Subgroups — #30854895

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₆₇⋊C₈	C₂	1	2	yes	maximal
2	#22560078	C₂	1	6	yes	maximal
3	#22560062	C₂	1	3	yes	maximal
4	#22559955	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₆₇⋊C₈	C₄	1	16	yes	complemented
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
8	C₂×#14523954	C₂²	1	6	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	6	yes	—
10	#16863151	C₂²	1	3	yes	—
11	#16863151	C₂²	1	6	yes	—
12	#16863151	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₇	C₄	1	6	yes	—
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
17	C₄³×C₆₇	C₂²	1	3	yes	—
18	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
19	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
20	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	24	yes	—
21	#14523954	—	4	6	no	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	24	yes	—
23	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₆₇	—	2	8	no	—
25	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
26	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
33	#95×C₆₇	C₂×C₄	1	6	yes	—
34	C₂×C₆₇⋊C₈	—	2	6	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	2	6	no	—
38	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
39	C₂×C₆₇⋊C₈	C₂²×C₄	1	12	yes	—
40	C₂×Q₈×C₆₇	C₄²	1	4	yes	—
41	C₂×Q₈×C₆₇	M₁₆	1	8	yes	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	—	2	12	no	—
48	C₄⋊C₄×C₆₇	C₄²	1	12	yes	—
49	C₄²×C₆₇	—	2	12	no	—
50	C₄²×C₆₇	—	2	12	no	—
51	C₄²×C₆₇	—	2	3	no	—
52	C₄²×C₆₇	—	2	3	no	—
53	C₄²×C₆₇	—	2	2	no	—
54	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
55	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
56	Q₈×C₆₇	#97	1	16	yes	complemented
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	6	no	—
60	C₂×C₄×C₆₇	—	2	6	no	—
61	C₂×C₄×C₆₇	—	2	6	no	—
62	C₂×C₄×C₆₇	—	2	6	no	—
63	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
64	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
65	C₂×C₄×C₆₇	C₂×M₁₆	1	12	yes	—
66	C₂×C₄×C₆₇	#131	1	4	yes	—
67	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
68	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
69	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
70	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
71	C₆₇⋊C₈	—	2	4	no	—
72	C₆₇⋊C₈	—	2	12	no	—
73	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
74	C₂²×C₆₇	#519	1	1	yes	characteristic
75	C₂²×C₆₇	#520	1	1	yes	characteristic
76	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
77	C₂²×C₆₇	C₂×#131	1	2	yes	—
78	C₄×C₆₇	—	2	6	no	—
79	C₄×C₆₇	—	2	4	no	—
80	C₄×C₆₇	—	4	6	no	—
81	C₄×C₆₇	C₂×#97	1	12	yes	—
82	C₄×C₆₇	#519	1	2	yes	—
83	C₄×C₆₇	#520	1	2	yes	—
84	#20572	—	67	1	no	maximal, Sylow
85	C₂×C₆₇	#2276	1	1	yes	characteristic
86	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
87	C₂×C₆₇	C₄×#131	1	1	yes	characteristic
88	C₂×C₆₇	#2984	1	2	yes	—
89	C₂×C₆₇	#2991	1	2	yes	—
90	C₂×C₈×Q₈	—	67	2	no	—
91	C₄²×Q₈	D₆₇	1	1	yes	characteristic
92	#1770	—	67	6	no	—
93	#1767	—	67	3	no	—
94	#1744	—	67	3	no	—
95	C₆₇	#20572	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×Q₈	Dic₆₇	1	2	yes	—
97	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
98	C₈×Q₈	—	67	16	no	—
99	C₂×C₄⋊C₈	—	67	6	no	—
100	C₂×#97	—	67	6	no	—
101	C₂×C₄×C₈	—	67	6	no	—
102	C₄×C₄⋊C₄	Dic₆₇	1	6	yes	—
103	C₄×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
104	C₄³	C₂×D₆₇	1	3	yes	—
105	#336	—	67	1	no	—
106	#336	—	67	3	no	—
107	#336	—	67	6	no	—
108	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
109	C₂²×C₈	—	67	2	no	—
110	C₂²×C₈	—	67	6	no	—
111	C₄×Q₈	—	2	8	no	—
112	C₄×Q₈	C₄×D₆₇	1	8	yes	—
113	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
114	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
115	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
116	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
117	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
118	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
119	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
120	C₄⋊C₈	—	67	24	no	—
121	#97	—	268	6	no	—
122	C₄×C₈	—	67	24	no	—
123	#95	C₂×Dic₆₇	1	6	yes	—
124	C₂×Q₈	C₄×Dic₆₇	1	4	yes	—
125	C₂×Q₈	#11761612	1	8	yes	—
126	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
127	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
128	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
129	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
130	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
131	C₂×C₈	—	67	4	no	—
132	C₂×C₈	—	67	12	no	—
133	C₂×C₈	—	134	6	no	—
134	C₂×C₈	—	134	2	no	—
135	C₂×C₈	—	134	2	no	—
136	C₂×C₈	—	134	6	no	—
137	C₄⋊C₄	—	2	12	no	—
138	C₄⋊C₄	C₄×Dic₆₇	1	12	yes	—
139	C₄²	—	2	12	no	—
140	C₄²	—	2	12	no	—
141	C₄²	—	2	3	no	—
142	C₄²	—	2	3	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂×C₄×D₆₇	1	12	yes	—
145	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
146	Q₈	#14523954	1	16	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	C₂×C₄×Dic₆₇	1	6	yes	—
154	C₂×C₄	C₂×C₄×Dic₆₇	1	6	yes	—
155	C₂×C₄	C₂×#11761612	1	12	yes	—
156	C₂×C₄	#14524050	1	4	yes	—
157	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
158	C₂×C₄	#14524089	1	1	yes	characteristic
159	C₂×C₄	#14524090	1	1	yes	characteristic
160	C₂×C₄	#14524091	1	2	yes	—
161	C₈	—	134	4	no	—
162	C₈	—	134	12	no	—
163	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
164	C₂²	#16863874	1	1	yes	characteristic
165	C₂²	#16863879	1	1	yes	characteristic
166	C₂²	C₂²×#11761612	1	2	yes	—
167	C₂²	C₂×#14524050	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	2	6	no	—
170	C₄	—	4	6	no	—
171	C₄	C₂×#14523954	1	12	yes	—
172	C₄	#16863872	1	2	yes	—
173	C₄	#16863878	1	2	yes	—
174	C₂	#22562574	1	1	yes	characteristic
175	C₂	C₂²×#14523954	1	1	yes	characteristic
176	C₂	C₄×#14524050	1	1	yes	characteristic
177	C₂	#22569298	1	2	yes	—
178	C₂	#22569341	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Fitting	C₄²×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂³×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×Q₈	D₆₇
hypocenter	C₆₇	#20572
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×C₆₇⋊C₈	C₄	semidirect
56. Q₈×C₆₇	#97	semidirect
95. C₆₇	#20572	semidirect
146. Q₈	#14523954	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₆₇⋊C₈	2
#22560078	2
#22560062	2
#22559955	2
C₄²×Q₈×C₆₇	2
#20572	67