Subgroups — #13372801

Order 1792.

642 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	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1136	C₂	1	2	yes	maximal
3	#11728228	C₂	1	6	yes	maximal
4	#11727545	C₂	1	3	yes	maximal
5	C₄×#6393	C₂	1	3	yes	maximal
6	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₇×C₈×Q₈	C₄	1	6	yes	—
8	C₇×C₈×Q₈	C₄	1	2	yes	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
11	C₂×#6393	C₂²	1	6	yes	—
12	#89796	—	2	6	no	—
13	Q₈×C₇⋊C₈	—	2	2	no	—
14	#89696	C₂²	1	6	yes	—
15	C₄×#1136	C₂²	1	6	yes	—
16	#89347	C₂²	1	6	yes	—
17	#89347	C₂²	1	1	yes	characteristic
18	#89347	C₂²	1	3	yes	—
19	#22746	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂³	1	3	yes	—
23	C₄×C₇×Q₈	C₂×C₄	1	3	yes	—
24	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
26	C₄×C₇×Q₈	D₄	1	3	yes	—
27	C₂×C₇×C₄⋊C₄	Q₈	1	3	yes	—
28	C₂×C₄²×C₇	C₂³	1	3	yes	—
29	C₂×#1136	C₂³	1	6	yes	—
30	C₂×#1136	C₂³	1	2	yes	—
31	C₇×C₄⋊C₈	C₂×C₄	1	6	yes	—
32	C₇×C₄⋊C₈	C₂×C₄	1	6	yes	—
33	C₄×C₇×C₈	C₂×C₄	1	6	yes	—
34	C₄×C₇×C₈	C₂×C₄	1	6	yes	—
35	#6393	—	2	12	no	—
36	C₄⋊(C₇⋊C₈)	—	2	6	no	—
37	C₄⋊(C₇⋊C₈)	—	2	6	no	—
38	#6377	—	2	6	no	—
39	C₄×C₇⋊C₈	—	2	6	no	—
40	Q₈×M₁₆	—	7	2	no	—
41	C₂×C₈×Q₈	D₇	1	1	yes	characteristic
42	#1950	—	7	6	no	—
43	#1837	—	7	3	no	—
44	C₄×#108	—	7	3	no	—
45	C₂×C₇×Q₈	—	2	4	no	—
46	C₂×C₇×Q₈	C₄⋊C₄	1	3	yes	—
47	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
49	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
50	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
51	C₂×C₇×C₈	—	2	6	no	—
52	C₂×C₇×C₈	—	2	2	no	—
53	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
54	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
55	C₇×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
56	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
57	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
58	C₄²×C₇	C₂²×C₄	1	3	yes	—
59	C₄²×C₇	C₂²×C₄	1	3	yes	—
60	C₄²×C₇	C₂×D₄	1	3	yes	—
61	C₄²×C₇	C₂×D₄	1	3	yes	—
62	#1136	—	2	2	no	—
63	#1136	—	2	6	no	—
64	#1136	—	2	6	no	—
65	#1136	#41	1	4	yes	—
66	C₂×C₇⋊C₈	—	2	2	no	—
67	C₂×C₇⋊C₈	—	2	6	no	—
68	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
69	#446	—	14	6	no	—
70	C₈×Q₈	—	14	2	no	—
71	C₈×Q₈	Dic₇	1	6	yes	—
72	C₈×Q₈	Dic₇	1	2	yes	—
73	C₂×#108	—	7	6	no	—
74	#423	—	7	6	no	—
75	C₂×C₄⋊C₈	C₂×D₇	1	3	yes	—
76	C₄×M₁₆	—	7	6	no	—
77	C₂×C₄×C₈	C₂×D₇	1	3	yes	—
78	#341	—	7	3	no	—
79	#341	—	7	6	no	—
80	#341	—	7	1	no	—
81	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
82	C₇×Q₈	—	2	6	no	—
83	C₇×Q₈	#108	1	4	yes	complemented
84	C₂×C₄×C₇	—	2	6	no	—
85	C₂×C₄×C₇	—	4	3	no	—
86	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
87	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
88	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
89	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂×#41	1	2	yes	—
93	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
94	C₇×C₈	—	2	6	no	—
95	C₇×C₈	—	2	2	no	—
96	C₇⋊C₈	—	2	4	no	—
97	C₇⋊C₈	—	4	6	no	—
98	C₂²×Q₈	Dic₁₄	1	1	yes	characteristic
99	C₂×M₁₆	—	7	2	no	—
100	C₂×M₁₆	—	7	6	no	—
101	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇	1	3	yes	—
103	C₄×Q₈	D₂₈	1	1	yes	characteristic
104	C₄×Q₈	D₂₈	1	3	yes	—
105	C₄×Q₈	C₂×Dic₇	1	3	yes	—
106	C₄×Q₈	C₂×Dic₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	Dic₁₄	1	3	yes	—
108	C₂×C₄²	C₂²×D₇	1	3	yes	—
109	#108	—	14	12	no	—
110	C₄⋊C₈	—	14	6	no	—
111	C₄⋊C₈	—	14	6	no	—
112	C₄⋊C₈	C₂×Dic₇	1	6	yes	—
113	C₄⋊C₈	C₂×Dic₇	1	6	yes	—
114	#97	—	14	6	no	—
115	C₄×C₈	—	14	6	no	—
116	C₄×C₈	C₂×Dic₇	1	6	yes	—
117	C₄×C₈	C₂×Dic₇	1	6	yes	—
118	C₂²×C₇	—	2	1	no	—
119	C₂²×C₇	—	2	1	no	—
120	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₇	#557	1	2	yes	—
122	C₄×C₇	—	2	3	no	—
123	C₄×C₇	—	2	3	no	—
124	C₄×C₇	—	2	3	no	—
125	C₄×C₇	C₂×#108	1	6	yes	—
126	C₄×C₇	C₄×#41	1	1	yes	characteristic
127	C₄×C₇	#519	1	1	yes	characteristic
128	C₄×C₇	#548	1	2	yes	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	C₄⋊Dic₇	1	3	yes	—
131	C₂×Q₈	C₄⋊Dic₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×Dic₁₄	1	3	yes	—
133	C₂²×C₄	C₂×Dic₁₄	1	3	yes	—
134	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
135	M₁₆	—	7	4	no	—
136	M₁₆	—	14	2	no	—
137	M₁₆	—	14	6	no	—
138	M₁₆	—	14	6	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	6	no	—
141	C₂×C₈	—	14	2	no	—
142	C₂×C₈	—	14	6	no	—
143	C₂×C₈	C₂²×Dic₇	1	2	yes	—
144	C₂×C₈	C₂²×Dic₇	1	6	yes	—
145	C₄⋊C₄	C₄⋊Dic₇	1	6	yes	—
146	C₄⋊C₄	C₄⋊Dic₇	1	3	yes	—
147	C₄⋊C₄	C₄⋊Dic₇	1	3	yes	—
148	C₄²	C₂×D₂₈	1	3	yes	—
149	C₄²	C₂×D₂₈	1	3	yes	—
150	C₄²	C₂²×Dic₇	1	3	yes	—
151	C₄²	C₂²×Dic₇	1	3	yes	—
152	C₂×C₇	—	2	2	no	—
153	C₂×C₇	#2351	1	1	yes	characteristic
154	C₂×C₇	C₂²×#108	1	1	yes	characteristic
155	C₂×C₇	#2916	1	1	yes	characteristic
156	C₂³	C₂²×Dic₁₄	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	#6393	1	4	yes	complemented
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	C₂×C₄⋊Dic₇	1	3	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₇	1	3	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₇	1	3	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₇	1	3	yes	—
165	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
166	C₂×C₄	#6551	1	2	yes	—
167	C₂×C₄	D₇×#41	1	2	yes	—
168	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	6	no	—
171	C₈	—	14	4	no	—
172	C₈	—	28	6	no	—
173	C₇	#22746	1	1	yes	characteristic, complemented, Sylow
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#90260	1	2	yes	—
177	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₂×#6393	1	6	yes	—
182	C₄	#90267	1	2	yes	—
183	C₄	#90503	1	1	yes	characteristic
184	C₄	#41×Dic₇	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#11739905	1	1	yes	characteristic
187	C₂	C₂²×#6393	1	1	yes	characteristic
188	C₂	#11741234	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₂×C₇×C₈×Q₈	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₇	C₂²×Q₈
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇
hypocenter	C₇	#22746
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
83. C₇×Q₈	#108	semidirect
158. Q₈	#6393	semidirect
173. C₇	#22746	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×Q₈	2
Q₈×#1136	2
#11728228	2
#11727545	2
C₄×#6393	2
#22746	7