Finite Group Databasebeta

← D₇×#135 #90458

Subgroups — D₇×#135 #90458

Order 448.

266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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	D₇×#41	C₂	1	2	yes	maximal, complemented
2	C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	#6513	C₂	1	2	yes	maximal, complemented
4	#6482	C₂	1	1	yes	maximal, characteristic, complemented
5	D₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#6479	C₂	1	2	yes	maximal, complemented
7	D₇×SD₁₆	C₂	1	2	yes	maximal, complemented
8	#6475	C₂	1	1	yes	maximal, characteristic, complemented
9	D₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#6467	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#41	C₂²	1	2	yes	complemented
13	#1161	C₂²	1	2	yes	complemented
14	Q₈×D₇	C₂²	1	2	yes	—
15	#1159	—	2	2	no	—
16	#1159	C₂²	1	2	yes	complemented
17	D₄×D₇	—	2	2	no	—
18	D₄×D₇	C₂²	1	2	yes	—
19	#1157	—	2	2	no	—
20	#1157	C₂²	1	2	yes	complemented
21	C₂×C₄×D₇	—	2	2	no	—
22	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
23	C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
24	C₇×SD₁₆	C₂²	1	2	yes	complemented
25	C₇×D₈	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
27	#1144	C₂²	1	2	yes	complemented
28	#1143	C₂²	1	2	yes	complemented
29	#1142	C₂²	1	2	yes	complemented
30	#1141	C₂²	1	2	yes	complemented
31	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	Dic₂₈	C₂²	1	1	yes	characteristic, complemented
33	D₅₆	C₂²	1	1	yes	characteristic, complemented
34	#1132	C₂²	1	2	yes	complemented
35	C₈×D₇	C₂²	1	1	yes	characteristic
36	C₈×D₇	C₂²	1	1	yes	characteristic
37	C₈×D₇	C₂²	1	1	yes	characteristic, complemented
38	C₈×D₇	C₂²	1	1	yes	characteristic, complemented
39	C₂×#135	—	7	1	no	maximal, Sylow
40	C₂²×D₇	—	2	2	no	—
41	C₂²×D₇	D₄	1	1	yes	characteristic
42	C₇×Q₈	C₂³	1	2	yes	—
43	C₇×D₄	—	2	2	no	—
44	C₇×D₄	C₂³	1	2	yes	—
45	C₂×C₄×C₇	—	2	2	no	—
46	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²⋊D₇	—	2	2	no	—
48	C₂²⋊D₇	—	2	2	no	—
49	C₂²⋊D₇	—	2	2	no	—
50	C₂×Dic₇	—	2	2	no	—
51	C₂×Dic₇	D₄	1	1	yes	characteristic
52	D₂₈	—	2	2	no	—
53	D₂₈	C₂³	1	2	yes	—
54	C₄×D₇	—	2	2	no	—
55	C₄×D₇	—	2	2	no	—
56	C₄×D₇	—	2	2	no	—
57	C₄×D₇	D₄	1	1	yes	characteristic
58	C₄×D₇	D₄	1	1	yes	characteristic
59	C₄×D₇	C₂³	1	1	yes	characteristic
60	C₄×D₇	C₂³	1	1	yes	characteristic
61	Dic₁₄	—	2	2	no	—
62	Dic₁₄	C₂³	1	2	yes	—
63	C₇×C₈	C₂³	1	1	yes	characteristic
64	C₇×C₈	C₂³	1	1	yes	characteristic
65	C₇⋊C₈	C₂³	1	1	yes	characteristic
66	C₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×#41	—	7	2	no	—
68	#135	—	7	2	no	—
69	#135	—	7	1	no	—
70	#135	—	7	2	no	—
71	#135	—	7	1	no	—
72	#135	—	7	1	no	—
73	#135	D₇	1	1	yes	characteristic, direct product factor
74	C₂×Q₁₆	—	7	1	no	—
75	C₂×SD₁₆	—	7	2	no	—
76	C₂×D₈	—	7	1	no	—
77	C₂²×C₈	—	7	1	no	—
78	C₂²×C₇	—	2	2	no	—
79	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
80	C₂×D₇	—	2	2	no	—
81	C₂×D₇	—	2	2	no	—
82	C₂×D₇	—	4	4	no	—
83	C₂×D₇	C₂×D₄	1	1	yes	characteristic
84	C₂×D₇	C₂×D₄	1	1	yes	characteristic
85	C₄×C₇	—	2	2	no	—
86	C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₄×C₇	C₂⁴	1	1	yes	characteristic
88	Dic₇	—	2	2	no	—
89	Dic₇	C₂×D₄	1	1	yes	characteristic
90	Dic₇	C₂×D₄	1	1	yes	characteristic
91	#41	—	7	2	no	—
92	#41	—	7	2	no	—
93	#41	—	7	2	no	—
94	#41	—	14	2	no	—
95	#41	—	14	2	no	—
96	#41	C₂×D₇	1	2	yes	complemented
97	C₂×Q₈	—	7	2	no	—
98	C₂×D₄	—	7	2	no	—
99	C₂×D₄	—	14	2	no	—
100	C₂²×C₄	—	7	1	no	—
101	C₂²×C₄	—	14	2	no	—
102	Q₁₆	—	7	2	no	—
103	Q₁₆	—	7	1	no	—
104	Q₁₆	C₂×D₇	1	1	yes	characteristic, complemented
105	SD₁₆	—	7	2	no	—
106	SD₁₆	—	7	2	no	—
107	SD₁₆	—	7	2	no	—
108	SD₁₆	C₂×D₇	1	2	yes	complemented
109	D₈	—	7	2	no	—
110	D₈	—	7	1	no	—
111	D₈	C₂×D₇	1	1	yes	characteristic, complemented
112	C₂×C₈	—	7	1	no	—
113	C₂×C₈	—	7	1	no	—
114	C₂×C₈	—	7	1	no	—
115	C₂×C₈	—	7	1	no	—
116	C₂×C₈	—	7	1	no	—
117	C₂×C₈	C₂×D₇	1	1	yes	characteristic, complemented
118	C₂×C₇	—	2	1	no	—
119	C₂×C₇	—	4	2	no	—
120	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
121	D₇	—	2	1	no	—
122	D₇	—	4	2	no	—
123	D₇	#135	1	2	yes	direct product factor
124	C₂³	—	7	1	no	—
125	C₂³	—	14	2	no	—
126	Q₈	—	7	2	no	—
127	Q₈	—	14	2	no	—
128	Q₈	C₂²×D₇	1	2	yes	—
129	D₄	—	2	2	no	—
130	D₄	—	7	2	no	—
131	D₄	—	14	2	no	—
132	D₄	—	14	2	no	—
133	D₄	—	14	2	no	—
134	D₄	—	14	2	no	—
135	D₄	C₂²×D₇	1	2	yes	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	7	1	no	—
138	C₂×C₄	—	7	1	no	—
139	C₂×C₄	—	7	1	no	—
140	C₂×C₄	—	7	1	no	—
141	C₂×C₄	—	7	1	no	—
142	C₂×C₄	—	14	2	no	—
143	C₂×C₄	—	14	2	no	—
144	C₂×C₄	—	14	2	no	—
145	C₂×C₄	—	14	2	no	—
146	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
147	C₈	—	7	1	no	—
148	C₈	—	7	1	no	—
149	C₈	C₂²×D₇	1	1	yes	characteristic
150	C₈	C₂²×D₇	1	1	yes	characteristic
151	C₇	C₂×#135	1	1	yes	characteristic, complemented, Sylow
152	C₂²	—	2	2	no	—
153	C₂²	—	7	1	no	—
154	C₂²	—	7	1	no	—
155	C₂²	—	14	2	no	—
156	C₂²	—	14	2	no	—
157	C₂²	—	28	4	no	—
158	C₂²	D₄×D₇	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	7	1	no	—
161	C₄	—	7	1	no	—
162	C₄	—	14	2	no	—
163	C₄	D₄×D₇	1	1	yes	characteristic
164	C₄	C₂³×D₇	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	4	2	no	—
167	C₂	—	7	2	no	—
168	C₂	—	14	1	no	—
169	C₂	—	28	2	no	—
170	C₂	C₂×D₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₇
commutator, cosocle	C₄×C₇	C₂⁴
Frattini	C₄	C₂³×D₇
Fitting	C₇×#135	C₂
socle	C₂×C₇	C₂²×D₄
hypercenter	#135	D₇
hypocenter	C₇	C₂×#135
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. D₇×#41	C₂	semidirect
2. C₇×#135	C₂	semidirect
3. #6513	C₂	semidirect
4. #6482	C₂	semidirect
5. D₇×Q₁₆	C₂	semidirect
6. #6479	C₂	semidirect
7. D₇×SD₁₆	C₂	semidirect
8. #6475	C₂	semidirect
9. D₇×D₈	C₂	semidirect
10. #6467	C₂	semidirect
11. C₂×C₈×D₇	C₂	semidirect
12. C₇×#41	C₂²	semidirect
13. #1161	C₂²	semidirect
16. #1159	C₂²	semidirect
20. #1157	C₂²	semidirect
23. C₇×Q₁₆	C₂²	semidirect
24. C₇×SD₁₆	C₂²	semidirect
25. C₇×D₈	C₂²	semidirect
26. C₂×C₇×C₈	C₂²	semidirect
27. #1144	C₂²	semidirect
28. #1143	C₂²	semidirect
29. #1142	C₂²	semidirect
30. #1141	C₂²	semidirect
31. C₂×C₇⋊C₈	C₂²	semidirect
32. Dic₂₈	C₂²	semidirect
33. D₅₆	C₂²	semidirect
34. #1132	C₂²	semidirect
37. C₈×D₇	C₂²	semidirect
38. C₈×D₇	C₂²	semidirect
73. #135	D₇	direct (N × H)
96. #41	C₂×D₇	semidirect
104. Q₁₆	C₂×D₇	semidirect
108. SD₁₆	C₂×D₇	semidirect
111. D₈	C₂×D₇	semidirect
117. C₂×C₈	C₂×D₇	semidirect
123. D₇	#135	direct (N × H)
151. C₇	C₂×#135	semidirect

Maximal subgroups

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

subgroup	index
D₇×#41	2
C₇×#135	2
#6513	2
#6482	2
D₇×Q₁₆	2
#6479	2
D₇×SD₁₆	2
#6475	2
D₇×D₈	2
#6467	2
C₂×C₈×D₇	2
C₂×#135	7