Finite Group Databasebeta

Subgroups — #90289

Order 448.

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

normal subgroups only

This group has 290 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	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
4	#6502	C₂	1	2	yes	maximal, complemented
5	#6499	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₇	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#6460	C₂	1	1	yes	maximal, characteristic, complemented
9	#6450	C₂	1	1	yes	maximal, characteristic, complemented
10	#6448	C₂	1	2	yes	maximal, complemented
11	#6441	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×D₄	C₂²	1	2	yes	—
13	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
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	4	no	—
20	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄	—	2	4	no	—
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	1	yes	characteristic
26	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂²⋊Dic₇	C₂²	1	2	yes	—
28	C₂²⋊Dic₇	C₂²	1	2	yes	—
29	C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
30	(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
31	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
32	C₄⋊Dic₇	C₂²	1	2	yes	—
33	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
34	Dic₇⋊C₄	C₂²	1	2	yes	—
35	C₄×Dic₇	C₂²	1	2	yes	—
36	C₄×Dic₇	C₂²	1	1	yes	characteristic
37	#548	—	7	1	no	maximal, Sylow
38	C₂³×C₇	C₂³	1	2	yes	—
39	C₂³×C₇	C₂³	1	1	yes	characteristic
40	C₂²×D₇	C₂³	1	1	yes	characteristic
41	C₇×D₄	—	2	4	no	—
42	C₇×D₄	—	2	2	no	—
43	C₂×C₄×C₇	—	2	2	no	—
44	C₂×C₄×C₇	C₂³	1	2	yes	—
45	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²⋊D₇	—	2	2	no	—
48	C₂²⋊D₇	—	2	4	no	—
49	C₂×Dic₇	—	2	4	no	—
50	C₂×Dic₇	—	2	4	no	—
51	C₂×Dic₇	C₂³	1	1	yes	characteristic
52	C₂×Dic₇	C₂³	1	1	yes	characteristic
53	C₂×Dic₇	C₂³	1	1	yes	characteristic
54	C₂×Dic₇	C₂³	1	2	yes	—
55	C₂×Dic₇	C₂³	1	2	yes	—
56	C₄×D₇	—	2	2	no	—
57	Dic₁₄	—	2	2	no	—
58	Dic₁₄	—	2	4	no	—
59	Dic₁₄	D₄	1	4	yes	complemented
60	C₂×#41	—	7	1	no	—
61	C₂²×Q₈	—	7	1	no	—
62	#124	—	7	1	no	—
63	#124	—	7	2	no	—
64	C₂²⋊Q₈	—	7	1	no	—
65	C₂²⋊Q₈	—	7	2	no	—
66	#121	—	7	2	no	—
67	#121	D₇	1	1	yes	characteristic, complemented
68	C₄×Q₈	—	7	1	no	—
69	C₄×D₄	—	7	2	no	—
70	C₄×D₄	—	7	1	no	—
71	C₂²×C₇	—	2	1	no	—
72	C₂²×C₇	—	2	1	no	—
73	C₂²×C₇	—	2	2	no	—
74	C₂²×C₇	—	2	2	no	—
75	C₂²×C₇	—	2	2	no	—
76	C₂²×C₇	#41	1	2	yes	complemented
77	C₂²×C₇	C₂⁴	1	1	yes	characteristic
78	C₂×D₇	—	2	1	no	—
79	C₂×D₇	—	2	1	no	—
80	C₂×D₇	—	2	1	no	—
81	C₄×C₇	—	2	1	no	—
82	C₄×C₇	—	2	2	no	—
83	C₄×C₇	C₂×D₄	1	2	yes	—
84	Dic₇	—	2	2	no	—
85	Dic₇	—	2	1	no	—
86	Dic₇	—	2	1	no	—
87	Dic₇	—	2	1	no	—
88	Dic₇	C₂×D₄	1	4	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×D₄	—	7	1	no	—
95	C₂×D₄	—	7	2	no	—
96	C₂×D₄	C₂×D₇	1	2	yes	—
97	C₂×D₄	C₂×D₇	1	1	yes	characteristic
98	C₂²×C₄	—	7	2	no	—
99	C₂²×C₄	—	7	2	no	—
100	C₂²×C₄	—	7	1	no	—
101	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
102	C₄⋊C₄	—	7	1	no	—
103	C₄⋊C₄	—	7	2	no	—
104	C₄⋊C₄	—	7	2	no	—
105	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
106	C₂²⋊C₄	—	7	1	no	—
107	C₂²⋊C₄	—	7	2	no	—
108	C₂²⋊C₄	—	7	2	no	—
109	C₂²⋊C₄	—	7	1	no	—
110	C₂²⋊C₄	—	7	2	no	—
111	C₂²⋊C₄	C₂×D₇	1	2	yes	—
112	C₄²	—	7	2	no	—
113	C₄²	—	7	1	no	—
114	C₂×C₇	—	2	2	no	—
115	C₂×C₇	—	4	2	no	—
116	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
119	D₇	—	4	1	no	—
120	C₂³	—	7	1	no	—
121	C₂³	C₂²×D₇	1	2	yes	—
122	C₂³	C₂²×D₇	1	1	yes	characteristic
123	Q₈	—	7	4	no	—
124	Q₈	—	14	4	no	—
125	Q₈	—	14	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	D₄	—	14	4	no	—
129	D₄	—	14	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	7	2	no	—
132	C₂×C₄	—	7	1	no	—
133	C₂×C₄	—	7	1	no	—
134	C₂×C₄	—	7	2	no	—
135	C₂×C₄	—	7	1	no	—
136	C₂×C₄	—	14	4	no	—
137	C₂×C₄	—	14	4	no	—
138	C₂×C₄	—	14	2	no	—
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	#548	1	1	yes	characteristic, complemented, Sylow
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	14	1	no	—
149	C₂²	—	14	1	no	—
150	C₂²	—	14	1	no	—
151	C₂²	#1159	1	2	yes	complemented
152	C₂²	C₂³×D₇	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	2	no	—
155	C₄	—	7	4	no	—
156	C₄	—	14	1	no	—
157	C₄	—	14	2	no	—
158	C₄	—	14	1	no	—
159	C₄	—	14	1	no	—
160	C₄	D₄×D₇	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	—	4	2	no	—
163	C₂	—	28	1	no	—
164	C₂	C₂×D₄×D₇	1	1	yes	characteristic
165	C₂	C₂×#1159	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×#121	C₂
hypercenter	#121	D₇
hypocenter	C₇	#548
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. C₂²×Dic₁₄	C₂	semidirect
3. C₇×#121	C₂	semidirect
4. #6502	C₂	semidirect
5. #6499	C₂	semidirect
6. D₄×Dic₇	C₂	semidirect
7. C₄×C₂²⋊D₇	C₂	semidirect
8. #6460	C₂	semidirect
9. #6450	C₂	semidirect
10. #6448	C₂	semidirect
11. #6441	C₂	semidirect
59. Dic₁₄	D₄	semidirect
67. #121	D₇	semidirect
76. C₂²×C₇	#41	semidirect
142. C₇	#548	semidirect
151. C₂²	#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
C₂²×Dic₁₄	2
C₇×#121	2
#6502	2
#6499	2
D₄×Dic₇	2
C₄×C₂²⋊D₇	2
#6460	2
#6450	2
#6448	2
#6441	2
#548	7