Subgroups — #13270231

Order 1792.

374 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×#89300	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic
4	#11728360	C₂	1	2	yes	maximal
5	C₂×#89285	C₂	1	2	yes	maximal
6	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₇×#107	C₂²	1	2	yes	—
8	C₂×#6392	C₂²	1	2	yes	—
9	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	#89748	—	2	2	no	—
12	C₂×#6381	—	2	2	no	—
13	C₇×#367	—	2	2	no	—
14	#89300	—	2	2	no	—
15	C₁₆×Dic₇	C₄	1	4	yes	complemented
16	#89285	—	2	4	no	—
17	#12906	—	7	1	no	maximal, Sylow
18	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄⋊Dic₇	—	2	2	no	—
21	C₂×Dic₇⋊C₄	—	4	4	no	—
22	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
23	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
24	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₇×C₁₆	D₄	1	2	yes	—
26	C₂×C₇×C₁₆	Q₈	1	2	yes	—
27	C₇×#107	—	2	2	no	—
28	C₇×#107	—	2	2	no	—
29	#6392	—	2	2	no	—
30	#6392	—	2	2	no	—
31	C₈×Dic₇	C₂×C₄	1	2	yes	—
32	C₈×Dic₇	Q₈	1	2	yes	—
33	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
35	C₂×C₇⋊C₁₆	Q₈	1	2	yes	—
36	#6381	—	4	4	no	—
37	C₂×#367	—	7	1	no	—
38	C₂×#367	—	7	2	no	—
39	C₂×#367	D₇	1	1	yes	characteristic
40	C₂×C₄×C₁₆	—	7	1	no	—
41	#1845	—	7	2	no	—
42	C₂²×C₄×C₇	—	4	2	no	—
43	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₇	—	4	2	no	—
45	C₂²×Dic₇	D₈	1	2	yes	—
46	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₇×C₄⋊C₄	—	4	2	no	—
53	C₇×C₄⋊C₄	—	4	2	no	—
54	C₄⋊Dic₇	—	4	2	no	—
55	C₄⋊Dic₇	—	4	2	no	—
56	Dic₇⋊C₄	—	8	8	no	—
57	C₄×Dic₇	C₄⋊C₄	1	2	yes	—
58	C₄×Dic₇	Q₁₆	1	2	yes	—
59	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂×C₇⋊C₈	D₈	1	2	yes	—
61	C₂×C₇⋊C₈	Q₁₆	1	2	yes	—
62	C₇×C₁₆	C₄⋊C₄	1	4	yes	complemented
63	C₇⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
64	C₂²×C₁₆	—	7	1	no	—
65	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
66	C₂×#107	—	7	2	no	—
67	C₂×#107	—	14	2	no	—
68	C₂×#107	C₂×D₇	1	2	yes	—
69	C₂×C₄×C₈	—	7	1	no	—
70	#384	—	14	2	no	—
71	#367	—	2	2	no	—
72	#367	—	14	4	no	—
73	#367	—	14	2	no	—
74	C₄×C₁₆	—	7	4	no	—
75	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₇	—	8	2	no	—
77	C₂×C₄×C₇	—	8	4	no	—
78	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇	#127	1	1	yes	characteristic
80	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×Dic₇	—	8	2	no	—
85	C₂×Dic₇	—	8	4	no	—
86	C₂×Dic₇	#107	1	4	yes	—
87	C₂×Dic₇	SD₃₂	1	8	yes	—
88	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
91	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
92	C₇⋊C₈	#107	1	4	yes	—
93	C₂²×C₈	—	7	1	no	—
94	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	14	2	no	—
97	C₂×C₄⋊C₄	—	28	4	no	—
98	C₂×C₄²	—	7	1	no	—
99	C₂×C₁₆	—	7	2	no	—
100	C₂×C₁₆	—	7	2	no	—
101	C₂×C₁₆	—	7	2	no	—
102	C₂×C₁₆	Dic₁₄	1	2	yes	—
103	C₂×C₁₆	C₄×D₇	1	2	yes	—
104	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
105	#107	—	2	2	no	—
106	#107	—	2	2	no	—
107	#107	—	14	2	no	—
108	#107	—	14	2	no	—
109	#107	—	28	4	no	—
110	C₄×C₈	—	7	2	no	—
111	C₄×C₈	—	7	2	no	—
112	C₂²×C₇	C₂×#107	1	1	yes	characteristic
113	C₂²×C₇	#493	1	1	yes	characteristic
114	C₂²×C₇	#500	1	1	yes	characteristic
115	C₂²×C₇	C₂×SD₃₂	1	4	yes	—
116	C₄×C₇	—	16	2	no	—
117	C₄×C₇	#384	1	1	yes	characteristic
118	C₄×C₇	C₂×#107	1	1	yes	characteristic
119	C₄×C₇	#494	1	1	yes	characteristic
120	C₄×C₇	#500	1	1	yes	characteristic
121	Dic₇	—	16	2	no	—
122	Dic₇	#367	1	8	yes	complemented
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	7	2	no	—
125	C₂²×C₄	—	28	2	no	—
126	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
127	C₂×C₈	—	7	2	no	—
128	C₂×C₈	—	7	2	no	—
129	C₂×C₈	—	7	2	no	—
130	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
132	C₂×C₈	D₄×D₇	1	1	yes	characteristic
133	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
134	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
135	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	28	2	no	—
139	C₄⋊C₄	—	28	2	no	—
140	C₄⋊C₄	—	56	8	no	—
141	C₄²	—	7	2	no	—
142	C₄²	—	7	2	no	—
143	C₁₆	—	7	4	no	—
144	C₁₆	Dic₇⋊C₄	1	4	yes	complemented
145	C₂×C₇	#1845	1	1	yes	characteristic
146	C₂×C₇	C₂×#367	1	2	yes	—
147	C₂×C₇	#2248	1	2	yes	—
148	C₂×C₇	#2254	1	2	yes	—
149	C₂³	D₇×D₈	1	1	yes	characteristic
150	C₂×C₄	—	7	4	no	—
151	C₂×C₄	—	7	8	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	4	no	—
154	C₂×C₄	—	56	2	no	—
155	C₂×C₄	—	56	4	no	—
156	C₂×C₄	#6451	1	1	yes	characteristic
157	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄	D₇×D₈	1	1	yes	characteristic
159	C₂×C₄	D₇×Q₁₆	1	1	yes	characteristic
160	C₂×C₄	D₇×Q₁₆	1	1	yes	characteristic
161	C₂×C₄	#6503	1	1	yes	characteristic
162	C₈	—	7	4	no	—
163	C₈	#6451	1	1	yes	characteristic
164	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
165	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
166	C₈	#6507	1	1	yes	characteristic
167	C₇	#12906	1	1	yes	characteristic, complemented, Sylow
168	C₂²	#89646	1	1	yes	characteristic
169	C₂²	D₇×#107	1	1	yes	characteristic
170	C₂²	D₇×SD₃₂	1	4	yes	—
171	C₂²	#89923	1	1	yes	characteristic
172	C₄	—	7	8	no	—
173	C₄	—	16	2	no	—
174	C₄	—	112	2	no	—
175	C₄	#89646	1	1	yes	characteristic
176	C₄	D₇×#107	1	1	yes	characteristic
177	C₄	#89748	1	1	yes	characteristic
178	C₄	#89953	1	1	yes	characteristic
179	C₂	#11727094	1	2	yes	—
180	C₂	D₇×#367	1	2	yes	—
181	C₂	#11728360	1	1	yes	characteristic
182	C₂	#11729233	1	2	yes	—

	subgroup	G/H
center	C₂³	D₇×D₈
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#367	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#367	D₇
hypocenter	C₇	#12906
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
15. C₁₆×Dic₇	C₄	semidirect
62. C₇×C₁₆	C₄⋊C₄	semidirect
63. C₇⋊C₁₆	C₄⋊C₄	semidirect
122. Dic₇	#367	semidirect
144. C₁₆	Dic₇⋊C₄	semidirect
167. C₇	#12906	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#367	2
C₂×#89300	2
C₂×C₁₆×Dic₇	2
#11728360	2
C₂×#89285	2
#12906	7