Subgroups — #13247489

Order 1792.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1450	C₂	1	1	yes	maximal, characteristic
2	#11727546	C₂	1	1	yes	maximal, characteristic
3	#11727520	C₂	1	1	yes	maximal, characteristic
4	#11725544	C₂	1	4	yes	maximal
5	C₇×#431	C₂²	1	2	yes	—
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×#6393	—	2	1	no	—
8	C₂×#6393	—	2	1	no	—
9	C₂×#6392	—	2	1	no	—
10	C₂×#6391	—	2	1	no	—
11	#89708	C₂²	1	2	yes	—
12	#89696	C₂²	1	2	yes	—
13	C₇×#322	C₄	1	2	yes	complemented
14	C₇×C₈²	C₄	1	2	yes	complemented
15	#8793	—	7	1	no	maximal, Sylow
16	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇	—	2	2	no	—
20	C₂×#1136	—	2	2	no	—
21	C₇×C₄⋊C₈	—	2	2	no	—
22	C₇×C₂²⋊C₈	—	2	2	no	—
23	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
27	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
28	C₇⋊#95	—	2	4	no	—
29	#6393	—	4	2	no	—
30	#6393	—	4	2	no	—
31	#6392	—	4	2	no	—
32	#6391	—	4	2	no	—
33	C₄⋊(C₇⋊C₈)	—	2	4	no	—
34	#1848	—	7	1	no	—
35	#1846	—	7	1	no	—
36	#1450	D₇	1	1	yes	characteristic
37	#1286	—	7	4	no	—
38	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	#41	1	2	yes	—
40	C₂²×Dic₇	—	4	2	no	—
41	C₂×C₇×C₈	—	2	4	no	—
42	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₇×C₈	D₈	1	2	yes	—
47	C₂×C₇×C₈	SD₁₆	1	2	yes	—
48	C₂×C₇×C₈	SD₁₆	1	2	yes	—
49	C₂×C₇×C₈	Q₁₆	1	2	yes	—
50	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₇	#41	1	2	yes	—
53	C₄⋊Dic₇	—	4	2	no	—
54	C₄⋊Dic₇	—	4	2	no	—
55	#1136	—	4	2	no	—
56	#1136	—	4	2	no	—
57	C₂×C₇⋊C₈	—	4	2	no	—
58	#431	C₂×D₇	1	2	yes	—
59	C₂×#108	—	14	1	no	—
60	C₂×#108	—	14	1	no	—
61	C₂×#107	—	14	1	no	—
62	C₂×#106	—	14	1	no	—
63	#423	—	7	2	no	—
64	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
65	#382	—	7	2	no	—
66	#322	Dic₇	1	2	yes	complemented
67	C₈²	Dic₇	1	2	yes	complemented
68	C₂³×C₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₇	—	4	2	no	—
70	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇	#117	1	2	yes	—
72	C₂×C₄×C₇	#124	1	1	yes	characteristic
73	C₂×C₄×C₇	#125	1	2	yes	—
74	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₇	—	8	4	no	—
79	C₂×Dic₇	—	8	2	no	—
80	C₇×C₈	—	4	4	no	—
81	C₇×C₈	#106	1	2	yes	—
82	C₇×C₈	#106	1	2	yes	—
83	C₇×C₈	#107	1	2	yes	—
84	C₇×C₈	#107	1	2	yes	—
85	C₇⋊C₈	—	8	2	no	—
86	C₂×M₁₆	—	14	2	no	—
87	C₂²×C₈	D₂₈	1	1	yes	characteristic
88	C₂²×C₈	D₂₈	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	14	2	no	—
90	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
91	#108	—	28	2	no	—
92	#108	—	28	2	no	—
93	#107	—	28	2	no	—
94	#106	—	28	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	14	4	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	C₄×C₈	Dic₁₄	1	1	yes	characteristic
99	C₄×C₈	Dic₁₄	1	1	yes	characteristic
100	C₄×C₈	C₂×Dic₇	1	4	yes	—
101	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
103	#95	—	14	4	no	—
104	C₂²×C₇	—	2	2	no	—
105	C₂²×C₇	#382	1	1	yes	characteristic
106	C₂²×C₇	#486	1	1	yes	characteristic
107	C₂²×C₇	#487	1	1	yes	characteristic
108	C₄×C₇	—	2	2	no	—
109	C₄×C₇	—	2	2	no	—
110	C₄×C₇	C₂×#106	1	1	yes	characteristic
111	C₄×C₇	C₂×#107	1	1	yes	characteristic
112	C₄×C₇	#486	1	1	yes	characteristic
113	C₄×C₇	#487	1	1	yes	characteristic
114	Dic₇	—	16	2	no	—
115	C₂²×C₄	—	28	2	no	—
116	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
117	C₂²×C₄	#1157	1	2	yes	—
118	M₁₆	—	28	2	no	—
119	M₁₆	—	28	2	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	28	2	no	—
122	C₂×C₈	#1132	1	2	yes	—
123	C₂×C₈	#1132	1	2	yes	—
124	C₂×C₈	D₅₆	1	2	yes	—
125	C₂×C₈	Dic₂₈	1	2	yes	—
126	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
129	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
130	C₄⋊C₄	—	28	2	no	—
131	C₄⋊C₄	—	28	2	no	—
132	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
133	C₄²	#1157	1	2	yes	—
134	C₄²	C₂²×Dic₇	1	1	yes	characteristic
135	C₂×C₇	—	2	1	no	—
136	C₂×C₇	—	2	1	no	—
137	C₂×C₇	#1833	1	2	yes	—
138	C₂×C₇	#1834	1	1	yes	characteristic
139	C₂³	#6438	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	56	4	no	—
142	C₂×C₄	—	56	2	no	—
143	C₂×C₄	#6433	1	2	yes	—
144	C₂×C₄	#6438	1	1	yes	characteristic
145	C₂×C₄	C₂×#1132	1	1	yes	characteristic
146	C₂×C₄	C₂×#1132	1	1	yes	characteristic
147	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
150	C₂×C₄	#6489	1	2	yes	—
151	C₈	—	4	4	no	—
152	C₈	—	56	2	no	—
153	C₈	#6391	1	2	yes	—
154	C₈	#6391	1	2	yes	—
155	C₈	#6392	1	2	yes	—
156	C₈	#6392	1	2	yes	—
157	C₇	#8793	1	1	yes	characteristic, complemented, Sylow
158	C₂²	—	2	2	no	—
159	C₂²	#89453	1	1	yes	characteristic
160	C₂²	#89466	1	1	yes	characteristic
161	C₂²	#89708	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	112	2	no	—
165	C₄	#89453	1	1	yes	characteristic
166	C₄	#89466	1	1	yes	characteristic
167	C₄	C₂×#6391	1	1	yes	characteristic
168	C₄	C₂×#6392	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#11727516	1	1	yes	characteristic
172	C₂	#11727650	1	2	yes	—

	subgroup	G/H
center	C₂²	#89708
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1450	C₂
socle	C₂²×C₇	#382
prime radical	C₂³×C₇	#124
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1450	D₇
hypocenter	C₇	#8793
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
13. C₇×#322	C₄	semidirect
14. C₇×C₈²	C₄	semidirect
66. #322	Dic₇	semidirect
67. C₈²	Dic₇	semidirect
157. C₇	#8793	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1450	2
#11727546	2
#11727520	2
#11725544	2
#8793	7