Finite Group Databasebeta

Subgroups — #11728473

Order 896.

256 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 89 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	#89911	C₂	1	2	yes	maximal
4	#89878	C₂	1	1	yes	maximal, characteristic, complemented
5	#89351	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	#6489	C₂²	1	2	yes	—
10	C₂×#1136	C₂²	1	1	yes	characteristic
11	#6464	—	2	1	no	—
12	#6464	—	2	1	no	—
13	#6464	C₄	1	2	yes	complemented
14	#6464	C₄	1	2	yes	complemented
15	C₂×#1131	—	2	1	no	—
16	C₂×C₈×D₇	—	2	1	no	—
17	C₇×#108	—	2	1	no	—
18	C₇×#108	—	2	1	no	—
19	#6399	—	2	4	no	—
20	#6392	—	2	1	no	—
21	#6391	—	2	1	no	—
22	#1814	—	7	1	no	maximal, Sylow
23	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊D₇	—	2	1	no	—
25	#1157	—	2	1	no	—
26	#1157	—	2	1	no	—
27	#1157	C₂×C₄	1	2	yes	—
28	#1157	D₄	1	2	yes	—
29	C₂×D₂₈	Q₈	1	1	yes	characteristic
30	C₂×C₄×D₇	—	2	1	no	—
31	C₂×Dic₁₄	Q₈	1	1	yes	characteristic
32	C₇×M₁₆	—	2	2	no	—
33	C₇×M₁₆	—	2	2	no	—
34	C₂×C₇×C₈	—	2	2	no	—
35	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₇×C₈	D₄	1	1	yes	characteristic
38	C₂×C₇×C₈	D₄	1	1	yes	characteristic
39	C₂×C₇×C₈	D₄	1	1	yes	characteristic
40	C₂×C₇×C₈	D₄	1	1	yes	characteristic
41	C₂²⋊Dic₇	—	2	2	no	—
42	C₄⋊Dic₇	—	2	2	no	—
43	C₄×Dic₇	—	2	2	no	—
44	#1136	—	2	1	no	—
45	#1136	C₂×C₄	1	2	yes	—
46	C₂×C₇⋊C₈	—	2	1	no	—
47	#1131	—	2	2	no	—
48	#1131	—	2	2	no	—
49	C₈×D₇	—	2	2	no	—
50	C₈×D₇	—	2	2	no	—
51	C₂×#131	—	7	1	no	—
52	C₂×#108	D₇	1	1	yes	characteristic, complemented
53	#427	—	7	1	no	—
54	#421	—	7	2	no	—
55	#344	—	7	2	no	—
56	C₂³×C₇	#41	1	1	yes	characteristic
57	C₂²×D₇	—	2	1	no	—
58	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇	#41	1	1	yes	characteristic
64	C₂²⋊D₇	—	2	2	no	—
65	C₂²⋊D₇	—	2	2	no	—
66	C₂×Dic₇	—	2	1	no	—
67	C₂×Dic₇	—	4	2	no	—
68	D₂₈	—	2	1	no	—
69	D₂₈	C₄⋊C₄	1	2	yes	—
70	C₄×D₇	—	2	2	no	—
71	C₄×D₇	—	2	2	no	—
72	Dic₁₄	—	2	1	no	—
73	Dic₁₄	C₄⋊C₄	1	2	yes	—
74	C₇×C₈	—	4	2	no	—
75	C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₇⋊C₈	—	2	2	no	—
80	C₂×#41	—	7	1	no	—
81	#131	—	7	2	no	—
82	#131	—	7	2	no	—
83	#131	—	14	1	no	—
84	#131	—	14	1	no	—
85	C₂×M₁₆	—	7	1	no	—
86	C₂×M₁₆	—	14	1	no	—
87	C₂×M₁₆	C₂×D₇	1	2	yes	—
88	C₂²×C₈	—	14	1	no	—
89	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
90	#117	—	7	2	no	—
91	#108	—	2	1	no	—
92	#108	—	2	1	no	—
93	#107	—	14	1	no	—
94	#106	—	14	1	no	—
95	#104	—	14	4	no	—
96	C₂²×C₇	—	4	1	no	—
97	C₂²×C₇	#117	1	1	yes	characteristic
98	C₂²×C₇	#121	1	1	yes	characteristic
99	C₂²×C₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×D₇	—	2	2	no	—
101	C₂×D₇	—	4	2	no	—
102	C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₄×C₇	#121	1	1	yes	characteristic
105	C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
106	Dic₇	—	2	2	no	—
107	Dic₇	—	8	2	no	—
108	#41	—	7	2	no	—
109	#41	—	7	2	no	—
110	#41	—	14	1	no	—
111	#41	—	14	1	no	—
112	C₂×Q₈	—	7	1	no	—
113	C₂×D₄	—	7	1	no	—
114	C₂×D₄	—	14	1	no	—
115	C₂²×C₄	—	14	1	no	—
116	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	2	2	no	—
119	M₁₆	—	7	2	no	—
120	M₁₆	—	14	1	no	—
121	M₁₆	—	14	2	no	—
122	M₁₆	—	14	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	14	1	no	—
125	C₂×C₈	—	14	2	no	—
126	C₂×C₈	—	14	2	no	—
127	C₂×C₈	C₄×D₇	1	1	yes	characteristic
128	C₂×C₈	C₄×D₇	1	1	yes	characteristic
129	C₂×C₈	D₂₈	1	1	yes	characteristic
130	C₂×C₈	D₂₈	1	1	yes	characteristic
131	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
132	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
133	C₄⋊C₄	—	14	2	no	—
134	C₂²⋊C₄	—	14	2	no	—
135	C₄²	—	14	2	no	—
136	C₂×C₇	—	2	1	no	—
137	C₂×C₇	—	2	1	no	—
138	C₂×C₇	—	2	1	no	—
139	C₂×C₇	#380	1	1	yes	characteristic
140	D₇	—	4	2	no	—
141	C₂³	—	14	1	no	—
142	C₂³	#1161	1	1	yes	characteristic
143	Q₈	—	7	2	no	—
144	Q₈	—	14	1	no	—
145	D₄	—	7	2	no	—
146	D₄	—	14	2	no	—
147	D₄	—	14	2	no	—
148	D₄	—	14	1	no	—
149	C₂×C₄	—	14	1	no	—
150	C₂×C₄	—	14	2	no	—
151	C₂×C₄	—	14	2	no	—
152	C₂×C₄	—	28	2	no	—
153	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
154	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
155	C₂×C₄	D₄×D₇	1	1	yes	characteristic
156	C₂×C₄	#1159	1	1	yes	characteristic
157	C₂×C₄	Q₈×D₇	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
159	C₈	—	4	2	no	—
160	C₈	—	14	2	no	—
161	C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
162	C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
163	C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
164	C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
165	C₇	#1814	1	1	yes	characteristic, complemented, Sylow
166	C₂²	—	4	1	no	—
167	C₂²	—	14	2	no	—
168	C₂²	—	28	2	no	—
169	C₂²	#6455	1	1	yes	characteristic
170	C₂²	#6458	1	1	yes	characteristic
171	C₂²	#6509	1	1	yes	characteristic
172	C₄	—	14	2	no	—
173	C₄	—	56	2	no	—
174	C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
175	C₄	#6460	1	1	yes	characteristic
176	C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
177	C₄	#6501	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	—	28	2	no	—
182	C₂	#89759	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#6501
commutator	C₂×C₄×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂×C₇×#108	C₂
socle	C₂×C₇	#380
prime radical	C₂×#1157	C₂²
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂×#108	D₇
hypocenter	C₇	#1814
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ 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. #89878	C₂	semidirect
5. #89351	C₂	semidirect
13. #6464	C₄	semidirect
14. #6464	C₄	semidirect
52. C₂×#108	D₇	semidirect
165. C₇	#1814	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
#89911	2
#89878	2
#89351	2
#1814	7