Subgroups — #13270235

Order 1792.

342 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1846	C₂	1	1	yes	maximal, characteristic
2	C₂×#89285	C₂	1	2	yes	maximal
3	C₂×#89284	C₂	1	2	yes	maximal
4	#11727520	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×#107	C₂²	1	2	yes	—
7	C₂×C₇×#106	—	2	1	no	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#382	—	2	1	no	—
10	C₂×#6392	C₂²	1	2	yes	—
11	C₂×#6391	—	2	1	no	—
12	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	#89708	—	2	1	no	—
15	#89285	—	2	4	no	—
16	#89284	—	2	4	no	—
17	C₄×C₇⋊C₁₆	C₄	1	4	yes	complemented
18	#12907	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇	—	2	2	no	—
24	C₇×#107	—	2	2	no	—
25	C₇×#107	—	2	2	no	—
26	C₇×#106	—	4	2	no	—
27	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₇×C₈	Q₈	1	2	yes	—
29	C₇×#95	—	4	2	no	—
30	C₇⋊#95	—	4	2	no	—
31	#6392	—	2	2	no	—
32	#6392	—	2	2	no	—
33	#6391	—	4	2	no	—
34	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
37	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
38	C₂×C₇⋊C₁₆	Q₈	1	2	yes	—
39	C₂×C₇⋊C₁₆	Q₈	1	2	yes	—
40	C₂×#367	—	7	2	no	—
41	C₂×#366	—	7	2	no	—
42	C₂×C₄×C₁₆	—	7	1	no	—
43	#1846	—	7	1	no	—
44	#1846	D₇	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	4	2	no	—
46	C₂²×C₄×C₇	D₈	1	2	yes	—
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₇	—	4	2	no	—
49	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₇×C₈	D₈	1	2	yes	—
51	C₂×C₇×C₈	Q₁₆	1	2	yes	—
52	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₇×C₄⋊C₄	—	4	2	no	—
59	C₇×C₄⋊C₄	—	4	2	no	—
60	C₄²×C₇	C₄⋊C₄	1	2	yes	—
61	C₄²×C₇	Q₁₆	1	2	yes	—
62	C₄⋊Dic₇	—	4	2	no	—
63	C₄⋊Dic₇	—	4	2	no	—
64	C₇⋊C₁₆	C₄⋊C₄	1	4	yes	—
65	C₇⋊C₁₆	C₄⋊C₄	1	4	yes	—
66	C₂²×C₁₆	—	7	1	no	—
67	C₂²×C₁₆	—	7	1	no	—
68	C₂×#107	—	7	2	no	—
69	C₂×#107	C₂×D₇	1	2	yes	—
70	C₂×#106	—	2	1	no	—
71	C₂×#106	—	14	1	no	—
72	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
73	#382	—	2	1	no	—
74	#382	—	14	1	no	—
75	#367	—	14	4	no	—
76	#366	—	14	4	no	—
77	C₄×C₁₆	—	7	4	no	—
78	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₇	—	2	4	no	—
80	C₂×C₄×C₇	—	8	2	no	—
81	C₂×C₄×C₇	—	8	4	no	—
82	C₂×C₄×C₇	#107	1	4	yes	—
83	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇	#127	1	1	yes	characteristic
85	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×Dic₇	—	8	2	no	—
90	C₂×Dic₇	—	8	4	no	—
91	C₇×C₈	#107	1	4	yes	—
92	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
95	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₄⋊C₄	—	14	2	no	—
100	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
101	C₂×C₁₆	—	7	2	no	—
102	C₂×C₁₆	—	7	2	no	—
103	C₂×C₁₆	—	7	2	no	—
104	C₂×C₁₆	—	7	2	no	—
105	C₂×C₁₆	—	7	2	no	—
106	C₂×C₁₆	—	7	2	no	—
107	#107	—	2	2	no	—
108	#107	—	2	2	no	—
109	#107	—	14	2	no	—
110	#107	—	14	2	no	—
111	#106	—	4	2	no	—
112	#106	—	28	2	no	—
113	C₄×C₈	Dic₁₄	1	2	yes	—
114	C₄×C₈	C₄×D₇	1	2	yes	—
115	#95	—	4	2	no	—
116	#95	—	28	2	no	—
117	C₂²×C₇	C₂×#107	1	1	yes	characteristic
118	C₂²×C₇	#493	1	1	yes	characteristic
119	C₂²×C₇	#500	1	1	yes	characteristic
120	C₂²×C₇	#508	1	4	yes	—
121	C₄×C₇	—	2	4	no	—
122	C₄×C₇	—	16	2	no	—
123	C₄×C₇	#384	1	1	yes	characteristic
124	C₄×C₇	C₂×#107	1	1	yes	characteristic
125	C₄×C₇	#494	1	1	yes	characteristic
126	C₄×C₇	#500	1	1	yes	characteristic
127	Dic₇	—	16	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	28	2	no	—
130	C₂²×C₄	#1141	1	2	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
132	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
133	C₂×C₈	#1141	1	2	yes	—
134	C₂×C₈	#1144	1	2	yes	—
135	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
137	C₂×C₈	D₄×D₇	1	1	yes	characteristic
138	C₂×C₈	D₄×D₇	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	28	2	no	—
144	C₄⋊C₄	—	28	2	no	—
145	C₄²	Dic₇⋊C₄	1	2	yes	—
146	C₄²	#1144	1	2	yes	—
147	C₁₆	—	7	4	no	—
148	C₁₆	—	7	4	no	—
149	C₂×C₇	#1845	1	1	yes	characteristic
150	C₂×C₇	#2158	1	2	yes	—
151	C₂×C₇	#2249	1	2	yes	—
152	C₂×C₇	#2253	1	2	yes	—
153	C₂³	C₂×#1141	1	1	yes	characteristic
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	56	4	no	—
158	C₂×C₄	—	56	2	no	—
159	C₂×C₄	#6381	1	4	yes	—
160	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
161	C₂×C₄	C₂×#1141	1	1	yes	characteristic
162	C₂×C₄	#6503	1	1	yes	characteristic
163	C₂×C₄	C₂×#1144	1	1	yes	characteristic
164	C₂×C₄	C₂×#1144	1	1	yes	characteristic
165	C₂×C₄	#6507	1	1	yes	characteristic
166	C₈	#6381	1	4	yes	—
167	C₈	#6451	1	1	yes	characteristic
168	C₈	#6451	1	1	yes	characteristic
169	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
170	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
171	C₇	#12907	1	1	yes	characteristic, complemented, Sylow
172	C₂²	C₂×#6381	1	1	yes	characteristic
173	C₂²	#89845	1	1	yes	characteristic
174	C₂²	#89850	1	1	yes	characteristic
175	C₂²	#89966	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	16	2	no	—
178	C₄	—	112	2	no	—
179	C₄	C₂×#6381	1	1	yes	characteristic
180	C₄	#89748	1	1	yes	characteristic
181	C₄	#89850	1	1	yes	characteristic
182	C₄	#89864	1	1	yes	characteristic
183	C₂	#11727331	1	1	yes	characteristic
184	C₂	#11728402	1	2	yes	—
185	C₂	#11729074	1	2	yes	—
186	C₂	#11729108	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1141
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1846	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1846	D₇
hypocenter	C₇	#12907
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
17. C₄×C₇⋊C₁₆	C₄	semidirect
171. C₇	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1846	2
C₂×#89285	2
C₂×#89284	2
#11727520	2
C₂×C₄×C₇⋊C₁₆	2
#12907	7