Subgroups — #13323020

Order 1792.

702 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	#11739957	C₂	1	2	yes	maximal
3	#11728503	C₂	1	6	yes	maximal, complemented
4	#11727484	C₂	1	3	yes	maximal, complemented
5	C₂×#89252	C₂	1	3	yes	maximal, complemented
6	C₄×C₇×#41	C₂²	1	1	yes	characteristic
7	C₂×#6511	—	2	2	no	—
8	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
10	C₇×C₈×D₄	C₄	1	3	yes	—
11	C₇×C₈×D₄	C₄	1	3	yes	—
12	C₇×#431	C₂²	1	3	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
14	#89771	—	2	6	no	—
15	#89696	C₂²	1	6	yes	—
16	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	6	yes	—
17	#89252	C₂²	1	6	yes	—
18	#89252	C₂²	1	6	yes	—
19	#89252	C₂²	1	1	yes	characteristic
20	#89252	C₂²	1	3	yes	—
21	#17826	—	7	1	no	maximal, Sylow
22	C₂×C₇×#41	D₄	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
24	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
25	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×Q₈	Q₈	1	1	yes	characteristic
27	C₄×C₇×D₄	C₂×C₄	1	3	yes	—
28	C₄×C₇×D₄	Q₈	1	3	yes	—
29	C₇×#117	D₄	1	3	yes	—
30	C₂×C₄²×C₇	C₂³	1	3	yes	—
31	#6511	—	4	8	no	—
32	C₂×#1136	—	2	6	no	—
33	C₂²×C₇⋊C₈	—	2	6	no	—
34	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
35	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
36	C₇×C₂²⋊C₈	C₂×C₄	1	6	yes	—
37	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
38	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
39	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₇×C₈	C₂³	1	3	yes	—
41	C₄⋊(C₇⋊C₈)	—	2	6	no	—
42	C₄⋊(C₇⋊C₈)	—	2	6	no	—
43	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
44	C₄⋊(C₇⋊C₈)	C₂³	1	6	yes	—
45	C₈×#41	D₇	1	1	yes	characteristic
46	#2906	—	7	2	no	—
47	#1570	—	7	6	no	—
48	#1566	—	7	3	no	—
49	C₂×#334	—	7	3	no	—
50	C₇×#41	SD₁₆	1	8	yes	—
51	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₇×D₄	C₄⋊C₄	1	3	yes	—
53	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
56	C₂×C₇×C₈	—	2	6	no	—
57	C₂×C₇×C₈	—	4	3	no	—
58	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
60	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
61	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
63	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
64	C₇×C₂²⋊C₄	C₄⋊C₄	1	6	yes	—
65	C₄²×C₇	C₂²×C₄	1	3	yes	—
66	C₄²×C₇	C₂×D₄	1	3	yes	—
67	C₄²×C₇	C₂×Q₈	1	3	yes	—
68	C₄²×C₇	C₂⁴	1	1	yes	characteristic
69	#1136	—	4	12	no	—
70	C₂×C₇⋊C₈	—	2	2	no	—
71	C₂×C₇⋊C₈	—	2	6	no	—
72	C₂×C₇⋊C₈	—	4	12	no	—
73	C₂×#131	—	14	2	no	—
74	C₄×#41	C₂×D₇	1	1	yes	characteristic
75	C₈×Q₈	Dic₇	1	1	yes	characteristic
76	C₈×Q₈	Dic₇	1	1	yes	characteristic
77	C₈×D₄	Dic₇	1	3	yes	—
78	C₈×D₄	Dic₇	1	3	yes	—
79	#431	C₂×D₇	1	3	yes	—
80	#424	—	14	6	no	—
81	#423	—	7	6	no	—
82	C₂×C₄⋊C₈	—	7	6	no	—
83	C₂×C₄×C₈	C₂×D₇	1	3	yes	—
84	#334	—	7	1	no	—
85	#334	—	7	3	no	—
86	#334	—	7	6	no	—
87	#334	—	7	6	no	—
88	C₂³×C₇	C₂×C₄⋊C₄	1	3	yes	—
89	C₇×Q₈	#106	1	4	yes	—
90	C₇×D₄	#106	1	12	yes	—
91	C₂×C₄×C₇	—	2	3	no	—
92	C₂×C₄×C₇	—	2	3	no	—
93	C₂×C₄×C₇	—	4	3	no	—
94	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
95	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
96	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
97	C₂×C₄×C₇	C₂×SD₁₆	1	12	yes	—
98	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
101	C₇×C₈	—	2	3	no	—
102	C₇×C₈	—	2	1	no	—
103	C₇×C₈	—	2	3	no	—
104	C₇×C₈	#131	1	2	yes	complemented
105	C₇⋊C₈	—	4	2	no	—
106	C₇⋊C₈	—	4	6	no	—
107	C₂×#41	D₂₈	1	1	yes	characteristic
108	#131	—	28	8	no	—
109	C₂×M₁₆	—	14	6	no	—
110	C₂²×C₈	—	14	6	no	—
111	C₂²×C₈	C₂×Dic₇	1	3	yes	—
112	C₂²×C₈	C₂×Dic₇	1	3	yes	—
113	C₄×Q₈	Dic₁₄	1	1	yes	characteristic
114	C₄×Q₈	C₂×Dic₇	1	1	yes	characteristic
115	C₄×D₄	Dic₁₄	1	3	yes	—
116	C₄×D₄	C₂×Dic₇	1	3	yes	—
117	#117	D₂₈	1	3	yes	—
118	C₂×C₄²	C₂²×D₇	1	3	yes	—
119	C₄⋊C₈	—	7	2	no	—
120	C₄⋊C₈	—	7	6	no	—
121	C₄⋊C₈	—	14	6	no	—
122	C₄⋊C₈	—	14	6	no	—
123	C₄⋊C₈	C₂×Dic₇	1	3	yes	—
124	C₄⋊C₈	C₂×Dic₇	1	3	yes	—
125	C₂²⋊C₈	C₂×Dic₇	1	6	yes	—
126	C₄×C₈	C₂×Dic₇	1	3	yes	—
127	C₄×C₈	C₂×Dic₇	1	3	yes	—
128	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
129	C₄×C₈	C₂²×D₇	1	3	yes	—
130	C₂²×C₇	—	2	3	no	—
131	C₂²×C₇	—	2	3	no	—
132	C₂²×C₇	C₂×#106	1	6	yes	—
133	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₇	—	2	3	no	—
135	C₄×C₇	—	2	1	no	—
136	C₄×C₇	—	2	3	no	—
137	C₄×C₇	C₂×#106	1	6	yes	—
138	C₄×C₇	C₂×#131	1	1	yes	characteristic
139	C₄×C₇	#568	1	1	yes	characteristic
140	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
141	#41	#1132	1	8	yes	—
142	C₂×Q₈	C₄⋊Dic₇	1	1	yes	characteristic
143	C₂×D₄	C₄⋊Dic₇	1	3	yes	—
144	C₂²×C₄	C₂×Dic₁₄	1	3	yes	—
145	C₂²×C₄	C₂×D₂₈	1	3	yes	—
146	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
147	M₁₆	—	28	12	no	—
148	C₂×C₈	—	2	6	no	—
149	C₂×C₈	—	4	3	no	—
150	C₂×C₈	—	14	2	no	—
151	C₂×C₈	—	14	6	no	—
152	C₂×C₈	—	28	12	no	—
153	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
154	C₂×C₈	C₂²×Dic₇	1	3	yes	—
155	C₂×C₈	C₂²×Dic₇	1	3	yes	—
156	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
157	C₄⋊C₄	C₄⋊Dic₇	1	3	yes	—
158	C₄⋊C₄	C₄⋊Dic₇	1	3	yes	—
159	C₂²⋊C₄	C₄⋊Dic₇	1	6	yes	—
160	C₄²	C₂×Dic₁₄	1	3	yes	—
161	C₄²	C₂×D₂₈	1	3	yes	—
162	C₄²	C₂²×Dic₇	1	3	yes	—
163	C₄²	C₂³×D₇	1	1	yes	characteristic
164	C₂×C₇	—	2	6	no	—
165	C₂×C₇	#2906	1	1	yes	characteristic
166	C₂×C₇	C₂²×#106	1	1	yes	characteristic
167	C₂×C₇	#2916	1	1	yes	characteristic
168	C₂³	C₂×C₄⋊Dic₇	1	3	yes	—
169	Q₈	#6391	1	4	yes	—
170	D₄	#6391	1	12	yes	—
171	C₂×C₄	—	2	3	no	—
172	C₂×C₄	—	2	3	no	—
173	C₂×C₄	—	4	3	no	—
174	C₂×C₄	C₂×#1132	1	12	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₇	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₇	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₇	1	3	yes	—
178	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
179	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
180	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
181	C₈	—	2	3	no	—
182	C₈	—	2	1	no	—
183	C₈	—	2	3	no	—
184	C₈	—	28	2	no	—
185	C₈	—	28	6	no	—
186	C₈	#6511	1	2	yes	complemented
187	C₇	#17826	1	1	yes	characteristic, complemented, Sylow
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×#6391	1	6	yes	—
191	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	3	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×#6391	1	6	yes	—
196	C₄	C₂²×#1132	1	2	yes	—
197	C₄	C₂×#6511	1	1	yes	characteristic
198	C₄	#90510	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	#11739957	1	1	yes	characteristic
201	C₂	C₂²×#6391	1	1	yes	characteristic
202	C₂	#11741234	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×C₈×#41	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂×C₇×#41	D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₈×#41	D₇
hypocenter	C₇	#17826
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
3. #11728503	C₂	semidirect
4. #11727484	C₂	semidirect
5. C₂×#89252	C₂	semidirect
104. C₇×C₈	#131	semidirect
186. C₈	#6511	semidirect
187. C₇	#17826	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₈×#41	2
#11739957	2
#11728503	2
#11727484	2
C₂×#89252	2
#17826	7