Subgroups — C₂²×C₈×Q₈×C₆₇ #30739436

Order 17152.

2478 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₈×C₆₇	C₂²
commutator	C₂	C₂⁴×C₈×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₆₇	C₂	direct (N × H)
12. C₈×Q₈×C₆₇	C₂²	direct (N × H)
15. C₂²×Q₈×C₆₇	C₈	direct (N × H)
21. C₂²×C₈×C₆₇	Q₈	direct (N × H)
32. C₂×Q₈×C₆₇	C₂×C₈	direct (N × H)
43. C₂×C₈×C₆₇	C₂×Q₈	direct (N × H)
53. Q₈×C₆₇	C₂²×C₈	direct (N × H)
64. C₈×C₆₇	C₂²×Q₈	direct (N × H)
65. C₂²×C₆₇	C₈×Q₈	direct (N × H)
75. C₂²×C₈×Q₈	C₆₇	direct (N × H)
76. C₂×C₆₇	C₂×C₈×Q₈	direct (N × H)
81. C₂×C₈×Q₈	C₂×C₆₇	direct (N × H)
84. C₆₇	C₂²×C₈×Q₈	direct (N × H)
92. C₈×Q₈	C₂²×C₆₇	direct (N × H)
95. C₂²×Q₈	C₈×C₆₇	direct (N × H)
101. C₂²×C₈	Q₈×C₆₇	direct (N × H)
112. C₂×Q₈	C₂×C₈×C₆₇	direct (N × H)
123. C₂×C₈	C₂×Q₈×C₆₇	direct (N × H)
133. Q₈	C₂²×C₈×C₆₇	direct (N × H)
144. C₈	C₂²×Q₈×C₆₇	direct (N × H)
145. C₂²	C₈×Q₈×C₆₇	direct (N × H)
155. C₂	C₂×C₈×Q₈×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
C₂²×C₄×Q₈×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂²×C₈×Q₈	67

Subgroups — C22×C8×Q8×C67 #30739436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×Q₈×C₆₇ #30739436