Finite Group Databasebeta

Subgroups — #11728474

Order 896.

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

normal subgroups only

This group has 264 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	C₂×#6464	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	#89905	C₂	1	2	yes	maximal
4	C₂×#6391	C₂	1	1	yes	maximal, characteristic, complemented
5	#89349	C₂	1	2	yes	maximal, complemented
6	C₂×#1157	C₂²	1	1	yes	characteristic
7	C₂×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
10	C₂×#1136	C₂²	1	1	yes	characteristic
11	#6464	—	2	2	no	—
12	#6464	C₄	1	4	yes	complemented
13	C₂×#1131	—	2	1	no	—
14	C₂×C₈×D₇	—	2	1	no	—
15	C₇×#108	—	2	2	no	—
16	#6395	—	2	2	no	—
17	#6391	—	2	1	no	—
18	#6391	—	2	1	no	—
19	#6390	—	2	2	no	—
20	#1815	—	7	1	no	maximal, Sylow
21	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₇	—	2	1	no	—
23	C₂²×Dic₇	—	2	2	no	—
24	#1157	—	2	1	no	—
25	#1157	—	2	1	no	—
26	#1157	C₂×C₄	1	2	yes	—
27	#1157	Q₈	1	2	yes	—
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	2	no	—
32	C₇×M₁₆	—	2	2	no	—
33	C₂×C₇×C₈	—	2	2	no	—
34	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₇×C₈	D₄	1	1	yes	characteristic
36	C₂×C₇×C₈	D₄	1	1	yes	characteristic
37	C₂×C₇×C₈	D₄	1	2	yes	—
38	C₄⋊Dic₇	—	2	2	no	—
39	C₄⋊Dic₇	—	2	2	no	—
40	#1136	—	2	1	no	—
41	#1136	C₂×C₄	1	2	yes	—
42	C₂×C₇⋊C₈	—	2	1	no	—
43	#1131	—	2	4	no	—
44	C₈×D₇	—	2	4	no	—
45	C₂×#131	—	7	1	no	—
46	C₂×#108	D₇	1	1	yes	characteristic, complemented
47	C₂×#106	—	7	1	no	—
48	#417	—	7	2	no	—
49	#343	—	7	2	no	—
50	C₂³×C₇	#41	1	1	yes	characteristic
51	C₂²×D₇	—	2	1	no	—
52	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇	#41	1	1	yes	characteristic
58	C₂²⋊D₇	—	2	2	no	—
59	C₂²⋊D₇	—	2	2	no	—
60	C₂×Dic₇	—	2	1	no	—
61	C₂×Dic₇	—	4	2	no	—
62	C₂×Dic₇	—	8	2	no	—
63	D₂₈	—	2	1	no	—
64	D₂₈	C₄⋊C₄	1	2	yes	—
65	C₄×D₇	—	2	2	no	—
66	C₄×D₇	—	2	2	no	—
67	Dic₁₄	—	2	1	no	—
68	Dic₁₄	C₄⋊C₄	1	2	yes	—
69	C₇×C₈	—	4	2	no	—
70	C₇×C₈	C₂²⋊C₄	1	2	yes	—
71	C₇×C₈	C₂²⋊C₄	1	2	yes	—
72	C₇⋊C₈	—	2	2	no	—
73	C₂×#41	—	7	1	no	—
74	#131	—	7	4	no	—
75	#131	—	14	2	no	—
76	C₂×M₁₆	—	7	1	no	—
77	C₂×M₁₆	—	14	1	no	—
78	C₂×M₁₆	C₂×D₇	1	2	yes	—
79	C₂²×C₈	—	14	1	no	—
80	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	7	2	no	—
82	#108	—	2	2	no	—
83	#106	—	14	1	no	—
84	#106	—	14	1	no	—
85	#103	—	14	2	no	—
86	#102	—	14	2	no	—
87	C₂²×C₇	—	2	2	no	—
88	C₂²×C₇	#117	1	1	yes	characteristic
89	C₂²×C₇	#121	1	1	yes	characteristic
90	C₂²×C₇	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×D₇	—	2	2	no	—
92	C₂×D₇	—	4	2	no	—
93	C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₇	#121	1	1	yes	characteristic
96	C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
97	Dic₇	—	2	2	no	—
98	Dic₇	—	8	2	no	—
99	#41	—	7	2	no	—
100	#41	—	7	2	no	—
101	#41	—	14	1	no	—
102	#41	—	14	1	no	—
103	C₂×Q₈	—	7	1	no	—
104	C₂×D₄	—	7	1	no	—
105	C₂×D₄	—	14	1	no	—
106	C₂²×C₄	—	14	1	no	—
107	C₂²×C₄	—	14	2	no	—
108	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	7	2	no	—
112	M₁₆	—	14	1	no	—
113	M₁₆	—	14	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	14	1	no	—
116	C₂×C₈	—	14	4	no	—
117	C₂×C₈	C₄×D₇	1	2	yes	—
118	C₂×C₈	D₂₈	1	1	yes	characteristic
119	C₂×C₈	D₂₈	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊D₇	1	2	yes	—
121	C₄⋊C₄	—	14	2	no	—
122	C₄⋊C₄	—	14	2	no	—
123	C₂×C₇	—	2	1	no	—
124	C₂×C₇	—	2	1	no	—
125	C₂×C₇	#380	1	1	yes	characteristic
126	C₂×C₇	#471	1	2	yes	—
127	D₇	—	4	2	no	—
128	C₂³	—	14	1	no	—
129	C₂³	#1161	1	1	yes	characteristic
130	Q₈	—	7	2	no	—
131	Q₈	—	14	1	no	—
132	D₄	—	7	2	no	—
133	D₄	—	14	2	no	—
134	D₄	—	14	2	no	—
135	D₄	—	14	1	no	—
136	C₂×C₄	—	14	1	no	—
137	C₂×C₄	—	14	2	no	—
138	C₂×C₄	—	14	2	no	—
139	C₂×C₄	—	28	2	no	—
140	C₂×C₄	—	56	2	no	—
141	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
142	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
143	C₂×C₄	D₄×D₇	1	1	yes	characteristic
144	C₂×C₄	#1159	1	1	yes	characteristic
145	C₂×C₄	Q₈×D₇	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
147	C₈	—	4	2	no	—
148	C₈	—	14	2	no	—
149	C₈	(C₂×D₇)⋊C₄	1	2	yes	—
150	C₈	(C₂×D₇)⋊C₄	1	2	yes	—
151	C₇	#1815	1	1	yes	characteristic, complemented, Sylow
152	C₂²	—	2	2	no	—
153	C₂²	—	14	2	no	—
154	C₂²	—	28	2	no	—
155	C₂²	#6455	1	1	yes	characteristic
156	C₂²	#6458	1	1	yes	characteristic
157	C₂²	#6509	1	1	yes	characteristic
158	C₄	—	14	2	no	—
159	C₄	—	56	2	no	—
160	C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
161	C₄	#6460	1	1	yes	characteristic
162	C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
163	C₄	#6501	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	—	28	2	no	—
167	C₂	#89670	1	2	yes	—
168	C₂	#89759	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#6458
commutator	C₂×C₄×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂×C₇×#108	C₂
socle	C₂²×C₇	#121
prime radical	C₂×#1157	C₂²
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂×#108	D₇
hypocenter	C₇	#1815
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#108	C₂	semidirect
4. C₂×#6391	C₂	semidirect
5. #89349	C₂	semidirect
12. #6464	C₄	semidirect
46. C₂×#108	D₇	semidirect
151. C₇	#1815	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6464	2
C₂×C₇×#108	2
#89905	2
C₂×#6391	2
#89349	2
#1815	7