Subgroups — C₂×#22558407 #30776321

Order 17152.

214 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863230	C₂	1	2	yes	maximal
2	#22558407	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863230	—	2	2	no	—
5	#16863230	C₂²	1	4	yes	—
6	C₂×C₆₇⋊C₃₂	—	2	2	no	—
7	C₂²×C₁₆×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
10	C₄×C₁₆×C₆₇	C₄	1	2	yes	—
11	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
12	C₆₇⋊C₃₂	—	2	4	no	—
13	C₂²×C₈×C₆₇	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
16	C₂×C₁₆×C₆₇	—	2	1	no	—
17	C₂×C₁₆×C₆₇	—	2	1	no	—
18	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
23	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
24	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₆₇	C₈	1	2	yes	—
26	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₆₇	—	2	1	no	—
28	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₆₇	C₁₆	1	2	yes	—
30	C₂×C₈×C₆₇	C₁₆	1	2	yes	—
31	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	C₂×C₈	1	2	yes	—
40	C₁₆×C₆₇	—	2	2	no	—
41	C₁₆×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₁₆×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂³×C₆₇	M₃₂	1	1	yes	characteristic
46	C₂×C₄×C₆₇	—	2	4	no	—
47	C₂×C₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
55	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
56	C₈×C₆₇	C₂×C₁₆	1	4	yes	—
57	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
59	C₂²×C₆₇	—	2	2	no	—
60	C₂²×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
61	C₂²×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
62	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
63	C₄×C₆₇	—	2	4	no	—
64	C₄×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
65	C₄×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
66	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
68	C₂×#1422	—	67	1	no	maximal, Sylow
69	C₂×C₆₇	—	2	1	no	—
70	C₂×C₆₇	—	2	1	no	—
71	C₂×C₆₇	#1422	1	2	yes	complemented
72	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
73	C₂×M₆₄	—	67	2	no	—
74	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
75	#1422	—	67	4	no	—
76	C₆₇	C₂×#1422	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₁₆	Dic₆₇	1	1	yes	characteristic
78	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
79	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
80	M₆₄	—	67	4	no	—
81	M₆₄	—	134	2	no	—
82	C₂×C₃₂	—	134	2	no	—
83	C₄×C₁₆	Dic₆₇	1	2	yes	—
84	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
85	C₂²×C₈	C₆₇⋊C₈	1	1	yes	characteristic
86	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
87	C₂×C₄²	C₆₇⋊C₈	1	1	yes	characteristic
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	Dic₁₃₄	1	1	yes	characteristic
91	C₂×C₁₆	Dic₁₃₄	1	1	yes	characteristic
92	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
93	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
94	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
95	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
96	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
97	C₄×C₈	C₆₇⋊C₈	1	2	yes	—
98	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
99	C₃₂	—	134	4	no	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
102	C₂×C₈	C₆₇⋊C₁₆	1	2	yes	—
103	C₂×C₈	C₆₇⋊C₁₆	1	2	yes	—
104	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
105	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
106	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
107	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
108	C₂×C₈	#11761612	1	1	yes	characteristic
109	C₂×C₈	#11761612	1	1	yes	characteristic
110	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
111	C₄²	—	2	1	no	—
112	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
113	C₁₆	—	2	2	no	—
114	C₁₆	C₄⋊Dic₆₇	1	1	yes	characteristic
115	C₁₆	C₄⋊Dic₆₇	1	1	yes	characteristic
116	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
117	C₁₆	C₂×Dic₁₃₄	1	1	yes	characteristic
118	C₂³	#14523983	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
122	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
123	C₂×C₄	C₂×C₆₇⋊C₁₆	1	1	yes	characteristic
124	C₂×C₄	C₂×C₆₇⋊C₁₆	1	1	yes	characteristic
125	C₂×C₄	#14523983	1	1	yes	characteristic
126	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
127	C₈	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
128	C₈	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
129	C₈	C₂×C₆₇⋊C₁₆	1	4	yes	—
130	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
131	C₈	C₂×#11761612	1	1	yes	characteristic
132	C₂²	—	2	2	no	—
133	C₂²	C₄⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
134	C₂²	C₄⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
135	C₂²	C₂×#14523983	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	C₄⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
138	C₄	C₄⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
139	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
140	C₄	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#22558407	1	2	yes	direct product factor
144	C₂	C₂×C₄⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₂₆₈
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₄⋊C₁₆
prime radical	C₂³×C₆₇	M₃₂
cosocle	C₂×C₁₆×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	C₂×#1422
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22558407	C₂	direct (N × H)
71. C₂×C₆₇	#1422	semidirect
76. C₆₇	C₂×#1422	semidirect
143. C₂	#22558407	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×#16863230	2
#22558407	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#1422	67

Subgroups — C2×#22558407 #30776321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558407 #30776321