Subgroups — C₂×#22586404 #31908994

Order 18176.

286 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	2	yes	maximal
2	#22586404	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈²×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₁₆	—	2	2	no	—
5	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
6	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₈²×C₇₁	C₄	1	2	yes	—
11	C₈²×C₇₁	C₂²	1	2	yes	—
12	C₂×C₇₁⋊C₁₆	—	2	4	no	—
13	C₂×C₇₁⋊C₁₆	—	4	4	no	—
14	C₂²×C₈×C₇₁	—	2	1	no	—
15	C₂²×C₈×C₇₁	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₇₁	—	2	1	no	—
21	C₄×C₈×C₇₁	C₈	1	2	yes	—
22	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	C₇₁⋊C₁₆	—	4	4	no	—
29	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₇₁	—	2	4	no	—
33	C₂×C₈×C₇₁	—	2	2	no	—
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	C₁₆	1	4	yes	complemented
37	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
42	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
47	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₇₁	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#106	1	2	yes	—
54	C₂×C₄×C₇₁	#108	1	2	yes	—
55	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
57	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
60	C₈×C₇₁	—	2	4	no	—
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	—	2	2	no	—
64	C₈×C₇₁	#106	1	2	yes	—
65	C₈×C₇₁	C₂×C₁₆	1	4	yes	complemented
66	C₈×C₇₁	C₂×SD₁₆	1	2	yes	—
67	C₂²×C₇₁	#334	1	2	yes	—
68	C₂²×C₇₁	C₄⋊C₁₆	1	2	yes	—
69	C₂²×C₇₁	#364	1	2	yes	—
70	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	#334	1	2	yes	—
74	C₄×C₇₁	C₄⋊C₁₆	1	2	yes	—
75	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
76	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
77	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
78	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
79	C₂×#1367	—	71	1	no	maximal, Sylow
80	C₂×C₇₁	#1367	1	4	yes	complemented
81	C₂×C₇₁	C₂×#334	1	1	yes	characteristic
82	C₂×C₇₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
83	C₂×C₇₁	C₂×#364	1	1	yes	characteristic
84	C₂×C₄⋊C₁₆	—	71	2	no	—
85	C₂×C₈²	D₇₁	1	1	yes	characteristic
86	#1367	—	71	4	no	—
87	C₇₁	C₂×#1367	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	142	2	no	—
89	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
90	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
92	C₄⋊C₁₆	—	71	4	no	—
93	C₄⋊C₁₆	—	142	2	no	—
94	C₈²	Dic₇₁	1	2	yes	—
95	C₈²	C₂×D₇₁	1	2	yes	—
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	C₇₁⋊C₈	1	1	yes	characteristic
98	C₂²×C₈	C₇₁⋊C₈	1	1	yes	characteristic
99	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
100	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
101	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
102	C₂×C₁₆	—	284	4	no	—
103	C₂×C₁₆	—	142	4	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₇₁⋊C₈	1	2	yes	—
106	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
107	C₄×C₈	D₂₈₄	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
109	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
111	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
114	C₂²×C₄	#11765849	1	1	yes	characteristic
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₇₁⋊C₁₆	1	4	yes	complemented
120	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
122	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
123	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
124	C₂×C₈	#11765854	1	4	yes	—
125	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
126	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
127	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
128	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
129	C₄²	#11765849	1	1	yes	characteristic
130	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
131	C₁₆	—	284	4	no	—
132	C₂³	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
135	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
136	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
137	C₂×C₄	#14556060	1	2	yes	—
138	C₂×C₄	#14556064	1	2	yes	—
139	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
140	C₂×C₄	#14556068	1	2	yes	—
141	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
143	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	2	2	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	#14556060	1	2	yes	—
149	C₈	C₂×C₇₁⋊C₁₆	1	4	yes	complemented
150	C₈	C₂×#11765854	1	2	yes	—
151	C₂²	#17985750	1	2	yes	—
152	C₂²	C₄⋊(C₇₁⋊C₁₆)	1	2	yes	—
153	C₂²	#17985820	1	2	yes	—
154	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	#17985750	1	2	yes	—
158	C₄	C₄⋊(C₇₁⋊C₁₆)	1	2	yes	—
159	C₄	C₂×#14556060	1	1	yes	characteristic
160	C₄	C₂×#14556064	1	1	yes	characteristic
161	C₄	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
162	C₄	C₂×#14556068	1	1	yes	characteristic
163	C₂	#22586404	1	4	yes	direct product factor
164	C₂	C₂×#17985750	1	1	yes	characteristic
165	C₂	C₂×C₄⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
166	C₂	C₂×#17985820	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₂₈₄
commutator	C₄×C₇₁	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×C₈²×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₄⋊C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×C₈²	D₇₁
hypocenter	C₇₁	C₂×#1367
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #22586404	C₂	direct (N × H)
36. C₂×C₈×C₇₁	C₁₆	semidirect
65. C₈×C₇₁	C₂×C₁₆	semidirect
80. C₂×C₇₁	#1367	semidirect
87. C₇₁	C₂×#1367	semidirect
119. C₂×C₈	C₇₁⋊C₁₆	semidirect
149. C₈	C₂×C₇₁⋊C₁₆	semidirect
163. C₂	#22586404	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×C₄⋊(C₇₁⋊C₁₆)	2
#22586404	2
C₂×C₈²×C₇₁	2
C₂×#1367	71

Subgroups — C2×#22586404 #31908994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586404 #31908994