Finite Group Databasebeta

Subgroups — #11726689

Order 896.

190 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 57 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	#89911	C₂	1	1	yes	maximal, characteristic
2	#89904	C₂	1	1	yes	maximal, characteristic
3	#89885	C₂	1	1	yes	maximal, characteristic
4	#89874	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	#89352	C₂	1	1	yes	maximal, characteristic, complemented
7	#89351	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#1157	C₂²	1	1	yes	characteristic
9	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
12	#6489	C₂²	1	1	yes	characteristic
13	#6489	C₂²	1	1	yes	characteristic
14	C₂×#1136	C₂²	1	1	yes	characteristic
15	#6399	—	2	2	no	—
16	#6398	—	2	1	no	—
17	#6397	—	2	1	no	—
18	#6395	—	2	1	no	—
19	#6390	—	2	1	no	—
20	Dic₇⋊C₈	—	2	2	no	—
21	#2080	—	7	1	no	maximal, Sylow
22	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₇	—	2	1	no	—
24	#1157	—	2	1	no	—
25	#1157	—	2	1	no	—
26	#1157	—	2	1	no	—
27	#1157	—	2	1	no	—
28	C₂×D₂₈	D₄	1	1	yes	characteristic
29	C₂×C₄×D₇	—	2	1	no	—
30	C₂×Dic₁₄	D₄	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	C₂×C₇×C₈	—	2	1	no	—
38	C₂×C₇×C₈	—	2	1	no	—
39	C₂×C₇×C₈	—	2	1	no	—
40	C₂²⋊Dic₇	—	2	1	no	—
41	C₂²⋊Dic₇	—	2	1	no	—
42	C₄⋊Dic₇	—	2	1	no	—
43	C₄⋊Dic₇	—	2	1	no	—
44	C₄×Dic₇	—	2	1	no	—
45	C₄×Dic₇	D₄	1	2	yes	—
46	#1136	—	2	1	no	—
47	#1136	—	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	—	7	1	no	—
53	#344	—	7	1	no	—
54	#341	D₇	1	1	yes	characteristic, complemented
55	H(3,ℤ/4)	—	7	1	no	—
56	C₂³×C₇	#41	1	1	yes	characteristic
57	C₂²×D₇	—	2	1	no	—
58	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇	#41	1	1	yes	characteristic
61	C₂×C₄×C₇	#41	1	1	yes	characteristic
62	C₂×C₄×C₇	#41	1	1	yes	characteristic
63	C₂×C₄×C₇	#41	1	1	yes	characteristic
64	C₂²⋊D₇	—	4	1	no	—
65	C₂²⋊D₇	—	4	1	no	—
66	C₂×Dic₇	—	2	2	no	—
67	C₂×Dic₇	—	2	1	no	—
68	C₂×Dic₇	—	4	1	no	—
69	D₂₈	—	2	1	no	—
70	D₂₈	—	2	1	no	—
71	C₄×D₇	—	4	1	no	—
72	C₄×D₇	—	4	1	no	—
73	Dic₁₄	—	2	1	no	—
74	Dic₁₄	—	2	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	—	7	1	no	—
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	C₂×D₇	1	1	yes	characteristic
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	—	14	1	no	—
91	#100	—	14	1	no	—
92	C₂²×C₇	—	4	1	no	—
93	C₂²×C₇	#123	1	1	yes	characteristic
94	C₂²×C₇	#123	1	1	yes	characteristic
95	C₂²×C₇	#126	1	1	yes	characteristic
96	C₂×D₇	—	4	1	no	—
97	C₂×D₇	—	8	1	no	—
98	C₄×C₇	#121	1	1	yes	characteristic
99	C₄×C₇	#123	1	1	yes	characteristic
100	C₄×C₇	#124	1	1	yes	characteristic
101	C₄×C₇	#126	1	1	yes	characteristic
102	Dic₇	—	4	2	no	—
103	Dic₇	—	4	1	no	—
104	Dic₇	—	8	1	no	—
105	#41	—	14	1	no	—
106	#41	—	14	1	no	—
107	#41	—	14	1	no	—
108	#41	—	14	1	no	—
109	C₂×Q₈	—	7	1	no	—
110	C₂×D₄	—	7	1	no	—
111	C₂×D₄	—	14	1	no	—
112	C₂²×C₄	—	14	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	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	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	#397	1	1	yes	characteristic
136	D₇	—	8	1	no	—
137	C₂³	—	14	1	no	—
138	C₂³	#1161	1	1	yes	characteristic
139	Q₈	—	14	1	no	—
140	Q₈	—	14	1	no	—
141	D₄	—	14	1	no	—
142	D₄	—	14	1	no	—
143	D₄	—	28	1	no	—
144	D₄	—	28	1	no	—
145	C₂×C₄	—	14	2	no	—
146	C₂×C₄	—	14	1	no	—
147	C₂×C₄	—	28	1	no	—
148	C₂×C₄	—	28	1	no	—
149	C₂×C₄	—	28	1	no	—
150	C₂×C₄	#1157	1	1	yes	characteristic
151	C₂×C₄	#1157	1	1	yes	characteristic
152	C₂×C₄	#1157	1	1	yes	characteristic
153	C₂×C₄	D₄×D₇	1	1	yes	characteristic
154	C₂×C₄	D₄×D₇	1	1	yes	characteristic
155	C₂×C₄	#1159	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	4	1	no	—
159	C₈	—	28	1	no	—
160	C₇	#2080	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	4	1	no	—
162	C₂²	—	28	1	no	—
163	C₂²	—	56	1	no	—
164	C₂²	#6457	1	1	yes	characteristic
165	C₂²	#6457	1	1	yes	characteristic
166	C₂²	#6461	1	1	yes	characteristic
167	C₄	—	28	2	no	—
168	C₄	—	28	1	no	—
169	C₄	—	56	1	no	—
170	C₄	#6439	1	1	yes	characteristic
171	C₄	#6446	1	1	yes	characteristic
172	C₄	#6447	1	1	yes	characteristic
173	C₄	#6448	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	56	1	no	—
178	C₂	#89447	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#6448
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#341	C₂
socle	C₂×C₇	#397
prime radical	C₂×#1157	C₂²
hypercenter	#341	D₇
hypocenter	C₇	#2080
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #341

Complements

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

N	H ≅ G/N	type
4. #89874	C₂	semidirect
5. C₇×#341	C₂	semidirect
6. #89352	C₂	semidirect
7. #89351	C₂	semidirect
54. #341	D₇	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
#89911	2
#89904	2
#89885	2
#89874	2
C₇×#341	2
#89352	2
#89351	2
#2080	7