Subgroups — #97×C₆₇⋊C₈ #30745182

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559955	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863117	C₂	1	2	yes	maximal
4	C₂×C₈×C₆₇⋊C₈	C₂	1	2	yes	maximal
5	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	#16863151	C₄	1	4	yes	—
11	#16863117	C₄	1	8	yes	complemented
12	C₈×C₆₇⋊C₈	C₄	1	8	yes	complemented
13	C₂×#97×C₆₇	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₄×C₆₇⋊C₈	—	2	8	no	—
20	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	#97×C₆₇	C₈	1	16	yes	complemented
30	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₆₇⋊C₈	—	2	4	no	—
33	C₂×C₆₇⋊C₈	—	2	8	no	—
34	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
35	C₂×C₆₇⋊C₈	C₄²	1	8	yes	—
36	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
37	C₂×C₆₇⋊C₈	M₁₆	1	8	yes	—
38	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₄²	1	8	yes	—
43	C₂×C₈×C₆₇	C₂×C₈	1	16	yes	—
44	C₄²×C₆₇	—	2	4	no	—
45	C₄²×C₆₇	—	2	4	no	—
46	C₄²×C₆₇	C₂×C₈	1	8	yes	—
47	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
48	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
49	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	2	4	no	—
51	C₂×C₄×C₆₇	—	2	4	no	—
52	C₂×C₄×C₆₇	C₄×C₈	1	8	yes	—
53	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
57	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
58	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₆₇	#131	1	4	yes	—
60	C₈×C₆₇	C₄×C₈	1	16	yes	complemented
61	C₆₇⋊C₈	—	2	4	no	—
62	C₆₇⋊C₈	—	2	8	no	—
63	C₆₇⋊C₈	#97	1	8	yes	direct product factor
64	C₂²×C₆₇	C₄³	1	1	yes	characteristic
65	C₂²×C₆₇	C₂×C₄×C₈	1	2	yes	—
66	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
67	C₂²×C₆₇	#405	1	2	yes	—
68	C₄×C₆₇	—	2	8	no	—
69	C₄×C₆₇	C₂×C₄×C₈	1	4	yes	—
70	C₄×C₆₇	C₂×#97	1	2	yes	—
71	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
72	C₄×C₆₇	#405	1	4	yes	—
73	C₈×#97	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	C₈×M₁₆	1	4	yes	—
75	C₂×C₆₇	C₄²×C₈	1	1	yes	characteristic
76	C₂×C₆₇	C₄×#97	1	1	yes	characteristic
77	C₂×C₆₇	#1726	1	1	yes	characteristic
78	#1744	—	67	1	no	—
79	C₄×#97	D₆₇	1	1	yes	characteristic
80	C₄²×C₈	—	67	1	no	—
81	C₂×#322	—	67	2	no	—
82	C₂×C₈²	—	67	2	no	—
83	C₆₇	C₈×#97	1	1	yes	characteristic, complemented, Sylow
84	C₂×#97	Dic₆₇	1	4	yes	—
85	C₂×C₄×C₈	—	67	1	no	—
86	C₂×C₄×C₈	—	67	2	no	—
87	C₂×C₄×C₈	—	67	1	no	—
88	C₂×C₄×C₈	—	67	4	no	—
89	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
90	C₄³	C₂×D₆₇	1	1	yes	characteristic
91	#336	—	67	4	no	—
92	#322	—	67	8	no	—
93	C₈²	—	67	8	no	—
94	C₂²×C₈	—	67	2	no	—
95	C₂²×C₈	—	67	4	no	—
96	C₂²×C₈	—	67	2	no	—
97	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
98	C₂×C₄²	C₄×D₆₇	1	2	yes	—
99	C₂×C₄²	C₄×D₆₇	1	2	yes	—
100	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
101	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
102	#97	C₆₇⋊C₈	1	16	yes	direct product factor
103	C₄×C₈	—	67	4	no	—
104	C₄×C₈	—	67	4	no	—
105	C₄×C₈	—	67	4	no	—
106	C₄×C₈	—	67	4	no	—
107	C₄×C₈	—	134	8	no	—
108	C₄×C₈	C₄×D₆₇	1	4	yes	—
109	C₄×C₈	C₄×D₆₇	1	4	yes	—
110	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
111	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
114	C₂×C₈	—	67	4	no	—
115	C₂×C₈	—	67	8	no	—
116	C₂×C₈	—	67	4	no	—
117	C₂×C₈	—	67	8	no	—
118	C₂×C₈	—	134	4	no	—
119	C₂×C₈	—	134	8	no	—
120	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
121	C₂×C₈	C₂×C₆₇⋊C₈	1	16	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	C₂×C₆₇⋊C₈	1	8	yes	—
125	C₄²	C₂×C₄×D₆₇	1	2	yes	—
126	C₄²	C₂×C₄×D₆₇	1	2	yes	—
127	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	C₄×C₆₇⋊C₈	1	8	yes	—
131	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
132	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
133	C₂×C₄	C₄²×D₆₇	1	2	yes	—
134	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
135	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
136	C₂×C₄	M₁₆×D₆₇	1	4	yes	—
137	C₂×C₄	#14524051	1	4	yes	—
138	C₈	—	67	8	no	—
139	C₈	—	134	4	no	—
140	C₈	—	134	8	no	—
141	C₈	C₄×C₆₇⋊C₈	1	16	yes	complemented
142	C₂²	C₄²×Dic₆₇	1	1	yes	characteristic
143	C₂²	C₂×C₄×C₆₇⋊C₈	1	2	yes	—
144	C₂²	M₁₆×Dic₆₇	1	2	yes	—
145	C₂²	#16863354	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₂×C₄×C₆₇⋊C₈	1	4	yes	—
148	C₄	#97×D₆₇	1	2	yes	—
149	C₄	#16863348	1	2	yes	—
150	C₄	#16863353	1	4	yes	—
151	C₂	M₁₆×C₆₇⋊C₈	1	4	yes	—
152	C₂	C₄²×C₆₇⋊C₈	1	1	yes	characteristic
153	C₂	#97×Dic₆₇	1	1	yes	characteristic
154	C₂	#22559897	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂×C₆₇	C₄²×C₈
Fitting	C₄×#97×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#97	D₆₇
hypocenter	C₆₇	C₈×#97
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
11. #16863117	C₄	semidirect
12. C₈×C₆₇⋊C₈	C₄	semidirect
29. #97×C₆₇	C₈	semidirect
60. C₈×C₆₇	C₄×C₈	semidirect
63. C₆₇⋊C₈	#97	direct (N × H)
83. C₆₇	C₈×#97	semidirect
102. #97	C₆₇⋊C₈	direct (N × H)
141. C₈	C₄×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22559955	2
C₄²×C₆₇⋊C₈	2
C₂×#16863117	2
C₂×C₈×C₆₇⋊C₈	2
C₄×#97×C₆₇	2
C₈×#97	67

Subgroups — #97×C67⋊C8 #30745182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #97×C₆₇⋊C₈ #30745182