Subgroups — Q₈×#14523983 #31009345

Order 17152.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562118	C₂	1	6	yes	maximal, complemented
2	Q₈×C₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	#22561965	C₂	1	3	yes	maximal
4	C₄×#14523983	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	3	yes	—
7	C₂×#14523983	C₂²	1	1	yes	characteristic
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	6	yes	—
9	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	6	yes	—
10	#16863172	C₂²	1	6	yes	—
11	C₄×C₆₇⋊C₁₆	C₂²	1	6	yes	—
12	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₆₇	C₄	1	3	yes	—
14	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
15	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₆₇	C₂²	1	3	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
19	#14523983	—	2	6	no	—
20	#14523983	Q₈	1	4	yes	direct product factor
21	C₂×C₆₇⋊C₁₆	C₂³	1	6	yes	—
22	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
23	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₄×Q₈×C₆₇	C₈	1	3	yes	—
27	C₄×Q₈×C₆₇	C₈	1	1	yes	characteristic
28	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
29	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₆₇	C₈	1	3	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₆₇	C₂×C₄	1	3	yes	—
33	C₄⋊C₈×C₆₇	C₂×C₄	1	6	yes	—
34	C₄⋊C₈×C₆₇	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₆₇	C₂³	1	3	yes	—
38	C₄×C₈×C₆₇	C₂³	1	3	yes	—
39	C₆₇⋊C₁₆	—	2	6	no	—
40	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
41	C₂×Q₈×C₆₇	—	2	4	no	—
42	C₂×Q₈×C₆₇	C₂×C₈	1	3	yes	—
43	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	6	no	—
48	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₆₇	#41	1	2	yes	—
53	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	C₂×C₈	1	6	yes	—
55	C₄⋊C₄×C₆₇	C₂×C₈	1	3	yes	—
56	C₄⋊C₄×C₆₇	C₂×C₈	1	3	yes	—
57	C₄²×C₆₇	C₂×C₈	1	3	yes	—
58	C₄²×C₆₇	C₂×C₈	1	3	yes	—
59	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
60	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
61	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
62	Q₈×C₆₇	—	2	6	no	—
63	Q₈×C₆₇	M₃₂	1	4	yes	complemented
64	C₂×C₄×C₆₇	—	2	6	no	—
65	C₂×C₄×C₆₇	—	4	3	no	—
66	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
72	C₂×C₄×C₆₇	#131	1	2	yes	—
73	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
74	C₈×C₆₇	—	2	3	no	—
75	C₈×C₆₇	—	2	3	no	—
76	C₈×C₆₇	C₄×Q₈	1	2	yes	—
77	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
79	C₂²×C₆₇	—	2	1	no	—
80	C₂²×C₆₇	—	2	1	no	—
81	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
82	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₆₇	—	2	3	no	—
84	C₄×C₆₇	—	2	3	no	—
85	C₄×C₆₇	—	2	3	no	—
86	C₄×C₆₇	C₈×Q₈	1	2	yes	—
87	C₄×C₆₇	C₂×M₃₂	1	6	yes	—
88	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
90	Q₈×M₃₂	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	2	2	no	—
92	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
93	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
94	C₂×C₆₇	#3407	1	1	yes	characteristic
95	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
96	#2183	—	67	6	no	—
97	Q₈×C₁₆	—	67	2	no	—
98	#2150	—	67	3	no	—
99	C₄×M₃₂	—	67	3	no	—
100	C₆₇	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
102	C₂×M₃₂	—	67	1	no	—
103	C₂×M₃₂	—	67	3	no	—
104	C₈×Q₈	Dic₆₇	1	3	yes	—
105	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
106	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
107	C₈×Q₈	C₂×D₆₇	1	3	yes	—
108	C₂×C₄⋊C₈	Dic₆₇	1	3	yes	—
109	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
110	C₄⋊C₁₆	—	67	6	no	—
111	C₄⋊C₁₆	—	67	6	no	—
112	#346	—	67	6	no	—
113	C₄×C₁₆	—	67	6	no	—
114	C₂²×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
116	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
117	C₄×Q₈	C₆₇⋊C₈	1	3	yes	—
118	C₄×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
119	C₄×Q₈	C₂×Dic₆₇	1	3	yes	—
120	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₆₇⋊C₈	1	3	yes	—
122	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
123	M₃₂	—	67	4	no	—
124	M₃₂	—	134	6	no	—
125	C₂×C₁₆	—	67	2	no	—
126	C₂×C₁₆	—	67	6	no	—
127	C₄⋊C₈	C₂×Dic₆₇	1	3	yes	—
128	C₄⋊C₈	C₂×Dic₆₇	1	6	yes	—
129	C₄⋊C₈	C₂×Dic₆₇	1	3	yes	—
130	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
131	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
132	C₄×C₈	C₂²×D₆₇	1	3	yes	—
133	C₄×C₈	C₂²×D₆₇	1	3	yes	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₂×C₆₇⋊C₈	1	3	yes	—
136	C₂×Q₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
138	C₂²×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
139	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
140	C₂×C₈	—	2	6	no	—
141	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
142	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
143	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₆₇	1	2	yes	—
145	C₂×C₈	#11761629	1	2	yes	—
146	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
147	C₄⋊C₄	C₂×C₆₇⋊C₈	1	6	yes	—
148	C₄⋊C₄	C₂×C₆₇⋊C₈	1	3	yes	—
149	C₄⋊C₄	C₂×C₆₇⋊C₈	1	3	yes	—
150	C₄²	C₂×C₆₇⋊C₈	1	3	yes	—
151	C₄²	C₂×C₆₇⋊C₈	1	3	yes	—
152	C₄²	C₂²×Dic₆₇	1	3	yes	—
153	C₄²	C₂²×Dic₆₇	1	3	yes	—
154	C₁₆	—	67	4	no	—
155	C₁₆	—	134	6	no	—
156	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	#14523983	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
162	C₂×C₄	C₂²×C₆₇⋊C₈	1	3	yes	—
163	C₂×C₄	C₂²×C₆₇⋊C₈	1	3	yes	—
164	C₂×C₄	C₂²×C₆₇⋊C₈	1	3	yes	—
165	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₂²×C₆₇⋊C₈	1	3	yes	—
167	C₂×C₄	#14524050	1	2	yes	—
168	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	2	3	no	—
171	C₈	Q₈×Dic₆₇	1	2	yes	—
172	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
173	C₈	C₂×#11761629	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	Q₈×C₆₇⋊C₈	1	2	yes	—
177	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	Q₈×C₆₇⋊C₈	1	2	yes	—
182	C₄	C₂×#14523983	1	6	yes	—
183	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
184	C₄	C₂×#14524050	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×Q₈×C₆₇⋊C₈	1	1	yes	characteristic
187	C₂	C₂²×#14523983	1	1	yes	characteristic
188	C₂	#22573312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₆₇	C₂
prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	Q₈×M₃₂
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #22562118	C₂	semidirect
2. Q₈×C₆₇⋊C₁₆	C₂	semidirect
20. #14523983	Q₈	direct (N × H)
40. C₆₇⋊C₁₆	C₂×Q₈	semidirect
63. Q₈×C₆₇	M₃₂	semidirect
100. C₆₇	Q₈×M₃₂	semidirect
158. Q₈	#14523983	direct (N × H)

Maximal subgroups

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

subgroup	index
#22562118	2
Q₈×C₆₇⋊C₁₆	2
#22561965	2
C₄×#14523983	2
C₂×C₈×Q₈×C₆₇	2
Q₈×M₃₂	67

Subgroups — Q8×#14523983 #31009345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14523983 #31009345