Subgroups — Q₈×#14523954 #30854500

Order 17152.

954 conjugacy classes of subgroups.

The subgroup lattice has 184 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	C₄×#14523954	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	3	yes	—
9	C₂×#14523954	C₂²	1	6	yes	—
10	C₂×#14523954	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	6	yes	—
12	#16863151	C₂²	1	6	yes	—
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	6	yes	—
19	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
20	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	24	yes	—
21	#14523954	—	2	12	no	—
22	#14523954	—	2	6	no	—
23	#14523954	Q₈	1	4	yes	direct product factor
24	C₄×C₆₇⋊C₈	C₂×C₄	1	24	yes	—
25	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₆₇	—	2	8	no	—
27	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	#95×C₆₇	C₂×C₄	1	6	yes	—
36	C₂×C₆₇⋊C₈	—	2	6	no	—
37	C₂×C₆₇⋊C₈	—	2	6	no	—
38	C₂×C₆₇⋊C₈	C₂²×C₄	1	12	yes	—
39	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
40	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
41	C₂×C₆₇⋊C₈	#41	1	4	yes	—
42	C₂×Q₈×C₆₇	C₄²	1	4	yes	—
43	C₂×Q₈×C₆₇	M₁₆	1	8	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₆₇	—	2	12	no	—
50	C₄⋊C₄×C₆₇	C₄²	1	12	yes	—
51	C₄²×C₆₇	—	2	12	no	—
52	C₄²×C₆₇	—	2	6	no	—
53	C₄²×C₆₇	—	4	6	no	—
54	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
55	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
56	C₄²×C₆₇	#41	1	2	yes	—
57	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
58	Q₈×C₆₇	#97	1	16	yes	complemented
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	6	no	—
62	C₂×C₄×C₆₇	—	2	6	no	—
63	C₂×C₄×C₆₇	—	2	6	no	—
64	C₂×C₄×C₆₇	—	2	6	no	—
65	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
66	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
67	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
68	C₂×C₄×C₆₇	C₂×M₁₆	1	12	yes	—
69	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
70	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
73	C₆₇⋊C₈	—	2	12	no	—
74	C₆₇⋊C₈	C₄×Q₈	1	8	yes	complemented
75	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
76	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
77	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
78	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
79	C₂²×C₆₇	#568	1	2	yes	—
80	C₄×C₆₇	—	2	6	no	—
81	C₄×C₆₇	—	2	4	no	—
82	C₄×C₆₇	—	4	6	no	—
83	C₄×C₆₇	C₂×#97	1	12	yes	—
84	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
85	C₄×C₆₇	C₄×#41	1	2	yes	—
86	Q₈×#97	—	67	1	no	maximal, Sylow
87	C₂×C₆₇	C₄²×Q₈	1	1	yes	characteristic
88	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
89	C₂×C₆₇	#2875	1	1	yes	characteristic
90	C₂×C₆₇	Q₈×M₁₆	1	2	yes	—
91	C₂×C₆₇	#2971	1	2	yes	—
92	C₂×C₈×Q₈	—	67	2	no	—
93	C₄²×Q₈	D₆₇	1	1	yes	characteristic
94	#1770	—	67	6	no	—
95	#1767	—	67	3	no	—
96	C₄×#97	—	67	3	no	—
97	C₆₇	Q₈×#97	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄×Q₈	Dic₆₇	1	2	yes	—
99	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
100	C₈×Q₈	—	67	16	no	—
101	C₂×C₄⋊C₈	—	67	6	no	—
102	C₂×#97	—	67	1	no	—
103	C₂×#97	—	67	3	no	—
104	C₂×#97	—	67	6	no	—
105	C₂×C₄×C₈	—	67	6	no	—
106	C₄×C₄⋊C₄	Dic₆₇	1	6	yes	—
107	C₄×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
108	C₄³	C₂×D₆₇	1	3	yes	—
109	#336	—	67	6	no	—
110	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
111	C₂²×C₈	—	67	2	no	—
112	C₂²×C₈	—	67	6	no	—
113	C₄×Q₈	—	2	8	no	—
114	C₄×Q₈	C₄×D₆₇	1	8	yes	—
115	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
116	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
117	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
118	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
119	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
120	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
121	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
122	C₄⋊C₈	—	67	24	no	—
123	#97	—	67	4	no	—
124	#97	—	134	12	no	—
125	#97	—	134	6	no	—
126	C₄×C₈	—	67	24	no	—
127	#95	C₂×Dic₆₇	1	6	yes	—
128	C₂×Q₈	C₄×Dic₆₇	1	4	yes	—
129	C₂×Q₈	#11761612	1	8	yes	—
130	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
131	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
132	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
133	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
134	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
135	C₂×C₈	—	67	4	no	—
136	C₂×C₈	—	67	4	no	—
137	C₂×C₈	—	67	4	no	—
138	C₂×C₈	—	67	12	no	—
139	C₂×C₈	—	134	6	no	—
140	C₂×C₈	—	134	6	no	—
141	C₄⋊C₄	—	2	12	no	—
142	C₄⋊C₄	C₄×Dic₆₇	1	12	yes	—
143	C₄²	—	2	12	no	—
144	C₄²	—	2	6	no	—
145	C₄²	—	4	6	no	—
146	C₄²	C₂×C₄×D₆₇	1	12	yes	—
147	C₄²	Q₈×D₆₇	1	2	yes	—
148	C₄²	#11761629	1	2	yes	—
149	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
150	Q₈	#14523954	1	16	yes	direct product factor
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	6	no	—
156	C₂×C₄	—	2	6	no	—
157	C₂×C₄	C₂×C₄×Dic₆₇	1	6	yes	—
158	C₂×C₄	C₂×C₄×Dic₆₇	1	6	yes	—
159	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
160	C₂×C₄	C₂×#11761612	1	12	yes	—
161	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
162	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
163	C₂×C₄	#41×D₆₇	1	2	yes	—
164	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
165	C₈	—	67	8	no	—
166	C₈	—	134	12	no	—
167	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
168	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
169	C₂²	#41×Dic₆₇	1	1	yes	characteristic
170	C₂²	C₂²×#11761612	1	2	yes	—
171	C₂²	#16864163	1	2	yes	—
172	C₄	—	2	6	no	—
173	C₄	—	2	4	no	—
174	C₄	—	4	6	no	—
175	C₄	C₂×#14523954	1	12	yes	—
176	C₄	C₄×Q₈×D₆₇	1	2	yes	—
177	C₄	C₄×#11761629	1	2	yes	—
178	C₂	C₄×Q₈×Dic₆₇	1	1	yes	characteristic
179	C₂	C₂²×#14523954	1	1	yes	characteristic
180	C₂	#22568637	1	1	yes	characteristic
181	C₂	Q₈×#11761612	1	2	yes	—
182	C₂	#22569229	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	Q₈×#97
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
23. #14523954	Q₈	direct (N × H)
58. Q₈×C₆₇	#97	semidirect
74. C₆₇⋊C₈	C₄×Q₈	semidirect
97. C₆₇	Q₈×#97	semidirect
150. Q₈	#14523954	direct (N × H)

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
C₄×#14523954	2
C₄²×Q₈×C₆₇	2
Q₈×#97	67

Subgroups — Q8×#14523954 #30854500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14523954 #30854500