Finite Group Databasebeta

Subgroups — #13268502

Order 1792.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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×#2241	C₂	1	1	yes	maximal, characteristic
2	#11729136	C₂	1	1	yes	maximal, characteristic
3	#11729115	C₂	1	1	yes	maximal, characteristic
4	#11729064	C₂	1	1	yes	maximal, characteristic
5	#11729052	C₂	1	1	yes	maximal, characteristic
6	#11726819	C₂	1	1	yes	maximal, characteristic
7	#11725304	C₂	1	1	yes	maximal, characteristic
8	C₇×#500	C₂²	1	1	yes	characteristic
9	C₇×#494	C₂²	1	1	yes	characteristic
10	#89865	—	2	1	no	—
11	#89853	—	2	1	no	—
12	#89850	—	2	1	no	—
13	#89850	—	2	1	no	—
14	#89471	C₂²	1	1	yes	characteristic
15	#89454	C₂²	1	1	yes	characteristic
16	C₇×#358	—	2	2	no	—
17	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
18	#89359	—	2	2	no	—
19	#89298	—	2	2	no	—
20	#89287	—	2	2	no	—
21	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
23	#12514	—	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	C₂×Dic₂₈	—	2	1	no	—
29	C₂×Dic₂₈	—	2	1	no	—
30	#6432	—	2	1	no	—
31	#6432	—	2	1	no	—
32	C₂×C₇×C₁₆	D₄	1	2	yes	—
33	C₇×#107	—	2	1	no	—
34	C₇×#107	—	2	1	no	—
35	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
36	#6409	—	4	2	no	—
37	#6392	—	2	1	no	—
38	#6392	—	2	1	no	—
39	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
40	#6384	—	4	2	no	—
41	#6381	—	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	#2241	—	7	1	no	—
46	#2241	—	7	1	no	—
47	#2241	—	7	1	no	—
48	#2241	D₇	1	1	yes	characteristic
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	C₇×Q₁₆	—	4	2	no	—
57	C₇×Q₁₆	—	4	2	no	—
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
60	C₇×C₄⋊C₄	—	4	1	no	—
61	C₇×C₄⋊C₄	—	4	1	no	—
62	C₇×C₄⋊C₄	—	4	1	no	—
63	C₇×C₄⋊C₄	—	4	1	no	—
64	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
65	C₄⋊Dic₇	—	4	1	no	—
66	C₄⋊Dic₇	—	4	1	no	—
67	C₄⋊Dic₇	—	4	1	no	—
68	C₄⋊Dic₇	—	4	1	no	—
69	C₂×C₇⋊C₈	D₈	1	2	yes	—
70	C₂×C₇⋊C₈	D₈	1	2	yes	—
71	Dic₂₈	—	4	2	no	—
72	Dic₂₈	—	4	2	no	—
73	C₇×C₁₆	—	2	2	no	—
74	C₇⋊C₁₆	—	2	2	no	—
75	#500	—	7	1	no	—
76	#500	—	14	1	no	—
77	#500	—	14	1	no	—
78	#500	C₂×D₇	1	1	yes	characteristic
79	#494	—	7	1	no	—
80	#494	C₂×D₇	1	1	yes	characteristic
81	#487	—	14	1	no	—
82	#487	—	14	1	no	—
83	#358	—	2	2	no	—
84	#358	—	14	2	no	—
85	#358	—	14	2	no	—
86	#358	—	14	2	no	—
87	C₄×C₁₆	—	7	1	no	—
88	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
89	C₈²	—	7	1	no	—
90	C₇×Q₈	—	8	1	no	—
91	C₇×Q₈	—	8	1	no	—
92	C₇×Q₈	—	8	2	no	—
93	C₂×C₄×C₇	—	8	1	no	—
94	C₂×C₄×C₇	—	8	1	no	—
95	C₂×C₄×C₇	#127	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
98	C₂×Dic₇	—	8	1	no	—
99	C₂×Dic₇	—	8	1	no	—
100	Dic₁₄	—	8	1	no	—
101	Dic₁₄	—	8	1	no	—
102	Dic₁₄	—	8	2	no	—
103	C₇×C₈	#135	1	2	yes	—
104	C₇×C₈	#135	1	2	yes	—
105	C₇⋊C₈	—	2	2	no	—
106	C₇⋊C₈	—	2	2	no	—
107	C₂×Q₁₆	—	2	1	no	—
108	C₂×Q₁₆	—	2	1	no	—
109	C₂×Q₁₆	—	14	1	no	—
110	C₂×Q₁₆	—	14	1	no	—
111	C₄⋊Q₈	—	2	1	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	14	1	no	—
114	C₄⋊Q₈	—	14	1	no	—
115	C₂×C₁₆	—	7	2	no	—
116	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
117	#107	—	2	1	no	—
118	#107	—	2	1	no	—
119	#107	—	14	1	no	—
120	#107	—	14	1	no	—
121	#107	—	28	2	no	—
122	#107	—	28	2	no	—
123	#103	—	28	2	no	—
124	#103	—	28	2	no	—
125	C₄×C₈	—	7	1	no	—
126	C₄×C₈	—	7	1	no	—
127	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
128	C₂²×C₇	#493	1	1	yes	characteristic
129	C₄×C₇	—	16	1	no	—
130	C₄×C₇	—	16	1	no	—
131	C₄×C₇	#495	1	1	yes	characteristic
132	C₄×C₇	#495	1	1	yes	characteristic
133	C₄×C₇	#510	1	2	yes	—
134	C₄×C₇	#510	1	2	yes	—
135	Dic₇	—	16	1	no	—
136	Dic₇	—	16	1	no	—
137	C₂×Q₈	—	4	1	no	—
138	C₂×Q₈	—	4	1	no	—
139	C₂×Q₈	—	28	1	no	—
140	C₂×Q₈	—	28	1	no	—
141	Q₁₆	—	4	2	no	—
142	Q₁₆	—	4	2	no	—
143	Q₁₆	—	28	2	no	—
144	Q₁₆	—	28	2	no	—
145	C₂×C₈	—	7	2	no	—
146	C₂×C₈	—	7	2	no	—
147	C₂×C₈	D₄×D₇	1	1	yes	characteristic
148	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	4	1	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	28	1	no	—
154	C₄⋊C₄	—	28	1	no	—
155	C₄⋊C₄	—	28	1	no	—
156	C₄⋊C₄	—	28	1	no	—
157	C₄²	D₄×D₇	1	1	yes	characteristic
158	C₁₆	—	2	2	no	—
159	C₁₆	—	14	2	no	—
160	C₂×C₇	#1709	1	1	yes	characteristic
161	C₂×C₇	#2251	1	1	yes	characteristic
162	C₂×C₇	#2251	1	1	yes	characteristic
163	Q₈	—	8	1	no	—
164	Q₈	—	8	2	no	—
165	Q₈	—	8	1	no	—
166	Q₈	—	56	1	no	—
167	Q₈	—	56	2	no	—
168	Q₈	—	56	1	no	—
169	C₂×C₄	—	8	1	no	—
170	C₂×C₄	—	8	1	no	—
171	C₂×C₄	—	56	1	no	—
172	C₂×C₄	—	56	1	no	—
173	C₂×C₄	D₇×D₈	1	1	yes	characteristic
174	C₂×C₄	D₇×D₈	1	1	yes	characteristic
175	C₂×C₄	#6503	1	1	yes	characteristic
176	C₈	—	14	2	no	—
177	C₈	—	14	2	no	—
178	C₈	#6479	1	2	yes	—
179	C₈	#6513	1	2	yes	—
180	C₇	#12514	1	1	yes	characteristic, complemented, Sylow
181	C₂²	#89923	1	1	yes	characteristic
182	C₄	—	16	1	no	—
183	C₄	—	16	1	no	—
184	C₄	—	112	1	no	—
185	C₄	—	112	1	no	—
186	C₄	#89687	1	2	yes	—
187	C₄	#89690	1	2	yes	—
188	C₄	#89831	1	1	yes	characteristic
189	C₄	#89940	1	1	yes	characteristic
190	C₂	#11728936	1	1	yes	characteristic
191	C₂	#11729236	1	1	yes	characteristic
192	C₂	#11729255	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89923
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#2241	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#2241	D₇
hypocenter	C₇	#12514
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
180. C₇	#12514	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2241	2
#11729136	2
#11729115	2
#11729064	2
#11729052	2
#11726819	2
#11725304	2
#12514	7