Subgroups — #13269109

Order 1792.

382 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1726	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	2	yes	maximal
3	C₂×#89256	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#11725463	C₂	1	2	yes	maximal
6	C₂×C₇×#97	C₄	1	1	yes	characteristic
7	C₂×C₇×#97	C₄	1	1	yes	characteristic
8	C₂×C₇×#97	C₄	1	1	yes	characteristic
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#376	C₄	1	1	yes	characteristic
13	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
15	C₄⋊(C₇⋊C₁₆)	C₄	1	8	yes	complemented
16	#89256	—	2	2	no	—
17	C₄×C₇⋊C₁₆	—	2	2	no	—
18	#12654	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₇×#97	—	2	2	no	—
27	C₇×#97	—	2	2	no	—
28	C₇×#97	C₈	1	4	yes	—
29	C₄×C₇×C₈	—	2	2	no	—
30	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
31	C₄×C₇×C₈	D₄	1	2	yes	—
32	C₄×C₇×C₈	Q₈	1	2	yes	—
33	C₇×#95	C₈	1	4	yes	—
34	C₂×C₇⋊C₁₆	—	2	2	no	—
35	C₂×C₇⋊C₁₆	—	2	2	no	—
36	C₂×C₇⋊C₁₆	—	2	2	no	—
37	C₂×C₇⋊C₁₆	—	2	2	no	—
38	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
39	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₁₆	—	7	2	no	—
41	C₂×#346	—	7	1	no	—
42	C₂×C₄×C₁₆	—	7	1	no	—
43	#1726	D₇	1	1	yes	characteristic
44	#1374	—	7	2	no	—
45	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	—	2	1	no	—
50	C₂×C₇×C₈	—	2	1	no	—
51	C₂×C₇×C₈	—	2	1	no	—
52	C₂×C₇×C₈	—	2	1	no	—
53	C₂×C₇×C₈	C₄²	1	2	yes	—
54	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
56	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
57	C₂×C₇×C₈	M₁₆	1	2	yes	—
58	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
59	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₇×C₈	#41	1	2	yes	—
62	C₄²×C₇	—	2	1	no	—
63	C₄²×C₇	—	2	1	no	—
64	C₄²×C₇	C₄²	1	2	yes	—
65	C₄²×C₇	C₄⋊C₄	1	2	yes	—
66	C₄²×C₇	C₂×C₈	1	2	yes	—
67	C₄²×C₇	M₁₆	1	2	yes	—
68	C₇⋊C₁₆	—	2	4	no	—
69	C₇⋊C₁₆	—	2	4	no	—
70	C₂²×C₁₆	—	7	2	no	—
71	C₂²×C₁₆	—	7	2	no	—
72	C₂×#97	Dic₇	1	1	yes	characteristic
73	C₂×#97	Dic₇	1	1	yes	characteristic
74	C₂×#97	Dic₇	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
78	#376	Dic₇	1	1	yes	characteristic
79	C₄⋊C₁₆	—	7	8	no	—
80	#346	—	14	2	no	—
81	C₄×C₁₆	—	14	2	no	—
82	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₇	—	2	4	no	—
84	C₂×C₄×C₇	—	2	4	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	C₄×C₈	1	4	yes	—
88	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
89	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	#117	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
95	C₇×C₈	—	2	2	no	—
96	C₇×C₈	—	2	2	no	—
97	C₇×C₈	C₄⋊C₈	1	4	yes	—
98	C₇×C₈	C₄×D₄	1	2	yes	—
99	C₇×C₈	C₄×Q₈	1	2	yes	—
100	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
101	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
102	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
104	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
106	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
107	C₂×C₁₆	—	7	4	no	—
108	C₂×C₁₆	—	7	4	no	—
109	C₂×C₁₆	—	14	2	no	—
110	C₂×C₁₆	—	14	2	no	—
111	C₂×C₁₆	—	14	2	no	—
112	C₂×C₁₆	—	14	2	no	—
113	#97	—	2	2	no	—
114	#97	—	2	2	no	—
115	#97	C₇⋊C₈	1	4	yes	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	Dic₁₄	1	2	yes	—
118	C₄×C₈	C₄×D₇	1	4	yes	—
119	C₄×C₈	D₂₈	1	2	yes	—
120	#95	C₇⋊C₈	1	4	yes	—
121	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
122	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂²×C₇	#431	1	1	yes	characteristic
124	C₂²×C₇	#504	1	4	yes	—
125	C₄×C₇	—	2	4	no	—
126	C₄×C₇	—	2	4	no	—
127	C₄×C₇	—	4	2	no	—
128	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
130	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₄×C₇	#431	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
133	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
134	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
135	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
141	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
142	C₂×C₈	#1136	1	2	yes	—
143	C₂×C₈	C₄×Dic₇	1	2	yes	—
144	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
145	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
146	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
147	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
148	C₂×C₈	#1157	1	2	yes	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	C₂×C₇⋊C₈	1	2	yes	—
152	C₄²	#1136	1	2	yes	—
153	C₄²	C₄×Dic₇	1	2	yes	—
154	C₄²	C₄⋊Dic₇	1	2	yes	—
155	C₁₆	—	14	4	no	—
156	C₁₆	—	14	4	no	—
157	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
158	C₂×C₇	#2108	1	2	yes	—
159	C₂×C₇	#2151	1	2	yes	—
160	C₂×C₇	#2164	1	2	yes	—
161	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	C₄×C₇⋊C₈	1	4	yes	—
167	C₂×C₄	C₄⋊(C₇⋊C₈)	1	4	yes	—
168	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
169	C₂×C₄	C₂×#1136	1	1	yes	characteristic
170	C₂×C₄	C₂×#1136	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
173	C₂×C₄	#6489	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	2	no	—
176	C₈	C₄⋊(C₇⋊C₈)	1	4	yes	—
177	C₈	C₄×Dic₁₄	1	2	yes	—
178	C₈	C₄×D₂₈	1	2	yes	—
179	C₇	#12654	1	1	yes	characteristic, complemented, Sylow
180	C₂²	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
181	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
182	C₂²	#89697	1	1	yes	characteristic
183	C₂²	#89912	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₄×#1136	1	1	yes	characteristic
188	C₄	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
189	C₄	#89697	1	1	yes	characteristic
190	C₄	C₄×C₄⋊Dic₇	1	1	yes	characteristic
191	C₂	C₄×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
192	C₂	#11727635	1	2	yes	—
193	C₂	#11727636	1	2	yes	—
194	C₂	#11727637	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇
commutator	C₂²×C₇	C₂×C₄×C₈
Fitting	C₇×#1726	C₂
socle, prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1726	D₇
hypocenter	C₇	#12654
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
15. C₄⋊(C₇⋊C₁₆)	C₄	semidirect
179. C₇	#12654	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1726	2
C₂×C₄⋊(C₇⋊C₁₆)	2
C₂×#89256	2
C₂×C₄×C₇⋊C₁₆	2
#11725463	2
#12654	7