×#125	—	2	12	no	—
29	C₄×Q₈×C₃₁	—	2	6	no	—
30	C₄×Q₈×C₃₁	—	2	2	no	—
31	C₄×Q₈×C₃₁	—	2	6	no	—
32	C₄×Q₈×C₃₁	—	2	6	no	—
33	C₄×Q₈×C₃₁	D₄	1	2	yes	—
34	C₄×Q₈×C₃₁	D₄	1	6	yes	—
35	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	6	yes	—
36	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	6	yes	—
37	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	6	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂×C₄	1	6	yes	—
41	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
42	C₃₁×C₄⋊C₈	—	2	6	no	—
43	C₃₁×C₄⋊C₈	—	2	6	no	—
44	C₃₁×#101	—	2	12	no	—
45	C₃₁×#101	—	4	6	no	—
46	C₃₁×#97	—	2	6	no	—
47	C₄×C₈×C₃₁	—	2	6	no	—
48	C₂×Q₈×C₃₁	—	2	4	no	—
49	C₂×Q₈×C₃₁	—	2	6	no	—
50	C₂×Q₈×C₃₁	—	2	6	no	—
51	C₂×Q₈×C₃₁	—	2	3	no	—
52	C₂×Q₈×C₃₁	—	2	3	no	—
53	C₂×Q₈×C₃₁	C₂²⋊C₄	1	3	yes	—
54	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
59	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₃₁	—	2	2	no	—
61	M₁₆×C₃₁	—	2	6	no	—
62	M₁₆×C₃₁	—	2	6	no	—
63	M₁₆×C₃₁	#41	1	4	yes	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	2	6	no	—
66	C₄⋊C₄×C₃₁	—	2	12	no	—
67	C₄⋊C₄×C₃₁	—	2	6	no	—
68	C₄⋊C₄×C₃₁	—	2	6	no	—
69	C₄⋊C₄×C₃₁	—	2	12	no	—
70	C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
72	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
73	C₄²×C₃₁	—	2	6	no	—
74	C₄²×C₃₁	—	2	6	no	—
75	C₄²×C₃₁	C₂×D₄	1	6	yes	—
76	C₄²×C₃₁	C₂×D₄	1	6	yes	—
77	#22538	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
79	Q₈×C₃₁	—	2	6	no	—
80	Q₈×C₃₁	—	4	6	no	—
81	Q₈×C₃₁	—	4	6	no	—
82	Q₈×C₃₁	#101	1	4	yes	complemented
83	C₂×C₄×C₃₁	—	2	6	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	6	no	—
86	C₂×C₄×C₃₁	—	2	6	no	—
87	C₂×C₄×C₃₁	—	2	6	no	—
88	C₂×C₄×C₃₁	—	4	3	no	—
89	C₂×C₄×C₃₁	—	4	2	no	—
90	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
91	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
92	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
93	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
95	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
96	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
97	C₈×C₃₁	—	2	4	no	—
98	C₈×C₃₁	—	4	6	no	—
99	C₂×#557	C₂×C₃₁	1	1	yes	characteristic
100	Q₈×M₁₆	C₂×C₃₁	1	2	yes	—
101	#1966	C₂×C₃₁	1	6	yes	—
102	#1801	C₂×C₃₁	1	3	yes	—
103	C₄×#101	C₂×C₃₁	1	3	yes	—
104	C₂²×C₃₁	—	2	1	no	—
105	C₂²×C₃₁	—	2	1	no	—
106	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
108	C₂²×C₃₁	#520	1	1	yes	characteristic
109	C₄×C₃₁	—	2	3	no	—
110	C₄×C₃₁	—	2	6	no	—
111	C₄×C₃₁	—	4	6	no	—
112	C₄×C₃₁	—	4	2	no	—
113	C₄×C₃₁	C₂×#101	1	6	yes	—
114	C₄×C₃₁	#548	1	2	yes	—
115	C₄×C₃₁	#550	1	2	yes	—
116	#557	—	2	2	no	—
117	#557	—	2	6	no	—
118	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
119	C₂×#125	C₄×C₃₁	1	6	yes	—
120	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	—
121	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
122	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
123	#446	—	2	6	no	—
124	C₈×Q₈	—	2	2	no	—
125	#423	C₂²×C₃₁	1	6	yes	—
126	C₂×#101	C₂²×C₃₁	1	6	yes	—
127	C₂×#101	C₂²×C₃₁	1	6	yes	—
128	C₄×M₁₆	C₂²×C₃₁	1	6	yes	—
129	#343	C₂²×C₃₁	1	3	yes	—
130	#343	C₂²×C₃₁	1	1	yes	characteristic
131	C₂×C₃₁	—	2	1	no	—
132	C₂×C₃₁	—	2	1	no	—
133	C₂×C₃₁	#2341	1	1	yes	characteristic
134	C₂×C₃₁	C₂²×#101	1	1	yes	characteristic
135	C₂×C₃₁	#2889	1	1	yes	characteristic
136	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
137	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
138	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
139	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
140	C₄⋊Q₈	—	2	6	no	—
141	C₄⋊Q₈	—	2	6	no	—
142	#125	—	2	12	no	—
143	#125	—	2	12	no	—
144	C₄×Q₈	—	2	6	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	6	no	—
147	C₄×Q₈	—	2	6	no	—
148	C₄×Q₈	D₄×C₃₁	1	2	yes	—
149	C₄×Q₈	D₄×C₃₁	1	6	yes	—
150	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	6	yes	—
151	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	6	yes	—
152	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	6	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
154	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
155	C₂×C₄²	C₂×C₄×C₃₁	1	6	yes	—
156	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
157	C₄⋊C₈	—	2	6	no	—
158	C₄⋊C₈	—	2	6	no	—
159	#101	—	2	12	no	—
160	#101	—	4	6	no	—
161	#97	—	2	6	no	—
162	C₄×C₈	—	2	6	no	—
163	C₃₁	#22538	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	6	no	—
166	C₂×Q₈	—	2	6	no	—
167	C₂×Q₈	—	2	3	no	—
168	C₂×Q₈	—	2	3	no	—
169	C₂×Q₈	C₂²⋊C₄×C₃₁	1	3	yes	—
170	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
172	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
173	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
175	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
176	M₁₆	—	2	2	no	—
177	M₁₆	—	2	6	no	—
178	M₁₆	—	2	6	no	—
179	M₁₆	#41×C₃₁	1	4	yes	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	6	no	—
182	C₄⋊C₄	—	2	12	no	—
183	C₄⋊C₄	—	2	6	no	—
184	C₄⋊C₄	—	2	6	no	—
185	C₄⋊C₄	—	2	12	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
188	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
189	C₄²	—	2	6	no	—
190	C₄²	—	2	6	no	—
191	C₄²	C₂×D₄×C₃₁	1	6	yes	—
192	C₄²	C₂×D₄×C₃₁	1	6	yes	—
193	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
194	Q₈	—	2	6	no	—
195	Q₈	—	4	6	no	—
196	Q₈	—	4	6	no	—
197	Q₈	C₃₁×#101	1	4	yes	complemented
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	6	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	4	3	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
207	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
208	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
209	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
210	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₈	—	4	6	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
217	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
218	C₂²	C₃₁×#520	1	1	yes	characteristic
219	C₄	—	2	3	no	—
220	C₄	—	2	6	no	—
221	C₄	—	4	6	no	—
222	C₄	—	4	2	no	—
223	C₄	C₂×C₃₁×#101	1	6	yes	—
224	C₄	C₃₁×#548	1	2	yes	—
225	C₄	C₃₁×#550	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	C₃₁×#2341	1	1	yes	characteristic
229	C₂	C₂²×C₃₁×#101	1	1	yes	characteristic
230	C₂	C₃₁×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	C₂³×C₄
LCS	G ▷ C₂³ ▷ 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
77. #22538	C₃₁	direct (N × H)
82. Q₈×C₃₁	#101	semidirect
163. C₃₁	#22538	direct (N × H)
197. Q₈	C₃₁×#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#557	2
Q₈×M₁₆×C₃₁	2
C₃₁×#1966	2
C₃₁×#1801	2
C₄×C₃₁×#101	2
#22538	31

Subgroups — C31×#22538 #21479297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22538 #21479297