Subgroups — C₂×C₃₁×#2310 #21490801

Order 7936.

5214 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	C₂⁴×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#117	C₂	1	2	yes	maximal
3	C₃₁×#2310	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#391	C₂	1	8	yes	maximal, complemented
5	C₂×C₃₁×#381	C₂	1	4	yes	maximal
6	C₂³×Q₈×C₃₁	—	2	16	no	—
7	C₂³×Q₈×C₃₁	C₄	1	16	yes	complemented
8	C₂³×Q₈×C₃₁	C₂²	1	4	yes	—
9	C₂³×Q₈×C₃₁	C₂²	1	8	yes	—
10	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#117	—	2	16	no	—
13	C₂×C₃₁×#117	C₂²	1	16	yes	—
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
16	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
17	C₃₁×#391	C₂²	1	64	yes	complemented
18	C₃₁×#381	C₂²	1	32	yes	—
19	C₂×C₃₁×#95	C₂²	1	16	yes	—
20	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₃₁	—	2	128	no	—
22	C₂²×Q₈×C₃₁	—	2	16	no	—
23	C₂²×Q₈×C₃₁	—	2	16	no	—
24	C₂²×Q₈×C₃₁	—	2	64	no	—
25	C₂²×Q₈×C₃₁	C₂×C₄	1	32	yes	—
26	C₂²×Q₈×C₃₁	C₂×C₄	1	64	yes	complemented
27	C₂²×Q₈×C₃₁	C₂³	1	16	yes	—
28	C₂³×C₄×C₃₁	—	2	16	no	—
29	C₂³×C₄×C₃₁	C₂×C₄	1	16	yes	—
30	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₃₁	D₄	1	16	yes	—
32	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
33	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
34	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
35	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
36	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
37	C₃₁×#117	—	2	64	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
40	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
41	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
42	C₂×C₄²×C₃₁	—	2	16	no	—
43	C₂×C₄²×C₃₁	C₂³	1	16	yes	—
44	C₃₁×#95	C₂³	1	64	yes	—
45	C₂⁴×C₃₁	—	2	8	no	—
46	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₃₁	C₂×D₄	1	8	yes	—
48	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
49	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₃₁	—	2	128	no	—
51	C₂×Q₈×C₃₁	—	2	256	no	—
52	C₂×Q₈×C₃₁	—	2	32	no	—
53	C₂×Q₈×C₃₁	—	2	32	no	—
54	C₂×Q₈×C₃₁	C₂²×C₄	1	64	yes	—
55	C₂²×C₄×C₃₁	—	2	64	no	—
56	C₂²×C₄×C₃₁	—	2	32	no	—
57	C₂²×C₄×C₃₁	—	2	16	no	—
58	C₂²×C₄×C₃₁	—	2	16	no	—
59	C₂²×C₄×C₃₁	—	2	32	no	—
60	C₂²×C₄×C₃₁	C₂²⋊C₄	1	64	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	32	yes	—
63	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	32	yes	—
68	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
69	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
70	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
71	C₄⋊C₄×C₃₁	—	2	32	no	—
72	C₂²⋊C₄×C₃₁	—	2	32	no	—
73	C₄²×C₃₁	—	2	32	no	—
74	C₂×#2310	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	—	2	16	no	—
76	C₂³×C₃₁	—	2	32	no	—
77	C₂³×C₃₁	—	2	8	no	—
78	C₂³×C₃₁	C₂×C₂²⋊C₄	1	16	yes	—
79	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
80	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
81	C₂³×C₃₁	C₂²×D₄	1	8	yes	—
82	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
83	C₂³×C₃₁	2₋¹⁺⁴	1	8	yes	—
84	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
85	Q₈×C₃₁	—	2	128	no	—
86	C₂×C₄×C₃₁	—	2	64	no	—
87	C₂×C₄×C₃₁	—	2	32	no	—
88	C₂×C₄×C₃₁	—	2	16	no	—
89	C₂×C₄×C₃₁	—	2	16	no	—
90	C₂×C₄×C₃₁	—	2	32	no	—
91	C₂×C₄×C₃₁	—	4	32	no	—
92	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	64	yes	—
93	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	32	yes	—
94	C₂×C₄×C₃₁	C₂³×C₄	1	16	yes	—
95	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
96	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
97	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
98	C₂⁴×Q₈	C₂×C₃₁	1	1	yes	characteristic
99	C₂²×#117	C₂×C₃₁	1	2	yes	—
100	#2310	C₂×C₃₁	1	16	yes	direct product factor
101	C₂×#391	C₂×C₃₁	1	8	yes	complemented
102	C₂×#381	C₂×C₃₁	1	4	yes	—
103	C₂²×C₃₁	—	2	32	no	—
104	C₂²×C₃₁	—	2	8	no	—
105	C₂²×C₃₁	—	2	16	no	—
106	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	8	yes	—
107	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	4	yes	—
108	C₂²×C₃₁	#519	1	8	yes	—
109	C₂²×C₃₁	#536	1	16	yes	—
110	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
111	C₂²×C₃₁	C₂³×D₄	1	2	yes	—
112	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	4	yes	—
113	C₄×C₃₁	—	2	16	no	—
114	C₄×C₃₁	—	4	16	no	—
115	C₄×C₃₁	C₂²×C₂²⋊C₄	1	16	yes	—
116	C₂³×Q₈	—	2	16	no	—
117	C₂³×Q₈	C₄×C₃₁	1	16	yes	complemented
118	C₂³×Q₈	C₂²×C₃₁	1	4	yes	—
119	C₂³×Q₈	C₂²×C₃₁	1	8	yes	—
120	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
121	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
122	C₂×#117	—	2	16	no	—
123	C₂×#117	C₂²×C₃₁	1	16	yes	—
124	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
125	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
126	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
127	#391	C₂²×C₃₁	1	64	yes	complemented
128	#381	C₂²×C₃₁	1	32	yes	—
129	C₂×#95	C₂²×C₃₁	1	16	yes	—
130	C₂×C₃₁	—	2	8	no	—
131	C₂×C₃₁	#2310	1	8	yes	direct product factor
132	C₂×C₃₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₃₁	C₂×#519	1	2	yes	—
134	C₂×C₃₁	C₂×#536	1	4	yes	—
135	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
136	C₂²×Q₈	—	2	128	no	—
137	C₂²×Q₈	—	2	16	no	—
138	C₂²×Q₈	—	2	16	no	—
139	C₂²×Q₈	—	2	64	no	—
140	C₂²×Q₈	C₂×C₄×C₃₁	1	32	yes	—
141	C₂²×Q₈	C₂×C₄×C₃₁	1	64	yes	complemented
142	C₂²×Q₈	C₂³×C₃₁	1	16	yes	—
143	C₂³×C₄	—	2	16	no	—
144	C₂³×C₄	C₂×C₄×C₃₁	1	16	yes	—
145	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	—
146	C₂³×C₄	D₄×C₃₁	1	16	yes	—
147	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
148	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
149	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
150	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
151	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
152	#117	—	2	64	no	—
153	C₂×C₄⋊C₄	—	2	16	no	—
154	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
155	C₂×C₂²⋊C₄	—	2	16	no	—
156	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
157	C₂×C₄²	—	2	16	no	—
158	C₂×C₄²	C₂³×C₃₁	1	16	yes	—
159	#95	C₂³×C₃₁	1	64	yes	—
160	C₃₁	C₂×#2310	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	—	2	8	no	—
162	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
163	C₂⁴	C₂×D₄×C₃₁	1	8	yes	—
164	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
165	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
166	C₂×Q₈	—	2	128	no	—
167	C₂×Q₈	—	2	256	no	—
168	C₂×Q₈	—	2	32	no	—
169	C₂×Q₈	—	2	32	no	—
170	C₂×Q₈	C₂²×C₄×C₃₁	1	64	yes	—
171	C₂²×C₄	—	2	64	no	—
172	C₂²×C₄	—	2	32	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	16	no	—
175	C₂²×C₄	—	2	32	no	—
176	C₂²×C₄	C₂²⋊C₄×C₃₁	1	64	yes	—
177	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
178	C₂²×C₄	C₂²×C₄×C₃₁	1	32	yes	—
179	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
180	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
181	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
182	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
183	C₂²×C₄	C₂×D₄×C₃₁	1	32	yes	—
184	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
185	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
186	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₂²⋊C₄	—	2	32	no	—
189	C₄²	—	2	32	no	—
190	C₂³	—	2	16	no	—
191	C₂³	—	2	32	no	—
192	C₂³	—	2	8	no	—
193	C₂³	C₂×C₂²⋊C₄×C₃₁	1	16	yes	—
194	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
195	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
196	C₂³	C₂²×D₄×C₃₁	1	8	yes	—
197	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
198	C₂³	C₃₁×2₋¹⁺⁴	1	8	yes	—
199	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
200	Q₈	—	2	128	no	—
201	C₂×C₄	—	2	64	no	—
202	C₂×C₄	—	2	32	no	—
203	C₂×C₄	—	2	16	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	32	no	—
206	C₂×C₄	—	4	32	no	—
207	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	64	yes	—
208	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	32	yes	—
209	C₂×C₄	C₂³×C₄×C₃₁	1	16	yes	—
210	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
211	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
212	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
213	C₂²	—	2	32	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	16	no	—
216	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	8	yes	—
217	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	4	yes	—
218	C₂²	C₃₁×#519	1	8	yes	—
219	C₂²	C₃₁×#536	1	16	yes	—
220	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
221	C₂²	C₂³×D₄×C₃₁	1	2	yes	—
222	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	4	yes	—
223	C₄	—	2	16	no	—
224	C₄	—	4	16	no	—
225	C₄	C₂²×C₂²⋊C₄×C₃₁	1	16	yes	—
226	C₂	—	2	8	no	—
227	C₂	C₃₁×#2310	1	8	yes	direct product factor
228	C₂	C₂³×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
229	C₂	C₂×C₃₁×#519	1	2	yes	—
230	C₂	C₂×C₃₁×#536	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2310	C₂	direct (N × H)
4. C₂×C₃₁×#391	C₂	semidirect
7. C₂³×Q₈×C₃₁	C₄	semidirect
17. C₃₁×#391	C₂²	semidirect
26. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
74. C₂×#2310	C₃₁	direct (N × H)
100. #2310	C₂×C₃₁	direct (N × H)
101. C₂×#391	C₂×C₃₁	semidirect
117. C₂³×Q₈	C₄×C₃₁	semidirect
127. #391	C₂²×C₃₁	semidirect
131. C₂×C₃₁	#2310	direct (N × H)
141. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
160. C₃₁	C₂×#2310	direct (N × H)
227. C₂	C₃₁×#2310	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₃₁	2
C₂²×C₃₁×#117	2
C₃₁×#2310	2
C₂×C₃₁×#391	2
C₂×C₃₁×#381	2
C₂×#2310	31

Subgroups — C2×C31×#2310 #21490801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2310 #21490801