Finite Group Databasebeta

Subgroups — #90319

Order 448.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#1157	C₂	1	1	yes	maximal, characteristic
3	C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6509	C₂	1	1	yes	maximal, characteristic, complemented
5	#6487	C₂	1	1	yes	maximal, characteristic, complemented
6	#6460	C₂	1	1	yes	maximal, characteristic, complemented
7	#6457	C₂	1	2	yes	maximal, complemented
8	#6455	C₂	1	1	yes	maximal, characteristic, complemented
9	#6451	C₂	1	2	yes	maximal, complemented
10	#6448	C₂	1	2	yes	maximal, complemented
11	#6446	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₇	C₂²	1	1	yes	characteristic
16	Q₈×D₇	—	2	4	no	—
17	#1157	—	2	4	no	—
18	C₂×D₂₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₇	C₂²	1	2	yes	—
20	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₄	C₂²	1	2	yes	—
23	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
25	C₇×C₄⋊C₄	C₂²	1	2	yes	—
26	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂²⋊Dic₇	C₂²	1	2	yes	—
29	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
30	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
31	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
32	C₄⋊Dic₇	C₂²	1	2	yes	—
33	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
34	Dic₇⋊C₄	C₂²	1	2	yes	—
35	Dic₇⋊C₄	C₂²	1	2	yes	—
36	C₄×Dic₇	C₂²	1	2	yes	—
37	#536	—	7	1	no	maximal, Sylow
38	C₂³×C₇	C₂³	1	1	yes	characteristic
39	C₂²×D₇	C₂³	1	1	yes	characteristic
40	C₂²×D₇	C₂³	1	1	yes	characteristic
41	C₇×Q₈	—	2	2	no	—
42	C₂×C₄×C₇	—	2	2	no	—
43	C₂×C₄×C₇	C₂³	1	2	yes	—
44	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₇	C₂³	1	2	yes	—
46	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²⋊D₇	—	2	2	no	—
48	C₂²⋊D₇	—	2	2	no	—
49	C₂×Dic₇	C₂³	1	2	yes	—
50	C₂×Dic₇	C₂³	1	1	yes	characteristic
51	C₂×Dic₇	C₂³	1	2	yes	—
52	C₂×Dic₇	C₂³	1	1	yes	characteristic
53	D₂₈	—	2	2	no	—
54	C₄×D₇	—	2	4	no	—
55	C₄×D₇	—	2	2	no	—
56	C₄×D₇	D₄	1	4	yes	—
57	Dic₁₄	—	2	2	no	—
58	Dic₁₄	—	2	2	no	—
59	Dic₁₄	—	2	4	no	—
60	C₂×#41	—	7	1	no	—
61	C₂²×Q₈	—	7	1	no	—
62	C₄⋊Q₈	—	7	2	no	—
63	#124	—	7	2	no	—
64	#123	—	7	2	no	—
65	#123	—	7	2	no	—
66	C₂²⋊Q₈	—	7	1	no	—
67	C₂²⋊Q₈	—	7	1	no	—
68	C₂²⋊Q₈	—	7	1	no	—
69	C₂²⋊Q₈	D₇	1	1	yes	characteristic, complemented
70	#117	—	7	1	no	—
71	C₂²×C₇	—	2	1	no	—
72	C₂²×C₇	—	2	1	no	—
73	C₂²×C₇	—	2	1	no	—
74	C₂²×C₇	C₂⁴	1	1	yes	characteristic
75	C₂×D₇	—	2	1	no	—
76	C₂×D₇	—	2	1	no	—
77	C₂×D₇	—	2	1	no	—
78	C₂×D₇	—	2	1	no	—
79	C₂×D₇	—	2	1	no	—
80	C₂×D₇	C₂×D₄	1	2	yes	—
81	C₄×C₇	—	2	2	no	—
82	C₄×C₇	—	2	1	no	—
83	C₄×C₇	—	2	2	no	—
84	C₄×C₇	C₂×D₄	1	2	yes	—
85	Dic₇	—	2	2	no	—
86	Dic₇	—	2	1	no	—
87	Dic₇	—	2	2	no	—
88	Dic₇	C₂×D₄	1	2	yes	—
89	#41	—	14	4	no	—
90	C₂×Q₈	—	7	1	no	—
91	C₂×Q₈	—	7	2	no	—
92	C₂×Q₈	—	7	1	no	—
93	C₂×Q₈	—	14	4	no	—
94	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
95	C₂×D₄	—	7	1	no	—
96	C₂×D₄	—	7	1	no	—
97	C₂×D₄	—	7	1	no	—
98	C₂²×C₄	—	7	1	no	—
99	C₂²×C₄	—	7	1	no	—
100	C₂²×C₄	—	7	2	no	—
101	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
102	C₄⋊C₄	—	7	2	no	—
103	C₄⋊C₄	—	7	2	no	—
104	C₄⋊C₄	—	7	1	no	—
105	C₄⋊C₄	—	7	2	no	—
106	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
107	C₄⋊C₄	C₂×D₇	1	2	yes	—
108	C₂²⋊C₄	—	7	2	no	—
109	C₂²⋊C₄	—	7	2	no	—
110	C₂²⋊C₄	—	7	2	no	—
111	C₂²⋊C₄	—	7	2	no	—
112	C₂²⋊C₄	C₂×D₇	1	2	yes	—
113	C₄²	—	7	2	no	—
114	C₂×C₇	—	4	1	no	—
115	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
118	D₇	—	2	2	no	—
119	D₇	—	4	1	no	—
120	C₂³	—	7	1	no	—
121	C₂³	—	7	1	no	—
122	C₂³	C₂²×D₇	1	1	yes	characteristic
123	Q₈	—	2	2	no	—
124	Q₈	—	14	4	no	—
125	Q₈	—	14	2	no	—
126	Q₈	—	14	2	no	—
127	D₄	—	14	2	no	—
128	D₄	—	14	2	no	—
129	D₄	—	14	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	7	1	no	—
132	C₂×C₄	—	7	4	no	—
133	C₂×C₄	—	7	2	no	—
134	C₂×C₄	—	7	2	no	—
135	C₂×C₄	—	7	1	no	—
136	C₂×C₄	—	14	4	no	—
137	C₂×C₄	—	14	2	no	—
138	C₂×C₄	C₂²×D₇	1	2	yes	—
139	C₂×C₄	C₂²×D₇	1	2	yes	—
140	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
141	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
142	C₇	#536	1	1	yes	characteristic, complemented, Sylow
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	7	2	no	—
147	C₂²	—	14	1	no	—
148	C₂²	—	14	1	no	—
149	C₂²	—	14	1	no	—
150	C₂²	—	14	1	no	—
151	C₂²	—	14	1	no	—
152	C₂²	C₂³×D₇	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	7	2	no	—
157	C₄	—	14	1	no	—
158	C₄	—	14	2	no	—
159	C₄	—	14	2	no	—
160	C₄	D₄×D₇	1	2	yes	—
161	C₂	—	4	1	no	—
162	C₂	—	14	2	no	—
163	C₂	—	28	1	no	—
164	C₂	C₂×D₄×D₇	1	1	yes	characteristic
165	C₂	#6551	1	1	yes	characteristic
166	C₂	#6554	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×C₂²⋊Q₈	C₂
prime radical	C₂×#1157	C₂
hypercenter	C₂²⋊Q₈	D₇
hypocenter	C₇	#536
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₇	C₂	semidirect
3. C₇×C₂²⋊Q₈	C₂	semidirect
4. #6509	C₂	semidirect
5. #6487	C₂	semidirect
6. #6460	C₂	semidirect
7. #6457	C₂	semidirect
8. #6455	C₂	semidirect
9. #6451	C₂	semidirect
10. #6448	C₂	semidirect
11. #6446	C₂	semidirect
69. C₂²⋊Q₈	D₇	semidirect
142. C₇	#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₇	2
C₂×#1157	2
C₇×C₂²⋊Q₈	2
#6509	2
#6487	2
#6460	2
#6457	2
#6455	2
#6451	2
#6448	2
#6446	2
#536	7