Subgroups — #13247533

Order 1792.

462 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1842	C₂	1	1	yes	maximal, characteristic
2	C₂×#89275	C₂	1	4	yes	maximal
3	#11727148	C₂	1	1	yes	maximal, characteristic
4	C₄²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×C₄⋊C₈	—	2	2	no	—
6	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄³×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₇⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
14	#89275	—	2	8	no	—
15	#8805	—	7	1	no	maximal, Sylow
16	C₂²×C₇×C₈	—	2	2	no	—
17	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₇	D₄	1	2	yes	—
20	C₂×C₄²×C₇	Q₈	1	2	yes	—
21	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
22	C₂²×C₇⋊C₈	—	2	2	no	—
23	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₇×C₄⋊C₈	—	2	4	no	—
26	C₇×C₄⋊C₈	—	4	4	no	—
27	C₄⋊(C₇⋊C₈)	—	2	4	no	—
28	C₄⋊(C₇⋊C₈)	—	4	4	no	—
29	C₄×C₇⋊C₈	—	2	2	no	—
30	C₄×C₇⋊C₈	—	2	4	no	—
31	C₄×C₇⋊C₈	C₈	1	4	yes	complemented
32	C₄×C₇⋊C₈	D₄	1	4	yes	—
33	C₄×C₇⋊C₈	Q₈	1	4	yes	—
34	#1842	—	7	1	no	—
35	#1842	D₇	1	1	yes	characteristic
36	C₄²×C₈	—	7	1	no	—
37	C₂×#334	—	7	4	no	—
38	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₇×C₈	—	4	2	no	—
44	C₂×C₇×C₈	—	4	4	no	—
45	C₄²×C₇	—	2	4	no	—
46	C₄²×C₇	—	2	2	no	—
47	C₄²×C₇	C₂×C₈	1	2	yes	—
48	C₄²×C₇	M₁₆	1	2	yes	—
49	C₄²×C₇	SD₁₆	1	4	yes	—
50	C₄²×C₇	SD₁₆	1	4	yes	—
51	C₄²×C₇	C₂×D₄	1	2	yes	—
52	C₄²×C₇	C₂×Q₈	1	2	yes	—
53	C₂×C₇⋊C₈	—	2	2	no	—
54	C₂×C₇⋊C₈	—	2	2	no	—
55	C₂×C₇⋊C₈	—	4	2	no	—
56	C₂×C₇⋊C₈	—	4	4	no	—
57	C₂×C₇⋊C₈	C₄⋊C₄	1	4	yes	—
58	C₂×C₇⋊C₈	C₄⋊C₄	1	4	yes	—
59	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
60	C₂×C₇⋊C₈	M₁₆	1	4	yes	—
61	C₂×C₄⋊C₈	—	2	2	no	—
62	C₂×C₄⋊C₈	—	7	2	no	—
63	C₂×C₄⋊C₈	—	14	2	no	—
64	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
65	C₂×C₄×C₈	—	7	1	no	—
66	C₂×C₄×C₈	—	7	2	no	—
67	C₂×C₄×C₈	—	7	2	no	—
68	C₂×C₄×C₈	—	7	1	no	—
69	C₄³	C₂×D₇	1	1	yes	characteristic
70	#334	—	14	8	no	—
71	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
76	C₂×C₄×C₇	#106	1	4	yes	—
77	C₂×C₄×C₇	#106	1	4	yes	—
78	C₂×C₄×C₇	#108	1	4	yes	—
79	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₇	#127	1	1	yes	characteristic
81	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
88	C₇×C₈	—	8	2	no	—
89	C₇⋊C₈	—	2	4	no	—
90	C₇⋊C₈	—	8	2	no	—
91	C₇⋊C₈	C₄⋊C₈	1	8	yes	complemented
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	7	2	no	—
94	C₂²×C₈	—	7	2	no	—
95	C₂²×C₈	—	14	2	no	—
96	C₂×C₄²	Dic₁₄	1	2	yes	—
97	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
98	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
100	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
101	C₄⋊C₈	—	2	4	no	—
102	C₄⋊C₈	—	4	4	no	—
103	C₄⋊C₈	—	14	4	no	—
104	C₄⋊C₈	—	28	4	no	—
105	C₄×C₈	—	7	4	no	—
106	C₄×C₈	—	7	4	no	—
107	C₄×C₈	—	7	4	no	—
108	C₄×C₈	—	14	4	no	—
109	C₄×C₈	—	14	2	no	—
110	C₂²×C₇	#384	1	1	yes	characteristic
111	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₇	#423	1	1	yes	characteristic
113	C₂²×C₇	C₂×#106	1	2	yes	—
114	C₂²×C₇	C₂×#108	1	2	yes	—
115	C₄×C₇	—	2	4	no	—
116	C₄×C₇	—	2	2	no	—
117	C₄×C₇	#334	1	8	yes	complemented
118	C₄×C₇	C₂×C₄⋊C₈	1	2	yes	—
119	C₄×C₇	#423	1	2	yes	—
120	C₄×C₇	#492	1	1	yes	characteristic
121	C₄×C₇	#492	1	1	yes	characteristic
122	C₄×C₇	#498	1	2	yes	—
123	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
124	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
125	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
128	C₂×C₈	—	4	4	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	7	4	no	—
131	C₂×C₈	—	7	4	no	—
132	C₂×C₈	—	7	4	no	—
133	C₂×C₈	—	7	4	no	—
134	C₂×C₈	—	14	2	no	—
135	C₂×C₈	—	14	2	no	—
136	C₂×C₈	—	28	2	no	—
137	C₂×C₈	—	28	4	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	4	no	—
140	C₄²	C₈×D₇	1	2	yes	—
141	C₄²	#1131	1	2	yes	—
142	C₄²	#1142	1	4	yes	—
143	C₄²	#1143	1	4	yes	—
144	C₄²	D₄×D₇	1	2	yes	—
145	C₄²	Q₈×D₇	1	2	yes	—
146	C₂×C₇	C₂×#334	1	2	yes	—
147	C₂×C₇	#1566	1	2	yes	—
148	C₂×C₇	#1842	1	1	yes	characteristic
149	C₂×C₇	#1844	1	1	yes	characteristic
150	C₂×C₇	#1848	1	1	yes	characteristic
151	C₂³	C₂×Dic₇⋊C₄	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	#6382	1	4	yes	—
156	C₂×C₄	#6382	1	4	yes	—
157	C₂×C₄	Dic₇⋊C₈	1	4	yes	—
158	C₂×C₄	#6396	1	4	yes	—
159	C₂×C₄	#6451	1	2	yes	—
160	C₂×C₄	D₇×C₄⋊C₄	1	2	yes	—
161	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
162	C₂×C₄	C₂×#1131	1	1	yes	characteristic
163	C₂×C₄	D₇×M₁₆	1	2	yes	—
164	C₂×C₄	C₂×#1142	1	2	yes	—
165	C₂×C₄	#6503	1	1	yes	characteristic
166	C₂×C₄	C₂×#1143	1	2	yes	—
167	C₂×C₄	#6507	1	1	yes	characteristic
168	C₈	—	7	8	no	—
169	C₈	—	8	2	no	—
170	C₈	—	14	4	no	—
171	C₈	—	56	2	no	—
172	C₇	#8805	1	1	yes	characteristic, complemented, Sylow
173	C₂²	C₂×#6382	1	2	yes	—
174	C₂²	#89748	1	1	yes	characteristic
175	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
176	C₂²	#89890	1	1	yes	characteristic
177	C₂²	C₂×#6396	1	2	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	#89275	1	8	yes	complemented
181	C₄	D₇×C₄⋊C₈	1	2	yes	—
182	C₄	#89609	1	2	yes	—
183	C₄	#89848	1	1	yes	characteristic
184	C₄	#89851	1	2	yes	—
185	C₄	#89857	1	1	yes	characteristic
186	C₂	#11727330	1	1	yes	characteristic
187	C₂	C₂×#89275	1	2	yes	—
188	C₂	#11728199	1	1	yes	characteristic
189	C₂	#11728244	1	2	yes	—
190	C₂	#11728272	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1842	C₂
socle, prime radical	C₂³×C₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1842	D₇
hypocenter	C₇	#8805
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
31. C₄×C₇⋊C₈	C₈	semidirect
91. C₇⋊C₈	C₄⋊C₈	semidirect
117. C₄×C₇	#334	semidirect
172. C₇	#8805	semidirect
180. C₄	#89275	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1842	2
C₂×#89275	2
#11727148	2
C₄²×C₇⋊C₈	2
#8805	7