Subgroups — C₃₁×#33224 #21489983

Order 7936.

1318 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	4	yes	maximal, complemented
2	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2867	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1957	C₂	1	8	yes	maximal, complemented
5	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃₁×#127	—	2	2	no	—
9	C₂×C₃₁×#127	—	2	2	no	—
10	C₂×C₃₁×#127	C₂²	1	2	yes	—
11	C₂×C₃₁×#127	C₂²	1	4	yes	—
12	C₃₁×#486	—	2	16	no	—
13	C₂×C₃₁×#102	C₂²	1	16	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#384	—	2	2	no	—
17	C₃₁×#384	C₂²	1	4	yes	—
18	C₄³×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₃₁	—	2	2	no	—
20	C₂²×D₄×C₃₁	—	2	8	no	—
21	C₂²×D₄×C₃₁	—	2	2	no	—
22	C₂²×D₄×C₃₁	—	2	2	no	—
23	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
24	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
25	C₃₁×C₄⋊Q₈	—	2	8	no	—
26	C₃₁×C₄⋊Q₈	—	4	8	no	—
27	C₃₁×#127	—	2	8	no	—
28	C₃₁×#127	—	2	16	no	—
29	C₃₁×#127	—	4	8	no	—
30	C₃₁×#127	—	4	8	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
34	C₂×C₄²×C₃₁	D₄	1	2	yes	—
35	C₂×C₄²×C₃₁	D₄	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
38	C₃₁×#102	—	2	32	no	—
39	C₄×C₈×C₃₁	—	2	8	no	—
40	C₄×C₈×C₃₁	D₄	1	8	yes	complemented
41	C₂⁴×C₃₁	—	2	4	no	—
42	C₂×Q₈×C₃₁	—	4	4	no	—
43	C₂×Q₈×C₃₁	—	4	8	no	—
44	C₂×D₄×C₃₁	—	2	8	no	—
45	C₂×D₄×C₃₁	—	2	16	no	—
46	C₂×D₄×C₃₁	—	4	16	no	—
47	C₂×D₄×C₃₁	—	4	4	no	—
48	C₂×D₄×C₃₁	—	4	4	no	—
49	C₂×D₄×C₃₁	—	4	16	no	—
50	C₂×D₄×C₃₁	—	4	8	no	—
51	C₂×D₄×C₃₁	—	4	16	no	—
52	C₂×D₄×C₃₁	—	4	8	no	—
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₃₁	—	2	8	no	—
59	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
60	C₄⋊C₄×C₃₁	—	2	8	no	—
61	C₄⋊C₄×C₃₁	—	4	4	no	—
62	C₄⋊C₄×C₃₁	—	4	16	no	—
63	C₄²×C₃₁	D₈	1	8	yes	—
64	C₄²×C₃₁	SD₁₆	1	8	yes	—
65	C₄²×C₃₁	C₂×D₄	1	4	yes	—
66	C₄²×C₃₁	#41	1	8	yes	—
67	#33224	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₃₁	—	4	8	no	—
69	C₂³×C₃₁	—	4	4	no	—
70	C₂³×C₃₁	—	4	8	no	—
71	C₂³×C₃₁	—	4	8	no	—
72	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
73	Q₈×C₃₁	—	8	8	no	—
74	D₄×C₃₁	—	4	16	no	—
75	D₄×C₃₁	—	8	32	no	—
76	D₄×C₃₁	—	8	8	no	—
77	D₄×C₃₁	—	8	8	no	—
78	C₂×C₄×C₃₁	—	4	4	no	—
79	C₂×C₄×C₃₁	—	4	8	no	—
80	C₂×C₄×C₃₁	#124	1	8	yes	—
81	C₂×C₄×C₃₁	#127	1	2	yes	—
82	C₂×C₄×C₃₁	#127	1	2	yes	—
83	C₂×C₄×C₃₁	C₂×D₈	1	8	yes	—
84	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
85	C₂×C₄×C₃₁	C₂×SD₁₆	1	8	yes	—
86	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
87	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
89	C₈×C₃₁	—	2	8	no	—
90	C₂×#486	C₂×C₃₁	1	4	yes	complemented
91	#2868	C₂×C₃₁	1	1	yes	characteristic, complemented
92	#2867	C₂×C₃₁	1	1	yes	characteristic
93	#1957	C₂×C₃₁	1	8	yes	complemented
94	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
95	C₂²×C₃₁	—	8	4	no	—
96	C₂²×C₃₁	—	8	16	no	—
97	C₂²×C₃₁	—	8	4	no	—
98	C₂²×C₃₁	—	8	4	no	—
99	C₂²×C₃₁	C₂×#124	1	2	yes	—
100	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
101	C₂²×C₃₁	C₂²×D₈	1	2	yes	—
102	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
103	C₄×C₃₁	—	8	4	no	—
104	C₄×C₃₁	#486	1	16	yes	complemented
105	C₄×C₃₁	#492	1	4	yes	—
106	C₄×C₃₁	#493	1	4	yes	—
107	C₄×C₃₁	#532	1	4	yes	—
108	C₂×C₄⋊Q₈	—	2	2	no	—
109	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
110	C₂×#127	—	2	2	no	—
111	C₂×#127	—	2	2	no	—
112	C₂×#127	C₂²×C₃₁	1	2	yes	—
113	C₂×#127	C₂²×C₃₁	1	4	yes	—
114	#486	—	2	16	no	—
115	C₂×#102	C₂²×C₃₁	1	16	yes	—
116	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
117	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
118	#384	—	2	2	no	—
119	#384	C₂²×C₃₁	1	4	yes	—
120	C₄³	C₂²×C₃₁	1	1	yes	characteristic
121	C₂×C₃₁	—	16	4	no	—
122	C₂×C₃₁	#2858	1	1	yes	characteristic
123	C₂×C₃₁	C₂×#486	1	4	yes	—
124	C₂×C₃₁	C₂×#492	1	1	yes	characteristic
125	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
126	C₂²×Q₈	—	2	2	no	—
127	C₂²×D₄	—	2	8	no	—
128	C₂²×D₄	—	2	2	no	—
129	C₂²×D₄	—	2	2	no	—
130	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
131	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
132	C₄⋊Q₈	—	2	8	no	—
133	C₄⋊Q₈	—	4	8	no	—
134	#127	—	2	8	no	—
135	#127	—	2	16	no	—
136	#127	—	4	8	no	—
137	#127	—	4	8	no	—
138	C₂×C₄⋊C₄	—	2	8	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
141	C₂×C₄²	D₄×C₃₁	1	2	yes	—
142	C₂×C₄²	D₄×C₃₁	1	2	yes	—
143	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
144	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
145	#102	—	2	32	no	—
146	C₄×C₈	—	2	8	no	—
147	C₄×C₈	D₄×C₃₁	1	8	yes	complemented
148	C₃₁	#33224	1	1	yes	characteristic, direct product factor, Sylow
149	C₂⁴	—	2	4	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	8	no	—
152	C₂×D₄	—	2	8	no	—
153	C₂×D₄	—	2	16	no	—
154	C₂×D₄	—	4	16	no	—
155	C₂×D₄	—	4	4	no	—
156	C₂×D₄	—	4	4	no	—
157	C₂×D₄	—	4	16	no	—
158	C₂×D₄	—	4	8	no	—
159	C₂×D₄	—	4	16	no	—
160	C₂×D₄	—	4	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
163	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
168	C₄⋊C₄	—	2	8	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	16	no	—
171	C₄²	D₈×C₃₁	1	8	yes	—
172	C₄²	SD₁₆×C₃₁	1	8	yes	—
173	C₄²	C₂×D₄×C₃₁	1	4	yes	—
174	C₄²	#41×C₃₁	1	8	yes	—
175	C₂³	—	4	8	no	—
176	C₂³	—	4	4	no	—
177	C₂³	—	4	8	no	—
178	C₂³	—	4	8	no	—
179	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
180	Q₈	—	8	8	no	—
181	D₄	—	4	16	no	—
182	D₄	—	8	32	no	—
183	D₄	—	8	8	no	—
184	D₄	—	8	8	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	C₃₁×#124	1	8	yes	—
188	C₂×C₄	C₃₁×#127	1	2	yes	—
189	C₂×C₄	C₃₁×#127	1	2	yes	—
190	C₂×C₄	C₂×D₈×C₃₁	1	8	yes	—
191	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
192	C₂×C₄	C₂×SD₁₆×C₃₁	1	8	yes	—
193	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
194	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
195	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
196	C₈	—	2	8	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	16	no	—
199	C₂²	—	8	4	no	—
200	C₂²	—	8	4	no	—
201	C₂²	C₂×C₃₁×#124	1	2	yes	—
202	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
203	C₂²	C₂²×D₈×C₃₁	1	2	yes	—
204	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
205	C₄	—	8	4	no	—
206	C₄	C₃₁×#486	1	16	yes	complemented
207	C₄	C₃₁×#492	1	4	yes	—
208	C₄	C₃₁×#493	1	4	yes	—
209	C₄	C₃₁×#532	1	4	yes	—
210	C₂	—	16	4	no	—
211	C₂	C₃₁×#2858	1	1	yes	characteristic
212	C₂	C₂×C₃₁×#486	1	4	yes	—
213	C₂	C₂×C₃₁×#492	1	1	yes	characteristic
214	C₂	C₂×C₃₁×#493	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2867	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#486	C₂	semidirect
2. C₃₁×#2868	C₂	semidirect
4. C₃₁×#1957	C₂	semidirect
5. C₄²×C₈×C₃₁	C₂	semidirect
40. C₄×C₈×C₃₁	D₄	semidirect
67. #33224	C₃₁	direct (N × H)
90. C₂×#486	C₂×C₃₁	semidirect
91. #2868	C₂×C₃₁	semidirect
93. #1957	C₂×C₃₁	semidirect
94. C₄²×C₈	C₂×C₃₁	semidirect
104. C₄×C₃₁	#486	semidirect
147. C₄×C₈	D₄×C₃₁	semidirect
148. C₃₁	#33224	direct (N × H)
206. C₄	C₃₁×#486	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#486	2
C₃₁×#2868	2
C₃₁×#2867	2
C₃₁×#1957	2
C₄²×C₈×C₃₁	2
#33224	31

Subgroups — C31×#33224 #21489983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33224 #21489983