Subgroups — C₂³×C₂²⋊(C₆₇⋊C₈) #31719258

Order 17152.

7646 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂²⋊(C₆₇⋊C₈)	C₂	1	28	yes	maximal, direct product factor
3	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₆₇⋊C₈	—	2	16	no	—
5	C₂³×C₆₇⋊C₈	C₂²	1	28	yes	complemented
6	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	112	yes	direct product factor
7	C₂⁶×C₆₇	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	—	2	16	no	—
9	C₂⁴×C₄×C₆₇	C₄	1	1	yes	characteristic
10	C₂⁴×C₄×C₆₇	C₄	1	14	yes	—
11	C₂⁴×C₄×C₆₇	C₂²	1	14	yes	—
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₆₇⋊C₈	—	2	112	no	—
14	C₂²×C₆₇⋊C₈	C₂³	1	56	yes	complemented
15	C₂²⋊(C₆₇⋊C₈)	C₂³	1	64	yes	direct product factor
16	C₂⁵×C₆₇	—	2	8	no	—
17	C₂⁵×C₆₇	—	2	8	no	—
18	C₂⁵×C₆₇	C₈	1	16	yes	complemented
19	C₂⁵×C₆₇	C₂×C₄	1	14	yes	—
20	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₇	—	2	8	no	—
22	C₂³×C₄×C₆₇	—	2	112	no	—
23	C₂³×C₄×C₆₇	—	2	112	no	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	14	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	84	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	7	yes	—
27	C₂³×C₄×C₆₇	D₄	1	16	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	28	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
30	C₂×C₆₇⋊C₈	—	2	112	no	—
31	C₂×C₆₇⋊C₈	C₂⁴	1	16	yes	complemented
32	C₂⁴×C₆₇	—	2	128	no	—
33	C₂⁴×C₆₇	—	2	56	no	—
34	C₂⁴×C₆₇	—	2	56	no	—
35	C₂⁴×C₆₇	C₂²⋊C₄	1	8	yes	—
36	C₂⁴×C₆₇	C₂×C₈	1	8	yes	—
37	C₂⁴×C₆₇	C₂×C₈	1	112	yes	complemented
38	C₂⁴×C₆₇	M₁₆	1	8	yes	—
39	C₂⁴×C₆₇	C₂²×C₄	1	28	yes	—
40	C₂⁴×C₆₇	C₂²×C₄	1	7	yes	—
41	C₂²×C₄×C₆₇	—	2	56	no	—
42	C₂²×C₄×C₆₇	—	2	336	no	—
43	C₂²×C₄×C₆₇	—	2	112	no	—
44	C₂²×C₄×C₆₇	C₂²⋊C₄	1	56	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	28	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	56	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	21	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	56	yes	—
49	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
50	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
51	C₂³×C₆₇	—	2	56	no	—
52	C₂³×C₆₇	—	2	56	no	—
53	C₂³×C₆₇	—	2	448	no	—
54	C₂³×C₆₇	C₂²⋊C₈	1	64	yes	complemented
55	C₂³×C₆₇	C₂×C₂²⋊C₄	1	28	yes	—
56	C₂³×C₆₇	C₂²×C₈	1	28	yes	—
57	C₂³×C₆₇	C₂²×C₈	1	112	yes	complemented
58	C₂³×C₆₇	C₂×M₁₆	1	28	yes	—
59	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
60	C₂³×C₆₇	C₂³×C₄	1	7	yes	—
61	C₂×C₄×C₆₇	—	2	56	no	—
62	C₂×C₄×C₆₇	—	2	112	no	—
63	C₂×C₄×C₆₇	—	2	16	no	—
64	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	84	yes	—
65	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
66	C₂×C₄×C₆₇	C₂³×C₄	1	7	yes	—
67	C₂×C₄×C₆₇	C₂²×D₄	1	28	yes	—
68	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
69	C₆₇⋊C₈	—	2	16	no	—
70	C₂²×C₆₇	—	2	224	no	—
71	C₂²×C₆₇	—	2	8	no	—
72	C₂²×C₆₇	—	2	8	no	—
73	C₂²×C₆₇	C₂×C₂²⋊C₈	1	112	yes	complemented
74	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	14	yes	—
75	C₂²×C₆₇	C₂³×C₈	1	14	yes	—
76	C₂²×C₆₇	C₂³×C₈	1	16	yes	complemented
77	C₂²×C₆₇	C₂²×M₁₆	1	14	yes	—
78	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
79	C₄×C₆₇	—	2	8	no	—
80	C₄×C₆₇	C₂²×C₂²⋊C₄	1	14	yes	—
81	C₄×C₆₇	C₂³×D₄	1	2	yes	—
82	C₂³×C₂²⋊C₈	—	67	1	no	maximal, Sylow
83	C₂×C₆₇	—	2	16	no	—
84	C₂×C₆₇	C₂²×C₂²⋊C₈	1	28	yes	complemented
85	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₆₇	C₂⁴×C₈	1	1	yes	characteristic
87	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
88	C₂⁵×C₄	D₆₇	1	1	yes	characteristic
89	C₂⁴×C₈	—	67	2	no	—
90	C₂²×C₂²⋊C₈	—	67	28	no	—
91	C₆₇	C₂³×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
92	C₂⁶	Dic₆₇	1	1	yes	characteristic
93	C₂⁴×C₄	—	2	16	no	—
94	C₂⁴×C₄	Dic₆₇	1	1	yes	characteristic
95	C₂⁴×C₄	Dic₆₇	1	14	yes	—
96	C₂⁴×C₄	C₂×D₆₇	1	14	yes	—
97	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
98	C₂³×C₈	—	67	28	no	—
99	C₂³×C₈	—	134	16	no	—
100	C₂×C₂²⋊C₈	—	67	112	no	—
101	C₂⁵	—	2	8	no	—
102	C₂⁵	—	2	8	no	—
103	C₂⁵	C₆₇⋊C₈	1	16	yes	complemented
104	C₂⁵	C₂×Dic₆₇	1	14	yes	—
105	C₂⁵	C₂×Dic₆₇	1	1	yes	characteristic
106	C₂³×C₄	—	2	8	no	—
107	C₂³×C₄	—	2	112	no	—
108	C₂³×C₄	—	2	112	no	—
109	C₂³×C₄	C₂×Dic₆₇	1	14	yes	—
110	C₂³×C₄	C₂×Dic₆₇	1	84	yes	—
111	C₂³×C₄	C₂×Dic₆₇	1	7	yes	—
112	C₂³×C₄	C₂²⋊D₆₇	1	16	yes	—
113	C₂³×C₄	C₂²×D₆₇	1	28	yes	—
114	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
115	C₂²×C₈	—	67	56	no	—
116	C₂²×C₈	—	134	112	no	—
117	C₂²⋊C₈	—	67	64	no	—
118	C₂⁴	—	2	128	no	—
119	C₂⁴	—	2	56	no	—
120	C₂⁴	—	2	56	no	—
121	C₂⁴	C₂²⋊Dic₆₇	1	8	yes	—
122	C₂⁴	C₂×C₆₇⋊C₈	1	8	yes	—
123	C₂⁴	C₂×C₆₇⋊C₈	1	112	yes	complemented
124	C₂⁴	#11761612	1	8	yes	—
125	C₂⁴	C₂²×Dic₆₇	1	28	yes	—
126	C₂⁴	C₂²×Dic₆₇	1	7	yes	—
127	C₂²×C₄	—	2	56	no	—
128	C₂²×C₄	—	2	336	no	—
129	C₂²×C₄	—	2	112	no	—
130	C₂²×C₄	C₂²⋊Dic₆₇	1	56	yes	—
131	C₂²×C₄	C₂²×Dic₆₇	1	28	yes	—
132	C₂²×C₄	C₂²×Dic₆₇	1	56	yes	—
133	C₂²×C₄	C₂²×Dic₆₇	1	21	yes	—
134	C₂²×C₄	C₂×C₂²⋊D₆₇	1	56	yes	—
135	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
136	C₂²×C₄	C₂³×D₆₇	1	7	yes	—
137	C₂×C₈	—	67	16	no	—
138	C₂×C₈	—	134	112	no	—
139	C₂³	—	2	56	no	—
140	C₂³	—	2	448	no	—
141	C₂³	—	2	56	no	—
142	C₂³	C₂²⋊(C₆₇⋊C₈)	1	64	yes	direct product factor
143	C₂³	C₂×C₂²⋊Dic₆₇	1	28	yes	—
144	C₂³	C₂²×C₆₇⋊C₈	1	28	yes	—
145	C₂³	C₂²×C₆₇⋊C₈	1	112	yes	complemented
146	C₂³	C₂×#11761612	1	28	yes	—
147	C₂³	C₂³×Dic₆₇	1	8	yes	—
148	C₂³	C₂³×Dic₆₇	1	7	yes	—
149	C₂×C₄	—	2	56	no	—
150	C₂×C₄	—	2	112	no	—
151	C₂×C₄	—	2	16	no	—
152	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	84	yes	—
153	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
154	C₂×C₄	C₂³×Dic₆₇	1	7	yes	—
155	C₂×C₄	C₂²×C₂²⋊D₆₇	1	28	yes	—
156	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
157	C₈	—	134	16	no	—
158	C₂²	—	2	8	no	—
159	C₂²	—	2	8	no	—
160	C₂²	—	2	224	no	—
161	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	112	yes	direct product factor
162	C₂²	C₂²×C₂²⋊Dic₆₇	1	14	yes	—
163	C₂²	C₂³×C₆₇⋊C₈	1	14	yes	—
164	C₂²	C₂³×C₆₇⋊C₈	1	16	yes	complemented
165	C₂²	C₂²×#11761612	1	14	yes	—
166	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
167	C₄	—	2	8	no	—
168	C₄	C₂²×C₂²⋊Dic₆₇	1	14	yes	—
169	C₄	C₂³×C₂²⋊D₆₇	1	2	yes	—
170	C₂	—	2	16	no	—
171	C₂	C₂²×C₂²⋊(C₆₇⋊C₈)	1	28	yes	direct product factor
172	C₂	C₂³×C₂²⋊Dic₆₇	1	1	yes	characteristic
173	C₂	C₂⁴×C₆₇⋊C₈	1	1	yes	characteristic
174	C₂	C₂³×#11761612	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₆₇
commutator	C₂×C₆₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁵×C₄×C₆₇	C₂
socle	C₂⁵×C₆₇	C₂×C₄
prime radical	C₂⁶×C₆₇	C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁵×C₄	D₆₇
hypocenter	C₆₇	C₂³×C₂²⋊C₈
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₆₇⋊C₈	C₂	semidirect
2. C₂²×C₂²⋊(C₆₇⋊C₈)	C₂	direct (N × H)
5. C₂³×C₆₇⋊C₈	C₂²	semidirect
6. C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	direct (N × H)
14. C₂²×C₆₇⋊C₈	C₂³	semidirect
15. C₂²⋊(C₆₇⋊C₈)	C₂³	direct (N × H)
18. C₂⁵×C₆₇	C₈	semidirect
31. C₂×C₆₇⋊C₈	C₂⁴	semidirect
37. C₂⁴×C₆₇	C₂×C₈	semidirect
54. C₂³×C₆₇	C₂²⋊C₈	semidirect
57. C₂³×C₆₇	C₂²×C₈	semidirect
73. C₂²×C₆₇	C₂×C₂²⋊C₈	semidirect
76. C₂²×C₆₇	C₂³×C₈	semidirect
84. C₂×C₆₇	C₂²×C₂²⋊C₈	semidirect
91. C₆₇	C₂³×C₂²⋊C₈	semidirect
103. C₂⁵	C₆₇⋊C₈	semidirect
123. C₂⁴	C₂×C₆₇⋊C₈	semidirect
142. C₂³	C₂²⋊(C₆₇⋊C₈)	direct (N × H)
145. C₂³	C₂²×C₆₇⋊C₈	semidirect
161. C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	direct (N × H)
164. C₂²	C₂³×C₆₇⋊C₈	semidirect
171. C₂	C₂²×C₂²⋊(C₆₇⋊C₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₆₇⋊C₈	2
C₂²×C₂²⋊(C₆₇⋊C₈)	2
C₂⁵×C₄×C₆₇	2
C₂³×C₂²⋊C₈	67

Subgroups — C23×C22⋊(C67⋊C8) #31719258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊(C₆₇⋊C₈) #31719258