Subgroups — C₆₇×#7154 #30686405

Order 17152.

242 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#335×C₆₇	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₆₇	C₄	1	2	yes	—
4	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
5	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₆₇	—	2	2	no	—
7	#335×C₆₇	—	2	2	no	—
8	#335×C₆₇	C₄	1	4	yes	—
9	C₂²×C₈×C₆₇	—	4	2	no	—
10	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
12	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₆₇	—	2	2	no	—
14	C₂×C₁₆×C₆₇	—	2	2	no	—
15	C₂×C₁₆×C₆₇	C₈	1	4	yes	complemented
16	C₄⋊C₈×C₆₇	—	2	4	no	—
17	C₄⋊C₈×C₆₇	—	4	2	no	—
18	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₆₇	D₄	1	2	yes	—
20	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₆₇	—	2	2	no	—
24	C₂×C₈×C₆₇	—	4	4	no	—
25	C₂×C₈×C₆₇	—	8	2	no	—
26	C₂×C₈×C₆₇	—	8	2	no	—
27	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
32	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
33	C₄²×C₆₇	SD₁₆	1	2	yes	—
34	C₁₆×C₆₇	—	2	4	no	—
35	C₂³×C₆₇	#107	1	1	yes	characteristic
36	C₂×C₄×C₆₇	—	2	2	no	—
37	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
38	C₂×C₄×C₆₇	#102	1	2	yes	—
39	C₂×C₄×C₆₇	#103	1	2	yes	—
40	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
42	C₂×C₄×C₆₇	#108	1	2	yes	—
43	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
44	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
45	C₂×C₄×C₆₇	Q₃₂	1	1	yes	characteristic
46	C₈×C₆₇	—	2	2	no	—
47	C₈×C₆₇	—	8	4	no	—
48	C₈×C₆₇	C₄×C₈	1	1	yes	characteristic
49	C₈×C₆₇	#97	1	1	yes	characteristic
50	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
51	C₂²×C₆₇	#335	1	2	yes	—
52	C₂²×C₆₇	#340	1	1	yes	characteristic
53	C₂²×C₆₇	#366	1	1	yes	characteristic
54	C₂²×C₆₇	#367	1	1	yes	characteristic
55	C₂²×C₆₇	#368	1	2	yes	—
56	C₄×C₆₇	—	2	2	no	—
57	C₄×C₆₇	#334	1	2	yes	—
58	C₄×C₆₇	#336	1	1	yes	characteristic
59	C₄×C₆₇	#341	1	1	yes	characteristic
60	C₄×C₆₇	#357	1	2	yes	—
61	C₄×C₆₇	#358	1	2	yes	—
62	#7154	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₆₇	#1277	1	1	yes	characteristic
64	C₂×C₆₇	#1371	1	2	yes	—
65	C₂×C₆₇	#1372	1	2	yes	—
66	C₂×C₆₇	#1380	1	1	yes	characteristic
67	C₂×C₆₇	#1381	1	1	yes	characteristic
68	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic
69	C₂×#335	C₂×C₆₇	1	2	yes	—
70	C₆₇	#7154	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×C₁₆	C₄×C₆₇	1	2	yes	—
72	C₂×C₄⋊C₈	—	2	2	no	—
73	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
74	C₄×C₁₆	—	2	2	no	—
75	#335	—	2	2	no	—
76	#335	C₄×C₆₇	1	4	yes	—
77	C₂²×C₈	—	4	2	no	—
78	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
79	C₂²×C₈	Q₈×C₆₇	1	1	yes	characteristic
80	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₈×C₆₇	1	4	yes	complemented
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
87	C₄×C₈	D₄×C₆₇	1	2	yes	—
88	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
89	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
90	C₂²×C₄	Q₁₆×C₆₇	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	—	8	2	no	—
94	C₂×C₈	—	8	2	no	—
95	C₂×C₈	C₄²×C₆₇	1	1	yes	characteristic
96	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
98	C₂×C₈	C₂×C₈×C₆₇	1	2	yes	—
99	C₂×C₈	M₁₆×C₆₇	1	2	yes	—
100	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
101	C₄²	SD₁₆×C₆₇	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂³	#107×C₆₇	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	#95×C₆₇	1	1	yes	characteristic
106	C₂×C₄	#102×C₆₇	1	2	yes	—
107	C₂×C₄	#103×C₆₇	1	2	yes	—
108	C₂×C₄	C₄⋊C₈×C₆₇	1	2	yes	—
109	C₂×C₄	#106×C₆₇	1	1	yes	characteristic
110	C₂×C₄	#108×C₆₇	1	2	yes	—
111	C₂×C₄	D₁₆×C₆₇	1	1	yes	characteristic
112	C₂×C₄	SD₃₂×C₆₇	1	2	yes	—
113	C₂×C₄	Q₃₂×C₆₇	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	8	4	no	—
116	C₈	C₄×C₈×C₆₇	1	1	yes	characteristic
117	C₈	#97×C₆₇	1	1	yes	characteristic
118	C₈	C₂²⋊C₈×C₆₇	1	2	yes	—
119	C₂²	#335×C₆₇	1	2	yes	—
120	C₂²	#340×C₆₇	1	1	yes	characteristic
121	C₂²	#366×C₆₇	1	1	yes	characteristic
122	C₂²	#367×C₆₇	1	1	yes	characteristic
123	C₂²	#368×C₆₇	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	#334×C₆₇	1	2	yes	—
126	C₄	#336×C₆₇	1	1	yes	characteristic
127	C₄	#341×C₆₇	1	1	yes	characteristic
128	C₄	#357×C₆₇	1	2	yes	—
129	C₄	#358×C₆₇	1	2	yes	—
130	C₂	C₆₇×#1277	1	1	yes	characteristic
131	C₂	C₆₇×#1371	1	2	yes	—
132	C₂	C₆₇×#1372	1	2	yes	—
133	C₂	C₆₇×#1380	1	1	yes	characteristic
134	C₂	C₆₇×#1381	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	D₈
commutator	C₈	C₄×C₈×C₆₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₇
socle, prime radical	C₂³×C₆₇	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂×C₁₆×C₆₇	C₈	semidirect
62. #7154	C₆₇	direct (N × H)
70. C₆₇	#7154	direct (N × H)
83. C₂×C₁₆	C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₆₇	2
C₂×#335×C₆₇	2
#7154	67

Subgroups — C67×#7154 #30686405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#7154 #30686405