Subgroups — C₂×C₄×#11761612 #22568618

Order 8576.

656 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11761612	C₂	1	2	yes	maximal
2	C₄×#11761612	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523954	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₄	1	16	yes	direct product factor
7	C₂×#11761612	C₂²	1	8	yes	—
8	C₂²×C₆₇⋊C₈	C₂²	1	4	yes	—
9	#14523954	C₂²	1	8	yes	complemented
10	C₄×C₆₇⋊C₈	C₂²	1	8	yes	complemented
11	C₂³×C₄×C₆₇	C₄	1	2	yes	—
12	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₆₇	C₄	1	4	yes	—
14	C₂×C₄²×C₆₇	C₄	1	2	yes	—
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#11761612	C₂×C₄	1	32	yes	direct product factor
19	C₂×C₆₇⋊C₈	C₂×C₄	1	16	yes	complemented
20	C₂×C₆₇⋊C₈	C₂³	1	8	yes	—
21	C₂⁴×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₇	—	2	8	no	—
23	C₂²×C₄×C₆₇	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
31	C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₄²×C₆₇	C₂×C₄	1	8	yes	—
33	C₄²×C₆₇	C₂×C₄	1	2	yes	—
34	C₄²×C₆₇	C₂³	1	2	yes	—
35	C₄²×C₆₇	C₂³	1	2	yes	—
36	C₂³×C₆₇	—	2	4	no	—
37	C₂³×C₆₇	C₄²	1	4	yes	—
38	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
39	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	—	2	4	no	—
41	C₂×C₄×C₆₇	—	2	16	no	—
42	C₂×C₄×C₆₇	C₄²	1	8	yes	—
43	C₂×C₄×C₆₇	C₄²	1	4	yes	—
44	C₂×C₄×C₆₇	M₁₆	1	8	yes	complemented
45	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₆₇	C₂²×C₄	1	8	yes	—
47	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
52	C₆₇⋊C₈	C₂²×C₄	1	16	yes	complemented
53	C₂²×C₆₇	—	2	4	no	—
54	C₂²×C₆₇	—	2	8	no	—
55	C₂²×C₆₇	C₂×C₄²	1	4	yes	—
56	C₂²×C₆₇	C₂×C₄²	1	2	yes	—
57	C₂²×C₆₇	C₂×M₁₆	1	4	yes	—
58	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₆₇	—	2	4	no	—
60	C₄×C₆₇	C₂×C₄²	1	2	yes	—
61	C₄×C₆₇	C₂×C₄²	1	4	yes	—
62	C₄×C₆₇	C₂×M₁₆	1	8	yes	complemented
63	C₄×C₆₇	C₂³×C₄	1	2	yes	—
64	C₂×C₆₇	—	2	4	no	—
65	C₂×C₆₇	C₄×M₁₆	1	4	yes	complemented
66	C₂×C₆₇	C₂²×C₄²	1	1	yes	characteristic
67	C₂×C₆₇	C₂²×M₁₆	1	2	yes	—
68	C₂×C₄×M₁₆	—	67	1	no	maximal, Sylow
69	C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
70	C₂²×M₁₆	—	67	2	no	—
71	C₂²×C₄²	D₆₇	1	1	yes	characteristic
72	C₄×M₁₆	—	67	8	no	—
73	C₂×#97	—	67	2	no	—
74	C₂×C₄×C₈	—	67	2	no	—
75	C₂³×C₄	Dic₆₇	1	2	yes	—
76	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
77	C₂×M₁₆	—	67	16	no	—
78	C₂×M₁₆	—	67	8	no	—
79	C₂²×C₈	—	67	4	no	—
80	C₂×C₄²	Dic₆₇	1	2	yes	—
81	C₂×C₄²	Dic₆₇	1	4	yes	—
82	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₆₇	1	4	yes	—
84	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
85	#97	—	67	8	no	—
86	C₄×C₈	—	67	8	no	—
87	C₂⁴	C₂×Dic₆₇	1	1	yes	characteristic
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	C₄×D₆₇	1	8	yes	—
90	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
91	C₂²×C₄	C₂×Dic₆₇	1	8	yes	—
92	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
95	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
97	M₁₆	—	67	32	no	—
98	C₂×C₈	—	67	16	no	—
99	C₂×C₈	—	67	8	no	—
100	C₄²	C₂×Dic₆₇	1	2	yes	—
101	C₄²	C₂×Dic₆₇	1	8	yes	—
102	C₄²	C₂×Dic₆₇	1	2	yes	—
103	C₄²	C₂²×D₆₇	1	2	yes	—
104	C₄²	C₂²×D₆₇	1	2	yes	—
105	C₂³	—	2	4	no	—
106	C₂³	C₄×Dic₆₇	1	4	yes	—
107	C₂³	C₂²×Dic₆₇	1	2	yes	—
108	C₂³	C₂²×Dic₆₇	1	1	yes	characteristic
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	16	no	—
111	C₂×C₄	C₄×Dic₆₇	1	8	yes	—
112	C₂×C₄	C₄×Dic₆₇	1	4	yes	—
113	C₂×C₄	#11761612	1	8	yes	direct product factor
114	C₂×C₄	C₂×C₄×D₆₇	1	4	yes	—
115	C₂×C₄	C₂×C₄×D₆₇	1	8	yes	—
116	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
117	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
118	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
119	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
120	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
121	C₈	—	67	16	no	—
122	C₂²	—	2	8	no	—
123	C₂²	—	2	4	no	—
124	C₂²	C₂×C₄×Dic₆₇	1	4	yes	—
125	C₂²	C₂×C₄×Dic₆₇	1	2	yes	—
126	C₂²	C₂×#11761612	1	4	yes	—
127	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
128	C₄	—	2	4	no	—
129	C₄	C₂×C₄×Dic₆₇	1	2	yes	—
130	C₄	C₂×C₄×Dic₆₇	1	4	yes	—
131	C₄	C₂×#11761612	1	8	yes	direct product factor
132	C₄	C₂²×C₄×D₆₇	1	2	yes	—
133	C₂	—	2	4	no	—
134	C₂	C₄×#11761612	1	4	yes	direct product factor
135	C₂	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
136	C₂	C₂²×#11761612	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₆₇
commutator	C₂×C₆₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂²×C₄²×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₄
hypercenter	C₂²×C₄²	D₆₇
hypocenter	C₆₇	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#11761612	C₂	direct (N × H)
3. C₂×#14523954	C₂	semidirect
4. C₂×C₄×C₆₇⋊C₈	C₂	semidirect
6. C₂×#11761612	C₄	direct (N × H)
9. #14523954	C₂²	semidirect
10. C₄×C₆₇⋊C₈	C₂²	semidirect
18. #11761612	C₂×C₄	direct (N × H)
19. C₂×C₆₇⋊C₈	C₂×C₄	semidirect
44. C₂×C₄×C₆₇	M₁₆	semidirect
52. C₆₇⋊C₈	C₂²×C₄	semidirect
62. C₄×C₆₇	C₂×M₁₆	semidirect
65. C₂×C₆₇	C₄×M₁₆	semidirect
69. C₆₇	C₂×C₄×M₁₆	semidirect
113. C₂×C₄	#11761612	direct (N × H)
131. C₄	C₂×#11761612	direct (N × H)
134. C₂	C₄×#11761612	direct (N × H)

Subgroups — C2×C4×#11761612 #22568618

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄×#11761612 #22568618