Subgroups — C₂²×#366×C₆₇ #30712656

Order 17152.

782 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₆₇	C₂	1	2	yes	maximal
3	C₂×#366×C₆₇	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
5	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
6	C₂²×C₁₆×C₆₇	C₄	1	8	yes	complemented
7	C₂²×C₁₆×C₆₇	C₂²	1	6	yes	—
8	C₂×#107×C₆₇	—	2	6	no	—
9	C₂×#107×C₆₇	C₂²	1	12	yes	—
10	#366×C₆₇	C₂²	1	16	yes	direct product factor
11	C₂³×C₄×C₆₇	—	4	2	no	—
12	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
14	C₂²×C₈×C₆₇	D₄	1	3	yes	—
15	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
16	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
17	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
18	C₂×C₄⋊C₄×C₆₇	—	4	6	no	—
19	C₂×C₁₆×C₆₇	C₂×C₄	1	24	yes	complemented
20	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
21	#107×C₆₇	—	2	12	no	—
22	#107×C₆₇	C₂³	1	8	yes	—
23	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	—	4	12	no	—
25	C₂²×C₄×C₆₇	—	8	8	no	—
26	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
27	C₂²×C₄×C₆₇	D₈	1	3	yes	—
28	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
29	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
30	C₂×C₈×C₆₇	C₄⋊C₄	1	12	yes	—
31	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
32	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
33	C₂×C₈×C₆₇	C₂×Q₈	1	6	yes	—
34	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
35	C₄⋊C₄×C₆₇	—	2	8	no	—
36	C₄⋊C₄×C₆₇	—	4	12	no	—
37	C₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
38	C₂³×C₆₇	#107	1	4	yes	—
39	C₂³×C₆₇	D₁₆	1	4	yes	—
40	C₂³×C₆₇	Q₃₂	1	4	yes	—
41	C₂³×C₆₇	C₂×D₈	1	3	yes	—
42	C₂×C₄×C₆₇	—	4	8	no	—
43	C₂×C₄×C₆₇	—	8	24	no	—
44	C₂×C₄×C₆₇	#107	1	12	yes	—
45	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
46	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
47	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
48	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
49	C₈×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
50	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
51	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₆₇	#366	1	16	yes	direct product factor
53	C₂²×C₆₇	C₂×#107	1	6	yes	—
54	C₂²×C₆₇	C₂×D₁₆	1	6	yes	—
55	C₂²×C₆₇	C₂×Q₃₂	1	6	yes	—
56	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
57	C₄×C₆₇	—	8	8	no	—
58	C₄×C₆₇	C₂×#107	1	6	yes	—
59	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
61	C₂²×#366	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₆₇	C₂×#366	1	12	yes	direct product factor
63	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
64	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
65	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
66	C₂³×C₁₆	C₂×C₆₇	1	1	yes	characteristic
67	C₂²×#107	C₂×C₆₇	1	2	yes	—
68	C₂×#366	C₂×C₆₇	1	12	yes	direct product factor
69	C₆₇	C₂²×#366	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
71	C₂²×C₄⋊C₄	—	2	2	no	—
72	C₂²×C₁₆	C₄×C₆₇	1	8	yes	complemented
73	C₂²×C₁₆	C₂²×C₆₇	1	6	yes	—
74	C₂×#107	—	2	6	no	—
75	C₂×#107	C₂²×C₆₇	1	12	yes	—
76	#366	C₂²×C₆₇	1	16	yes	direct product factor
77	C₂³×C₄	—	4	2	no	—
78	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
79	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
80	C₂²×C₈	D₄×C₆₇	1	3	yes	—
81	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
82	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
83	C₂×C₄⋊C₄	—	2	12	no	—
84	C₂×C₄⋊C₄	—	4	6	no	—
85	C₂×C₁₆	C₂×C₄×C₆₇	1	24	yes	complemented
86	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
87	#107	—	2	12	no	—
88	#107	C₂³×C₆₇	1	8	yes	—
89	C₂⁴	D₈×C₆₇	1	1	yes	characteristic
90	C₂²×C₄	—	4	12	no	—
91	C₂²×C₄	—	8	8	no	—
92	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
93	C₂²×C₄	D₈×C₆₇	1	3	yes	—
94	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
95	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
96	C₂×C₈	C₄⋊C₄×C₆₇	1	12	yes	—
97	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
98	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
99	C₂×C₈	C₂×Q₈×C₆₇	1	6	yes	—
100	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
101	C₄⋊C₄	—	2	8	no	—
102	C₄⋊C₄	—	4	12	no	—
103	C₁₆	C₂²×C₄×C₆₇	1	8	yes	complemented
104	C₂³	#107×C₆₇	1	4	yes	—
105	C₂³	D₁₆×C₆₇	1	4	yes	—
106	C₂³	Q₃₂×C₆₇	1	4	yes	—
107	C₂³	C₂×D₈×C₆₇	1	3	yes	—
108	C₂×C₄	—	4	8	no	—
109	C₂×C₄	—	8	24	no	—
110	C₂×C₄	#107×C₆₇	1	12	yes	—
111	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
112	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
113	C₂×C₄	C₂×Q₁₆×C₆₇	1	6	yes	—
114	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
115	C₈	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
116	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
117	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
118	C₂²	#366×C₆₇	1	16	yes	direct product factor
119	C₂²	C₂×#107×C₆₇	1	6	yes	—
120	C₂²	C₂×D₁₆×C₆₇	1	6	yes	—
121	C₂²	C₂×Q₃₂×C₆₇	1	6	yes	—
122	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
123	C₄	—	8	8	no	—
124	C₄	C₂×#107×C₆₇	1	6	yes	—
125	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
126	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
127	C₂	C₂×#366×C₆₇	1	12	yes	direct product factor
128	C₂	C₂²×#107×C₆₇	1	1	yes	characteristic
129	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic
130	C₂	C₂²×Q₃₂×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#366×C₆₇	C₂	direct (N × H)
6. C₂²×C₁₆×C₆₇	C₄	semidirect
10. #366×C₆₇	C₂²	direct (N × H)
19. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
37. C₁₆×C₆₇	C₂²×C₄	semidirect
52. C₂²×C₆₇	#366	direct (N × H)
61. C₂²×#366	C₆₇	direct (N × H)
62. C₂×C₆₇	C₂×#366	direct (N × H)
68. C₂×#366	C₂×C₆₇	direct (N × H)
69. C₆₇	C₂²×#366	direct (N × H)
72. C₂²×C₁₆	C₄×C₆₇	semidirect
76. #366	C₂²×C₆₇	direct (N × H)
85. C₂×C₁₆	C₂×C₄×C₆₇	semidirect
103. C₁₆	C₂²×C₄×C₆₇	semidirect
118. C₂²	#366×C₆₇	direct (N × H)
127. C₂	C₂×#366×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₆₇	2
C₂²×#107×C₆₇	2
C₂×#366×C₆₇	2
C₂²×#366	67

Subgroups — C22×#366×C67 #30712656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#366×C₆₇ #30712656