Finite Group Databasebeta

Subgroups — #11727886

Order 896.

190 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 59 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×#413	C₂	1	1	yes	maximal, characteristic
2	#90008	C₂	1	1	yes	maximal, characteristic
3	#90005	C₂	1	1	yes	maximal, characteristic
4	#89891	C₂	1	1	yes	maximal, characteristic, complemented
5	#89354	C₂	1	1	yes	maximal, characteristic, complemented
6	#89353	C₂	1	1	yes	maximal, characteristic, complemented
7	#89352	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#41	C₂²	1	1	yes	characteristic
9	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
11	#6489	C₂²	1	1	yes	characteristic
12	#6489	C₂²	1	1	yes	characteristic
13	C₂×#1136	C₂²	1	1	yes	characteristic
14	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₇×#101	—	2	1	no	—
16	C₇×#100	—	2	1	no	—
17	#6411	—	2	2	no	—
18	#6409	—	2	1	no	—
19	#6406	—	2	1	no	—
20	Dic₇⋊C₈	—	2	2	no	—
21	#2080	—	7	1	no	maximal, Sylow
22	C₇×#41	—	2	1	no	—
23	C₇×#41	—	2	1	no	—
24	C₇×#41	—	2	1	no	—
25	C₇×#41	—	2	1	no	—
26	C₂×C₇×Q₈	D₄	1	1	yes	characteristic
27	C₂×C₇×D₄	—	2	1	no	—
28	C₂×C₇×D₄	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇	—	2	1	no	—
30	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₇×M₁₆	—	2	1	no	—
32	C₇×M₁₆	—	2	1	no	—
33	C₇×M₁₆	—	2	1	no	—
34	C₇×M₁₆	—	2	1	no	—
35	C₂×C₇×C₈	—	2	1	no	—
36	C₂×C₇×C₈	—	2	1	no	—
37	C₂²⋊Dic₇	—	2	1	no	—
38	C₂²⋊Dic₇	—	2	1	no	—
39	C₄⋊Dic₇	—	2	1	no	—
40	C₄⋊Dic₇	—	2	1	no	—
41	C₄×Dic₇	—	2	1	no	—
42	C₄×Dic₇	D₄	1	2	yes	—
43	#1136	—	2	1	no	—
44	#1136	—	2	1	no	—
45	C₂×C₇⋊C₈	—	2	1	no	—
46	C₂×C₇⋊C₈	—	2	1	no	—
47	C₂×C₇⋊C₈	—	2	1	no	—
48	C₂×C₇⋊C₈	—	2	1	no	—
49	#424	—	7	1	no	—
50	#421	—	7	1	no	—
51	#416	—	7	1	no	—
52	#413	D₇	1	1	yes	characteristic
53	#344	—	7	1	no	—
54	#341	—	7	1	no	—
55	H(3,ℤ/4)	—	7	1	no	—
56	C₂³×C₇	—	2	1	no	—
57	C₂³×C₇	#41	1	1	yes	characteristic
58	C₇×Q₈	—	2	1	no	—
59	C₇×Q₈	—	2	1	no	—
60	C₇×D₄	—	2	1	no	—
61	C₇×D₄	—	2	1	no	—
62	C₇×D₄	—	4	1	no	—
63	C₇×D₄	—	4	1	no	—
64	C₂×C₄×C₇	—	2	1	no	—
65	C₂×C₄×C₇	—	4	1	no	—
66	C₂×C₄×C₇	—	4	1	no	—
67	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇	#41	1	1	yes	characteristic
70	C₂×C₄×C₇	#41	1	1	yes	characteristic
71	C₂×C₄×C₇	#41	1	1	yes	characteristic
72	C₂×C₄×C₇	#41	1	1	yes	characteristic
73	C₂×Dic₇	—	2	2	no	—
74	C₂×Dic₇	—	4	1	no	—
75	C₇×C₈	—	4	1	no	—
76	C₇×C₈	—	4	1	no	—
77	C₇⋊C₈	—	4	1	no	—
78	C₇⋊C₈	—	4	1	no	—
79	C₂×#41	C₂×D₇	1	1	yes	characteristic
80	C₂×M₁₆	—	7	1	no	—
81	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
82	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
83	C₂²×C₈	—	7	1	no	—
84	#117	—	7	1	no	—
85	#117	—	7	1	no	—
86	C₄⋊C₈	—	14	2	no	—
87	#104	—	14	2	no	—
88	#103	—	14	1	no	—
89	#102	—	14	1	no	—
90	#101	—	2	1	no	—
91	#100	—	2	1	no	—
92	C₂²×C₇	—	4	1	no	—
93	C₂²×C₇	—	4	1	no	—
94	C₂²×C₇	—	8	1	no	—
95	C₂²×C₇	#123	1	1	yes	characteristic
96	C₂²×C₇	#123	1	1	yes	characteristic
97	C₂²×C₇	#126	1	1	yes	characteristic
98	C₄×C₇	—	4	1	no	—
99	C₄×C₇	#121	1	1	yes	characteristic
100	C₄×C₇	#123	1	1	yes	characteristic
101	C₄×C₇	#124	1	1	yes	characteristic
102	C₄×C₇	#126	1	1	yes	characteristic
103	Dic₇	—	4	2	no	—
104	Dic₇	—	8	1	no	—
105	#41	—	2	1	no	—
106	#41	—	2	1	no	—
107	#41	—	2	1	no	—
108	#41	—	2	1	no	—
109	C₂×Q₈	C₂²⋊D₇	1	1	yes	characteristic
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	C₂²⋊D₇	1	1	yes	characteristic
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	2	1	no	—
118	M₁₆	—	14	1	no	—
119	M₁₆	—	14	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	14	1	no	—
123	C₂×C₈	—	14	1	no	—
124	C₂×C₈	—	14	1	no	—
125	C₂×C₈	—	14	1	no	—
126	C₄⋊C₄	—	14	1	no	—
127	C₄⋊C₄	—	14	1	no	—
128	C₂²⋊C₄	—	14	1	no	—
129	C₂²⋊C₄	—	14	1	no	—
130	C₄²	—	7	2	no	—
131	C₄²	—	14	1	no	—
132	C₂×C₇	—	2	1	no	—
133	C₂×C₇	—	2	1	no	—
134	C₂×C₇	—	2	1	no	—
135	C₂×C₇	—	8	1	no	—
136	C₂×C₇	#397	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	C₂³	#1157	1	1	yes	characteristic
139	Q₈	—	2	1	no	—
140	Q₈	—	2	1	no	—
141	D₄	—	2	1	no	—
142	D₄	—	2	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	4	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	14	2	no	—
149	C₂×C₄	—	28	1	no	—
150	C₂×C₄	#1157	1	1	yes	characteristic
151	C₂×C₄	D₄×D₇	1	1	yes	characteristic
152	C₂×C₄	#1159	1	1	yes	characteristic
153	C₂×C₄	#1159	1	1	yes	characteristic
154	C₂×C₄	#1159	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	28	1	no	—
159	C₈	—	28	1	no	—
160	C₇	#2080	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	4	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	#6442	1	1	yes	characteristic
165	C₂²	#6446	1	1	yes	characteristic
166	C₂²	#6492	1	1	yes	characteristic
167	C₄	—	4	1	no	—
168	C₄	—	28	2	no	—
169	C₄	—	56	1	no	—
170	C₄	#6442	1	1	yes	characteristic
171	C₄	#6448	1	1	yes	characteristic
172	C₄	#6498	1	1	yes	characteristic
173	C₄	#6502	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	#89724	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#6448
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#413	C₂
socle	C₂×C₇	#397
prime radical	C₂×C₇×#41	C₂²
hypercenter	#413	D₇
hypocenter	C₇	#2080
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

Complements

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

N	H ≅ G/N	type
4. #89891	C₂	semidirect
5. #89354	C₂	semidirect
6. #89353	C₂	semidirect
7. #89352	C₂	semidirect
160. C₇	#2080	semidirect

Maximal subgroups

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

subgroup	index
C₇×#413	2
#90008	2
#90005	2
#89891	2
#89354	2
#89353	2
#89352	2
#2080	7