Finite Group Databasebeta

Subgroups — #13268590

Order 1792.

238 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₇×#1707	C₂	1	1	yes	maximal, characteristic
2	#11729139	C₂	1	1	yes	maximal, characteristic
3	#11729138	C₂	1	1	yes	maximal, characteristic
4	#11729064	C₂	1	1	yes	maximal, characteristic
5	#11729064	C₂	1	1	yes	maximal, characteristic
6	#11725818	C₂	1	1	yes	maximal, characteristic
7	#11725244	C₂	1	1	yes	maximal, characteristic
8	C₇×#500	C₂²	1	1	yes	characteristic
9	C₇×#498	—	2	1	no	—
10	C₇×#494	C₂²	1	1	yes	characteristic
11	C₇×#487	—	2	1	no	—
12	#89454	C₂²	1	1	yes	characteristic
13	#89454	C₂²	1	1	yes	characteristic
14	#89453	—	2	1	no	—
15	#89453	—	2	1	no	—
16	C₇×C₈²	C₂²	1	1	yes	characteristic
17	#89359	—	2	2	no	—
18	#89359	—	2	2	no	—
19	#89285	—	2	2	no	—
20	#89284	—	2	2	no	—
21	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
23	#12533	—	7	1	no	maximal, Sylow
24	C₂×C₇×Q₁₆	—	2	1	no	—
25	C₂×C₇×Q₁₆	—	2	1	no	—
26	C₇×C₄⋊Q₈	—	2	1	no	—
27	C₇×C₄⋊Q₈	—	2	1	no	—
28	#6432	—	2	1	no	—
29	#6432	—	2	1	no	—
30	C₇×#107	—	2	1	no	—
31	C₇×#107	—	2	1	no	—
32	C₇×#106	—	4	2	no	—
33	C₇×#103	—	4	2	no	—
34	C₄×C₇×C₈	D₄	1	1	yes	characteristic
35	C₄×C₇×C₈	D₄	1	1	yes	characteristic
36	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
37	#6392	—	2	1	no	—
38	#6392	—	2	1	no	—
39	#6392	—	2	1	no	—
40	#6392	—	2	1	no	—
41	#6390	—	4	2	no	—
42	#6390	—	4	2	no	—
43	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
44	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
45	#2254	—	7	1	no	—
46	#2252	—	7	1	no	—
47	#2241	—	7	1	no	—
48	#2241	—	7	1	no	—
49	#1710	—	7	1	no	—
50	#1707	D₇	1	1	yes	characteristic
51	#1312	—	7	1	no	—
52	C₂×C₇×Q₈	—	4	1	no	—
53	C₂×C₇×Q₈	—	4	1	no	—
54	C₂×Dic₁₄	—	4	1	no	—
55	C₂×Dic₁₄	—	4	1	no	—
56	C₇×Q₁₆	—	4	2	no	—
57	C₇×Q₁₆	—	4	2	no	—
58	C₂×C₇×C₈	D₈	1	2	yes	—
59	C₂×C₇×C₈	D₈	1	2	yes	—
60	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
62	C₇×C₄⋊C₄	—	4	1	no	—
63	C₇×C₄⋊C₄	—	4	1	no	—
64	C₇×C₄⋊C₄	—	4	1	no	—
65	C₇×C₄⋊C₄	—	4	1	no	—
66	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
67	C₄⋊Dic₇	—	4	1	no	—
68	C₄⋊Dic₇	—	4	1	no	—
69	C₄⋊Dic₇	—	4	1	no	—
70	C₄⋊Dic₇	—	4	1	no	—
71	C₇⋊C₁₆	—	2	2	no	—
72	C₇⋊C₁₆	—	2	2	no	—
73	#500	—	7	1	no	—
74	#500	—	7	1	no	—
75	#500	C₂×D₇	1	1	yes	characteristic
76	#498	—	2	1	no	—
77	#494	C₂×D₇	1	1	yes	characteristic
78	#487	—	2	1	no	—
79	#487	—	14	1	no	—
80	#487	—	14	1	no	—
81	#367	—	14	2	no	—
82	#366	—	14	2	no	—
83	#358	—	14	2	no	—
84	#358	—	14	2	no	—
85	C₄×C₁₆	—	7	1	no	—
86	C₄×C₁₆	—	7	1	no	—
87	C₈²	C₂×D₇	1	1	yes	characteristic
88	C₇×Q₈	—	8	1	no	—
89	C₇×Q₈	—	8	1	no	—
90	C₇×Q₈	—	8	2	no	—
91	C₂×C₄×C₇	—	8	1	no	—
92	C₂×C₄×C₇	—	8	1	no	—
93	C₂×C₄×C₇	#127	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
96	C₂×Dic₇	—	8	1	no	—
97	C₂×Dic₇	—	8	1	no	—
98	Dic₁₄	—	8	2	no	—
99	Dic₁₄	—	8	2	no	—
100	C₇×C₈	—	2	2	no	—
101	C₇×C₈	—	2	2	no	—
102	C₇×C₈	#137	1	2	yes	—
103	C₇×C₈	#137	1	2	yes	—
104	C₂×Q₁₆	—	2	1	no	—
105	C₂×Q₁₆	—	2	1	no	—
106	C₄⋊Q₈	—	2	1	no	—
107	C₄⋊Q₈	—	2	1	no	—
108	C₄⋊Q₈	—	14	1	no	—
109	C₄⋊Q₈	—	14	1	no	—
110	C₂×C₁₆	—	7	2	no	—
111	C₂×C₁₆	—	7	2	no	—
112	#107	—	2	1	no	—
113	#107	—	2	1	no	—
114	#107	—	14	1	no	—
115	#107	—	14	1	no	—
116	#107	—	14	1	no	—
117	#107	—	14	1	no	—
118	#106	—	4	2	no	—
119	#103	—	4	2	no	—
120	#103	—	28	2	no	—
121	#103	—	28	2	no	—
122	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
123	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
125	C₂²×C₇	#493	1	1	yes	characteristic
126	C₄×C₇	—	16	1	no	—
127	C₄×C₇	—	16	1	no	—
128	C₄×C₇	#497	1	1	yes	characteristic
129	C₄×C₇	#497	1	1	yes	characteristic
130	C₄×C₇	#508	1	2	yes	—
131	C₄×C₇	#510	1	2	yes	—
132	Dic₇	—	16	1	no	—
133	Dic₇	—	16	1	no	—
134	C₂×Q₈	—	4	1	no	—
135	C₂×Q₈	—	4	1	no	—
136	C₂×Q₈	—	28	1	no	—
137	C₂×Q₈	—	28	1	no	—
138	Q₁₆	—	4	2	no	—
139	Q₁₆	—	4	2	no	—
140	C₂×C₈	#1141	1	2	yes	—
141	C₂×C₈	#1141	1	2	yes	—
142	C₂×C₈	D₄×D₇	1	1	yes	characteristic
143	C₂×C₈	D₄×D₇	1	1	yes	characteristic
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	28	1	no	—
149	C₄⋊C₄	—	28	1	no	—
150	C₄⋊C₄	—	28	1	no	—
151	C₄⋊C₄	—	28	1	no	—
152	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
153	C₁₆	—	14	2	no	—
154	C₁₆	—	14	2	no	—
155	C₂×C₇	#1722	1	1	yes	characteristic
156	C₂×C₇	#2249	1	1	yes	characteristic
157	C₂×C₇	#2251	1	1	yes	characteristic
158	Q₈	—	8	1	no	—
159	Q₈	—	8	1	no	—
160	Q₈	—	8	2	no	—
161	Q₈	—	56	2	no	—
162	Q₈	—	56	2	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	56	1	no	—
166	C₂×C₄	—	56	1	no	—
167	C₂×C₄	C₂×#1141	1	1	yes	characteristic
168	C₂×C₄	C₂×#1141	1	1	yes	characteristic
169	C₂×C₄	#6503	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	#6478	1	2	yes	—
173	C₈	#6478	1	2	yes	—
174	C₇	#12533	1	1	yes	characteristic, complemented, Sylow
175	C₂²	#89845	1	1	yes	characteristic
176	C₄	—	16	1	no	—
177	C₄	—	16	1	no	—
178	C₄	—	112	1	no	—
179	C₄	—	112	1	no	—
180	C₄	#89939	1	1	yes	characteristic
181	C₄	#89939	1	1	yes	characteristic
182	C₄	#89951	1	2	yes	—
183	C₄	#89966	1	2	yes	—
184	C₂	#11729017	1	1	yes	characteristic
185	C₂	#11729068	1	1	yes	characteristic
186	C₂	#11729074	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89845
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1707	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#1707	D₇
hypocenter	C₇	#12533
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

Complements

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

N	H ≅ G/N	type
174. C₇	#12533	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1707	2
#11729139	2
#11729138	2
#11729064	2
#11725818	2
#11725244	2
#12533	7