Finite Group Databasebeta

Subgroups — #90296

Order 448.

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

normal subgroups only

This group has 346 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×D₄×D₇	C₂	1	2	yes	maximal, complemented
2	C₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	#6502	C₂	1	2	yes	maximal, complemented
4	#6500	C₂	1	2	yes	maximal, complemented
5	#6493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#6457	C₂	1	2	yes	maximal, complemented
8	#6446	C₂	1	2	yes	maximal, complemented
9	#6445	C₂	1	2	yes	maximal, complemented
10	C₂³×D₇	C₂²	1	1	yes	characteristic
11	C₂³×D₇	C₂²	1	2	yes	complemented
12	C₂×C₇×D₄	C₂²	1	2	yes	complemented
13	C₂×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₇	C₂²	1	2	yes	complemented
17	C₂×C₂²⋊D₇	C₂²	1	2	yes	complemented
18	C₂²×Dic₇	C₂²	1	1	yes	characteristic, complemented
19	D₄×D₇	—	2	8	no	—
20	C₂×D₂₈	C₂²	1	2	yes	—
21	C₂×D₂₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₄×D₇	C₂²	1	2	yes	complemented
23	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
24	C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
25	C₂²⋊Dic₇	C₂²	1	2	yes	complemented
26	(C₂×D₇)⋊C₄	C₂²	1	2	yes	complemented
27	(C₂×D₇)⋊C₄	C₂²	1	2	yes	complemented
28	(C₂×D₇)⋊C₄	C₂²	1	2	yes	complemented
29	(C₂×D₇)⋊C₄	C₂²	1	2	yes	complemented
30	C₄⋊Dic₇	C₂²	1	1	yes	characteristic, complemented
31	Dic₇⋊C₄	C₂²	1	2	yes	complemented
32	#534	—	7	1	no	maximal, Sylow
33	C₂³×C₇	C₂³	1	2	yes	—
34	C₂³×C₇	C₂³	1	1	yes	characteristic
35	C₂²×D₇	—	2	4	no	—
36	C₂²×D₇	—	4	1	no	—
37	C₂²×D₇	—	4	1	no	—
38	C₂²×D₇	—	4	2	no	—
39	C₂²×D₇	—	4	2	no	—
40	C₂²×D₇	D₄	1	4	yes	complemented
41	C₂²×D₇	C₂³	1	2	yes	—
42	C₂²×D₇	C₂³	1	2	yes	—
43	C₇×D₄	—	2	4	no	—
44	C₇×D₄	—	4	1	no	—
45	C₂×C₄×C₇	—	4	1	no	—
46	C₂×C₄×C₇	C₂³	1	2	yes	—
47	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂²⋊D₇	—	2	4	no	—
50	C₂²⋊D₇	—	2	4	no	—
51	C₂²⋊D₇	—	4	2	no	—
52	C₂×Dic₇	—	4	1	no	—
53	C₂×Dic₇	C₂³	1	2	yes	—
54	C₂×Dic₇	C₂³	1	2	yes	—
55	D₂₈	—	2	4	no	—
56	D₂₈	—	4	1	no	—
57	C₄×D₇	—	2	4	no	—
58	C₂²×D₄	—	7	2	no	—
59	#123	—	7	2	no	—
60	#123	—	7	2	no	—
61	#121	—	7	2	no	—
62	#121	—	7	1	no	—
63	#121	D₇	1	1	yes	characteristic, complemented
64	#120	—	7	2	no	—
65	#120	—	7	2	no	—
66	C₂×C₂²⋊C₄	—	7	1	no	—
67	C₂²×C₇	—	2	1	no	—
68	C₂²×C₇	—	2	1	no	—
69	C₂²×C₇	—	2	2	no	—
70	C₂²×C₇	—	2	2	no	—
71	C₂²×C₇	—	2	2	no	—
72	C₂²×C₇	C₂×D₄	1	2	yes	—
73	C₂²×C₇	C₂⁴	1	1	yes	characteristic
74	C₂×D₇	—	2	2	no	—
75	C₂×D₇	—	2	2	no	—
76	C₂×D₇	—	2	2	no	—
77	C₂×D₇	—	2	2	no	—
78	C₂×D₇	—	2	2	no	—
79	C₂×D₇	—	4	4	no	—
80	C₂×D₇	—	4	8	no	—
81	C₂×D₇	C₂×D₄	1	4	yes	—
82	C₄×C₇	—	2	2	no	—
83	C₄×C₇	—	2	1	no	—
84	C₄×C₇	—	2	1	no	—
85	Dic₇	—	2	2	no	—
86	Dic₇	—	2	2	no	—
87	C₂⁴	—	7	1	no	—
88	C₂⁴	—	7	2	no	—
89	C₂×D₄	—	7	2	no	—
90	C₂×D₄	—	7	2	no	—
91	C₂×D₄	—	7	2	no	—
92	C₂×D₄	—	7	2	no	—
93	C₂×D₄	—	7	1	no	—
94	C₂×D₄	—	14	8	no	—
95	C₂×D₄	C₂×D₇	1	2	yes	complemented
96	C₂×D₄	C₂×D₇	1	1	yes	characteristic, complemented
97	C₂²×C₄	—	7	1	no	—
98	C₂²×C₄	—	7	2	no	—
99	C₂²×C₄	C₂×D₇	1	1	yes	characteristic, complemented
100	C₄⋊C₄	—	7	2	no	—
101	C₄⋊C₄	—	7	1	no	—
102	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic, complemented
103	C₂²⋊C₄	—	7	2	no	—
104	C₂²⋊C₄	—	7	2	no	—
105	C₂²⋊C₄	—	7	2	no	—
106	C₂²⋊C₄	—	7	2	no	—
107	C₂²⋊C₄	—	7	2	no	—
108	C₂²⋊C₄	C₂×D₇	1	2	yes	complemented
109	C₂×C₇	—	2	2	no	—
110	C₂×C₇	—	4	2	no	—
111	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
114	D₇	—	2	4	no	—
115	D₇	—	4	2	no	—
116	C₂³	—	7	2	no	—
117	C₂³	—	7	4	no	—
118	C₂³	—	7	2	no	—
119	C₂³	—	14	4	no	—
120	C₂³	—	28	1	no	—
121	C₂³	—	28	1	no	—
122	C₂³	—	28	2	no	—
123	C₂³	—	28	2	no	—
124	C₂³	C₂²×D₇	1	2	yes	—
125	C₂³	C₂²×D₇	1	1	yes	characteristic
126	D₄	—	2	4	no	—
127	D₄	—	4	1	no	—
128	D₄	—	14	4	no	—
129	D₄	—	14	4	no	—
130	D₄	—	14	4	no	—
131	D₄	—	28	1	no	—
132	D₄	—	28	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	7	2	no	—
135	C₂×C₄	—	7	2	no	—
136	C₂×C₄	—	14	4	no	—
137	C₂×C₄	—	28	1	no	—
138	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
139	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
140	C₂×C₄	C₂²×D₇	1	2	yes	—
141	C₇	#534	1	1	yes	characteristic, complemented, Sylow
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	7	4	no	—
148	C₂²	—	14	2	no	—
149	C₂²	—	14	2	no	—
150	C₂²	—	14	2	no	—
151	C₂²	—	14	2	no	—
152	C₂²	—	14	2	no	—
153	C₂²	—	28	4	no	—
154	C₂²	—	28	8	no	—
155	C₂²	D₄×D₇	1	2	yes	—
156	C₂²	C₂³×D₇	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	14	2	no	—
161	C₄	—	14	2	no	—
162	C₂	—	2	2	no	—
163	C₂	—	4	2	no	—
164	C₂	—	14	4	no	—
165	C₂	—	28	2	no	—
166	C₂	C₂×D₄×D₇	1	1	yes	characteristic
167	C₂	#6548	1	1	yes	characteristic
168	C₂	#6553	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	#534
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×D₄×D₇	C₂	semidirect
2. C₇×#121	C₂	semidirect
3. #6502	C₂	semidirect
4. #6500	C₂	semidirect
5. #6493	C₂	semidirect
6. C₂×(C₂×D₇)⋊C₄	C₂	semidirect
7. #6457	C₂	semidirect
8. #6446	C₂	semidirect
9. #6445	C₂	semidirect
11. C₂³×D₇	C₂²	semidirect
12. C₂×C₇×D₄	C₂²	semidirect
13. C₂×C₇×D₄	C₂²	semidirect
14. C₂²×C₄×C₇	C₂²	semidirect
16. C₂×C₂²⋊D₇	C₂²	semidirect
17. C₂×C₂²⋊D₇	C₂²	semidirect
18. C₂²×Dic₇	C₂²	semidirect
21. C₂×D₂₈	C₂²	semidirect
22. C₂×C₄×D₇	C₂²	semidirect
23. C₇×C₄⋊C₄	C₂²	semidirect
24. C₇×C₂²⋊C₄	C₂²	semidirect
25. C₂²⋊Dic₇	C₂²	semidirect
26. (C₂×D₇)⋊C₄	C₂²	semidirect
27. (C₂×D₇)⋊C₄	C₂²	semidirect
28. (C₂×D₇)⋊C₄	C₂²	semidirect
29. (C₂×D₇)⋊C₄	C₂²	semidirect
30. C₄⋊Dic₇	C₂²	semidirect
31. Dic₇⋊C₄	C₂²	semidirect
40. C₂²×D₇	D₄	semidirect
63. #121	D₇	semidirect
95. C₂×D₄	C₂×D₇	semidirect
96. C₂×D₄	C₂×D₇	semidirect
99. C₂²×C₄	C₂×D₇	semidirect
102. C₄⋊C₄	C₂×D₇	semidirect
108. C₂²⋊C₄	C₂×D₇	semidirect
141. C₇	#534	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₇	2
C₇×#121	2
#6502	2
#6500	2
#6493	2
C₂×(C₂×D₇)⋊C₄	2
#6457	2
#6446	2
#6445	2
#534	7