Subgroups — C₆₇×#33312 #30712563

Order 17152.

1006 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	4	yes	maximal
2	C₆₇×#2868	C₂	1	2	yes	maximal
3	C₆₇×#1845	C₂	1	8	yes	maximal
4	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₇	—	2	4	no	—
6	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
7	#500×C₆₇	—	2	16	no	—
8	C₂×#107×C₆₇	C₂²	1	16	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
11	#384×C₆₇	—	2	4	no	—
12	#384×C₆₇	C₂²	1	8	yes	—
13	C₄³×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₆₇	—	2	4	no	—
15	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
16	C₄⋊Q₈×C₆₇	—	2	16	no	—
17	C₄⋊Q₈×C₆₇	—	4	16	no	—
18	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
19	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
20	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
21	C₂×C₄²×C₆₇	D₄	1	2	yes	—
22	C₂×C₄²×C₆₇	D₄	1	2	yes	—
23	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
25	#107×C₆₇	—	2	32	no	—
26	C₄×C₈×C₆₇	D₄	1	8	yes	—
27	C₄×C₈×C₆₇	Q₈	1	16	yes	complemented
28	C₂×Q₈×C₆₇	—	4	8	no	—
29	C₂×Q₈×C₆₇	—	4	16	no	—
30	C₂²×C₄×C₆₇	—	2	8	no	—
31	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
34	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
36	C₂×C₈×C₆₇	C₂×Q₈	1	16	yes	—
37	C₄⋊C₄×C₆₇	—	2	16	no	—
38	C₄⋊C₄×C₆₇	—	4	8	no	—
39	C₄⋊C₄×C₆₇	—	4	32	no	—
40	C₄²×C₆₇	D₈	1	8	yes	—
41	C₄²×C₆₇	Q₁₆	1	8	yes	—
42	C₄²×C₆₇	C₂×D₄	1	4	yes	—
43	C₄²×C₆₇	C₂×Q₈	1	8	yes	—
44	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
45	Q₈×C₆₇	—	8	16	no	—
46	C₂×C₄×C₆₇	—	4	8	no	—
47	C₂×C₄×C₆₇	—	4	16	no	—
48	C₂×C₄×C₆₇	#127	1	2	yes	—
49	C₂×C₄×C₆₇	#127	1	2	yes	—
50	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
51	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
52	C₂×C₄×C₆₇	C₂×D₈	1	8	yes	—
53	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
54	C₂×C₄×C₆₇	C₂×Q₁₆	1	8	yes	—
55	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
56	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
57	C₈×C₆₇	C₄⋊Q₈	1	16	yes	complemented
58	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
59	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
60	C₂²×C₆₇	C₂²×D₈	1	2	yes	—
61	C₂²×C₆₇	C₂²×Q₁₆	1	2	yes	—
62	C₄×C₆₇	—	8	8	no	—
63	C₄×C₆₇	#493	1	4	yes	—
64	C₄×C₆₇	#494	1	4	yes	—
65	C₄×C₆₇	#500	1	16	yes	complemented
66	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
67	#33312	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₆₇	#2868	1	1	yes	characteristic
69	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
70	C₂×C₆₇	C₂×#494	1	1	yes	characteristic
71	C₂×C₆₇	C₂×#500	1	4	yes	—
72	C₂×#500	C₂×C₆₇	1	4	yes	—
73	#2868	C₂×C₆₇	1	2	yes	—
74	#1845	C₂×C₆₇	1	8	yes	—
75	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic
76	C₆₇	#33312	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×C₄⋊Q₈	—	2	4	no	—
78	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
79	#500	—	2	16	no	—
80	C₂×#107	C₂²×C₆₇	1	16	yes	—
81	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
82	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
83	#384	—	2	4	no	—
84	#384	C₂²×C₆₇	1	8	yes	—
85	C₄³	C₂²×C₆₇	1	1	yes	characteristic
86	C₂²×Q₈	—	2	4	no	—
87	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
88	C₄⋊Q₈	—	2	16	no	—
89	C₄⋊Q₈	—	4	16	no	—
90	C₂×C₄⋊C₄	—	2	16	no	—
91	C₂×C₄⋊C₄	—	2	4	no	—
92	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
93	C₂×C₄²	D₄×C₆₇	1	2	yes	—
94	C₂×C₄²	D₄×C₆₇	1	2	yes	—
95	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
96	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
97	#107	—	2	32	no	—
98	C₄×C₈	D₄×C₆₇	1	8	yes	—
99	C₄×C₈	Q₈×C₆₇	1	16	yes	complemented
100	C₂×Q₈	—	4	8	no	—
101	C₂×Q₈	—	4	16	no	—
102	C₂²×C₄	—	2	8	no	—
103	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
104	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
106	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
108	C₂×C₈	C₂×Q₈×C₆₇	1	16	yes	—
109	C₄⋊C₄	—	2	16	no	—
110	C₄⋊C₄	—	4	8	no	—
111	C₄⋊C₄	—	4	32	no	—
112	C₄²	D₈×C₆₇	1	8	yes	—
113	C₄²	Q₁₆×C₆₇	1	8	yes	—
114	C₄²	C₂×D₄×C₆₇	1	4	yes	—
115	C₄²	C₂×Q₈×C₆₇	1	8	yes	—
116	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
117	Q₈	—	8	16	no	—
118	C₂×C₄	—	4	8	no	—
119	C₂×C₄	—	4	16	no	—
120	C₂×C₄	#127×C₆₇	1	2	yes	—
121	C₂×C₄	#127×C₆₇	1	2	yes	—
122	C₂×C₄	C₄⋊Q₈×C₆₇	1	8	yes	—
123	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
124	C₂×C₄	C₂×D₈×C₆₇	1	8	yes	—
125	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
126	C₂×C₄	C₂×Q₁₆×C₆₇	1	8	yes	—
127	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
128	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
129	C₈	C₄⋊Q₈×C₆₇	1	16	yes	complemented
130	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
131	C₂²	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
132	C₂²	C₂²×D₈×C₆₇	1	2	yes	—
133	C₂²	C₂²×Q₁₆×C₆₇	1	2	yes	—
134	C₄	—	8	8	no	—
135	C₄	#493×C₆₇	1	4	yes	—
136	C₄	#494×C₆₇	1	4	yes	—
137	C₄	#500×C₆₇	1	16	yes	complemented
138	C₄	C₂×C₄⋊Q₈×C₆₇	1	4	yes	—
139	C₂	C₆₇×#2868	1	1	yes	characteristic
140	C₂	C₂×#493×C₆₇	1	1	yes	characteristic
141	C₂	C₂×#494×C₆₇	1	1	yes	characteristic
142	C₂	C₂×#500×C₆₇	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₄×C₈×C₆₇	Q₈	semidirect
57. C₈×C₆₇	C₄⋊Q₈	semidirect
65. C₄×C₆₇	#500	semidirect
67. #33312	C₆₇	direct (N × H)
76. C₆₇	#33312	direct (N × H)
99. C₄×C₈	Q₈×C₆₇	semidirect
129. C₈	C₄⋊Q₈×C₆₇	semidirect
137. C₄	#500×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₆₇×#2868	2
C₆₇×#1845	2
C₄²×C₈×C₆₇	2
#33312	67

Subgroups — C67×#33312 #30712563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33312 #30712563