Subgroups — Q₈×#6372 #13507306

Order 1792.

530 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₇	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11727038	C₂	1	3	yes	maximal
5	#11727035	C₂	1	3	yes	maximal, complemented
6	C₄×#6372	C₂	1	3	yes	maximal
7	#11726802	C₂	1	3	yes	maximal, complemented
8	C₄×Q₈×D₇	C₄	1	1	yes	characteristic
9	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
10	Q₈×C₇⋊C₈	C₄	1	1	yes	characteristic
11	C₂×#6372	C₂²	1	3	yes	—
12	C₂×#6372	C₂²	1	1	yes	characteristic
13	D₇×C₄⋊C₈	C₄	1	3	yes	—
14	#89602	C₄	1	3	yes	—
15	C₄×C₈×D₇	C₂²	1	3	yes	—
16	C₈×Dic₁₄	C₂²	1	3	yes	—
17	C₇×C₄⋊C₁₆	C₂²	1	3	yes	—
18	C₄×C₇×C₁₆	C₂²	1	3	yes	—
19	#89297	C₂²	1	3	yes	—
20	#89297	C₂²	1	3	yes	—
21	Dic₇⋊C₁₆	C₂²	1	6	yes	—
22	C₄⋊(C₇⋊C₁₆)	C₂²	1	3	yes	—
23	C₄×C₇⋊C₁₆	C₂²	1	3	yes	—
24	Q₈×M₃₂	—	7	1	no	maximal, Sylow
25	C₂×Q₈×D₇	C₈	1	1	yes	characteristic
26	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
27	Q₈×Dic₇	C₈	1	1	yes	characteristic
28	C₂×C₈×D₇	C₂×C₄	1	3	yes	—
29	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
30	D₇×C₄⋊C₄	C₈	1	3	yes	—
31	#6450	C₈	1	3	yes	—
32	C₄²×D₇	C₂×C₄	1	3	yes	—
33	C₄×Dic₁₄	C₂×C₄	1	3	yes	—
34	C₂×C₇×C₁₆	C₂³	1	3	yes	—
35	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
36	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
37	C₄×C₇×C₈	C₂³	1	3	yes	—
38	Dic₇⋊C₈	C₂×C₄	1	6	yes	—
39	C₈×Dic₇	C₂×C₄	1	3	yes	—
40	C₈×Dic₇	C₂³	1	3	yes	—
41	C₂×C₇⋊C₁₆	C₂³	1	3	yes	—
42	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
43	C₄⋊(C₇⋊C₈)	C₂×C₄	1	3	yes	—
44	C₄×C₇⋊C₈	C₂×C₄	1	3	yes	—
45	#6372	—	2	6	no	—
46	#6372	Q₈	1	4	yes	direct product factor
47	C₂×C₈×Q₈	—	7	1	no	—
48	#2183	—	7	3	no	—
49	#2183	—	7	3	no	—
50	Q₈×C₁₆	—	7	1	no	—
51	Q₈×C₁₆	D₇	1	1	yes	characteristic, complemented
52	#2150	—	7	3	no	—
53	C₄×M₃₂	—	7	3	no	—
54	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
55	Q₈×D₇	—	2	4	no	—
56	C₂×C₄×D₇	C₂×C₈	1	3	yes	—
57	C₂×C₄×D₇	C₂×C₈	1	3	yes	—
58	C₂×C₄×D₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×Dic₁₄	C₂×C₈	1	3	yes	—
60	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
61	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
62	C₇×C₄⋊C₄	C₂×C₈	1	3	yes	—
63	C₄²×C₇	C₂²×C₄	1	3	yes	—
64	C₄⋊Dic₇	C₂×C₈	1	3	yes	—
65	Dic₇⋊C₄	C₂×C₈	1	6	yes	—
66	C₄×Dic₇	C₂×C₈	1	3	yes	—
67	C₄×Dic₇	C₂×C₈	1	3	yes	—
68	C₄×Dic₇	C₂²×C₄	1	3	yes	—
69	C₂×C₇⋊C₈	C₂²×C₄	1	3	yes	—
70	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
71	C₈×D₇	—	2	6	no	—
72	C₈×D₇	C₂×Q₈	1	2	yes	—
73	C₈×D₇	#41	1	2	yes	—
74	C₇×C₁₆	—	2	3	no	—
75	C₇×C₁₆	C₂×Q₈	1	2	yes	complemented
76	C₇⋊C₁₆	—	2	3	no	—
77	C₇⋊C₁₆	C₂×Q₈	1	2	yes	complemented
78	C₂×C₄×Q₈	—	7	1	no	—
79	C₂×M₃₂	—	7	1	no	—
80	C₂×M₃₂	—	7	3	no	—
81	C₈×Q₈	—	7	1	no	—
82	C₈×Q₈	—	7	3	no	—
83	C₈×Q₈	—	7	3	no	—
84	C₈×Q₈	C₂×D₇	1	1	yes	characteristic
85	C₂×C₄⋊C₈	—	7	3	no	—
86	C₂×C₄×C₈	—	7	3	no	—
87	C₄⋊C₁₆	—	7	6	no	—
88	C₄⋊C₁₆	—	7	3	no	—
89	C₄⋊C₁₆	C₂×D₇	1	3	yes	—
90	#346	—	7	3	no	—
91	#346	—	7	3	no	—
92	C₄×C₁₆	—	7	3	no	—
93	C₄×C₁₆	C₂×D₇	1	3	yes	—
94	C₂²×D₇	C₂²×C₈	1	1	yes	characteristic
95	C₇×Q₈	M₃₂	1	4	yes	complemented
96	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
97	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
98	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×Dic₇	C₂²×C₈	1	3	yes	—
100	C₂×Dic₇	C₂²×C₈	1	3	yes	—
101	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
102	C₄×D₇	—	2	6	no	—
103	C₄×D₇	—	4	3	no	—
104	C₄×D₇	C₄×Q₈	1	2	yes	—
105	C₄×D₇	#131	1	2	yes	—
106	Dic₁₄	—	2	6	no	—
107	C₇×C₈	—	2	3	no	—
108	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
109	C₇×C₈	C₂×#41	1	1	yes	characteristic
110	C₇⋊C₈	—	2	3	no	—
111	C₇⋊C₈	C₄×Q₈	1	2	yes	—
112	C₂²×Q₈	—	7	1	no	—
113	C₂²×C₈	—	7	1	no	—
114	C₂²×C₈	—	7	3	no	—
115	C₄×Q₈	—	7	1	no	—
116	C₄×Q₈	—	7	3	no	—
117	C₄×Q₈	—	7	3	no	—
118	C₄×Q₈	C₄×D₇	1	1	yes	characteristic
119	C₂×C₄⋊C₄	—	7	3	no	—
120	C₂×C₄²	—	7	3	no	—
121	M₃₂	—	7	4	no	—
122	M₃₂	—	14	6	no	—
123	C₂×C₁₆	—	7	1	no	—
124	C₂×C₁₆	—	7	3	no	—
125	C₂×C₁₆	C₂²×D₇	1	3	yes	—
126	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
127	C₄⋊C₈	—	7	6	no	—
128	C₄⋊C₈	—	7	3	no	—
129	C₄⋊C₈	C₄×D₇	1	3	yes	—
130	C₄×C₈	—	7	3	no	—
131	C₄×C₈	—	7	3	no	—
132	C₄×C₈	—	7	3	no	—
133	C₄×C₈	C₂²×D₇	1	3	yes	—
134	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
135	C₂×D₇	—	2	1	no	—
136	C₂×D₇	—	2	1	no	—
137	C₂×D₇	C₈×Q₈	1	2	yes	—
138	C₄×C₇	—	2	3	no	—
139	C₄×C₇	C₂×M₃₂	1	6	yes	—
140	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₄×C₇	C₂×#131	1	1	yes	characteristic
142	Dic₇	—	2	3	no	—
143	Dic₇	—	2	3	no	—
144	Dic₇	C₈×Q₈	1	2	yes	—
145	C₂×Q₈	—	7	3	no	—
146	C₂×Q₈	—	14	4	no	—
147	C₂×Q₈	C₈×D₇	1	1	yes	characteristic
148	C₂²×C₄	—	7	1	no	—
149	C₂²×C₄	—	7	3	no	—
150	C₂²×C₄	—	7	3	no	—
151	C₂×C₈	—	7	1	no	—
152	C₂×C₈	—	7	2	no	—
153	C₂×C₈	—	7	2	no	—
154	C₂×C₈	—	7	3	no	—
155	C₂×C₈	—	14	6	no	—
156	C₂×C₈	C₂×C₄×D₇	1	3	yes	—
157	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
158	C₄⋊C₄	—	7	6	no	—
159	C₄⋊C₄	—	7	3	no	—
160	C₄⋊C₄	C₈×D₇	1	3	yes	—
161	C₄²	—	7	3	no	—
162	C₄²	—	7	3	no	—
163	C₄²	—	7	3	no	—
164	C₄²	C₂×C₄×D₇	1	3	yes	—
165	C₁₆	—	2	3	no	—
166	C₁₆	—	7	2	no	—
167	C₁₆	—	14	3	no	—
168	C₁₆	Q₈×D₇	1	2	yes	complemented
169	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
170	C₂×C₇	C₂²×M₃₂	1	1	yes	characteristic
171	C₂×C₇	#3407	1	1	yes	characteristic
172	D₇	—	2	2	no	—
173	C₂³	—	7	1	no	—
174	Q₈	—	14	6	no	—
175	Q₈	#6372	1	4	yes	direct product factor
176	C₂×C₄	—	7	1	no	—
177	C₂×C₄	—	7	2	no	—
178	C₂×C₄	—	7	2	no	—
179	C₂×C₄	—	7	3	no	—
180	C₂×C₄	—	7	3	no	—
181	C₂×C₄	—	14	6	no	—
182	C₂×C₄	—	28	3	no	—
183	C₂×C₄	C₂×C₈×D₇	1	3	yes	—
184	C₂×C₄	C₂×C₈×D₇	1	3	yes	—
185	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
186	C₈	—	2	3	no	—
187	C₈	—	7	2	no	—
188	C₈	—	14	3	no	—
189	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
190	C₈	D₇×#41	1	1	yes	characteristic
191	C₇	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
192	C₂²	—	7	2	no	—
193	C₂²	—	14	1	no	—
194	C₂²	—	14	1	no	—
195	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	7	2	no	—
198	C₄	—	14	3	no	—
199	C₄	—	14	3	no	—
200	C₄	C₂×#6372	1	6	yes	—
201	C₄	C₄×Q₈×D₇	1	1	yes	characteristic
202	C₄	D₇×#131	1	1	yes	characteristic
203	C₂	—	14	2	no	—
204	C₂	C₈×Q₈×D₇	1	1	yes	characteristic
205	C₂	C₂²×#6372	1	1	yes	characteristic
206	C₂	#11739457	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₇×Q₈×C₁₆	C₂
prime radical	C₂²×D₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	Q₈×C₁₆	D₇
hypocenter	C₇	Q₈×M₃₂
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₇×Q₈×C₁₆	C₂	semidirect
3. Q₈×C₇⋊C₁₆	C₂	semidirect
5. #11727035	C₂	semidirect
7. #11726802	C₂	semidirect
46. #6372	Q₈	direct (N × H)
51. Q₈×C₁₆	D₇	semidirect
75. C₇×C₁₆	C₂×Q₈	semidirect
77. C₇⋊C₁₆	C₂×Q₈	semidirect
95. C₇×Q₈	M₃₂	semidirect
168. C₁₆	Q₈×D₇	semidirect
175. Q₈	#6372	direct (N × H)
191. C₇	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₇	2
C₇×Q₈×C₁₆	2
Q₈×C₇⋊C₁₆	2
#11727038	2
#11727035	2
C₄×#6372	2
#11726802	2
Q₈×M₃₂	7

Subgroups — Q8×#6372 #13507306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6372 #13507306