Finite Group Databasebeta

Subgroups — #13274615

Order 1792.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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×#2263	C₂	1	1	yes	maximal, characteristic, complemented
2	#11729265	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729263	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729218	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725631	C₂	1	1	yes	maximal, characteristic, complemented
6	#11725410	C₂	1	1	yes	maximal, characteristic, complemented
7	#11725405	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#508	C₂²	1	1	yes	characteristic
9	C₂×C₇×D₁₆	C₂²	1	1	yes	characteristic
10	#89968	—	2	1	no	—
11	#89923	—	2	1	no	—
12	C₂×#6400	—	2	1	no	—
13	C₂×D₁₁₂	C₂²	1	1	yes	characteristic
14	C₇×SD₆₄	—	2	1	no	—
15	C₇×D₃₂	—	2	1	no	—
16	C₇×M₆₄	C₂²	1	1	yes	characteristic
17	#89362	—	2	1	no	—
18	#89316	—	2	1	no	—
19	#89314	—	2	1	no	—
20	#89301	C₂²	1	1	yes	characteristic
21	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
22	#89294	C₂²	1	1	yes	characteristic
23	#13692	—	7	1	no	maximal, Sylow
24	C₇×#135	—	2	1	no	—
25	C₂×C₇×D₈	—	2	1	no	—
26	#6511	—	4	1	no	—
27	#6503	—	4	1	no	—
28	C₂×#1141	—	4	1	no	—
29	C₂×D₅₆	—	2	1	no	—
30	C₇×Q₃₂	D₄	1	1	yes	characteristic, complemented
31	C₇×SD₃₂	—	2	1	no	—
32	C₇×D₁₆	—	2	1	no	—
33	C₇×D₁₆	—	2	1	no	—
34	C₇×D₁₆	D₄	1	1	yes	characteristic, complemented
35	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
36	#6411	—	4	1	no	—
37	#6400	—	4	2	no	—
38	#6393	—	2	1	no	—
39	C₈×Dic₇	D₄	1	1	yes	characteristic
40	C₂×C₇⋊C₁₆	D₄	1	1	yes	characteristic
41	D₁₁₂	—	2	1	no	—
42	D₁₁₂	—	2	1	no	—
43	C₇×C₃₂	—	2	1	no	—
44	C₇⋊C₃₂	—	2	1	no	—
45	#2263	—	7	1	no	—
46	#2263	D₇	1	1	yes	characteristic, complemented
47	#2246	—	7	1	no	—
48	#2178	—	7	1	no	—
49	#1422	—	7	1	no	—
50	#1419	—	7	1	no	—
51	#1419	—	7	1	no	—
52	C₇×#41	—	4	1	no	—
53	C₂×C₇×D₄	—	4	1	no	—
54	C₂×C₂²⋊D₇	—	8	2	no	—
55	C₂×D₂₈	—	4	1	no	—
56	C₇×Q₁₆	—	2	1	no	—
57	C₇×SD₁₆	—	4	1	no	—
58	C₇×D₈	—	2	1	no	—
59	C₇×D₈	—	4	1	no	—
60	C₇×D₈	—	4	1	no	—
61	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
62	#1141	—	8	1	no	—
63	#1141	—	8	1	no	—
64	C₄×Dic₇	D₈	1	1	yes	characteristic
65	#1136	—	4	1	no	—
66	#1136	—	8	1	no	—
67	C₂×C₇⋊C₈	—	8	1	no	—
68	C₂×C₇⋊C₈	D₈	1	1	yes	characteristic
69	D₅₆	—	4	1	no	—
70	D₅₆	—	4	1	no	—
71	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
72	C₇×C₁₆	#41	1	1	yes	characteristic
73	C₇⋊C₁₆	—	2	1	no	—
74	#508	C₂×D₇	1	1	yes	characteristic
75	C₂×D₁₆	—	7	1	no	—
76	C₂×D₁₆	—	14	1	no	—
77	C₂×D₁₆	C₂×D₇	1	1	yes	characteristic
78	#493	—	14	1	no	—
79	#443	—	14	1	no	—
80	SD₆₄	—	2	1	no	—
81	SD₆₄	—	14	1	no	—
82	D₃₂	—	2	1	no	—
83	D₃₂	—	14	1	no	—
84	M₆₄	—	7	1	no	—
85	M₆₄	C₂×D₇	1	1	yes	characteristic
86	#368	—	7	1	no	—
87	#359	—	14	1	no	—
88	C₄×C₁₆	—	7	1	no	—
89	C₂³×C₇	—	8	1	no	—
90	C₂²×D₇	—	8	1	no	—
91	C₇×Q₈	—	4	1	no	—
92	C₇×D₄	—	4	1	no	—
93	C₇×D₄	—	8	1	no	—
94	C₇×D₄	—	8	1	no	—
95	C₇×D₄	—	8	1	no	—
96	C₂×C₄×C₇	—	8	1	no	—
97	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
98	C₂²⋊D₇	—	16	4	no	—
99	C₂×Dic₇	—	2	1	no	—
100	D₂₈	—	8	1	no	—
101	D₂₈	—	8	1	no	—
102	C₇×C₈	#121	1	1	yes	characteristic
103	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₇⋊C₈	—	8	1	no	—
105	C₇⋊C₈	D₁₆	1	1	yes	characteristic
106	C₇⋊C₈	D₁₆	1	1	yes	characteristic
107	#135	—	2	1	no	—
108	C₂×D₈	—	2	1	no	—
109	C₂×D₈	—	14	1	no	—
110	C₂×D₈	—	28	1	no	—
111	#131	—	28	1	no	—
112	#127	—	28	1	no	—
113	Q₃₂	C₂²⋊D₇	1	1	yes	characteristic, complemented
114	SD₃₂	—	2	1	no	—
115	D₁₆	—	2	1	no	—
116	D₁₆	—	2	1	no	—
117	D₁₆	—	14	1	no	—
118	D₁₆	—	14	1	no	—
119	D₁₆	—	28	2	no	—
120	D₁₆	C₂²⋊D₇	1	1	yes	characteristic, complemented
121	C₂×C₁₆	—	7	1	no	—
122	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
123	#108	—	14	1	no	—
124	#104	—	28	1	no	—
125	C₄×C₈	—	7	1	no	—
126	C₃₂	—	2	1	no	—
127	C₃₂	—	14	1	no	—
128	C₂²×C₇	—	8	1	no	—
129	C₂²×C₇	—	16	2	no	—
130	C₂²×C₇	—	16	1	no	—
131	C₂²×C₇	#509	1	1	yes	characteristic
132	C₂×D₇	—	16	2	no	—
133	C₂×D₇	—	16	1	no	—
134	C₄×C₇	—	8	1	no	—
135	C₄×C₇	#459	1	1	yes	characteristic
136	C₄×C₇	C₂×D₁₆	1	1	yes	characteristic
137	Dic₇	—	2	2	no	—
138	#41	—	4	1	no	—
139	C₂×D₄	—	4	1	no	—
140	C₂×D₄	—	28	1	no	—
141	C₂×D₄	—	56	2	no	—
142	Q₁₆	—	2	1	no	—
143	SD₁₆	—	4	1	no	—
144	D₈	—	2	1	no	—
145	D₈	—	4	1	no	—
146	D₈	—	4	1	no	—
147	D₈	—	28	1	no	—
148	D₈	—	28	1	no	—
149	D₈	—	56	1	no	—
150	D₈	—	56	1	no	—
151	M₁₆	—	28	1	no	—
152	M₁₆	—	56	1	no	—
153	C₂×C₈	—	7	1	no	—
154	C₂×C₈	—	56	1	no	—
155	C₂×C₈	D₄×D₇	1	1	yes	characteristic
156	C₄²	—	7	1	no	—
157	C₁₆	—	14	1	no	—
158	C₁₆	#1159	1	1	yes	characteristic
159	C₁₆	C₂×C₂²⋊D₇	1	1	yes	characteristic
160	C₂×C₇	—	2	1	no	—
161	C₂×C₇	—	16	1	no	—
162	C₂×C₇	—	32	1	no	—
163	C₂×C₇	#2206	1	1	yes	characteristic
164	D₇	—	32	1	no	—
165	C₂³	—	8	1	no	—
166	C₂³	—	56	1	no	—
167	Q₈	—	4	1	no	—
168	D₄	—	4	1	no	—
169	D₄	—	8	1	no	—
170	D₄	—	8	1	no	—
171	D₄	—	8	1	no	—
172	D₄	—	56	1	no	—
173	D₄	—	56	1	no	—
174	D₄	—	112	4	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	14	1	no	—
177	C₂×C₄	D₇×D₈	1	1	yes	characteristic
178	C₈	—	7	1	no	—
179	C₈	—	7	1	no	—
180	C₈	—	56	1	no	—
181	C₈	#6474	1	1	yes	characteristic
182	C₈	#6502	1	1	yes	characteristic
183	C₇	#13692	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	8	1	no	—
185	C₂²	—	16	2	no	—
186	C₂²	—	16	1	no	—
187	C₂²	—	112	2	no	—
188	C₂²	—	112	1	no	—
189	C₂²	#89683	1	1	yes	characteristic
190	C₄	—	8	1	no	—
191	C₄	—	14	2	no	—
192	C₄	D₇×D₁₆	1	1	yes	characteristic
193	C₄	#89922	1	1	yes	characteristic
194	C₂	—	2	1	no	—
195	C₂	—	16	1	no	—
196	C₂	—	32	1	no	—
197	C₂	—	224	1	no	—
198	C₂	#11729216	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11729216
commutator, cosocle	C₂×C₇×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₇
Fitting	C₇×#2263	C₂
socle	C₂×C₇	#2206
hypercenter	#2263	D₇
hypocenter	C₇	#13692
LCS	G ▷ C₂×C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2263

Complements

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

N	H ≅ G/N	type
1. C₇×#2263	C₂	semidirect
2. #11729265	C₂	semidirect
3. #11729263	C₂	semidirect
4. #11729218	C₂	semidirect
5. #11725631	C₂	semidirect
6. #11725410	C₂	semidirect
7. #11725405	C₂	semidirect
30. C₇×Q₃₂	D₄	semidirect
34. C₇×D₁₆	D₄	semidirect
46. #2263	D₇	semidirect
113. Q₃₂	C₂²⋊D₇	semidirect
120. D₁₆	C₂²⋊D₇	semidirect
183. C₇	#13692	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2263	2
#11729265	2
#11729263	2
#11729218	2
#11725631	2
#11725410	2
#11725405	2
#13692	7