Subgroups — #13270232

Order 1792.

374 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1845	C₂	1	1	yes	maximal, characteristic
2	C₂×#89285	C₂	1	4	yes	maximal
3	#11727519	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×#107	—	2	1	no	—
6	C₂×C₇×#107	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×#384	—	2	1	no	—
9	C₂×#6392	—	2	1	no	—
10	C₂×#6392	C₂²	1	2	yes	—
11	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
12	#89707	—	2	1	no	—
13	#89285	—	2	8	no	—
14	C₄×C₇⋊C₁₆	C₄	1	4	yes	complemented
15	#12906	—	7	1	no	maximal, Sylow
16	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	—	2	2	no	—
19	C₂×C₇×C₄⋊C₄	—	4	2	no	—
20	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇	—	2	2	no	—
22	C₂×C₄⋊Dic₇	—	4	2	no	—
23	C₇×#107	—	2	4	no	—
24	C₇×#107	—	4	2	no	—
25	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇×C₈	Q₈	1	2	yes	—
27	#6392	—	2	4	no	—
28	#6392	—	4	2	no	—
29	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
31	C₂×C₇⋊C₁₆	Q₈	1	4	yes	—
32	C₂×#367	—	7	4	no	—
33	C₂×C₄×C₁₆	—	7	1	no	—
34	#1845	—	7	1	no	—
35	#1845	D₇	1	1	yes	characteristic
36	C₂²×C₄×C₇	—	4	2	no	—
37	C₂²×C₄×C₇	D₈	1	2	yes	—
38	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₇	—	4	2	no	—
40	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
41	C₂×C₇×C₈	D₈	1	2	yes	—
42	C₂×C₇×C₈	Q₁₆	1	2	yes	—
43	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
45	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
46	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₇×C₄⋊C₄	—	4	2	no	—
48	C₇×C₄⋊C₄	—	4	2	no	—
49	C₇×C₄⋊C₄	—	8	4	no	—
50	C₄²×C₇	C₄⋊C₄	1	2	yes	—
51	C₄²×C₇	Q₁₆	1	2	yes	—
52	C₄⋊Dic₇	—	4	2	no	—
53	C₄⋊Dic₇	—	4	2	no	—
54	C₄⋊Dic₇	—	8	4	no	—
55	C₇⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
56	C₂²×C₁₆	—	7	2	no	—
57	C₂×#107	—	2	1	no	—
58	C₂×#107	—	7	2	no	—
59	C₂×#107	—	14	1	no	—
60	C₂×#107	C₂×D₇	1	2	yes	—
61	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
62	#384	—	2	1	no	—
63	#384	—	14	1	no	—
64	#367	—	14	8	no	—
65	C₄×C₁₆	—	7	4	no	—
66	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇	—	8	2	no	—
68	C₂×C₄×C₇	—	8	4	no	—
69	C₂×C₄×C₇	#107	1	4	yes	—
70	C₂×C₄×C₇	SD₃₂	1	4	yes	—
71	C₂×C₄×C₇	SD₃₂	1	4	yes	—
72	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇	#127	1	1	yes	characteristic
74	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₇	—	8	2	no	—
79	C₂×Dic₇	—	8	4	no	—
80	C₇×C₈	#107	1	4	yes	—
81	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
82	C₇×C₈	C₄⋊Q₈	1	2	yes	—
83	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
84	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	4	2	no	—
87	C₂×C₄⋊C₄	—	14	2	no	—
88	C₂×C₄⋊C₄	—	28	2	no	—
89	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
90	C₂×C₁₆	—	7	4	no	—
91	C₂×C₁₆	—	7	4	no	—
92	C₂×C₁₆	—	7	4	no	—
93	#107	—	2	4	no	—
94	#107	—	4	2	no	—
95	#107	—	14	4	no	—
96	#107	—	28	2	no	—
97	C₄×C₈	Dic₁₄	1	2	yes	—
98	C₄×C₈	C₄×D₇	1	2	yes	—
99	C₂²×C₇	C₂×#107	1	1	yes	characteristic
100	C₂²×C₇	#493	1	1	yes	characteristic
101	C₂²×C₇	#500	1	1	yes	characteristic
102	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
103	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
104	C₄×C₇	—	16	2	no	—
105	C₄×C₇	#367	1	8	yes	complemented
106	C₄×C₇	#384	1	1	yes	characteristic
107	C₄×C₇	C₂×#107	1	1	yes	characteristic
108	C₄×C₇	#494	1	1	yes	characteristic
109	C₄×C₇	#500	1	1	yes	characteristic
110	Dic₇	—	16	2	no	—
111	C₂²×C₄	—	4	2	no	—
112	C₂²×C₄	—	28	2	no	—
113	C₂²×C₄	#1141	1	2	yes	—
114	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
115	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
116	C₂×C₈	#1141	1	2	yes	—
117	C₂×C₈	#1144	1	2	yes	—
118	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
119	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
120	C₂×C₈	D₄×D₇	1	2	yes	—
121	C₂×C₈	Q₈×D₇	1	2	yes	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	8	4	no	—
125	C₄⋊C₄	—	28	2	no	—
126	C₄⋊C₄	—	28	2	no	—
127	C₄⋊C₄	—	56	4	no	—
128	C₄²	Dic₇⋊C₄	1	2	yes	—
129	C₄²	#1144	1	2	yes	—
130	C₁₆	—	7	8	no	—
131	C₂×C₇	#1845	1	1	yes	characteristic
132	C₂×C₇	C₂×#367	1	2	yes	—
133	C₂×C₇	#2248	1	1	yes	characteristic
134	C₂×C₇	#2248	1	1	yes	characteristic
135	C₂×C₇	#2254	1	2	yes	—
136	C₂³	C₂×#1141	1	1	yes	characteristic
137	C₂×C₄	—	8	4	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	56	4	no	—
140	C₂×C₄	—	56	2	no	—
141	C₂×C₄	#6381	1	4	yes	—
142	C₂×C₄	#6401	1	4	yes	—
143	C₂×C₄	#6402	1	4	yes	—
144	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₂×#1141	1	1	yes	characteristic
146	C₂×C₄	#6503	1	1	yes	characteristic
147	C₂×C₄	C₂×#1144	1	1	yes	characteristic
148	C₂×C₄	C₂×#1144	1	1	yes	characteristic
149	C₂×C₄	#6507	1	1	yes	characteristic
150	C₈	#6381	1	4	yes	—
151	C₈	#6451	1	2	yes	—
152	C₈	D₇×C₄⋊C₄	1	2	yes	—
153	C₇	#12906	1	1	yes	characteristic, complemented, Sylow
154	C₂²	C₂×#6381	1	1	yes	characteristic
155	C₂²	#89845	1	1	yes	characteristic
156	C₂²	#89850	1	1	yes	characteristic
157	C₂²	C₂×#6401	1	2	yes	—
158	C₂²	C₂×#6402	1	2	yes	—
159	C₄	—	16	2	no	—
160	C₄	—	112	2	no	—
161	C₄	#89285	1	8	yes	complemented
162	C₄	C₂×#6381	1	1	yes	characteristic
163	C₄	#89748	1	1	yes	characteristic
164	C₄	#89850	1	1	yes	characteristic
165	C₄	#89864	1	1	yes	characteristic
166	C₂	#11727331	1	1	yes	characteristic
167	C₂	C₂×#89285	1	2	yes	—
168	C₂	#11729045	1	1	yes	characteristic
169	C₂	#11729067	1	1	yes	characteristic
170	C₂	#11729139	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1141
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1845	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1845	D₇
hypocenter	C₇	#12906
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
14. C₄×C₇⋊C₁₆	C₄	semidirect
55. C₇⋊C₁₆	C₄⋊C₄	semidirect
105. C₄×C₇	#367	semidirect
153. C₇	#12906	semidirect
161. C₄	#89285	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1845	2
C₂×#89285	2
#11727519	2
C₂×C₄×C₇⋊C₁₆	2
#12906	7