Subgroups — C₈×2₋¹⁺⁴×C₆₇ #30739580

Order 17152.

2190 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2988	C₂	1	10	yes	maximal, complemented
3	C₆₇×#2982	C₂	1	5	yes	maximal, complemented
4	C₈×#41×C₆₇	C₂	1	10	yes	maximal, complemented
5	C₂×C₈×Q₈×C₆₇	C₂	1	5	yes	maximal, complemented
6	C₂×2₋¹⁺⁴×C₆₇	C₄	1	1	yes	characteristic
7	#520×C₆₇	C₄	1	10	yes	—
8	#519×C₆₇	C₄	1	5	yes	—
9	C₄×#41×C₆₇	C₂²	1	10	yes	—
10	C₂×C₄×Q₈×C₆₇	C₂²	1	5	yes	—
11	C₈×Q₈×C₆₇	C₂²	1	10	yes	—
12	C₈×Q₈×C₆₇	C₂²	1	20	yes	—
13	C₈×Q₈×C₆₇	C₂²	1	10	yes	—
14	C₈×D₄×C₆₇	C₂²	1	10	yes	—
15	C₈×D₄×C₆₇	C₂²	1	30	yes	—
16	#431×C₆₇	C₂²	1	30	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	15	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	15	yes	—
19	2₋¹⁺⁴×C₆₇	C₈	1	16	yes	direct product factor
20	C₂×#41×C₆₇	C₂×C₄	1	10	yes	—
21	C₂²×Q₈×C₆₇	C₂×C₄	1	5	yes	—
22	C₂²×C₈×C₆₇	C₂³	1	5	yes	—
23	C₂²×C₈×C₆₇	C₂³	1	15	yes	—
24	C₄×Q₈×C₆₇	C₂×C₄	1	20	yes	—
25	C₄×Q₈×C₆₇	C₂×C₄	1	10	yes	—
26	C₄×Q₈×C₆₇	C₂³	1	10	yes	—
27	C₄×D₄×C₆₇	C₂×C₄	1	30	yes	—
28	C₄×D₄×C₆₇	C₂³	1	10	yes	—
29	#117×C₆₇	C₂×C₄	1	30	yes	—
30	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	15	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	15	yes	—
32	C₄⋊C₈×C₆₇	C₂³	1	30	yes	—
33	C₄⋊C₈×C₆₇	C₂³	1	10	yes	—
34	C₂²⋊C₈×C₆₇	C₂³	1	20	yes	—
35	C₄×C₈×C₆₇	C₂³	1	30	yes	—
36	C₄×C₈×C₆₇	C₂³	1	10	yes	—
37	#41×C₆₇	C₂×C₈	1	80	yes	complemented
38	C₂×Q₈×C₆₇	C₂×C₈	1	40	yes	complemented
39	C₂×Q₈×C₆₇	C₂²×C₄	1	10	yes	—
40	C₂×D₄×C₆₇	C₂²×C₄	1	10	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	15	yes	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	15	yes	—
43	C₂²×C₄×C₆₇	C₂⁴	1	5	yes	—
44	C₂×C₈×C₆₇	—	2	10	no	—
45	C₂×C₈×C₆₇	—	4	15	no	—
46	C₂×C₈×C₆₇	C₂⁴	1	10	yes	—
47	C₂×C₈×C₆₇	C₂⁴	1	5	yes	—
48	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₆₇	C₂²×C₄	1	30	yes	—
50	C₄⋊C₄×C₆₇	C₂²×C₄	1	10	yes	—
51	C₂²⋊C₄×C₆₇	C₂²×C₄	1	20	yes	—
52	C₄²×C₆₇	C₂²×C₄	1	30	yes	—
53	C₄²×C₆₇	C₂⁴	1	10	yes	—
54	C₂³×C₆₇	C₂³×C₄	1	5	yes	—
55	Q₈×C₆₇	C₂²×C₈	1	40	yes	—
56	D₄×C₆₇	C₂²×C₈	1	40	yes	—
57	C₂×C₄×C₆₇	—	2	5	no	—
58	C₂×C₄×C₆₇	—	2	5	no	—
59	C₂×C₄×C₆₇	—	4	15	no	—
60	C₂×C₄×C₆₇	C₂²×C₈	1	60	yes	—
61	C₂×C₄×C₆₇	C₂³×C₄	1	5	yes	—
62	C₂×C₄×C₆₇	C₂³×C₄	1	10	yes	—
63	C₂×C₄×C₆₇	C₂³×C₄	1	10	yes	—
64	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
65	C₈×C₆₇	—	2	5	no	—
66	C₈×C₆₇	—	2	10	no	—
67	C₈×C₆₇	2₋¹⁺⁴	1	2	yes	direct product factor
68	C₂²×C₆₇	—	2	5	no	—
69	C₂²×C₆₇	—	2	5	no	—
70	C₂²×C₆₇	C₂³×C₈	1	10	yes	—
71	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
72	C₄×C₆₇	—	2	5	no	—
73	C₄×C₆₇	—	2	10	no	—
74	C₄×C₆₇	C₂³×C₈	1	20	yes	—
75	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
76	C₄×C₆₇	#585	1	1	yes	characteristic
77	C₈×2₋¹⁺⁴	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₆₇	—	2	10	no	—
79	C₂×C₆₇	C₄×2₋¹⁺⁴	1	1	yes	characteristic
80	C₂×C₆₇	C₂⁴×C₈	1	1	yes	characteristic
81	C₂×C₆₇	#3573	1	1	yes	characteristic
82	C₄×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic
83	#2988	C₂×C₆₇	1	10	yes	complemented
84	#2982	C₂×C₆₇	1	5	yes	complemented
85	C₈×#41	C₂×C₆₇	1	10	yes	complemented
86	C₂×C₈×Q₈	C₂×C₆₇	1	5	yes	complemented
87	C₆₇	C₈×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×2₋¹⁺⁴	C₄×C₆₇	1	1	yes	characteristic
89	#520	C₄×C₆₇	1	10	yes	—
90	#519	C₄×C₆₇	1	5	yes	—
91	C₄×#41	C₂²×C₆₇	1	10	yes	—
92	C₂×C₄×Q₈	C₂²×C₆₇	1	5	yes	—
93	C₈×Q₈	C₂²×C₆₇	1	10	yes	—
94	C₈×Q₈	C₂²×C₆₇	1	20	yes	—
95	C₈×Q₈	C₂²×C₆₇	1	10	yes	—
96	C₈×D₄	C₂²×C₆₇	1	10	yes	—
97	C₈×D₄	C₂²×C₆₇	1	30	yes	—
98	#431	C₂²×C₆₇	1	30	yes	—
99	C₂×C₄⋊C₈	C₂²×C₆₇	1	15	yes	—
100	C₂×C₄×C₈	C₂²×C₆₇	1	15	yes	—
101	2₋¹⁺⁴	C₈×C₆₇	1	16	yes	direct product factor
102	C₂×#41	C₂×C₄×C₆₇	1	10	yes	—
103	C₂²×Q₈	C₂×C₄×C₆₇	1	5	yes	—
104	C₂²×C₈	C₂³×C₆₇	1	5	yes	—
105	C₂²×C₈	C₂³×C₆₇	1	15	yes	—
106	C₄×Q₈	C₂×C₄×C₆₇	1	20	yes	—
107	C₄×Q₈	C₂×C₄×C₆₇	1	10	yes	—
108	C₄×Q₈	C₂³×C₆₇	1	10	yes	—
109	C₄×D₄	C₂×C₄×C₆₇	1	30	yes	—
110	C₄×D₄	C₂³×C₆₇	1	10	yes	—
111	#117	C₂×C₄×C₆₇	1	30	yes	—
112	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	15	yes	—
113	C₂×C₄²	C₂³×C₆₇	1	15	yes	—
114	C₄⋊C₈	C₂³×C₆₇	1	30	yes	—
115	C₄⋊C₈	C₂³×C₆₇	1	10	yes	—
116	C₂²⋊C₈	C₂³×C₆₇	1	20	yes	—
117	C₄×C₈	C₂³×C₆₇	1	10	yes	—
118	C₄×C₈	C₂³×C₆₇	1	30	yes	—
119	#41	C₂×C₈×C₆₇	1	80	yes	complemented
120	C₂×Q₈	C₂×C₈×C₆₇	1	40	yes	complemented
121	C₂×Q₈	C₂²×C₄×C₆₇	1	10	yes	—
122	C₂×D₄	C₂²×C₄×C₆₇	1	10	yes	—
123	C₂²×C₄	C₂²×C₄×C₆₇	1	15	yes	—
124	C₂²×C₄	C₂²×C₄×C₆₇	1	15	yes	—
125	C₂²×C₄	C₂⁴×C₆₇	1	5	yes	—
126	C₂×C₈	—	2	10	no	—
127	C₂×C₈	—	4	15	no	—
128	C₂×C₈	C₂⁴×C₆₇	1	5	yes	—
129	C₂×C₈	C₂⁴×C₆₇	1	10	yes	—
130	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
131	C₄⋊C₄	C₂²×C₄×C₆₇	1	30	yes	—
132	C₄⋊C₄	C₂²×C₄×C₆₇	1	10	yes	—
133	C₂²⋊C₄	C₂²×C₄×C₆₇	1	20	yes	—
134	C₄²	C₂²×C₄×C₆₇	1	30	yes	—
135	C₄²	C₂⁴×C₆₇	1	10	yes	—
136	C₂³	C₂³×C₄×C₆₇	1	5	yes	—
137	Q₈	C₂²×C₈×C₆₇	1	40	yes	—
138	D₄	C₂²×C₈×C₆₇	1	40	yes	—
139	C₂×C₄	—	2	5	no	—
140	C₂×C₄	—	2	5	no	—
141	C₂×C₄	—	4	15	no	—
142	C₂×C₄	C₂²×C₈×C₆₇	1	60	yes	—
143	C₂×C₄	C₂³×C₄×C₆₇	1	5	yes	—
144	C₂×C₄	C₂³×C₄×C₆₇	1	10	yes	—
145	C₂×C₄	C₂³×C₄×C₆₇	1	10	yes	—
146	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
147	C₈	—	2	5	no	—
148	C₈	—	2	10	no	—
149	C₈	2₋¹⁺⁴×C₆₇	1	2	yes	direct product factor
150	C₂²	—	2	5	no	—
151	C₂²	—	2	5	no	—
152	C₂²	C₂³×C₈×C₆₇	1	10	yes	—
153	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
154	C₄	—	2	10	no	—
155	C₄	—	2	5	no	—
156	C₄	C₂³×C₈×C₆₇	1	20	yes	—
157	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
158	C₄	#585×C₆₇	1	1	yes	characteristic
159	C₂	—	2	10	no	—
160	C₂	C₄×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
161	C₂	C₂⁴×C₈×C₆₇	1	1	yes	characteristic
162	C₂	C₆₇×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂	C₂⁴×C₈×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂⁴×C₄
prime radical	C₂×2₋¹⁺⁴×C₆₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2988	C₂	semidirect
3. C₆₇×#2982	C₂	semidirect
4. C₈×#41×C₆₇	C₂	semidirect
5. C₂×C₈×Q₈×C₆₇	C₂	semidirect
19. 2₋¹⁺⁴×C₆₇	C₈	direct (N × H)
37. #41×C₆₇	C₂×C₈	semidirect
38. C₂×Q₈×C₆₇	C₂×C₈	semidirect
67. C₈×C₆₇	2₋¹⁺⁴	direct (N × H)
77. C₈×2₋¹⁺⁴	C₆₇	direct (N × H)
83. #2988	C₂×C₆₇	semidirect
84. #2982	C₂×C₆₇	semidirect
85. C₈×#41	C₂×C₆₇	semidirect
86. C₂×C₈×Q₈	C₂×C₆₇	semidirect
87. C₆₇	C₈×2₋¹⁺⁴	direct (N × H)
101. 2₋¹⁺⁴	C₈×C₆₇	direct (N × H)
119. #41	C₂×C₈×C₆₇	semidirect
120. C₂×Q₈	C₂×C₈×C₆₇	semidirect
149. C₈	2₋¹⁺⁴×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
C₄×2₋¹⁺⁴×C₆₇	2
C₆₇×#2988	2
C₆₇×#2982	2
C₈×#41×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₈×2₋¹⁺⁴	67

Subgroups — C8×2−1+4×C67 #30739580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×2₋¹⁺⁴×C₆₇ #30739580