Finite Group Databasebeta

Subgroups — #13268543

Order 1792.

246 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2254	C₂	1	1	yes	maximal, characteristic
2	#11729139	C₂	1	1	yes	maximal, characteristic
3	#11729138	C₂	1	1	yes	maximal, characteristic
4	#11729116	C₂	1	1	yes	maximal, characteristic
5	#11729115	C₂	1	1	yes	maximal, characteristic
6	#11726798	C₂	1	1	yes	maximal, characteristic
7	#11725304	C₂	1	1	yes	maximal, characteristic
8	C₇×#500	C₂²	1	1	yes	characteristic
9	C₇×#500	C₂²	1	1	yes	characteristic
10	#89865	—	2	1	no	—
11	#89864	—	2	1	no	—
12	#89853	—	2	1	no	—
13	#89850	—	2	1	no	—
14	#89454	C₂²	1	1	yes	characteristic
15	#89454	C₂²	1	1	yes	characteristic
16	C₇×#367	—	2	2	no	—
17	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
18	#89299	—	2	2	no	—
19	#89285	—	2	2	no	—
20	#89284	—	2	2	no	—
21	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
23	#12524	—	7	1	no	maximal, Sylow
24	C₇×C₄⋊Q₈	—	2	1	no	—
25	C₇×C₄⋊Q₈	—	2	1	no	—
26	C₂×#1144	—	4	2	no	—
27	#6432	—	2	1	no	—
28	#6432	—	2	1	no	—
29	C₂×C₇×C₁₆	D₄	1	2	yes	—
30	C₇×#107	—	2	1	no	—
31	C₇×#107	—	2	1	no	—
32	C₇×#107	—	2	1	no	—
33	C₇×#107	—	2	1	no	—
34	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
35	#6409	—	4	2	no	—
36	#6392	—	2	1	no	—
37	#6392	—	2	1	no	—
38	#6392	—	2	1	no	—
39	#6392	—	2	1	no	—
40	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
41	#6384	—	4	2	no	—
42	#6381	—	4	2	no	—
43	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
44	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
45	#2254	—	7	1	no	—
46	#2254	D₇	1	1	yes	characteristic
47	#2252	—	7	1	no	—
48	#2252	—	7	1	no	—
49	#1713	—	7	1	no	—
50	#1710	—	7	1	no	—
51	#1311	—	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	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
58	C₇×C₄⋊C₄	—	4	1	no	—
59	C₇×C₄⋊C₄	—	4	1	no	—
60	C₇×C₄⋊C₄	—	4	1	no	—
61	C₇×C₄⋊C₄	—	4	1	no	—
62	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
63	#1144	—	8	4	no	—
64	C₄⋊Dic₇	—	4	1	no	—
65	C₄⋊Dic₇	—	4	1	no	—
66	C₄⋊Dic₇	—	4	1	no	—
67	C₄⋊Dic₇	—	4	1	no	—
68	C₂×C₇⋊C₈	D₈	1	2	yes	—
69	C₂×C₇⋊C₈	D₈	1	2	yes	—
70	C₇×C₁₆	Q₁₆	1	4	yes	complemented
71	C₇⋊C₁₆	Q₁₆	1	4	yes	complemented
72	#500	—	7	1	no	—
73	#500	—	7	1	no	—
74	#500	—	14	1	no	—
75	#500	C₂×D₇	1	1	yes	characteristic
76	#500	C₂×D₇	1	1	yes	characteristic
77	#494	—	14	1	no	—
78	#487	—	14	1	no	—
79	#487	—	14	1	no	—
80	#367	—	2	2	no	—
81	#367	—	14	2	no	—
82	#366	—	14	2	no	—
83	#366	—	14	2	no	—
84	C₄×C₁₆	—	7	1	no	—
85	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
86	C₈²	—	7	1	no	—
87	C₇×Q₈	—	8	2	no	—
88	C₇×Q₈	—	8	2	no	—
89	C₂×C₄×C₇	—	8	1	no	—
90	C₂×C₄×C₇	—	8	1	no	—
91	C₂×C₄×C₇	#127	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
94	C₂×Dic₇	—	8	1	no	—
95	C₂×Dic₇	—	8	1	no	—
96	Dic₁₄	—	8	2	no	—
97	Dic₁₄	—	8	2	no	—
98	C₇×C₈	C₂×Q₁₆	1	2	yes	—
99	C₇×C₈	C₂×Q₁₆	1	2	yes	—
100	C₇⋊C₈	—	2	2	no	—
101	C₇⋊C₈	—	2	2	no	—
102	C₂×Q₁₆	—	28	2	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	—	2	1	no	—
105	C₄⋊Q₈	—	14	1	no	—
106	C₄⋊Q₈	—	14	1	no	—
107	C₂×C₁₆	—	7	2	no	—
108	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
109	#107	—	2	1	no	—
110	#107	—	2	1	no	—
111	#107	—	2	1	no	—
112	#107	—	2	1	no	—
113	#107	—	14	1	no	—
114	#107	—	14	1	no	—
115	#107	—	14	1	no	—
116	#107	—	14	1	no	—
117	#107	—	28	2	no	—
118	#103	—	28	2	no	—
119	#103	—	28	2	no	—
120	C₄×C₈	—	7	1	no	—
121	C₄×C₈	—	7	1	no	—
122	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
123	C₂²×C₇	#493	1	1	yes	characteristic
124	C₄×C₇	—	16	1	no	—
125	C₄×C₇	—	16	1	no	—
126	C₄×C₇	#494	1	1	yes	characteristic
127	C₄×C₇	#494	1	1	yes	characteristic
128	C₄×C₇	#509	1	2	yes	—
129	C₄×C₇	#510	1	2	yes	—
130	Dic₇	—	16	1	no	—
131	Dic₇	—	16	1	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	28	1	no	—
135	C₂×Q₈	—	28	1	no	—
136	Q₁₆	—	56	4	no	—
137	C₂×C₈	—	7	2	no	—
138	C₂×C₈	—	7	2	no	—
139	C₂×C₈	D₄×D₇	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	28	1	no	—
146	C₄⋊C₄	—	28	1	no	—
147	C₄⋊C₄	—	28	1	no	—
148	C₄⋊C₄	—	28	1	no	—
149	C₄²	D₄×D₇	1	1	yes	characteristic
150	C₁₆	—	7	4	no	—
151	C₁₆	#1144	1	4	yes	complemented
152	C₂×C₇	#1713	1	1	yes	characteristic
153	C₂×C₇	#2250	1	1	yes	characteristic
154	C₂×C₇	#2251	1	1	yes	characteristic
155	Q₈	—	8	2	no	—
156	Q₈	—	8	2	no	—
157	Q₈	—	56	2	no	—
158	Q₈	—	56	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	56	1	no	—
162	C₂×C₄	—	56	1	no	—
163	C₂×C₄	D₇×D₈	1	1	yes	characteristic
164	C₂×C₄	D₇×D₈	1	1	yes	characteristic
165	C₂×C₄	#6503	1	1	yes	characteristic
166	C₈	—	14	2	no	—
167	C₈	—	14	2	no	—
168	C₈	D₇×Q₁₆	1	2	yes	—
169	C₈	C₂×#1144	1	2	yes	—
170	C₇	#12524	1	1	yes	characteristic, complemented, Sylow
171	C₂²	#89923	1	1	yes	characteristic
172	C₄	—	16	1	no	—
173	C₄	—	16	1	no	—
174	C₄	—	112	1	no	—
175	C₄	—	112	1	no	—
176	C₄	#89686	1	2	yes	—
177	C₄	#89687	1	2	yes	—
178	C₄	#89864	1	1	yes	characteristic
179	C₄	#89953	1	1	yes	characteristic
180	C₂	#11729116	1	1	yes	characteristic
181	C₂	#11729235	1	1	yes	characteristic
182	C₂	#11729236	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89923
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#2254	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#2254	D₇
hypocenter	C₇	#12524
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
70. C₇×C₁₆	Q₁₆	semidirect
71. C₇⋊C₁₆	Q₁₆	semidirect
151. C₁₆	#1144	semidirect
170. C₇	#12524	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2254	2
#11729139	2
#11729138	2
#11729116	2
#11729115	2
#11726798	2
#11725304	2
#12524	7