Subgroups — C₂×C₃₁×#2177 #21490209

Order 7936.

798 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	C₂²×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2177	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#357	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₃₁	—	2	2	no	—
9	C₂×C₃₁×D₁₆	C₄	1	8	yes	complemented
10	C₂×C₃₁×D₁₆	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×D₈×C₃₁	—	2	4	no	—
14	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#102	—	2	2	no	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#367	C₂²	1	4	yes	complemented
19	C₃₁×#357	C₂²	1	8	yes	complemented
20	C₃₁×#346	C₂²	1	4	yes	complemented
21	C₂²×D₄×C₃₁	—	2	2	no	—
22	C₂³×C₄×C₃₁	—	4	2	no	—
23	C₂×D₈×C₃₁	—	2	2	no	—
24	C₂×D₈×C₃₁	—	2	4	no	—
25	C₂×D₈×C₃₁	C₂×C₄	1	8	yes	complemented
26	C₂×D₈×C₃₁	C₂³	1	4	yes	—
27	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₄×D₄×C₃₁	—	2	8	no	—
30	C₄×D₄×C₃₁	—	4	4	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₃₁×D₁₆	C₂×C₄	1	16	yes	complemented
35	C₂×C₁₆×C₃₁	—	2	2	no	—
36	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
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×#102	—	2	8	no	—
42	C₄×C₈×C₃₁	D₄	1	2	yes	—
43	C₄×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂⁴×C₃₁	—	4	2	no	—
45	C₂×D₄×C₃₁	—	2	8	no	—
46	C₂×D₄×C₃₁	—	2	4	no	—
47	C₂×D₄×C₃₁	—	4	2	no	—
48	C₂×D₄×C₃₁	—	4	4	no	—
49	C₂²×C₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	—	4	8	no	—
51	C₂²×C₄×C₃₁	—	8	8	no	—
52	C₂²×C₄×C₃₁	D₈	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	D₈×C₃₁	—	2	8	no	—
55	D₈×C₃₁	—	2	4	no	—
56	D₈×C₃₁	C₂²×C₄	1	8	yes	complemented
57	C₂×C₈×C₃₁	—	2	2	no	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	#41	1	2	yes	—
62	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₂²⋊C₄×C₃₁	—	4	8	no	—
66	C₄²×C₃₁	D₈	1	2	yes	—
67	C₄²×C₃₁	C₂×D₄	1	2	yes	—
68	C₁₆×C₃₁	—	2	2	no	—
69	C₁₆×C₃₁	C₂²×C₄	1	4	yes	complemented
70	C₂×#2177	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	—	4	4	no	—
72	C₂³×C₃₁	—	4	8	no	—
73	C₂³×C₃₁	—	8	4	no	—
74	C₂³×C₃₁	—	8	4	no	—
75	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
76	D₄×C₃₁	—	2	8	no	—
77	D₄×C₃₁	—	4	4	no	—
78	D₄×C₃₁	—	4	8	no	—
79	C₂×C₄×C₃₁	—	2	4	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	8	16	no	—
82	C₂×C₄×C₃₁	—	8	4	no	—
83	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
85	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	#135	1	2	yes	—
87	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₃₁	—	2	2	no	—
89	C₈×C₃₁	C₄×D₄	1	2	yes	—
90	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
92	C₂²×D₁₆	C₂×C₃₁	1	1	yes	characteristic
93	C₂×C₄×D₈	C₂×C₃₁	1	2	yes	complemented
94	#2177	C₂×C₃₁	1	8	yes	direct product factor
95	C₂×#367	C₂×C₃₁	1	1	yes	characteristic, complemented
96	C₂×#357	C₂×C₃₁	1	2	yes	complemented
97	C₂×#346	C₂×C₃₁	1	1	yes	characteristic, complemented
98	C₂²×C₃₁	—	4	8	no	—
99	C₂²×C₃₁	—	8	4	no	—
100	C₂²×C₃₁	—	8	4	no	—
101	C₂²×C₃₁	—	8	16	no	—
102	C₂²×C₃₁	C₄×D₈	1	2	yes	—
103	C₂²×C₃₁	#509	1	4	yes	—
104	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
105	C₄×C₃₁	—	2	4	no	—
106	C₄×C₃₁	—	8	4	no	—
107	C₄×C₃₁	C₄×D₈	1	2	yes	—
108	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
109	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
110	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
111	C₂×C₄×D₄	—	2	2	no	—
112	C₂×D₁₆	C₄×C₃₁	1	8	yes	complemented
113	C₂×D₁₆	C₂²×C₃₁	1	4	yes	—
114	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
115	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
116	C₄×D₈	—	2	4	no	—
117	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
118	C₂×#107	C₂²×C₃₁	1	2	yes	—
119	C₂×#102	—	2	2	no	—
120	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
121	#367	C₂²×C₃₁	1	4	yes	complemented
122	#357	C₂²×C₃₁	1	8	yes	complemented
123	#346	C₂²×C₃₁	1	4	yes	complemented
124	C₂×C₃₁	—	8	8	no	—
125	C₂×C₃₁	#2177	1	4	yes	direct product factor
126	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
127	C₂×C₃₁	C₂×#509	1	2	yes	—
128	C₂²×D₄	—	2	2	no	—
129	C₂³×C₄	—	4	2	no	—
130	C₂×D₈	—	2	2	no	—
131	C₂×D₈	—	2	4	no	—
132	C₂×D₈	C₂×C₄×C₃₁	1	8	yes	complemented
133	C₂×D₈	C₂³×C₃₁	1	4	yes	—
134	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
136	C₄×D₄	—	2	8	no	—
137	C₄×D₄	—	4	4	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₂²⋊C₄	—	4	2	no	—
140	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
141	D₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
144	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
145	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
146	#107	—	2	2	no	—
147	#107	C₂³×C₃₁	1	4	yes	—
148	#102	—	2	8	no	—
149	C₄×C₈	D₄×C₃₁	1	2	yes	—
150	C₄×C₈	C₂³×C₃₁	1	2	yes	—
151	C₃₁	C₂×#2177	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	—	4	2	no	—
153	C₂×D₄	—	2	8	no	—
154	C₂×D₄	—	2	4	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	4	8	no	—
159	C₂²×C₄	—	8	8	no	—
160	C₂²×C₄	D₈×C₃₁	1	2	yes	—
161	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
162	D₈	—	2	8	no	—
163	D₈	—	2	4	no	—
164	D₈	C₂²×C₄×C₃₁	1	8	yes	complemented
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
167	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
168	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
169	C₂×C₈	#41×C₃₁	1	2	yes	—
170	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₂²⋊C₄	—	4	8	no	—
174	C₄²	D₈×C₃₁	1	2	yes	—
175	C₄²	C₂×D₄×C₃₁	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂²×C₄×C₃₁	1	4	yes	complemented
178	C₂³	—	4	8	no	—
179	C₂³	—	4	4	no	—
180	C₂³	—	8	4	no	—
181	C₂³	—	8	4	no	—
182	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
183	D₄	—	2	8	no	—
184	D₄	—	4	8	no	—
185	D₄	—	4	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	8	16	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
191	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
192	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
193	C₂×C₄	C₃₁×#135	1	2	yes	—
194	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	C₄×D₄×C₃₁	1	2	yes	—
197	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
198	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
199	C₂²	—	4	8	no	—
200	C₂²	—	8	4	no	—
201	C₂²	—	8	4	no	—
202	C₂²	—	8	16	no	—
203	C₂²	C₄×D₈×C₃₁	1	2	yes	—
204	C₂²	C₃₁×#509	1	4	yes	—
205	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
206	C₄	—	2	4	no	—
207	C₄	—	8	4	no	—
208	C₄	C₄×D₈×C₃₁	1	2	yes	—
209	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
210	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
211	C₂	—	8	8	no	—
212	C₂	C₃₁×#2177	1	4	yes	direct product factor
213	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
214	C₂	C₂×C₃₁×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×C₃₁×D₁₆	C₂
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
2. C₂×C₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2177	C₂	direct (N × H)
4. C₂×C₃₁×#367	C₂	semidirect
5. C₂×C₃₁×#357	C₂	semidirect
6. C₂×C₃₁×#346	C₂	semidirect
9. C₂×C₃₁×D₁₆	C₄	semidirect
14. C₄×D₈×C₃₁	C₂²	semidirect
18. C₃₁×#367	C₂²	semidirect
19. C₃₁×#357	C₂²	semidirect
20. C₃₁×#346	C₂²	semidirect
25. C₂×D₈×C₃₁	C₂×C₄	semidirect
34. C₃₁×D₁₆	C₂×C₄	semidirect
36. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
56. D₈×C₃₁	C₂²×C₄	semidirect
69. C₁₆×C₃₁	C₂²×C₄	semidirect
70. C₂×#2177	C₃₁	direct (N × H)
93. C₂×C₄×D₈	C₂×C₃₁	semidirect
94. #2177	C₂×C₃₁	direct (N × H)
95. C₂×#367	C₂×C₃₁	semidirect
96. C₂×#357	C₂×C₃₁	semidirect
97. C₂×#346	C₂×C₃₁	semidirect
112. C₂×D₁₆	C₄×C₃₁	semidirect
117. C₄×D₈	C₂²×C₃₁	semidirect
121. #367	C₂²×C₃₁	semidirect
122. #357	C₂²×C₃₁	semidirect
123. #346	C₂²×C₃₁	semidirect
125. C₂×C₃₁	#2177	direct (N × H)
132. C₂×D₈	C₂×C₄×C₃₁	semidirect
141. D₁₆	C₂×C₄×C₃₁	semidirect
143. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
151. C₃₁	C₂×#2177	direct (N × H)
164. D₈	C₂²×C₄×C₃₁	semidirect
177. C₁₆	C₂²×C₄×C₃₁	semidirect
212. C₂	C₃₁×#2177	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×D₁₆	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2177	2
C₂×C₃₁×#367	2
C₂×C₃₁×#357	2
C₂×C₃₁×#346	2
C₂×#2177	31

Subgroups — C2×C31×#2177 #21490209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2177 #21490209