Subgroups — C₂×#22590635 #32155562

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986415	C₂	1	1	yes	maximal, characteristic
2	#22590635	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17985832	C₂	1	2	yes	maximal, complemented
4	C₂²×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×D₇₁	C₂	1	2	yes	maximal, complemented
6	C₂×M₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556134	C₄	1	1	yes	characteristic
8	#17986415	C₄	1	4	yes	—
9	#17986415	C₂²	1	4	yes	—
10	C₂×#14556068	C₄	1	1	yes	characteristic
11	C₂×#14556067	C₄	1	2	yes	—
12	C₂×C₁₆×D₇₁	C₂²	1	2	yes	—
13	#17985832	C₂²	1	8	yes	complemented
14	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₇₁⋊C₃₂	C₂²	1	4	yes	complemented
17	C₃₂×D₇₁	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	M₆₄×C₇₁	C₂²	1	4	yes	complemented
20	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
21	C₂×#11765865	C₈	1	1	yes	characteristic
22	#14556134	C₈	1	4	yes	—
23	#14556134	C₂×C₄	1	4	yes	—
24	C₂×#11765849	C₈	1	1	yes	characteristic
25	C₂×#11765848	C₈	1	2	yes	—
26	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
27	#14556068	C₂×C₄	1	4	yes	—
28	#14556067	C₂×C₄	1	8	yes	—
29	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
30	C₁₆×D₇₁	C₂×C₄	1	4	yes	—
31	C₁₆×D₇₁	C₂³	1	4	yes	—
32	C₇₁⋊C₃₂	C₂³	1	2	yes	—
33	C₇₁⋊C₃₂	C₂³	1	2	yes	—
34	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
37	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
38	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
39	C₃₂×C₇₁	C₂³	1	4	yes	—
40	#11765865	C₁₆	1	4	yes	—
41	#11765865	C₂×C₈	1	4	yes	—
42	C₂×Dic₁₄₂	C₁₆	1	1	yes	characteristic
43	C₂×D₂₈₄	C₁₆	1	1	yes	characteristic
44	C₂×C₂²⋊D₇₁	C₁₆	1	2	yes	—
45	C₂×C₄×D₇₁	C₂×C₈	1	2	yes	—
46	#11765849	C₂×C₈	1	4	yes	—
47	#11765848	C₂×C₈	1	8	yes	—
48	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
49	C₈×D₇₁	C₂×C₈	1	4	yes	—
50	C₈×D₇₁	C₂²×C₄	1	4	yes	—
51	C₇₁⋊C₁₆	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
57	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
59	C₁₆×C₇₁	C₂⁴	1	1	yes	characteristic
60	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
61	C₂²×D₇₁	C₂×C₁₆	1	2	yes	—
62	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
66	C₂²⋊D₇₁	C₂×C₁₆	1	8	yes	—
67	C₂×Dic₇₁	C₂×C₁₆	1	2	yes	—
68	D₂₈₄	C₂×C₁₆	1	4	yes	—
69	C₄×D₇₁	C₂×C₁₆	1	4	yes	—
70	C₄×D₇₁	C₂²×C₈	1	4	yes	—
71	Dic₁₄₂	C₂×C₁₆	1	4	yes	—
72	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
73	C₈×C₇₁	C₂²×C₈	1	2	yes	—
74	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
75	C₇₁⋊C₈	C₂²×C₈	1	4	yes	—
76	C₂²×C₇₁	—	2	2	no	—
77	C₂²×C₇₁	C₂²×C₁₆	1	2	yes	—
78	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
79	C₂×D₇₁	—	2	4	no	—
80	C₂×D₇₁	C₂²×C₁₆	1	4	yes	—
81	C₄×C₇₁	C₂²×C₁₆	1	2	yes	—
82	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
83	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
84	Dic₇₁	C₂²×C₁₆	1	4	yes	—
85	C₂×#2258	—	71	1	no	maximal, Sylow
86	C₂×C₇₁	—	2	2	no	—
87	C₂×C₇₁	#2258	1	2	yes	complemented
88	C₂×C₇₁	C₂³×C₁₆	1	1	yes	characteristic
89	D₇₁	—	2	4	no	—
90	C₂×#504	—	71	1	no	—
91	#2258	—	71	8	no	—
92	C₂×M₆₄	—	71	2	no	—
93	C₂×M₆₄	D₇₁	1	1	yes	characteristic, complemented
94	C₂²×C₃₂	—	71	1	no	—
95	C₂²×C₃₂	—	71	2	no	—
96	C₇₁	C₂×#2258	1	1	yes	characteristic, complemented, Sylow
97	C₂×#131	—	71	1	no	—
98	#504	—	71	4	no	—
99	#504	—	71	4	no	—
100	C₂×M₃₂	—	71	1	no	—
101	C₂×M₃₂	—	71	2	no	—
102	C₂²×C₁₆	—	71	2	no	—
103	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
104	M₆₄	—	71	8	no	—
105	M₆₄	C₂×D₇₁	1	4	yes	complemented
106	C₂×C₃₂	—	71	4	no	—
107	C₂×C₃₂	—	71	1	no	—
108	C₂×C₃₂	—	71	1	no	—
109	C₂×C₃₂	—	71	8	no	—
110	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
111	C₂×#41	—	71	1	no	—
112	#131	—	71	4	no	—
113	#131	—	71	4	no	—
114	C₂×M₁₆	—	71	1	no	—
115	C₂×M₁₆	—	71	2	no	—
116	C₂²×C₈	—	71	2	no	—
117	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
118	M₃₂	—	71	4	no	—
119	M₃₂	—	71	8	no	—
120	C₂×C₁₆	—	71	2	no	—
121	C₂×C₁₆	—	71	4	no	—
122	C₂×C₁₆	—	71	4	no	—
123	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
124	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
125	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
126	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
127	C₃₂	—	71	2	no	—
128	C₃₂	—	71	2	no	—
129	C₃₂	C₂²×D₇₁	1	4	yes	—
130	#41	—	71	4	no	—
131	#41	—	71	4	no	—
132	C₂×Q₈	—	71	1	no	—
133	C₂×D₄	—	71	2	no	—
134	C₂×D₄	—	71	1	no	—
135	C₂²×C₄	—	71	2	no	—
136	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
137	M₁₆	—	71	4	no	—
138	M₁₆	—	71	8	no	—
139	C₂×C₈	—	71	2	no	—
140	C₂×C₈	—	71	4	no	—
141	C₂×C₈	—	71	4	no	—
142	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
143	C₂×C₈	C₈×D₇₁	1	2	yes	—
144	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
145	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
146	C₁₆	—	71	4	no	—
147	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
148	C₁₆	C₂×C₄×D₇₁	1	2	yes	—
149	C₁₆	C₂³×D₇₁	1	1	yes	characteristic
150	C₂³	—	71	2	no	—
151	C₂³	C₁₆×D₇₁	1	1	yes	characteristic
152	Q₈	—	71	4	no	—
153	D₄	—	71	8	no	—
154	D₄	—	71	4	no	—
155	C₂×C₄	—	71	2	no	—
156	C₂×C₄	—	71	4	no	—
157	C₂×C₄	—	71	4	no	—
158	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
159	C₂×C₄	C₁₆×D₇₁	1	2	yes	—
160	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
162	C₈	—	71	4	no	—
163	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
164	C₈	C₂×C₈×D₇₁	1	2	yes	—
165	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
166	C₂²	—	2	2	no	—
167	C₂²	—	71	4	no	—
168	C₂²	—	142	4	no	—
169	C₂²	C₂×C₁₆×D₇₁	1	2	yes	—
170	C₂²	C₂×C₁₆×D₇₁	1	1	yes	characteristic
171	C₄	—	71	4	no	—
172	C₄	C₂×C₁₆×D₇₁	1	1	yes	characteristic
173	C₄	C₂×C₁₆×D₇₁	1	2	yes	—
174	C₄	C₂²×C₈×D₇₁	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	142	4	no	—
177	C₂	#22590635	1	2	yes	direct product factor
178	C₂	C₂²×C₁₆×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇₁
commutator	C₂×C₇₁	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₇₁
Fitting	C₂×M₆₄×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₁₆
prime radical	C₂×#11765865	C₈
cosocle	C₁₆×C₇₁	C₂⁴
hypercenter	C₂×M₆₄	D₇₁
hypocenter	C₇₁	C₂×#2258
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. #22590635	C₂	direct (N × H)
3. C₂×#17985832	C₂	semidirect
4. C₂²×C₇₁⋊C₃₂	C₂	semidirect
5. C₂×C₃₂×D₇₁	C₂	semidirect
6. C₂×M₆₄×C₇₁	C₂	semidirect
13. #17985832	C₂²	semidirect
16. C₂×C₇₁⋊C₃₂	C₂²	semidirect
17. C₃₂×D₇₁	C₂²	semidirect
19. M₆₄×C₇₁	C₂²	semidirect
87. C₂×C₇₁	#2258	semidirect
93. C₂×M₆₄	D₇₁	semidirect
96. C₇₁	C₂×#2258	semidirect
105. M₆₄	C₂×D₇₁	semidirect
177. C₂	#22590635	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17986415	2
#22590635	2
C₂×#17985832	2
C₂²×C₇₁⋊C₃₂	2
C₂×C₃₂×D₇₁	2
C₂×M₆₄×C₇₁	2
C₂×#2258	71

Subgroups — C2×#22590635 #32155562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590635 #32155562