Subgroups — C₂×C₃₁×#2176 #21490208

Order 7936.

670 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	2	yes	maximal
3	C₃₁×#2176	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#367	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#358	C₂	1	2	yes	maximal
6	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
9	C₂×C₃₁×Q₃₂	C₄	1	8	yes	complemented
10	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×Q₁₆×C₃₁	—	2	4	no	—
14	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	—
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#103	—	2	2	no	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#367	C₂²	1	4	yes	—
19	C₃₁×#358	C₂²	1	8	yes	—
20	C₃₁×#346	C₂²	1	4	yes	—
21	C₂²×Q₈×C₃₁	—	2	2	no	—
22	C₂×Q₁₆×C₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₃₁	—	2	4	no	—
24	C₂×Q₁₆×C₃₁	C₂×C₄	1	8	yes	—
25	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₃₁	—	2	8	no	—
29	C₄×Q₈×C₃₁	—	4	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
32	C₂×C₄²×C₃₁	—	4	2	no	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₃₁×Q₃₂	C₂×C₄	1	16	yes	complemented
35	C₂×C₁₆×C₃₁	—	2	2	no	—
36	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
38	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
39	C₃₁×#107	—	2	2	no	—
40	C₃₁×#107	C₂³	1	4	yes	—
41	C₃₁×#103	—	2	8	no	—
42	C₄×C₈×C₃₁	D₄	1	2	yes	—
43	C₄×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂×Q₈×C₃₁	—	2	8	no	—
45	C₂×Q₈×C₃₁	—	2	4	no	—
46	C₂×Q₈×C₃₁	—	4	2	no	—
47	C₂×Q₈×C₃₁	—	4	4	no	—
48	C₂²×C₄×C₃₁	—	4	2	no	—
49	C₂²×C₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	D₈	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
52	Q₁₆×C₃₁	—	2	8	no	—
53	Q₁₆×C₃₁	—	2	4	no	—
54	Q₁₆×C₃₁	C₂²×C₄	1	8	yes	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	#41	1	2	yes	—
60	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	—	2	4	no	—
62	C₄⋊C₄×C₃₁	—	4	2	no	—
63	C₄⋊C₄×C₃₁	—	4	8	no	—
64	C₄²×C₃₁	—	4	8	no	—
65	C₄²×C₃₁	D₈	1	2	yes	—
66	C₄²×C₃₁	C₂×D₄	1	2	yes	—
67	C₁₆×C₃₁	—	2	2	no	—
68	C₁₆×C₃₁	C₂²×C₄	1	4	yes	—
69	C₂×#2176	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
71	Q₈×C₃₁	—	2	8	no	—
72	Q₈×C₃₁	—	4	4	no	—
73	Q₈×C₃₁	—	4	8	no	—
74	C₂×C₄×C₃₁	—	2	4	no	—
75	C₂×C₄×C₃₁	—	4	4	no	—
76	C₂×C₄×C₃₁	—	4	8	no	—
77	C₂×C₄×C₃₁	—	4	4	no	—
78	C₂×C₄×C₃₁	—	8	4	no	—
79	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₃₁	#135	1	2	yes	—
83	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
84	C₈×C₃₁	—	2	2	no	—
85	C₈×C₃₁	C₄×D₄	1	2	yes	—
86	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
88	C₂²×Q₃₂	C₂×C₃₁	1	1	yes	characteristic
89	C₂×C₄×Q₁₆	C₂×C₃₁	1	2	yes	—
90	#2176	C₂×C₃₁	1	8	yes	direct product factor
91	C₂×#367	C₂×C₃₁	1	1	yes	characteristic
92	C₂×#358	C₂×C₃₁	1	2	yes	—
93	C₂×#346	C₂×C₃₁	1	1	yes	characteristic
94	C₂²×C₃₁	C₄×D₈	1	2	yes	—
95	C₂²×C₃₁	#510	1	4	yes	—
96	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₃₁	—	2	4	no	—
98	C₄×C₃₁	—	4	8	no	—
99	C₄×C₃₁	—	8	4	no	—
100	C₄×C₃₁	C₄×D₈	1	2	yes	—
101	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
102	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
103	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
104	C₂×C₄×Q₈	—	2	2	no	—
105	C₂×Q₃₂	C₄×C₃₁	1	8	yes	complemented
106	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	—
107	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
108	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
109	C₄×Q₁₆	—	2	4	no	—
110	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	—
111	C₂×#107	C₂²×C₃₁	1	2	yes	—
112	C₂×#103	—	2	2	no	—
113	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
114	#367	C₂²×C₃₁	1	4	yes	—
115	#358	C₂²×C₃₁	1	8	yes	—
116	#346	C₂²×C₃₁	1	4	yes	—
117	C₂×C₃₁	#2176	1	4	yes	direct product factor
118	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
119	C₂×C₃₁	C₂×#510	1	2	yes	—
120	C₂²×Q₈	—	2	2	no	—
121	C₂×Q₁₆	—	2	2	no	—
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	C₂×C₄×C₃₁	1	8	yes	—
124	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
125	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
127	C₄×Q₈	—	2	8	no	—
128	C₄×Q₈	—	4	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	2	no	—
131	C₂×C₄²	—	4	2	no	—
132	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
133	Q₃₂	C₂×C₄×C₃₁	1	16	yes	complemented
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
136	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
137	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
138	#107	—	2	2	no	—
139	#107	C₂³×C₃₁	1	4	yes	—
140	#103	—	2	8	no	—
141	C₄×C₈	D₄×C₃₁	1	2	yes	—
142	C₄×C₈	C₂³×C₃₁	1	2	yes	—
143	C₃₁	C₂×#2176	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×Q₈	—	2	8	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	D₈×C₃₁	1	2	yes	—
151	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
152	Q₁₆	—	2	8	no	—
153	Q₁₆	—	2	4	no	—
154	Q₁₆	C₂²×C₄×C₃₁	1	8	yes	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
157	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
158	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
159	C₂×C₈	#41×C₃₁	1	2	yes	—
160	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	8	no	—
164	C₄²	—	4	8	no	—
165	C₄²	D₈×C₃₁	1	2	yes	—
166	C₄²	C₂×D₄×C₃₁	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	C₂²×C₄×C₃₁	1	4	yes	—
169	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
170	Q₈	—	2	8	no	—
171	Q₈	—	4	8	no	—
172	Q₈	—	4	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	4	no	—
178	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
179	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
180	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
181	C₂×C₄	C₃₁×#135	1	2	yes	—
182	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	C₄×D₄×C₃₁	1	2	yes	—
185	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
186	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
187	C₂²	C₄×D₈×C₃₁	1	2	yes	—
188	C₂²	C₃₁×#510	1	4	yes	—
189	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	4	8	no	—
192	C₄	—	8	4	no	—
193	C₄	C₄×D₈×C₃₁	1	2	yes	—
194	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
195	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
196	C₂	C₃₁×#2176	1	4	yes	direct product factor
197	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
198	C₂	C₂×C₃₁×#510	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2176	C₂	direct (N × H)
9. C₂×C₃₁×Q₃₂	C₄	semidirect
34. C₃₁×Q₃₂	C₂×C₄	semidirect
69. C₂×#2176	C₃₁	direct (N × H)
90. #2176	C₂×C₃₁	direct (N × H)
105. C₂×Q₃₂	C₄×C₃₁	semidirect
117. C₂×C₃₁	#2176	direct (N × H)
133. Q₃₂	C₂×C₄×C₃₁	semidirect
143. C₃₁	C₂×#2176	direct (N × H)
196. C₂	C₃₁×#2176	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×Q₃₂	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#2176	2
C₂×C₃₁×#367	2
C₂×C₃₁×#358	2
C₂×C₃₁×#346	2
C₂×#2176	31

Subgroups — C2×C31×#2176 #21490208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2176 #21490208