Finite Group Databasebeta

Subgroups — #11740384

Order 896.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₇×#124	C₂	1	1	yes	maximal, characteristic
2	#90161	C₂	1	1	yes	maximal, characteristic, complemented
3	#89997	C₂	1	2	yes	maximal, complemented
4	#89987	C₂	1	2	yes	maximal
5	#89828	C₂	1	4	yes	maximal, complemented
6	#89827	C₂	1	4	yes	maximal, complemented
7	C₄×#1136	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₇×#124	—	2	2	no	—
11	C₇×#124	C₂²	1	2	yes	—
12	C₇×#124	C₂²	1	2	yes	—
13	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
14	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
15	#6489	C₂²	1	2	yes	—
16	#6487	—	2	2	no	—
17	C₂×#1136	C₂²	1	2	yes	—
18	#6433	C₂²	1	2	yes	complemented
19	#6432	C₂²	1	2	yes	complemented
20	C₄×Dic₁₄	—	2	2	no	—
21	#6409	C₂²	1	8	yes	complemented
22	#6406	C₂²	1	8	yes	—
23	#6377	C₂²	1	2	yes	complemented
24	C₄×C₇⋊C₈	C₂²	1	2	yes	complemented
25	#3140	—	7	1	no	maximal, Sylow
26	C₂⁴×C₇	—	2	1	no	—
27	C₂×C₇×Q₈	—	2	1	no	—
28	C₂×C₇×Q₈	—	2	4	no	—
29	C₂×C₇×Q₈	C₂³	1	2	yes	—
30	C₂×C₇×D₄	—	2	1	no	—
31	C₂×C₇×D₄	—	2	4	no	—
32	C₂×C₇×D₄	C₂³	1	2	yes	—
33	C₂²×C₄×C₇	—	2	1	no	—
34	C₂²×C₄×C₇	D₄	1	2	yes	—
35	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₂×Dic₁₄	—	2	2	no	—
37	C₇×C₂²⋊C₄	—	2	4	no	—
38	C₇×C₂²⋊C₄	—	2	4	no	—
39	C₄²×C₇	D₄	1	2	yes	—
40	C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₂²⋊Dic₇	—	2	2	no	—
43	C₄⋊Dic₇	—	2	2	no	—
44	C₄⋊Dic₇	C₂³	1	4	yes	—
45	Dic₇⋊C₄	—	2	4	no	—
46	C₄×Dic₇	—	2	2	no	—
47	#1136	—	2	4	no	—
48	C₂×C₇⋊C₈	C₂³	1	4	yes	—
49	#533	—	7	1	no	—
50	C₂×#124	D₇	1	1	yes	characteristic
51	#489	—	7	4	no	—
52	#488	—	7	4	no	—
53	#419	—	7	2	no	—
54	#418	—	7	2	no	—
55	C₄×M₁₆	—	7	1	no	—
56	C₂³×C₇	—	2	2	no	—
57	C₂³×C₇	—	4	4	no	—
58	C₂³×C₇	—	4	2	no	—
59	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
60	C₇×Q₈	—	2	4	no	—
61	C₇×Q₈	—	4	2	no	—
62	C₇×D₄	—	2	4	no	—
63	C₇×D₄	—	4	2	no	—
64	C₂×C₄×C₇	—	2	2	no	—
65	C₂×C₄×C₇	—	4	2	no	—
66	C₂×C₄×C₇	—	4	2	no	—
67	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇	#41	1	4	yes	—
71	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₇	—	2	4	no	—
73	Dic₁₄	—	4	4	no	—
74	C₇⋊C₈	—	2	4	no	—
75	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
76	C₂²×D₄	C₂×D₇	1	1	yes	characteristic
77	C₂×M₁₆	—	7	2	no	—
78	C₄⋊Q₈	—	7	2	no	—
79	#125	—	7	2	no	—
80	#124	—	2	2	no	—
81	#124	C₂×D₇	1	2	yes	—
82	#124	C₂×D₇	1	2	yes	—
83	C₂²⋊Q₈	—	14	2	no	—
84	C₄×Q₈	—	14	2	no	—
85	#117	—	7	2	no	—
86	C₂×C₂²⋊C₄	—	2	2	no	—
87	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
88	#103	—	7	8	no	—
89	#102	—	7	8	no	—
90	#97	—	7	2	no	—
91	C₄×C₈	—	7	2	no	—
92	C₂²×C₇	—	2	2	no	—
93	C₂²×C₇	—	4	2	no	—
94	C₂²×C₇	—	4	4	no	—
95	C₂²×C₇	—	8	4	no	—
96	C₂²×C₇	#124	1	2	yes	—
97	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₇	—	2	2	no	—
99	C₄×C₇	—	2	2	no	—
100	C₄×C₇	—	4	2	no	—
101	C₄×C₇	#124	1	2	yes	—
102	C₄×C₇	C₂×#41	1	2	yes	—
103	Dic₇	—	4	4	no	—
104	C₂⁴	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	4	no	—
107	C₂×Q₈	—	14	2	no	—
108	C₂×Q₈	C₂²×D₇	1	2	yes	—
109	C₂×D₄	—	2	4	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	C₂²×D₇	1	2	yes	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
114	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
115	M₁₆	—	14	4	no	—
116	C₂×C₈	—	7	4	no	—
117	C₄⋊C₄	—	7	4	no	—
118	C₄⋊C₄	—	14	4	no	—
119	C₄⋊C₄	—	14	2	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₂²⋊C₄	—	14	2	no	—
123	C₄²	—	14	2	no	—
124	C₄²	C₂²⋊D₇	1	2	yes	—
125	C₄²	C₂²×D₇	1	1	yes	characteristic
126	C₄²	C₂²×D₇	1	1	yes	characteristic
127	C₂×C₇	—	2	2	no	—
128	C₂×C₇	—	8	2	no	—
129	C₂×C₇	C₂×#124	1	1	yes	characteristic
130	C₂×C₇	#575	1	2	yes	—
131	C₂³	—	2	2	no	—
132	C₂³	—	4	4	no	—
133	C₂³	—	4	2	no	—
134	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	4	2	no	—
137	Q₈	—	28	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	14	4	no	—
144	C₂×C₄	#1159	1	4	yes	—
145	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
148	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
149	C₈	—	14	4	no	—
150	C₇	#3140	1	1	yes	characteristic, complemented, Sylow
151	C₂²	—	2	2	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	8	4	no	—
155	C₂²	#6499	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	2	no	—
160	C₄	—	28	4	no	—
161	C₄	#6499	1	2	yes	—
162	C₄	C₂×#1159	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	8	2	no	—
165	C₂	C₂×#6499	1	1	yes	characteristic
166	C₂	#90513	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₂×C₇×#124	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂²×C₇×D₄	C₂²
hypercenter	C₂×#124	D₇
hypocenter	C₇	#3140
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
2. #90161	C₂	semidirect
3. #89997	C₂	semidirect
5. #89828	C₂	semidirect
6. #89827	C₂	semidirect
7. C₄×#1136	C₂	semidirect
18. #6433	C₂²	semidirect
19. #6432	C₂²	semidirect
21. #6409	C₂²	semidirect
23. #6377	C₂²	semidirect
24. C₄×C₇⋊C₈	C₂²	semidirect
150. C₇	#3140	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#124	2
#90161	2
#89997	2
#89987	2
#89828	2
#89827	2
C₄×#1136	2
#3140	7