Subgroups — #13244993

Order 1792.

446 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#89253	C₂	1	1	yes	maximal, characteristic
3	C₂×#89252	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
5	#11725460	C₂	1	2	yes	maximal
6	C₂×C₇×#97	C₄	1	2	yes	—
7	C₂×C₇×#97	C₄	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₄³×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄×C₇⋊C₈	—	2	2	no	—
14	#89319	—	2	2	no	—
15	#89253	C₄	1	4	yes	complemented
16	#89252	C₄	1	4	yes	complemented
17	#8275	—	7	1	no	maximal, Sylow
18	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇	D₄	1	2	yes	—
23	C₂×C₄²×C₇	Q₈	1	2	yes	—
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂²×C₇⋊C₈	—	2	2	no	—
26	C₂²×C₇⋊C₈	—	2	2	no	—
27	C₇×#97	—	2	4	no	—
28	C₇×#97	C₈	1	8	yes	complemented
29	C₄×C₇×C₈	—	2	1	no	—
30	C₄×C₇×C₈	—	2	1	no	—
31	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
33	C₄⋊(C₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
36	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
37	C₄×C₇⋊C₈	—	4	4	no	—
38	C₄×C₄⋊C₈	—	7	2	no	—
39	C₄×#97	D₇	1	1	yes	characteristic
40	C₂×#335	—	7	1	no	—
41	C₂×#334	—	7	1	no	—
42	#1277	—	7	2	no	—
43	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₇×C₈	—	2	2	no	—
48	C₂×C₇×C₈	—	2	2	no	—
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	C₄²	1	2	yes	—
51	C₂×C₇×C₈	C₄²	1	2	yes	—
52	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
53	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
54	C₄²×C₇	—	2	2	no	—
55	C₄²×C₇	—	2	2	no	—
56	C₄²×C₇	—	4	2	no	—
57	C₄²×C₇	C₂×C₈	1	4	yes	—
58	C₄²×C₇	M₁₆	1	4	yes	—
59	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₇	#41	1	1	yes	characteristic
62	C₄²×C₇	#41	1	1	yes	characteristic
63	C₂×C₇⋊C₈	—	2	4	no	—
64	C₂×C₇⋊C₈	—	2	4	no	—
65	C₂×C₇⋊C₈	—	4	2	no	—
66	C₂×C₇⋊C₈	—	4	2	no	—
67	C₂×C₇⋊C₈	—	4	2	no	—
68	C₂×C₇⋊C₈	—	4	2	no	—
69	C₂×C₄⋊C₈	—	7	2	no	—
70	C₂×C₄⋊C₈	—	7	2	no	—
71	C₂×#97	Dic₇	1	2	yes	—
72	C₂×#97	Dic₇	1	2	yes	—
73	C₂×C₄×C₈	—	14	2	no	—
74	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
76	C₄³	C₂×D₇	1	1	yes	characteristic
77	#336	—	14	2	no	—
78	#335	—	7	4	no	—
79	#334	—	7	4	no	—
80	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	—	2	4	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	C₄⋊C₈	1	8	yes	—
85	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₇	#117	1	1	yes	characteristic
87	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
94	C₂×C₄×C₇	#137	1	2	yes	—
95	C₇×C₈	—	2	2	no	—
96	C₇×C₈	—	2	4	no	—
97	C₇×C₈	C₄×C₈	1	4	yes	complemented
98	C₇⋊C₈	—	4	4	no	—
99	C₇⋊C₈	—	4	4	no	—
100	C₂²×C₈	—	14	2	no	—
101	C₂²×C₈	—	14	2	no	—
102	C₂²×C₈	C₂×Dic₇	1	2	yes	—
103	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
105	C₂×C₄²	Dic₁₄	1	2	yes	—
106	C₂×C₄²	D₂₈	1	2	yes	—
107	C₂×C₄²	C₂×Dic₇	1	2	yes	—
108	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
109	C₄⋊C₈	—	7	4	no	—
110	C₄⋊C₈	—	7	4	no	—
111	C₄⋊C₈	—	14	2	no	—
112	C₄⋊C₈	—	14	2	no	—
113	#97	—	2	4	no	—
114	#97	C₇⋊C₈	1	8	yes	complemented
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	—	2	1	no	—
117	C₄×C₈	—	28	4	no	—
118	C₄×C₈	C₄×D₇	1	2	yes	—
119	C₄×C₈	C₄×D₇	1	2	yes	—
120	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₇	C₂×C₄⋊C₈	1	2	yes	—
122	C₂²×C₇	#428	1	2	yes	—
123	C₂²×C₇	#430	1	2	yes	—
124	C₄×C₇	—	2	8	no	—
125	C₄×C₇	—	2	1	no	—
126	C₄×C₇	—	2	1	no	—
127	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
128	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
129	C₄×C₇	#431	1	2	yes	—
130	C₄×C₇	#440	1	2	yes	—
131	C₄×C₇	#441	1	1	yes	characteristic
132	C₄×C₇	#442	1	1	yes	characteristic
133	C₂²×C₄	C₄⋊Dic₇	1	4	yes	—
134	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
136	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	14	4	no	—
141	C₂×C₈	—	14	4	no	—
142	C₂×C₈	—	28	2	no	—
143	C₂×C₈	—	28	2	no	—
144	C₂×C₈	—	28	2	no	—
145	C₂×C₈	—	28	2	no	—
146	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
147	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
148	C₂×C₈	C₄×Dic₇	1	2	yes	—
149	C₂×C₈	C₄×Dic₇	1	2	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	4	2	no	—
153	C₄²	C₂×C₇⋊C₈	1	4	yes	—
154	C₄²	#1136	1	4	yes	—
155	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
157	C₄²	#1157	1	1	yes	characteristic
158	C₄²	#1157	1	1	yes	characteristic
159	C₂×C₇	#1565	1	4	yes	—
160	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
161	C₂×C₇	#1776	1	1	yes	characteristic
162	C₂×C₇	#1778	1	1	yes	characteristic
163	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	C₄⋊(C₇⋊C₈)	1	8	yes	—
168	C₂×C₄	C₄×Dic₁₄	1	1	yes	characteristic
169	C₂×C₄	C₄×Dic₁₄	1	1	yes	characteristic
170	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
171	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
172	C₂×C₄	#6471	1	2	yes	—
173	C₂×C₄	#6472	1	2	yes	—
174	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
175	C₂×C₄	C₂×#1136	1	2	yes	—
176	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
177	C₂×C₄	#6489	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	2	4	no	—
180	C₈	—	28	4	no	—
181	C₈	—	28	4	no	—
182	C₈	C₄×C₇⋊C₈	1	4	yes	complemented
183	C₇	#8275	1	1	yes	characteristic, complemented, Sylow
184	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	2	yes	—
185	C₂²	C₄×C₄⋊Dic₇	1	1	yes	characteristic
186	C₂²	#89893	1	2	yes	—
187	C₂²	#89897	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	8	no	—
191	C₄	#89482	1	2	yes	—
192	C₄	#89483	1	1	yes	characteristic
193	C₄	#89488	1	1	yes	characteristic
194	C₄	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
195	C₄	C₄×#1136	1	1	yes	characteristic
196	C₄	#89697	1	2	yes	—
197	C₂	C₄×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
198	C₂	#11727568	1	1	yes	characteristic
199	C₂	#11727594	1	1	yes	characteristic
200	C₂	#11727604	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₄×C₇×#97	C₂
socle, prime radical	C₂³×C₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₄×#97	D₇
hypocenter	C₇	#8275
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
15. #89253	C₄	semidirect
16. #89252	C₄	semidirect
28. C₇×#97	C₈	semidirect
97. C₇×C₈	C₄×C₈	semidirect
114. #97	C₇⋊C₈	semidirect
182. C₈	C₄×C₇⋊C₈	semidirect
183. C₇	#8275	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×#97	2
C₂×#89253	2
C₂×#89252	2
C₄×C₄⋊(C₇⋊C₈)	2
#11725460	2
#8275	7