Finite Group Databasebeta

Subgroups — #90302

Order 448.

292 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 107 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×#1159	C₂	1	2	yes	maximal, complemented
2	C₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	#6501	C₂	1	1	yes	maximal, characteristic, complemented
4	#6498	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
8	D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#6451	C₂	1	1	yes	maximal, characteristic, complemented
10	#6446	C₂	1	2	yes	maximal, complemented
11	#6441	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₇×D₄	C₂²	1	2	yes	—
14	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₇	C₂²	1	2	yes	—
18	C₂²×Dic₇	C₂²	1	2	yes	—
19	#1159	—	2	8	no	—
20	C₂×C₄×D₇	C₂²	1	2	yes	—
21	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₄	C₂²	1	2	yes	—
23	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂²⋊Dic₇	C₂²	1	2	yes	—
26	C₂²⋊Dic₇	C₂²	1	2	yes	—
27	C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
28	(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₇	C₂²	1	2	yes	—
30	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
31	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
32	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
33	Dic₇⋊C₄	C₂²	1	2	yes	—
34	Dic₇⋊C₄	C₂²	1	2	yes	—
35	C₄×Dic₇	C₂²	1	1	yes	characteristic
36	#547	—	7	1	no	maximal, Sylow
37	C₂³×C₇	C₂³	1	2	yes	—
38	C₂³×C₇	C₂³	1	1	yes	characteristic
39	C₂²×D₇	C₂³	1	1	yes	characteristic
40	C₇×D₄	—	2	4	no	—
41	C₇×D₄	—	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	1	yes	characteristic
46	C₂²⋊D₇	—	2	4	no	—
47	C₂²⋊D₇	D₄	1	4	yes	complemented
48	C₂×Dic₇	—	2	4	no	—
49	C₂×Dic₇	—	2	4	no	—
50	C₂×Dic₇	C₂³	1	1	yes	characteristic
51	C₂×Dic₇	C₂³	1	1	yes	characteristic
52	C₂×Dic₇	C₂³	1	2	yes	—
53	C₂×Dic₇	C₂³	1	2	yes	—
54	C₂×Dic₇	C₂³	1	1	yes	characteristic
55	C₄×D₇	—	2	4	no	—
56	C₄×D₇	—	2	2	no	—
57	Dic₁₄	—	2	4	no	—
58	C₂×#41	—	7	2	no	—
59	C₄⋊Q₈	—	7	1	no	—
60	#123	—	7	2	no	—
61	#123	—	7	2	no	—
62	C₂²⋊Q₈	—	7	2	no	—
63	#121	—	7	1	no	—
64	#121	D₇	1	1	yes	characteristic, complemented
65	C₄×D₄	—	7	1	no	—
66	C₄×D₄	—	7	1	no	—
67	C₂×C₄⋊C₄	—	7	1	no	—
68	C₂×C₄⋊C₄	—	7	1	no	—
69	C₂²×C₇	—	2	1	no	—
70	C₂²×C₇	—	2	1	no	—
71	C₂²×C₇	—	2	2	no	—
72	C₂²×C₇	—	2	2	no	—
73	C₂²×C₇	—	2	2	no	—
74	C₂²×C₇	C₂×D₄	1	2	yes	—
75	C₂²×C₇	C₂⁴	1	1	yes	characteristic
76	C₂×D₇	—	2	1	no	—
77	C₂×D₇	—	2	1	no	—
78	C₂×D₇	C₂×D₄	1	2	yes	—
79	C₄×C₇	—	2	2	no	—
80	C₄×C₇	—	2	1	no	—
81	C₄×C₇	#41	1	2	yes	complemented
82	Dic₇	—	2	2	no	—
83	Dic₇	—	2	2	no	—
84	Dic₇	—	2	1	no	—
85	Dic₇	—	2	1	no	—
86	Dic₇	C₂×D₄	1	2	yes	—
87	#41	—	14	8	no	—
88	C₂×Q₈	—	7	2	no	—
89	C₂×D₄	—	7	1	no	—
90	C₂×D₄	—	7	2	no	—
91	C₂×D₄	C₂×D₇	1	1	yes	characteristic
92	C₂×D₄	C₂×D₇	1	2	yes	—
93	C₂²×C₄	—	7	2	no	—
94	C₂²×C₄	—	7	2	no	—
95	C₂²×C₄	—	7	1	no	—
96	C₂²×C₄	—	7	2	no	—
97	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
98	C₄⋊C₄	—	7	2	no	—
99	C₄⋊C₄	—	7	1	no	—
100	C₄⋊C₄	—	7	2	no	—
101	C₄⋊C₄	—	7	1	no	—
102	C₄⋊C₄	—	7	1	no	—
103	C₄⋊C₄	—	7	2	no	—
104	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
105	C₂²⋊C₄	—	7	1	no	—
106	C₂²⋊C₄	—	7	2	no	—
107	C₂²⋊C₄	—	7	2	no	—
108	C₂²⋊C₄	—	7	1	no	—
109	C₂²⋊C₄	C₂×D₇	1	2	yes	—
110	C₄²	—	7	1	no	—
111	C₂×C₇	—	2	2	no	—
112	C₂×C₇	—	4	2	no	—
113	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
116	D₇	—	2	2	no	—
117	C₂³	—	7	1	no	—
118	C₂³	C₂²×D₇	1	2	yes	—
119	C₂³	C₂²×D₇	1	1	yes	characteristic
120	Q₈	—	14	4	no	—
121	D₄	—	2	4	no	—
122	D₄	—	2	2	no	—
123	D₄	—	7	4	no	—
124	D₄	—	14	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	7	1	no	—
127	C₂×C₄	—	7	1	no	—
128	C₂×C₄	—	7	1	no	—
129	C₂×C₄	—	7	2	no	—
130	C₂×C₄	—	7	2	no	—
131	C₂×C₄	—	14	4	no	—
132	C₂×C₄	—	14	4	no	—
133	C₂×C₄	—	14	4	no	—
134	C₂×C₄	—	14	2	no	—
135	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
136	C₂×C₄	C₂²×D₇	1	2	yes	—
137	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
138	C₇	#547	1	1	yes	characteristic, complemented, Sylow
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	7	2	no	—
145	C₂²	—	14	1	no	—
146	C₂²	—	14	1	no	—
147	C₂²	D₄×D₇	1	2	yes	—
148	C₂²	C₂³×D₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	7	2	no	—
152	C₄	—	14	2	no	—
153	C₄	—	14	2	no	—
154	C₄	—	14	1	no	—
155	C₄	—	14	1	no	—
156	C₄	#1159	1	2	yes	complemented
157	C₂	—	2	2	no	—
158	C₂	—	4	2	no	—
159	C₂	—	14	2	no	—
160	C₂	C₂×D₄×D₇	1	1	yes	characteristic
161	C₂	C₂×#1159	1	1	yes	characteristic
162	C₂	#6554	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#121	C₂
hypercenter	#121	D₇
hypocenter	C₇	#547
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂×#1159	C₂	semidirect
2. C₇×#121	C₂	semidirect
3. #6501	C₂	semidirect
4. #6498	C₂	semidirect
5. D₄×Dic₇	C₂	semidirect
6. C₄×C₂²⋊D₇	C₂	semidirect
7. C₂×C₄⋊Dic₇	C₂	semidirect
8. D₇×C₄⋊C₄	C₂	semidirect
9. #6451	C₂	semidirect
10. #6446	C₂	semidirect
11. #6441	C₂	semidirect
47. C₂²⋊D₇	D₄	semidirect
64. #121	D₇	semidirect
81. C₄×C₇	#41	semidirect
138. C₇	#547	semidirect
156. C₄	#1159	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1159	2
C₇×#121	2
#6501	2
#6498	2
D₄×Dic₇	2
C₄×C₂²⋊D₇	2
C₂×C₄⋊Dic₇	2
D₇×C₄⋊C₄	2
#6451	2
#6446	2
#6441	2
#547	7