Finite Group Databasebeta

Subgroups — #13274372

Order 1792.

202 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 67 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	Dic₇×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#89316	C₂	1	1	yes	maximal, characteristic
3	#11727071	C₂	1	1	yes	maximal, characteristic
4	C₇×#1416	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725402	C₂	1	1	yes	maximal, characteristic
6	Dic₇×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#11725382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×Q₃₂	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₇	—	2	1	no	—
10	C₂×D₁₁₂	C₂²	1	1	yes	characteristic
11	#89644	—	2	1	no	—
12	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
13	C₇×#366	C₂²	1	1	yes	characteristic
14	#89359	—	2	1	no	—
15	#89316	C₄	1	4	yes	complemented
16	#89299	C₂²	1	1	yes	characteristic
17	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₃₂	C₂²	1	1	yes	characteristic
19	#89286	—	2	1	no	—
20	C₄×SD₆₄	—	7	1	no	maximal, Sylow
21	C₂×C₇×Q₁₆	—	2	1	no	—
22	Q₈×Dic₇	—	4	1	no	—
23	C₂×D₅₆	—	2	1	no	—
24	#6456	—	4	1	no	—
25	C₇×Q₃₂	—	2	1	no	—
26	C₇×Q₃₂	C₂×C₄	1	2	yes	complemented
27	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
28	C₇×#107	—	2	1	no	—
29	#6409	—	4	1	no	—
30	#6392	—	2	1	no	—
31	C₈×Dic₇	D₄	1	1	yes	characteristic
32	C₂×C₇⋊C₁₆	D₄	1	1	yes	characteristic
33	#6383	—	4	1	no	—
34	D₁₁₂	—	2	1	no	—
35	D₁₁₂	C₂×C₄	1	2	yes	—
36	C₇×C₃₂	—	2	1	no	—
37	C₇⋊C₃₂	C₂×C₄	1	2	yes	complemented
38	C₂×SD₆₄	—	7	1	no	—
39	C₄×Q₃₂	—	7	1	no	—
40	C₄×D₁₆	—	7	1	no	—
41	#1424	—	7	1	no	—
42	#1416	D₇	1	1	yes	characteristic, complemented
43	#1415	—	7	1	no	—
44	C₄×C₃₂	—	7	1	no	—
45	C₂×C₇×Q₈	—	4	1	no	—
46	C₂×D₂₈	—	4	1	no	—
47	C₂×C₄×D₇	—	8	1	no	—
48	C₇×Q₁₆	—	2	2	no	—
49	C₇×Q₁₆	—	4	1	no	—
50	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
51	C₇×C₄⋊C₄	—	4	1	no	—
52	(C₂×D₇)⋊C₄	—	8	1	no	—
53	C₄⋊Dic₇	—	4	1	no	—
54	C₄⋊Dic₇	—	8	1	no	—
55	C₄×Dic₇	—	8	1	no	—
56	C₄×Dic₇	D₈	1	1	yes	characteristic
57	C₂×C₇⋊C₈	D₈	1	1	yes	characteristic
58	D₅₆	—	2	2	no	—
59	D₅₆	—	4	1	no	—
60	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
61	C₇×C₁₆	#41	1	1	yes	characteristic
62	C₇⋊C₁₆	—	2	1	no	—
63	C₂×Q₃₂	C₂×D₇	1	1	yes	characteristic
64	C₂×D₁₆	—	7	1	no	—
65	C₄×Q₁₆	—	14	1	no	—
66	C₄×D₈	—	14	1	no	—
67	SD₆₄	—	7	4	no	—
68	C₂×C₃₂	—	7	1	no	—
69	C₂×C₃₂	C₂×D₇	1	1	yes	characteristic
70	#366	—	7	1	no	—
71	#366	C₂×D₇	1	1	yes	characteristic
72	#358	—	14	1	no	—
73	#357	—	14	1	no	—
74	C₄×C₁₆	—	7	1	no	—
75	C₂²×D₇	—	8	1	no	—
76	C₇×Q₈	—	4	2	no	—
77	C₇×Q₈	—	8	1	no	—
78	C₂×C₄×C₇	—	8	1	no	—
79	C₂×C₄×C₇	—	8	1	no	—
80	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
81	C₂×Dic₇	—	8	1	no	—
82	C₂×Dic₇	D₁₆	1	1	yes	characteristic
83	C₂×Dic₇	D₁₆	1	1	yes	characteristic
84	D₂₈	—	4	2	no	—
85	D₂₈	—	8	1	no	—
86	C₄×D₇	—	16	2	no	—
87	C₇×C₈	C₄×D₄	1	1	yes	characteristic
88	C₇×C₈	#135	1	1	yes	characteristic
89	C₇⋊C₈	—	2	1	no	—
90	C₂×Q₁₆	—	2	1	no	—
91	C₂×D₈	—	14	1	no	—
92	C₄×Q₈	—	28	1	no	—
93	C₄×D₄	—	28	1	no	—
94	Q₃₂	—	2	1	no	—
95	Q₃₂	C₄×D₇	1	2	yes	complemented
96	D₁₆	—	7	2	no	—
97	D₁₆	—	14	1	no	—
98	C₂×C₁₆	—	7	1	no	—
99	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
100	#107	—	2	1	no	—
101	#107	—	14	1	no	—
102	#103	—	28	1	no	—
103	#102	—	28	1	no	—
104	C₄×C₈	—	7	1	no	—
105	C₃₂	—	2	1	no	—
106	C₃₂	—	7	2	no	—
107	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
108	C₂×D₇	—	8	2	no	—
109	C₂×D₇	—	16	1	no	—
110	C₂×D₇	—	16	1	no	—
111	C₄×C₇	—	8	2	no	—
112	C₄×C₇	—	16	1	no	—
113	C₄×C₇	C₄×D₈	1	1	yes	characteristic
114	C₄×C₇	#508	1	1	yes	characteristic
115	Dic₇	—	2	1	no	—
116	Dic₇	—	16	1	no	—
117	Dic₇	SD₆₄	1	2	yes	complemented
118	C₂×Q₈	—	4	1	no	—
119	C₂×D₄	—	28	1	no	—
120	C₂²×C₄	—	56	1	no	—
121	Q₁₆	—	2	2	no	—
122	Q₁₆	—	4	1	no	—
123	D₈	—	14	2	no	—
124	D₈	—	28	1	no	—
125	C₂×C₈	—	7	1	no	—
126	C₂×C₈	D₄×D₇	1	1	yes	characteristic
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	28	1	no	—
129	C₄⋊C₄	—	56	1	no	—
130	C₂²⋊C₄	—	56	1	no	—
131	C₄²	—	7	1	no	—
132	C₄²	—	56	1	no	—
133	C₁₆	—	14	1	no	—
134	C₁₆	C₂×C₄×D₇	1	1	yes	characteristic
135	C₁₆	#1159	1	1	yes	characteristic
136	C₂×C₇	C₄×D₁₆	1	1	yes	characteristic
137	C₂×C₇	C₂×SD₆₄	1	1	yes	characteristic
138	C₂×C₇	#2262	1	1	yes	characteristic
139	D₇	—	16	2	no	—
140	C₂³	—	56	1	no	—
141	Q₈	—	4	2	no	—
142	Q₈	—	8	1	no	—
143	D₄	—	28	2	no	—
144	D₄	—	56	1	no	—
145	C₂×C₄	—	7	1	no	—
146	C₂×C₄	—	7	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	56	1	no	—
150	C₂×C₄	—	112	2	no	—
151	C₂×C₄	D₇×D₈	1	1	yes	characteristic
152	C₈	—	14	1	no	—
153	C₈	(C₂²⋊D₇)⋊C₄	1	1	yes	characteristic
154	C₈	#6479	1	1	yes	characteristic
155	C₇	C₄×SD₆₄	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	56	2	no	—
157	C₂²	—	112	1	no	—
158	C₂²	—	112	1	no	—
159	C₂²	D₇×D₁₆	1	1	yes	characteristic
160	C₄	—	7	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	14	1	no	—
163	C₄	—	16	1	no	—
164	C₄	—	112	1	no	—
165	C₄	#89528	1	1	yes	characteristic
166	C₄	#89688	1	1	yes	characteristic
167	C₂	—	112	2	no	—
168	C₂	#11726928	1	1	yes	characteristic
169	C₂	D₇×SD₆₄	1	1	yes	characteristic
170	C₂	#11727137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₇×D₁₆
commutator	C₇×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇
Fitting	C₇×#1416	C₂
socle	C₂²×C₇	C₂×D₁₆
prime radical	C₂×D₁₁₂	C₂²
cosocle	C₂×C₇×C₁₆	C₂³
hypercenter	#1416	D₇
hypocenter	C₇	C₄×SD₆₄
LCS	G ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

Complements

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

N	H ≅ G/N	type
1. Dic₇×Q₃₂	C₂	semidirect
4. C₇×#1416	C₂	semidirect
6. Dic₇×C₃₂	C₂	semidirect
7. #11725382	C₂	semidirect
15. #89316	C₄	semidirect
26. C₇×Q₃₂	C₂×C₄	semidirect
37. C₇⋊C₃₂	C₂×C₄	semidirect
42. #1416	D₇	semidirect
95. Q₃₂	C₄×D₇	semidirect
117. Dic₇	SD₆₄	semidirect
155. C₇	C₄×SD₆₄	semidirect

Maximal subgroups

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

subgroup	index
Dic₇×Q₃₂	2
C₂×#89316	2
#11727071	2
C₇×#1416	2
#11725402	2
Dic₇×C₃₂	2
#11725382	2
C₄×SD₆₄	7