Finite Group Databasebeta

Subgroups — #90464

Order 448.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	1	yes	maximal, characteristic, complemented
2	C₂×#1159	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	#6514	C₂	1	1	yes	maximal, characteristic, complemented
5	#6495	C₂	1	1	yes	maximal, characteristic, complemented
6	#6479	C₂	1	2	yes	maximal, complemented
7	#6478	C₂	1	2	yes	maximal, complemented
8	#6475	C₂	1	2	yes	maximal, complemented
9	#6474	C₂	1	2	yes	maximal, complemented
10	#6472	C₂	1	1	yes	maximal, characteristic, complemented
11	D₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#41	C₂²	1	1	yes	characteristic
13	C₂×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊D₇	—	2	1	no	—
15	C₂²×Dic₇	—	2	1	no	—
16	#1161	C₂²	1	1	yes	characteristic, complemented
17	Q₈×D₇	C₂²	1	1	yes	characteristic, complemented
18	#1159	—	2	1	no	—
19	#1159	—	2	1	no	—
20	#1159	—	2	1	no	—
21	#1159	C₂²	1	2	yes	—
22	#1159	C₂²	1	1	yes	characteristic, complemented
23	#1159	C₂²	1	2	yes	complemented
24	D₄×D₇	—	2	1	no	—
25	D₄×D₇	C₂²	1	1	yes	characteristic, complemented
26	#1157	—	2	1	no	—
27	#1157	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₇	—	2	1	no	—
29	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
30	C₂×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
31	C₇×SD₁₆	C₂²	1	2	yes	complemented
32	C₇×D₈	C₂²	1	2	yes	complemented
33	C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
34	#1144	C₂²	1	2	yes	complemented
35	#1142	C₂²	1	2	yes	complemented
36	#1142	C₂²	1	2	yes	complemented
37	#1141	C₂²	1	2	yes	complemented
38	#1136	C₂²	1	1	yes	characteristic, complemented
39	Dic₂₈	C₂²	1	2	yes	complemented
40	#1132	C₂²	1	2	yes	complemented
41	#1131	C₂²	1	2	yes	complemented
42	C₈×D₇	C₂²	1	2	yes	complemented
43	#575	—	7	1	no	maximal, Sylow
44	C₂³×C₇	—	2	1	no	—
45	C₂²×D₇	—	2	1	no	—
46	C₂²×D₇	D₄	1	1	yes	characteristic
47	C₇×Q₈	C₂³	1	1	yes	characteristic
48	C₇×D₄	—	2	1	no	—
49	C₇×D₄	—	2	1	no	—
50	C₇×D₄	C₂³	1	2	yes	—
51	C₇×D₄	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₇	—	2	1	no	—
53	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂²⋊D₇	—	2	1	no	—
55	C₂²⋊D₇	—	2	1	no	—
56	C₂²⋊D₇	—	2	2	no	—
57	C₂²⋊D₇	—	2	2	no	—
58	C₂²⋊D₇	—	2	1	no	—
59	C₂×Dic₇	—	2	1	no	—
60	C₂×Dic₇	—	2	1	no	—
61	C₂×Dic₇	—	2	2	no	—
62	C₂×Dic₇	—	2	2	no	—
63	C₂×Dic₇	D₄	1	1	yes	characteristic
64	D₂₈	—	2	1	no	—
65	D₂₈	C₂³	1	1	yes	characteristic
66	C₄×D₇	—	2	1	no	—
67	C₄×D₇	—	2	1	no	—
68	C₄×D₇	—	2	1	no	—
69	C₄×D₇	D₄	1	1	yes	characteristic
70	C₄×D₇	D₄	1	1	yes	characteristic
71	C₄×D₇	C₂³	1	1	yes	characteristic
72	C₄×D₇	C₂³	1	1	yes	characteristic
73	Dic₁₄	—	2	1	no	—
74	Dic₁₄	—	2	1	no	—
75	Dic₁₄	C₂³	1	2	yes	—
76	Dic₁₄	C₂³	1	1	yes	characteristic
77	C₇×C₈	C₂³	1	2	yes	—
78	C₇⋊C₈	C₂³	1	2	yes	—
79	C₂×#41	—	7	1	no	—
80	C₂×#41	—	7	1	no	—
81	#137	—	7	1	no	—
82	#137	—	7	2	no	—
83	#137	—	7	1	no	—
84	AGL(1,ℤ/8)	—	7	1	no	—
85	AGL(1,ℤ/8)	—	7	2	no	—
86	AGL(1,ℤ/8)	D₇	1	1	yes	characteristic, complemented
87	#135	—	7	2	no	—
88	#135	—	7	2	no	—
89	C₂×M₁₆	—	7	1	no	—
90	C₂²×C₇	—	2	2	no	—
91	C₂²×C₇	—	2	1	no	—
92	C₂²×C₇	—	4	2	no	—
93	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
94	C₂×D₇	—	2	1	no	—
95	C₂×D₇	—	4	1	no	—
96	C₂×D₇	—	4	2	no	—
97	C₂×D₇	C₂×D₄	1	1	yes	characteristic
98	C₂×D₇	C₂×D₄	1	1	yes	characteristic
99	C₄×C₇	—	2	1	no	—
100	C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₄×C₇	C₂⁴	1	1	yes	characteristic
102	Dic₇	—	2	1	no	—
103	Dic₇	—	2	2	no	—
104	Dic₇	C₂×D₄	1	1	yes	characteristic
105	Dic₇	C₂×D₄	1	1	yes	characteristic
106	#41	—	7	1	no	—
107	#41	—	7	2	no	—
108	#41	—	7	1	no	—
109	#41	—	7	2	no	—
110	#41	—	7	1	no	—
111	#41	—	14	1	no	—
112	#41	—	14	1	no	—
113	#41	—	14	1	no	—
114	#41	—	14	1	no	—
115	#41	C₂×D₇	1	1	yes	characteristic
116	C₂×Q₈	—	7	1	no	—
117	C₂×Q₈	—	7	1	no	—
118	C₂×D₄	—	7	1	no	—
119	C₂×D₄	—	14	1	no	—
120	C₂×D₄	—	14	1	no	—
121	C₂×D₄	C₂×D₇	1	1	yes	characteristic, complemented
122	C₂²×C₄	—	7	1	no	—
123	C₂²×C₄	—	14	1	no	—
124	C₂²×C₄	—	14	1	no	—
125	Q₁₆	—	7	2	no	—
126	Q₁₆	—	7	2	no	—
127	SD₁₆	—	7	2	no	—
128	SD₁₆	—	7	2	no	—
129	SD₁₆	—	7	2	no	—
130	SD₁₆	C₂×D₇	1	2	yes	complemented
131	D₈	—	7	2	no	—
132	D₈	C₂×D₇	1	2	yes	complemented
133	M₁₆	—	7	1	no	—
134	M₁₆	—	7	2	no	—
135	M₁₆	C₂×D₇	1	1	yes	characteristic, complemented
136	C₂×C₈	—	7	2	no	—
137	C₂×C₇	—	2	1	no	—
138	C₂×C₇	—	4	1	no	—
139	C₂×C₇	—	4	2	no	—
140	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
141	D₇	—	2	1	no	—
142	D₇	—	2	1	no	—
143	D₇	—	4	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	—	7	1	no	—
146	C₂³	—	14	1	no	—
147	Q₈	—	7	2	no	—
148	Q₈	—	7	1	no	—
149	Q₈	—	14	1	no	—
150	Q₈	—	14	1	no	—
151	Q₈	C₂²×D₇	1	1	yes	characteristic
152	D₄	—	2	1	no	—
153	D₄	—	2	1	no	—
154	D₄	—	7	1	no	—
155	D₄	—	14	1	no	—
156	D₄	—	14	1	no	—
157	D₄	—	14	1	no	—
158	D₄	—	14	2	no	—
159	D₄	—	14	2	no	—
160	D₄	—	14	1	no	—
161	D₄	C₂²×D₇	1	2	yes	—
162	D₄	C₂²×D₇	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	7	1	no	—
165	C₂×C₄	—	7	1	no	—
166	C₂×C₄	—	7	1	no	—
167	C₂×C₄	—	7	1	no	—
168	C₂×C₄	—	7	1	no	—
169	C₂×C₄	—	14	1	no	—
170	C₂×C₄	—	14	1	no	—
171	C₂×C₄	—	14	1	no	—
172	C₂×C₄	—	14	2	no	—
173	C₂×C₄	—	14	2	no	—
174	C₂×C₄	—	14	1	no	—
175	C₂×C₄	—	14	1	no	—
176	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
177	C₈	—	7	2	no	—
178	C₈	C₂²×D₇	1	2	yes	—
179	C₇	#575	1	1	yes	characteristic, complemented, Sylow
180	C₂²	—	2	1	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	7	1	no	—
184	C₂²	—	7	1	no	—
185	C₂²	—	14	1	no	—
186	C₂²	—	28	1	no	—
187	C₂²	—	28	2	no	—
188	C₂²	D₄×D₇	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	7	1	no	—
191	C₄	—	7	1	no	—
192	C₄	—	14	1	no	—
193	C₄	—	14	2	no	—
194	C₄	D₄×D₇	1	1	yes	characteristic
195	C₄	C₂³×D₇	1	1	yes	characteristic
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	—	4	1	no	—
199	C₂	—	14	1	no	—
200	C₂	—	14	1	no	—
201	C₂	—	28	1	no	—
202	C₂	C₂×D₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₇
commutator, cosocle	C₄×C₇	C₂⁴
Frattini	C₄	C₂³×D₇
Fitting	C₇×AGL(1,ℤ/8)	C₂
socle	C₂×C₇	C₂²×D₄
hypercenter	AGL(1,ℤ/8)	D₇
hypocenter	C₇	#575
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. D₇×#41	C₂	semidirect
2. C₂×#1159	C₂	semidirect
3. C₇×AGL(1,ℤ/8)	C₂	semidirect
4. #6514	C₂	semidirect
5. #6495	C₂	semidirect
6. #6479	C₂	semidirect
7. #6478	C₂	semidirect
8. #6475	C₂	semidirect
9. #6474	C₂	semidirect
10. #6472	C₂	semidirect
11. D₇×M₁₆	C₂	semidirect
13. C₂×C₇×D₄	C₂²	semidirect
16. #1161	C₂²	semidirect
17. Q₈×D₇	C₂²	semidirect
22. #1159	C₂²	semidirect
23. #1159	C₂²	semidirect
25. D₄×D₇	C₂²	semidirect
30. C₂×Dic₁₄	C₂²	semidirect
31. C₇×SD₁₆	C₂²	semidirect
32. C₇×D₈	C₂²	semidirect
33. C₇×M₁₆	C₂²	semidirect
34. #1144	C₂²	semidirect
35. #1142	C₂²	semidirect
36. #1142	C₂²	semidirect
37. #1141	C₂²	semidirect
38. #1136	C₂²	semidirect
39. Dic₂₈	C₂²	semidirect
40. #1132	C₂²	semidirect
41. #1131	C₂²	semidirect
42. C₈×D₇	C₂²	semidirect
86. AGL(1,ℤ/8)	D₇	semidirect
121. C₂×D₄	C₂×D₇	semidirect
130. SD₁₆	C₂×D₇	semidirect
132. D₈	C₂×D₇	semidirect
135. M₁₆	C₂×D₇	semidirect
179. C₇	#575	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
C₂×#1159	2
C₇×AGL(1,ℤ/8)	2
#6514	2
#6495	2
#6479	2
#6478	2
#6475	2
#6474	2
#6472	2
D₇×M₁₆	2
#575	7