Finite Group Databasebeta

Subgroups — #11729607

Order 896.

362 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#535	C₂	1	1	yes	maximal, characteristic
2	#90489	C₂	1	1	yes	maximal, characteristic, complemented
3	#90008	C₂	1	2	yes	maximal, complemented
4	C₂×#6407	C₂	1	1	yes	maximal, characteristic, complemented
5	#89326	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#41	C₂²	1	1	yes	characteristic
7	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₇×#127	C₄	1	2	yes	complemented
9	C₇×#124	C₄	1	2	yes	complemented
10	C₇×#121	—	2	2	no	—
11	C₇×#120	—	2	2	no	—
12	C₇×#117	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Dic₇	—	2	2	no	—
14	D₄×Dic₇	—	2	4	no	—
15	#6489	C₂²	1	2	yes	—
16	C₂×#1136	C₂²	1	2	yes	—
17	#6411	—	2	4	no	—
18	#6407	—	2	2	no	—
19	#1911	—	7	1	no	maximal, Sylow
20	C₂⁴×C₇	D₄	1	2	yes	—
21	C₇×#41	—	2	2	no	—
22	C₇×#41	—	2	2	no	—
23	C₂×C₇×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
24	C₂×C₇×D₄	—	2	1	no	—
25	C₂×C₇×D₄	—	2	4	no	—
26	C₂×C₇×D₄	—	2	2	no	—
27	C₂×C₇×D₄	—	2	2	no	—
28	C₂×C₇×D₄	C₂×C₄	1	2	yes	—
29	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic, complemented
30	C₂×C₇×D₄	D₄	1	2	yes	—
31	C₂²×C₄×C₇	—	2	1	no	—
32	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₇	—	2	4	no	—
34	C₇×C₄⋊C₄	—	2	1	no	—
35	C₇×C₂²⋊C₄	—	2	4	no	—
36	C₇×C₂²⋊C₄	—	2	1	no	—
37	C₄²×C₇	C₂×C₄	1	2	yes	complemented
38	C₂²⋊Dic₇	—	2	4	no	—
39	C₂²⋊Dic₇	—	2	2	no	—
40	C₄⋊Dic₇	—	2	2	no	—
41	C₄×Dic₇	D₄	1	4	yes	complemented
42	#1136	—	2	2	no	—
43	#1136	—	2	2	no	—
44	C₂×C₇⋊C₈	—	2	2	no	—
45	#535	D₇	1	1	yes	characteristic
46	#518	—	7	1	no	—
47	#421	—	7	2	no	—
48	C₂×#100	—	7	1	no	—
49	H(3,ℤ/4)	—	7	2	no	—
50	C₂³×C₇	—	2	2	no	—
51	C₂³×C₇	—	2	2	no	—
52	C₂³×C₇	—	2	1	no	—
53	C₂³×C₇	—	8	2	no	—
54	C₂³×C₇	C₂²⋊C₄	1	4	yes	—
55	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
56	C₇×Q₈	—	2	2	no	—
57	C₇×D₄	—	2	4	no	—
58	C₇×D₄	—	2	2	no	—
59	C₇×D₄	—	2	4	no	—
60	C₇×D₄	—	4	1	no	—
61	C₇×D₄	—	4	1	no	—
62	C₇×D₄	—	8	4	no	—
63	C₂×C₄×C₇	—	2	2	no	—
64	C₂×C₄×C₇	—	2	1	no	—
65	C₂×C₄×C₇	—	4	2	no	—
66	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇	#41	1	2	yes	—
70	C₂×Dic₇	—	2	4	no	—
71	C₂×Dic₇	—	8	4	no	—
72	C₇⋊C₈	—	4	2	no	—
73	C₂×#41	C₂×D₇	1	1	yes	characteristic
74	C₂²×D₄	C₂×D₇	1	1	yes	characteristic
75	C₂×M₁₆	—	7	2	no	—
76	#127	Dic₇	1	2	yes	complemented
77	#124	Dic₇	1	2	yes	complemented
78	#121	—	2	2	no	—
79	#120	—	2	2	no	—
80	C₄×D₄	—	14	4	no	—
81	#117	—	7	2	no	—
82	#117	C₂×D₇	1	1	yes	characteristic
83	C₂×C₂²⋊C₄	—	14	2	no	—
84	#104	—	14	4	no	—
85	#100	—	14	2	no	—
86	C₂²×C₇	—	2	4	no	—
87	C₂²×C₇	—	4	1	no	—
88	C₂²×C₇	—	4	4	no	—
89	C₂²×C₇	—	4	4	no	—
90	C₂²×C₇	—	4	2	no	—
91	C₂²×C₇	—	4	1	no	—
92	C₂²×C₇	—	8	2	no	—
93	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₇	#120	1	1	yes	characteristic
95	C₂²×C₇	#123	1	1	yes	characteristic
96	C₄×C₇	—	4	2	no	—
97	C₄×C₇	—	4	1	no	—
98	C₄×C₇	C₄×D₄	1	2	yes	—
99	C₄×C₇	#121	1	2	yes	—
100	Dic₇	—	4	4	no	—
101	C₂⁴	C₂²⋊D₇	1	2	yes	—
102	#41	—	2	2	no	—
103	#41	—	2	2	no	—
104	C₂×Q₈	C₂×Dic₇	1	1	yes	characteristic, complemented
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	4	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂×Dic₇	1	2	yes	—
110	C₂×D₄	C₂×Dic₇	1	1	yes	characteristic, complemented
111	C₂×D₄	C₂²⋊D₇	1	2	yes	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	14	4	no	—
114	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
115	M₁₆	—	14	2	no	—
116	M₁₆	—	14	2	no	—
117	C₂×C₈	—	14	2	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	14	2	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	14	4	no	—
123	C₂²⋊C₄	—	14	2	no	—
124	C₄²	—	7	4	no	—
125	C₄²	C₂×Dic₇	1	2	yes	complemented
126	C₂×C₇	—	2	1	no	—
127	C₂×C₇	—	2	1	no	—
128	C₂×C₇	—	2	1	no	—
129	C₂×C₇	—	4	4	no	—
130	C₂×C₇	—	8	1	no	—
131	C₂×C₇	#386	1	1	yes	characteristic
132	C₂³	—	2	2	no	—
133	C₂³	—	2	2	no	—
134	C₂³	—	2	1	no	—
135	C₂³	—	8	2	no	—
136	C₂³	C₂²⋊Dic₇	1	4	yes	—
137	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
138	Q₈	—	2	2	no	—
139	D₄	—	2	4	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	4	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	8	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	14	4	no	—
149	C₂×C₄	—	56	4	no	—
150	C₂×C₄	D₄×D₇	1	2	yes	—
151	C₂×C₄	#1159	1	2	yes	—
152	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
154	C₈	—	28	2	no	—
155	C₇	#1911	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	2	4	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	4	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	8	2	no	—
163	C₂²	#6498	1	1	yes	characteristic
164	C₂²	#6500	1	1	yes	characteristic
165	C₂²	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
166	C₄	—	4	2	no	—
167	C₄	—	4	1	no	—
168	C₄	—	28	4	no	—
169	C₄	D₄×Dic₇	1	2	yes	—
170	C₄	#6502	1	2	yes	—
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	—	4	4	no	—
175	C₂	—	8	1	no	—
176	C₂	#89992	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#89992
commutator	C₂×C₄×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting, prime radical	C₇×#535	C₂
socle	C₂×C₇	#386
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	#535	D₇
hypocenter	C₇	#1911
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

Complements

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

N	H ≅ G/N	type
2. #90489	C₂	semidirect
3. #90008	C₂	semidirect
4. C₂×#6407	C₂	semidirect
5. #89326	C₂	semidirect
8. C₇×#127	C₄	semidirect
9. C₇×#124	C₄	semidirect
23. C₂×C₇×Q₈	C₂×C₄	semidirect
29. C₂×C₇×D₄	C₂×C₄	semidirect
37. C₄²×C₇	C₂×C₄	semidirect
41. C₄×Dic₇	D₄	semidirect
76. #127	Dic₇	semidirect
77. #124	Dic₇	semidirect
104. C₂×Q₈	C₂×Dic₇	semidirect
110. C₂×D₄	C₂×Dic₇	semidirect
125. C₄²	C₂×Dic₇	semidirect
155. C₇	#1911	semidirect

Maximal subgroups

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

subgroup	index
C₇×#535	2
#90489	2
#90008	2
C₂×#6407	2
#89326	2
#1911	7