Finite Group Databasebeta

Subgroups — #13268113

Order 1792.

206 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 63 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	#11729143	C₂	1	1	yes	maximal, characteristic
2	#11729115	C₂	1	1	yes	maximal, characteristic
3	#11727050	C₂	1	1	yes	maximal, characteristic
4	#11726311	C₂	1	1	yes	maximal, characteristic
5	C₇×#1364	C₂	1	1	yes	maximal, characteristic
6	#11725361	C₂	1	1	yes	maximal, characteristic
7	#11725282	C₂	1	1	yes	maximal, characteristic
8	C₇×#500	C₂²	1	1	yes	characteristic
9	#89865	—	2	1	no	—
10	#89853	—	2	1	no	—
11	#89602	—	2	1	no	—
12	C₄×Dic₂₈	C₂²	1	1	yes	characteristic
13	#89454	C₂²	1	1	yes	characteristic
14	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₇×#335	C₂²	1	1	yes	characteristic
16	#89299	—	2	1	no	—
17	#89298	—	2	1	no	—
18	#89287	—	2	1	no	—
19	#89285	—	2	1	no	—
20	#89279	—	2	1	no	—
21	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
22	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
23	#12450	—	7	1	no	maximal, Sylow
24	C₇×C₄⋊Q₈	—	2	1	no	—
25	C₂×Dic₂₈	D₄	1	1	yes	characteristic
26	#6432	—	2	1	no	—
27	C₄×Dic₁₄	—	2	1	no	—
28	C₂×C₇×C₁₆	—	2	1	no	—
29	C₇×#107	—	2	1	no	—
30	C₇×#107	—	2	1	no	—
31	C₇×C₄⋊C₈	—	2	1	no	—
32	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
33	#6409	—	4	1	no	—
34	#6409	—	4	1	no	—
35	#6392	—	2	1	no	—
36	#6392	—	2	1	no	—
37	#6392	D₄	1	1	yes	characteristic
38	#6390	—	2	1	no	—
39	Dic₇⋊C₈	—	4	1	no	—
40	C₈×Dic₇	—	4	1	no	—
41	C₂×C₇⋊C₁₆	—	2	1	no	—
42	#6384	—	4	2	no	—
43	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
44	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
45	#2227	—	7	1	no	—
46	#2225	—	7	1	no	—
47	#1710	—	7	1	no	—
48	C₈×Q₁₆	—	7	1	no	—
49	#1370	—	7	1	no	—
50	#1365	—	7	1	no	—
51	#1364	D₇	1	1	yes	characteristic
52	C₂×C₇×Q₈	—	4	1	no	—
53	C₂×Dic₁₄	—	2	1	no	—
54	C₂×Dic₁₄	—	4	1	no	—
55	C₂×C₇×C₈	—	4	1	no	—
56	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₇×C₈	#41	1	1	yes	characteristic
58	C₇×C₄⋊C₄	—	4	1	no	—
59	C₇×C₄⋊C₄	—	4	1	no	—
60	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₇	—	2	1	no	—
62	C₄⋊Dic₇	—	4	1	no	—
63	C₄⋊Dic₇	—	4	1	no	—
64	Dic₇⋊C₄	—	4	1	no	—
65	C₄×Dic₇	—	4	1	no	—
66	C₂×C₇⋊C₈	—	2	1	no	—
67	C₂×C₇⋊C₈	D₈	1	1	yes	characteristic
68	C₂×C₇⋊C₈	D₈	1	1	yes	characteristic
69	Dic₂₈	—	2	1	no	—
70	Dic₂₈	Q₁₆	1	2	yes	—
71	C₇×C₁₆	—	4	1	no	—
72	C₇⋊C₁₆	—	4	1	no	—
73	#500	—	7	1	no	—
74	#500	C₂×D₇	1	1	yes	characteristic
75	#487	—	14	1	no	—
76	#487	—	14	1	no	—
77	C₈×Q₈	—	14	1	no	—
78	C₄×Q₁₆	—	7	1	no	—
79	#367	—	14	1	no	—
80	#366	—	14	1	no	—
81	C₄⋊C₁₆	—	7	1	no	—
82	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
83	#358	—	14	1	no	—
84	#358	—	14	1	no	—
85	#335	C₂×D₇	1	1	yes	characteristic
86	Q₈⋊C₈	—	14	1	no	—
87	C₈²	—	7	1	no	—
88	C₇×Q₈	—	8	1	no	—
89	C₇×Q₈	—	8	1	no	—
90	C₂×C₄×C₇	—	8	1	no	—
91	C₂×C₄×C₇	#121	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×Dic₇	—	4	1	no	—
95	C₂×Dic₇	—	4	1	no	—
96	C₂×Dic₇	—	8	1	no	—
97	Dic₁₄	—	2	2	no	—
98	Dic₁₄	—	4	1	no	—
99	Dic₁₄	—	8	2	no	—
100	C₇×C₈	—	2	1	no	—
101	C₇×C₈	—	8	1	no	—
102	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
103	C₇×C₈	#135	1	1	yes	characteristic
104	C₇⋊C₈	—	2	1	no	—
105	C₇⋊C₈	—	2	1	no	—
106	C₇⋊C₈	—	4	1	no	—
107	C₂×Q₁₆	—	7	1	no	—
108	C₄⋊Q₈	—	2	1	no	—
109	C₄⋊Q₈	—	14	1	no	—
110	C₄×Q₈	—	14	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	14	1	no	—
113	#107	—	2	1	no	—
114	#107	—	2	1	no	—
115	#107	—	7	1	no	—
116	#107	—	14	1	no	—
117	#107	—	14	1	no	—
118	C₄⋊C₈	—	2	1	no	—
119	C₄⋊C₈	—	28	1	no	—
120	#103	—	14	1	no	—
121	#103	—	28	2	no	—
122	#103	—	28	1	no	—
123	#103	—	28	1	no	—
124	C₄×C₈	—	7	1	no	—
125	C₄×C₈	—	7	1	no	—
126	C₄×C₈	—	28	1	no	—
127	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
128	C₂²×C₇	#459	1	1	yes	characteristic
129	C₄×C₇	—	2	1	no	—
130	C₄×C₇	—	16	1	no	—
131	C₄×C₇	#467	1	1	yes	characteristic
132	C₄×C₇	#473	1	1	yes	characteristic
133	C₄×C₇	#508	1	1	yes	characteristic
134	C₄×C₇	#510	1	1	yes	characteristic
135	Dic₇	—	4	2	no	—
136	Dic₇	—	8	1	no	—
137	Dic₇	—	16	1	no	—
138	C₂×Q₈	—	4	1	no	—
139	C₂×Q₈	—	14	1	no	—
140	C₂×Q₈	—	28	1	no	—
141	Q₁₆	—	7	2	no	—
142	Q₁₆	—	14	1	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	7	1	no	—
145	C₂×C₈	—	7	1	no	—
146	C₂×C₈	—	14	1	no	—
147	C₂×C₈	#1157	1	1	yes	characteristic
148	C₂×C₈	D₄×D₇	1	1	yes	characteristic
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	14	1	no	—
152	C₄⋊C₄	—	28	1	no	—
153	C₄⋊C₄	—	28	1	no	—
154	C₄⋊C₄	—	28	1	no	—
155	C₄²	—	28	1	no	—
156	C₄²	D₄×D₇	1	1	yes	characteristic
157	C₁₆	—	4	1	no	—
158	C₁₆	—	28	1	no	—
159	C₂×C₇	#1669	1	1	yes	characteristic
160	C₂×C₇	#2211	1	1	yes	characteristic
161	C₂×C₇	#2212	1	1	yes	characteristic
162	Q₈	—	8	1	no	—
163	Q₈	—	8	1	no	—
164	Q₈	—	14	2	no	—
165	Q₈	—	28	1	no	—
166	Q₈	—	56	2	no	—
167	C₂×C₄	—	8	1	no	—
168	C₂×C₄	—	28	1	no	—
169	C₂×C₄	—	28	1	no	—
170	C₂×C₄	—	56	1	no	—
171	C₂×C₄	#6447	1	1	yes	characteristic
172	C₂×C₄	D₇×D₈	1	1	yes	characteristic
173	C₂×C₄	#6477	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	8	1	no	—
176	C₈	—	14	1	no	—
177	C₈	—	14	1	no	—
178	C₈	—	28	1	no	—
179	C₈	#6479	1	1	yes	characteristic
180	C₈	D₇×Q₁₆	1	1	yes	characteristic
181	C₇	#12450	1	1	yes	characteristic, complemented, Sylow
182	C₂²	#89558	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	16	1	no	—
185	C₄	—	28	2	no	—
186	C₄	—	56	1	no	—
187	C₄	—	112	1	no	—
188	C₄	#89527	1	1	yes	characteristic
189	C₄	#89585	1	1	yes	characteristic
190	C₄	#89684	1	1	yes	characteristic
191	C₄	#89687	1	1	yes	characteristic
192	C₂	#11726195	1	1	yes	characteristic
193	C₂	#11726959	1	1	yes	characteristic
194	C₂	#11727026	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89558
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1364	C₂
socle, prime radical	C₂²×C₇	#459
hypercenter	#1364	D₇
hypocenter	C₇	#12450
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1364

Complements

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

N	H ≅ G/N	type
181. C₇	#12450	semidirect

Maximal subgroups

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

subgroup	index
#11729143	2
#11729115	2
#11727050	2
#11726311	2
C₇×#1364	2
#11725361	2
#11725282	2
#12450	7