Finite Group Databasebeta

Subgroups — #13274720

Order 1792.

186 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 55 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	#11729223	C₂	1	1	yes	maximal, characteristic
2	D₇×#366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#1415	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725630	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₇)⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#11725397	C₂	1	1	yes	maximal, characteristic, complemented
7	#11725381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×D₁₆	C₂²	1	1	yes	characteristic
9	#89929	—	2	1	no	—
10	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
11	D₇×#107	—	2	1	no	—
12	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
13	C₇×#366	C₂²	1	1	yes	characteristic
14	#89357	—	2	1	no	—
15	#89299	C₂²	1	1	yes	characteristic
16	#89299	C₂²	1	1	yes	characteristic
17	C₂×C₇⋊C₃₂	C₂²	1	1	yes	characteristic
18	#89284	—	2	1	no	—
19	#13711	—	7	1	no	maximal, Sylow
20	C₂×C₇×D₈	—	2	1	no	—
21	#6501	—	4	1	no	—
22	C₂×C₈×D₇	D₄	1	1	yes	characteristic
23	D₇×C₄⋊C₄	—	4	1	no	—
24	C₇×D₁₆	—	2	1	no	—
25	C₇×D₁₆	—	2	1	no	—
26	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
27	C₇×#107	—	2	1	no	—
28	#6406	—	4	1	no	—
29	#6392	—	2	1	no	—
30	#6392	—	2	1	no	—
31	C₂×C₇⋊C₁₆	D₄	1	1	yes	characteristic
32	#6381	—	4	1	no	—
33	D₇×C₁₆	—	2	1	no	—
34	D₇×C₁₆	—	2	1	no	—
35	C₇×C₃₂	—	2	1	no	—
36	C₇⋊C₃₂	—	2	1	no	—
37	#2215	—	7	1	no	—
38	C₂×#366	—	7	1	no	—
39	#1423	—	7	1	no	—
40	#1423	—	7	1	no	—
41	#1415	—	7	1	no	—
42	#1415	D₇	1	1	yes	characteristic, complemented
43	C₂²⋊C₃₂	—	7	1	no	—
44	C₂×C₇×D₄	—	4	1	no	—
45	C₂×C₂²⋊D₇	—	8	1	no	—
46	C₂×C₄×D₇	—	8	1	no	—
47	C₂×C₄×D₇	D₈	1	1	yes	characteristic
48	C₇×D₈	—	4	1	no	—
49	C₇×D₈	—	4	1	no	—
50	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
51	C₇×C₄⋊C₄	—	4	1	no	—
52	C₂²⋊Dic₇	—	8	1	no	—
53	C₄⋊Dic₇	—	4	1	no	—
54	C₄⋊Dic₇	—	4	1	no	—
55	Dic₇⋊C₄	—	8	1	no	—
56	C₂×C₇⋊C₈	D₈	1	1	yes	characteristic
57	C₈×D₇	—	2	1	no	—
58	C₈×D₇	—	2	1	no	—
59	C₇×C₁₆	#41	1	1	yes	characteristic
60	C₇×C₁₆	#41	1	1	yes	characteristic
61	C₇⋊C₁₆	—	2	1	no	—
62	C₂×D₁₆	C₂×D₇	1	1	yes	characteristic
63	C₂²×C₁₆	—	7	1	no	—
64	#466	—	14	1	no	—
65	C₂×#107	—	14	1	no	—
66	C₂×C₃₂	—	7	1	no	—
67	C₂×C₃₂	C₂×D₇	1	1	yes	characteristic
68	#366	—	7	1	no	—
69	#366	—	7	1	no	—
70	#366	—	14	1	no	—
71	#366	C₂×D₇	1	1	yes	characteristic
72	#357	—	14	1	no	—
73	C₂³×C₇	—	8	1	no	—
74	C₂²×D₇	D₁₆	1	1	yes	characteristic
75	C₇×D₄	—	8	1	no	—
76	C₇×D₄	—	8	1	no	—
77	C₂×C₄×C₇	—	8	1	no	—
78	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
79	C₂²⋊D₇	—	16	2	no	—
80	C₂×Dic₇	—	8	1	no	—
81	C₂×Dic₇	—	8	1	no	—
82	C₂×Dic₇	D₁₆	1	1	yes	characteristic
83	C₄×D₇	—	2	1	no	—
84	C₄×D₇	—	2	1	no	—
85	C₄×D₇	—	16	2	no	—
86	C₇×C₈	#123	1	1	yes	characteristic
87	C₇×C₈	#137	1	1	yes	characteristic
88	C₇⋊C₈	—	2	1	no	—
89	C₂×D₈	—	2	1	no	—
90	C₂²×C₈	—	7	1	no	—
91	#121	—	28	1	no	—
92	C₂×C₄⋊C₄	—	28	1	no	—
93	D₁₆	—	2	1	no	—
94	D₁₆	—	2	1	no	—
95	C₂×C₁₆	—	7	1	no	—
96	C₂×C₁₆	—	14	1	no	—
97	C₂×C₁₆	—	14	1	no	—
98	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
99	#107	—	2	1	no	—
100	#107	—	14	1	no	—
101	#107	—	14	1	no	—
102	#107	—	28	1	no	—
103	#102	—	28	1	no	—
104	C₃₂	—	2	1	no	—
105	C₃₂	—	14	1	no	—
106	C₂²×C₇	—	16	1	no	—
107	C₂²×C₇	—	16	1	no	—
108	C₂²×C₇	—	16	1	no	—
109	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
110	C₂×D₇	—	2	1	no	—
111	C₂×D₇	—	2	1	no	—
112	C₂×D₇	D₃₂	1	2	yes	—
113	C₄×C₇	—	16	1	no	—
114	C₄×C₇	#480	1	1	yes	characteristic
115	C₄×C₇	#510	1	1	yes	characteristic
116	Dic₇	—	2	1	no	—
117	Dic₇	—	16	1	no	—
118	Dic₇	—	16	1	no	—
119	C₂×D₄	—	4	1	no	—
120	C₂×D₄	—	56	1	no	—
121	C₂²×C₄	—	7	1	no	—
122	C₂²×C₄	—	56	1	no	—
123	D₈	—	4	1	no	—
124	D₈	—	4	1	no	—
125	C₂×C₈	—	7	1	no	—
126	C₂×C₈	—	14	1	no	—
127	C₂×C₈	—	14	1	no	—
128	C₂×C₈	D₄×D₇	1	1	yes	characteristic
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	28	1	no	—
131	C₄⋊C₄	—	28	1	no	—
132	C₄⋊C₄	—	56	1	no	—
133	C₂²⋊C₄	—	56	1	no	—
134	C₁₆	—	14	1	no	—
135	C₁₆	#1157	1	1	yes	characteristic
136	C₁₆	#1159	1	1	yes	characteristic
137	C₂×C₇	—	32	1	no	—
138	C₂×C₇	#2232	1	1	yes	characteristic
139	C₂×C₇	C₂×D₃₂	1	1	yes	characteristic
140	C₂×C₇	#2264	1	1	yes	characteristic
141	D₇	—	2	2	no	—
142	C₂³	—	7	1	no	—
143	C₂³	—	8	1	no	—
144	D₄	—	8	1	no	—
145	D₄	—	8	1	no	—
146	D₄	—	112	2	no	—
147	C₂×C₄	—	7	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	14	1	no	—
150	C₂×C₄	—	14	1	no	—
151	C₂×C₄	—	56	1	no	—
152	C₂×C₄	—	56	1	no	—
153	C₂×C₄	—	112	2	no	—
154	C₂×C₄	D₇×D₈	1	1	yes	characteristic
155	C₈	—	14	1	no	—
156	C₈	#6446	1	1	yes	characteristic
157	C₈	#6478	1	1	yes	characteristic
158	C₇	#13711	1	1	yes	characteristic, complemented, Sylow
159	C₂²	—	7	2	no	—
160	C₂²	—	14	1	no	—
161	C₂²	—	14	1	no	—
162	C₂²	—	16	1	no	—
163	C₂²	—	16	1	no	—
164	C₂²	—	16	1	no	—
165	C₂²	D₇×D₁₆	1	1	yes	characteristic
166	C₄	—	14	1	no	—
167	C₄	—	16	1	no	—
168	C₄	—	112	1	no	—
169	C₄	—	112	1	no	—
170	C₄	#89546	1	1	yes	characteristic
171	C₄	#89687	1	1	yes	characteristic
172	C₂	—	14	2	no	—
173	C₂	—	32	1	no	—
174	C₂	#11726953	1	1	yes	characteristic
175	C₂	D₇×D₃₂	1	1	yes	characteristic
176	C₂	#11727135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₇×D₁₆
commutator, cosocle	C₂×C₇×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₇
Fitting	C₇×#1415	C₂
socle	C₂²×C₇	C₂×D₁₆
prime radical	#11729223	C₂
hypercenter	#1415	D₇
hypocenter	C₇	#13711
LCS	G ▷ C₂×C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1415

Complements

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

N	H ≅ G/N	type
2. D₇×#366	C₂	semidirect
3. C₇×#1415	C₂	semidirect
4. #11725630	C₂	semidirect
5. (C₂×D₇)⋊C₃₂	C₂	semidirect
6. #11725397	C₂	semidirect
7. #11725381	C₂	semidirect
42. #1415	D₇	semidirect
158. C₇	#13711	semidirect

Maximal subgroups

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

subgroup	index
#11729223	2
D₇×#366	2
C₇×#1415	2
#11725630	2
(C₂×D₇)⋊C₃₂	2
#11725397	2
#11725381	2
#13711	7