Finite Group Databasebeta

Subgroups — #13274315

Order 1792.

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

normal subgroups only

This group has 182 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	D₇×M₆₄	C₂	1	1	yes	maximal, characteristic
2	D₇×#368	C₂	1	1	yes	maximal, characteristic
3	#11727099	C₂	1	1	yes	maximal, characteristic
4	C₇×#1426	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725407	C₂	1	1	yes	maximal, characteristic, complemented
6	#11725388	C₂	1	1	yes	maximal, characteristic, complemented
7	#11725385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
9	D₇×#108	—	2	1	no	—
10	#89652	—	2	1	no	—
11	C₇×M₆₄	C₂²	1	1	yes	characteristic
12	C₇×#368	C₂²	1	1	yes	characteristic
13	C₇×#367	C₂²	1	1	yes	characteristic
14	#89301	C₂²	1	1	yes	characteristic
15	#89300	C₂²	1	1	yes	characteristic
16	#89294	C₂²	1	1	yes	characteristic
17	#89288	—	2	1	no	—
18	#89284	—	2	1	no	—
19	#89242	C₄	1	2	yes	complemented
20	D₇×C₃₂	C₄	1	2	yes	complemented
21	#13638	—	7	1	no	maximal, Sylow
22	D₇×M₁₆	—	4	1	no	—
23	C₂×C₈×D₇	D₄	1	1	yes	characteristic
24	#6455	—	4	1	no	—
25	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
26	C₇×#108	—	2	1	no	—
27	C₇×#107	—	2	1	no	—
28	#6396	—	4	1	no	—
29	#6393	—	2	1	no	—
30	#6392	—	2	1	no	—
31	C₂×C₇⋊C₁₆	D₄	1	1	yes	characteristic
32	#6382	—	4	1	no	—
33	D₇×C₁₆	C₂×C₄	1	1	yes	characteristic
34	D₇×C₁₆	C₂×C₄	1	1	yes	characteristic
35	D₇×C₁₆	Q₈	1	1	yes	characteristic
36	D₇×C₁₆	Q₈	1	1	yes	characteristic
37	C₇×C₃₂	C₂×C₄	1	2	yes	—
38	C₇⋊C₃₂	C₂×C₄	1	2	yes	—
39	C₂×M₆₄	—	7	1	no	—
40	C₂×#368	—	7	1	no	—
41	#2158	—	7	1	no	—
42	#1426	—	7	1	no	—
43	#1426	—	7	1	no	—
44	#1426	—	7	1	no	—
45	#1426	D₇	1	1	yes	characteristic, complemented
46	C₂×C₄×D₇	D₈	1	1	yes	characteristic
47	C₇×M₁₆	—	4	1	no	—
48	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
49	C₇×C₄⋊C₄	—	4	1	no	—
50	(C₂×D₇)⋊C₄	—	8	1	no	—
51	C₄⋊Dic₇	—	4	1	no	—
52	C₄×Dic₇	—	8	1	no	—
53	#1136	—	4	1	no	—
54	C₂×C₇⋊C₈	D₈	1	1	yes	characteristic
55	#1131	—	8	1	no	—
56	C₈×D₇	—	8	1	no	—
57	C₈×D₇	C₄⋊C₄	1	1	yes	characteristic
58	C₈×D₇	C₄⋊C₄	1	1	yes	characteristic
59	C₈×D₇	Q₁₆	1	1	yes	characteristic
60	C₈×D₇	Q₁₆	1	1	yes	characteristic
61	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
62	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
63	C₇⋊C₁₆	C₄⋊C₄	1	1	yes	characteristic
64	C₇⋊C₁₆	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₁₆	—	7	1	no	—
66	C₂×#108	—	14	1	no	—
67	#427	—	14	1	no	—
68	M₆₄	—	7	1	no	—
69	M₆₄	—	7	2	no	—
70	M₆₄	C₂×D₇	1	1	yes	characteristic
71	C₂×C₃₂	—	7	2	no	—
72	#368	—	7	1	no	—
73	#368	—	14	1	no	—
74	#368	C₂×D₇	1	1	yes	characteristic
75	#367	—	7	1	no	—
76	#367	C₂×D₇	1	1	yes	characteristic
77	#366	—	14	1	no	—
78	C₂²×D₇	SD₃₂	1	1	yes	characteristic
79	C₂×C₄×C₇	—	8	1	no	—
80	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
81	C₂×Dic₇	—	8	1	no	—
82	C₂×Dic₇	SD₃₂	1	1	yes	characteristic
83	C₄×D₇	#107	1	1	yes	characteristic
84	C₄×D₇	#107	1	1	yes	characteristic
85	C₄×D₇	SD₃₂	1	1	yes	characteristic
86	C₄×D₇	SD₃₂	1	1	yes	characteristic
87	C₇×C₈	—	8	1	no	—
88	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
90	C₇⋊C₈	—	8	1	no	—
91	C₇⋊C₈	#107	1	1	yes	characteristic
92	C₇⋊C₈	#107	1	1	yes	characteristic
93	C₂×M₁₆	—	28	1	no	—
94	C₂²×C₈	—	7	1	no	—
95	#117	—	28	1	no	—
96	C₂×C₁₆	—	7	1	no	—
97	C₂×C₁₆	—	7	1	no	—
98	C₂×C₁₆	—	7	1	no	—
99	C₂×C₁₆	—	7	1	no	—
100	C₂×C₁₆	—	7	1	no	—
101	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
102	#108	—	2	1	no	—
103	#108	—	14	1	no	—
104	#108	—	28	1	no	—
105	#107	—	2	1	no	—
106	#107	—	14	1	no	—
107	#106	—	28	1	no	—
108	C₃₂	—	7	2	no	—
109	C₃₂	C₄×D₇	1	2	yes	—
110	C₂²×C₇	C₂×SD₃₂	1	1	yes	characteristic
111	C₂×D₇	—	4	1	no	—
112	C₂×D₇	#367	1	1	yes	characteristic
113	C₂×D₇	#367	1	1	yes	characteristic
114	C₄×C₇	—	16	1	no	—
115	C₄×C₇	C₂×#107	1	1	yes	characteristic
116	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
117	Dic₇	—	16	1	no	—
118	Dic₇	#367	1	1	yes	characteristic
119	Dic₇	#367	1	1	yes	characteristic
120	C₂²×C₄	—	7	1	no	—
121	M₁₆	—	4	1	no	—
122	M₁₆	—	28	1	no	—
123	M₁₆	—	56	1	no	—
124	C₂×C₈	—	7	1	no	—
125	C₂×C₈	—	7	1	no	—
126	C₂×C₈	—	7	1	no	—
127	C₂×C₈	—	7	1	no	—
128	C₂×C₈	—	7	1	no	—
129	C₂×C₈	—	56	1	no	—
130	C₂×C₈	D₄×D₇	1	1	yes	characteristic
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	28	1	no	—
133	C₂²⋊C₄	—	56	1	no	—
134	C₄²	—	56	1	no	—
135	C₁₆	—	7	1	no	—
136	C₁₆	—	7	1	no	—
137	C₁₆	C₂×C₄×D₇	1	1	yes	characteristic
138	C₁₆	Q₈×D₇	1	1	yes	characteristic
139	C₂×C₇	—	2	1	no	—
140	C₂×C₇	C₂×#367	1	1	yes	characteristic
141	D₇	—	2	1	no	—
142	D₇	—	2	1	no	—
143	C₂³	—	7	1	no	—
144	C₂×C₄	—	7	1	no	—
145	C₂×C₄	—	7	1	no	—
146	C₂×C₄	—	7	1	no	—
147	C₂×C₄	—	7	1	no	—
148	C₂×C₄	—	7	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	56	1	no	—
151	C₂×C₄	D₇×D₈	1	1	yes	characteristic
152	C₈	—	7	1	no	—
153	C₈	—	7	1	no	—
154	C₈	—	8	1	no	—
155	C₈	—	56	1	no	—
156	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
157	C₈	D₇×Q₁₆	1	1	yes	characteristic
158	C₇	#13638	1	1	yes	characteristic, complemented, Sylow
159	C₂²	—	7	1	no	—
160	C₂²	—	7	1	no	—
161	C₂²	—	28	1	no	—
162	C₂²	D₇×SD₃₂	1	1	yes	characteristic
163	C₄	—	7	1	no	—
164	C₄	—	7	1	no	—
165	C₄	—	16	1	no	—
166	C₄	—	112	1	no	—
167	C₄	D₇×#107	1	1	yes	characteristic
168	C₄	D₇×SD₃₂	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	14	1	no	—
171	C₂	—	14	1	no	—
172	C₂	D₇×#367	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₇×#367
commutator	C₇×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇
Fitting	C₇×#1426	C₂
socle	C₂×C₇	C₂×#367
prime radical	C₂²×D₇	SD₃₂
cosocle	C₂×C₇×C₁₆	C₂³
hypercenter	#1426	D₇
hypocenter	C₇	#13638
LCS	G ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1426

Complements

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

N	H ≅ G/N	type
4. C₇×#1426	C₂	semidirect
5. #11725407	C₂	semidirect
6. #11725388	C₂	semidirect
7. #11725385	C₂	semidirect
19. #89242	C₄	semidirect
20. D₇×C₃₂	C₄	semidirect
45. #1426	D₇	semidirect
158. C₇	#13638	semidirect

Maximal subgroups

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

subgroup	index
D₇×M₆₄	2
D₇×#368	2
#11727099	2
C₇×#1426	2
#11725407	2
#11725388	2
#11725385	2
#13638	7