Subgroups — #13274349

Order 1792.

204 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×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#89294	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊(C₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊(C₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
7	#11725314	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₇×C₈×D₄	C₄	1	1	yes	characteristic
11	C₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic
12	C₇×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
13	C₇×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
14	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
15	#89294	—	2	2	no	—
16	C₂×C₇⋊C₃₂	—	2	2	no	—
17	C₂×C₇⋊C₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₃₂	C₂²	1	1	yes	characteristic
19	C₂×C₇⋊C₃₂	C₂²	1	1	yes	characteristic
20	C₂×C₇⋊C₃₂	C₂²	1	1	yes	characteristic
21	#13645	—	7	1	no	maximal, Sylow
22	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×D₄	C₈	1	1	yes	characteristic
25	C₂×C₇×C₁₆	—	2	1	no	—
26	C₂×C₇×C₁₆	—	2	1	no	—
27	C₂×C₇×C₁₆	—	2	1	no	—
28	C₂×C₇×C₁₆	—	2	1	no	—
29	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
33	C₇×C₄⋊C₈	C₈	1	1	yes	characteristic
34	C₇×C₂²⋊C₈	C₈	1	1	yes	characteristic
35	C₇×C₂²⋊C₈	C₈	1	1	yes	characteristic
36	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₇⋊C₃₂	—	2	1	no	—
38	C₇⋊C₃₂	—	2	1	no	—
39	C₇⋊C₃₂	—	2	1	no	—
40	C₇⋊C₃₂	D₄	1	2	yes	complemented
41	C₂×M₆₄	—	7	1	no	—
42	C₂²×C₃₂	—	7	1	no	—
43	D₄×C₁₆	D₇	1	1	yes	characteristic
44	C₄⋊C₃₂	—	7	1	no	—
45	C₂²⋊C₃₂	—	7	1	no	—
46	C₂²⋊C₃₂	—	7	1	no	—
47	#1397	—	7	1	no	—
48	C₂×C₇×D₄	C₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₇×C₈	—	2	1	no	—
52	C₂×C₇×C₈	—	2	1	no	—
53	C₂×C₇×C₈	—	2	1	no	—
54	C₂×C₇×C₈	—	2	1	no	—
55	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₇×C₄⋊C₄	C₁₆	1	1	yes	characteristic
60	C₇×C₂²⋊C₄	C₁₆	1	1	yes	characteristic
61	C₇×C₂²⋊C₄	C₁₆	1	1	yes	characteristic
62	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
63	C₇×C₁₆	—	2	1	no	—
64	C₇×C₁₆	—	2	1	no	—
65	C₇×C₁₆	—	2	1	no	—
66	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
67	C₇×C₁₆	#41	1	1	yes	characteristic
68	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
69	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
70	C₈×D₄	Dic₇	1	1	yes	characteristic
71	M₆₄	—	14	2	no	—
72	C₂×C₃₂	—	7	1	no	—
73	C₂×C₃₂	—	7	1	no	—
74	C₂×C₃₂	—	7	1	no	—
75	C₂×C₃₂	—	7	1	no	—
76	C₂×C₃₂	—	14	2	no	—
77	C₄⋊C₁₆	Dic₇	1	1	yes	characteristic
78	C₂²⋊C₁₆	Dic₇	1	1	yes	characteristic
79	C₂²⋊C₁₆	Dic₇	1	1	yes	characteristic
80	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
81	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
82	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
83	C₇×D₄	—	2	2	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
93	C₇×C₈	—	2	1	no	—
94	C₇×C₈	—	2	1	no	—
95	C₇×C₈	—	2	1	no	—
96	C₇×C₈	C₄×D₄	1	1	yes	characteristic
97	C₇×C₈	#131	1	1	yes	characteristic
98	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
99	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
100	C₄×D₄	C₇⋊C₈	1	1	yes	characteristic
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	C₂×Dic₇	1	1	yes	characteristic
106	C₂×C₁₆	C₂×Dic₇	1	1	yes	characteristic
107	C₂×C₁₆	C₂×Dic₇	1	1	yes	characteristic
108	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
109	C₄⋊C₈	C₇⋊C₈	1	1	yes	characteristic
110	C₂²⋊C₈	C₇⋊C₈	1	1	yes	characteristic
111	C₂²⋊C₈	C₇⋊C₈	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
113	C₃₂	—	7	2	no	—
114	C₃₂	—	14	1	no	—
115	C₃₂	—	14	1	no	—
116	C₃₂	—	14	1	no	—
117	C₂²×C₇	—	2	1	no	—
118	C₂²×C₇	—	2	1	no	—
119	C₂²×C₇	—	2	1	no	—
120	C₂²×C₇	—	2	1	no	—
121	C₂²×C₇	—	2	1	no	—
122	C₂²×C₇	M₆₄	1	2	yes	complemented
123	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
124	C₄×C₇	—	2	1	no	—
125	C₄×C₇	—	2	1	no	—
126	C₄×C₇	—	2	1	no	—
127	C₄×C₇	—	2	1	no	—
128	C₄×C₇	C₈×D₄	1	1	yes	characteristic
129	C₄×C₇	#504	1	1	yes	characteristic
130	C₂×D₄	C₇⋊C₁₆	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
138	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
139	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
141	C₄⋊C₄	C₇⋊C₁₆	1	1	yes	characteristic
142	C₂²⋊C₄	C₇⋊C₁₆	1	1	yes	characteristic
143	C₂²⋊C₄	C₇⋊C₁₆	1	1	yes	characteristic
144	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
145	C₁₆	—	2	1	no	—
146	C₁₆	—	2	1	no	—
147	C₁₆	—	2	1	no	—
148	C₁₆	D₄×D₇	1	1	yes	characteristic
149	C₁₆	#1159	1	1	yes	characteristic
150	C₂×C₇	—	2	2	no	—
151	C₂×C₇	—	4	1	no	—
152	C₂×C₇	D₄×C₁₆	1	1	yes	characteristic
153	C₂×C₇	C₂×M₆₄	1	1	yes	characteristic
154	C₂×C₇	#2258	1	1	yes	characteristic
155	C₂³	C₂×C₇⋊C₁₆	1	1	yes	characteristic
156	C₂³	C₂×C₇⋊C₁₆	1	1	yes	characteristic
157	D₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	C₂×C₇⋊C₁₆	1	1	yes	characteristic
163	C₂×C₄	C₂×C₇⋊C₁₆	1	1	yes	characteristic
164	C₂×C₄	C₂×C₇⋊C₁₆	1	1	yes	characteristic
165	C₂×C₄	C₂×C₇⋊C₁₆	1	1	yes	characteristic
166	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	D₄×Dic₇	1	1	yes	characteristic
171	C₈	#6511	1	1	yes	characteristic
172	C₇	#13645	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#89294	1	2	yes	complemented
179	C₂²	C₂²×C₇⋊C₁₆	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	D₄×C₇⋊C₈	1	1	yes	characteristic
185	C₄	#89912	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	—	4	1	no	—
188	C₂	D₄×C₇⋊C₁₆	1	1	yes	characteristic
189	C₂	C₂×#89294	1	1	yes	characteristic
190	C₂	#11729205	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₁₆
Fitting	C₇×D₄×C₁₆	C₂
prime radical	C₂×C₇×D₄	C₁₆
cosocle	C₂×C₇×C₁₆	C₂³
hypercenter	D₄×C₁₆	D₇
hypocenter	C₇	#13645
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#89294	C₂	semidirect
3. C₂²×C₇⋊C₃₂	C₂	semidirect
4. C₂²⋊(C₇⋊C₃₂)	C₂	semidirect
5. C₂²⋊(C₇⋊C₃₂)	C₂	semidirect
6. C₄⋊(C₇⋊C₃₂)	C₂	semidirect
7. #11725314	C₂	semidirect
40. C₇⋊C₃₂	D₄	semidirect
122. C₂²×C₇	M₆₄	semidirect
172. C₇	#13645	semidirect
178. C₂²	#89294	semidirect

Maximal subgroups

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

subgroup	index
C₇×D₄×C₁₆	2
C₂×#89294	2
C₂²×C₇⋊C₃₂	2
C₂²⋊(C₇⋊C₃₂)	2
C₄⋊(C₇⋊C₃₂)	2
#11725314	2
#13645	7