Subgroups — M₃₂×C₆₇⋊C₈ #30764808

Order 17152.

354 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22558125	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₄	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
8	#16863172	C₄	1	2	yes	—
9	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₈×C₆₇⋊C₈	C₄	1	2	yes	—
11	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₆₇	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#346×C₆₇	C₄	1	2	yes	—
16	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₆₇⋊C₈	C₈	1	4	yes	—
18	#14523983	C₈	1	8	yes	complemented
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
20	C₄×C₆₇⋊C₈	C₈	1	4	yes	—
21	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₆₇	C₈	1	8	yes	complemented
27	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₆₇⋊C₈	—	2	8	no	—
33	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
34	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
35	C₆₇⋊C₁₆	C₂×C₈	1	8	yes	complemented
36	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₄²	1	1	yes	characteristic
45	C₄²×C₆₇	C₂×C₈	1	2	yes	—
46	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₆₇	C₂×C₈	1	8	yes	complemented
48	C₂³×C₆₇	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	4	no	—
50	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
51	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
54	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
56	C₈×C₆₇	C₄×C₈	1	2	yes	—
57	C₈×C₆₇	C₄×C₈	1	2	yes	—
58	C₈×C₆₇	C₂²×C₈	1	2	yes	—
59	C₈×C₆₇	C₂²×C₈	1	2	yes	—
60	C₆₇⋊C₈	—	2	4	no	—
61	C₆₇⋊C₈	M₃₂	1	8	yes	direct product factor
62	C₂²×C₆₇	—	2	2	no	—
63	C₂²×C₆₇	C₈²	1	2	yes	—
64	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₆₇	—	2	2	no	—
66	C₄×C₆₇	C₈²	1	2	yes	—
67	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₆₇	C₂×M₃₂	1	4	yes	—
70	C₈×M₃₂	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	—	2	2	no	—
72	C₂×C₆₇	C₂×C₈²	1	1	yes	characteristic
73	C₂×C₆₇	C₄×M₃₂	1	2	yes	—
74	C₄×M₃₂	—	67	1	no	—
75	C₄×M₃₂	D₆₇	1	1	yes	characteristic
76	C₂×C₈²	—	67	1	no	—
77	#1311	—	67	2	no	—
78	C₈×C₁₆	—	67	2	no	—
79	C₆₇	C₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	67	2	no	—
81	C₂×M₃₂	Dic₆₇	1	2	yes	—
82	C₂×C₄×C₈	—	67	2	no	—
83	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
84	#346	—	67	2	no	—
85	#346	Dic₆₇	1	2	yes	—
86	C₄×C₁₆	—	67	2	no	—
87	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
88	C₈²	—	67	2	no	—
89	C₈²	—	67	1	no	—
90	C₈²	—	67	1	no	—
91	C₂²×C₈	—	67	4	no	—
92	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
95	M₃₂	—	67	8	no	—
96	M₃₂	C₆₇⋊C₈	1	8	yes	direct product factor
97	C₂×C₁₆	—	67	4	no	—
98	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
99	C₄×C₈	—	67	4	no	—
100	C₄×C₈	—	67	2	no	—
101	C₄×C₈	—	67	2	no	—
102	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
103	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
106	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
107	C₂²×C₄	C₈×D₆₇	1	2	yes	—
108	C₂×C₈	—	67	4	no	—
109	C₂×C₈	—	67	4	no	—
110	C₂×C₈	—	134	8	no	—
111	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
112	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
113	C₂×C₈	C₈×D₆₇	1	4	yes	—
114	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
115	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
116	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
117	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
118	C₄²	C₈×D₆₇	1	2	yes	—
119	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
120	C₁₆	—	67	8	no	—
121	C₁₆	C₂×C₆₇⋊C₈	1	8	yes	complemented
122	C₂³	C₈×Dic₆₇	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
125	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
126	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
127	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
128	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
130	C₈	—	67	8	no	—
131	C₈	—	134	4	no	—
132	C₈	C₈×Dic₆₇	1	2	yes	—
133	C₈	C₄×C₆₇⋊C₈	1	2	yes	—
134	C₈	C₂×C₈×D₆₇	1	2	yes	—
135	C₈	C₂²×C₆₇⋊C₈	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	C₈×C₆₇⋊C₈	1	2	yes	—
138	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	C₈×C₆₇⋊C₈	1	2	yes	—
141	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
142	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
143	C₄	M₃₂×D₆₇	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×C₈×C₆₇⋊C₈	1	1	yes	characteristic
146	C₂	M₃₂×Dic₆₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₈²
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×C₈
prime radical	C₂³×C₆₇	C₄×C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	C₈×M₃₂
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22558125	C₂	semidirect
4. C₁₆×C₆₇⋊C₈	C₂	semidirect
18. #14523983	C₈	semidirect
26. M₃₂×C₆₇	C₈	semidirect
35. C₆₇⋊C₁₆	C₂×C₈	semidirect
47. C₁₆×C₆₇	C₂×C₈	semidirect
61. C₆₇⋊C₈	M₃₂	direct (N × H)
79. C₆₇	C₈×M₃₂	semidirect
96. M₃₂	C₆₇⋊C₈	direct (N × H)
121. C₁₆	C₂×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523983	2
C₂×C₈×C₆₇⋊C₈	2
#22558125	2
C₁₆×C₆₇⋊C₈	2
C₄×M₃₂×C₆₇	2
C₈×M₃₂	67

Subgroups — M32×C67⋊C8 #30764808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×C₆₇⋊C₈ #30764808