Subgroups — #13269119

Order 1792.

358 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#89306	C₂	1	2	yes	maximal
3	#11727503	C₂	1	1	yes	maximal, characteristic
4	C₄×#6386	C₂	1	1	yes	maximal, characteristic
5	#11725616	C₂	1	2	yes	maximal
6	C₂×C₇×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6389	C₄	1	1	yes	characteristic
9	C₂×#6389	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×#6386	C₂²	1	2	yes	—
13	#89704	C₄	1	1	yes	characteristic
14	C₂×#6377	C₄	1	1	yes	characteristic
15	C₇×#346	—	2	1	no	—
16	C₄×C₇×C₁₆	—	2	1	no	—
17	#89306	C₄	1	8	yes	complemented
18	#89256	—	2	1	no	—
19	C₄×C₇⋊C₁₆	—	2	1	no	—
20	#12656	—	7	1	no	maximal, Sylow
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×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₇×M₃₂	—	2	2	no	—
29	C₇×M₃₂	C₂×C₄	1	4	yes	—
30	C₂×C₇×C₁₆	—	2	2	no	—
31	C₄×C₇×C₈	D₄	1	1	yes	characteristic
32	C₄×C₇×C₈	D₄	1	1	yes	characteristic
33	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
34	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
35	C₇⋊#95	—	2	2	no	—
36	#6389	—	2	2	no	—
37	#6389	C₈	1	4	yes	—
38	C₈×Dic₇	—	2	2	no	—
39	C₈×Dic₇	C₂×C₄	1	4	yes	—
40	#6386	—	2	2	no	—
41	#6386	C₂×C₄	1	4	yes	—
42	C₂×C₇⋊C₁₆	—	2	2	no	—
43	#6377	C₈	1	4	yes	—
44	C₂×#364	—	7	2	no	—
45	C₄×M₃₂	—	7	1	no	—
46	C₄×M₃₂	D₇	1	1	yes	characteristic
47	#1726	—	7	1	no	—
48	#1376	—	7	2	no	—
49	C₂²×C₄×C₇	M₁₆	1	2	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₇	—	2	1	no	—
52	C₂²×Dic₇	—	2	1	no	—
53	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
56	C₂×C₇×C₈	M₁₆	1	2	yes	—
57	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₇×C₈	#41	1	2	yes	—
61	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
63	C₄²×C₇	C₂×C₈	1	2	yes	—
64	C₄×Dic₇	—	2	1	no	—
65	C₄×Dic₇	—	2	1	no	—
66	C₄×Dic₇	C₄²	1	2	yes	—
67	C₄×Dic₇	C₂×C₈	1	2	yes	—
68	C₂×C₇⋊C₈	—	2	1	no	—
69	C₂×C₇⋊C₈	—	2	1	no	—
70	C₂×C₇⋊C₈	C₄²	1	2	yes	—
71	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
72	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
73	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
74	C₇×C₁₆	—	2	4	no	—
75	C₇⋊C₁₆	—	2	4	no	—
76	C₂×M₃₂	—	7	2	no	—
77	C₂×M₃₂	C₂×D₇	1	2	yes	—
78	C₂×#97	—	7	1	no	—
79	C₂×#97	—	7	1	no	—
80	C₂×#97	—	7	1	no	—
81	C₂×C₄×C₈	—	7	1	no	—
82	C₂×C₄×C₈	—	7	1	no	—
83	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
84	#376	—	7	1	no	—
85	#364	—	7	8	no	—
86	#346	—	2	1	no	—
87	#346	—	14	1	no	—
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	—	14	1	no	—
90	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇	—	4	2	no	—
92	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇	#117	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₇	—	2	2	no	—
101	C₂×Dic₇	—	2	2	no	—
102	C₂×Dic₇	—	4	2	no	—
103	C₂×Dic₇	—	4	2	no	—
104	C₇×C₈	C₄⋊C₈	1	2	yes	—
105	C₇×C₈	C₄⋊C₈	1	2	yes	—
106	C₇×C₈	C₄×D₄	1	2	yes	—
107	C₇×C₈	C₄×Q₈	1	2	yes	—
108	C₇⋊C₈	—	2	2	no	—
109	C₇⋊C₈	C₄×C₈	1	4	yes	—
110	C₂²×C₈	—	7	1	no	—
111	C₂²×C₈	—	7	1	no	—
112	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
114	C₂×C₄²	—	7	1	no	—
115	C₂×C₄²	—	7	1	no	—
116	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	7	4	no	—
119	M₃₂	—	14	2	no	—
120	M₃₂	C₄×D₇	1	4	yes	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	14	2	no	—
123	#97	—	7	4	no	—
124	#97	—	7	4	no	—
125	#97	—	14	2	no	—
126	C₄×C₈	—	7	4	no	—
127	C₄×C₈	—	14	2	no	—
128	C₄×C₈	Dic₁₄	1	1	yes	characteristic
129	C₄×C₈	Dic₁₄	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
132	#95	—	14	2	no	—
133	C₂²×C₇	—	2	2	no	—
134	C₂²×C₇	C₄×M₁₆	1	1	yes	characteristic
135	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
136	C₂²×C₇	#431	1	1	yes	characteristic
137	C₄×C₇	—	2	2	no	—
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₄×C₇	#431	1	1	yes	characteristic
143	Dic₇	—	4	2	no	—
144	Dic₇	—	4	2	no	—
145	C₂²×C₄	—	14	1	no	—
146	C₂²×C₄	—	14	1	no	—
147	C₂²×C₄	#1131	1	2	yes	—
148	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
149	C₂×C₈	—	7	2	no	—
150	C₂×C₈	—	7	2	no	—
151	C₂×C₈	—	7	2	no	—
152	C₂×C₈	—	7	2	no	—
153	C₂×C₈	—	14	1	no	—
154	C₂×C₈	—	14	1	no	—
155	C₂×C₈	C₈×D₇	1	2	yes	—
156	C₂×C₈	#1131	1	2	yes	—
157	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
158	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
161	C₂×C₈	#1157	1	2	yes	—
162	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
163	C₄²	—	7	2	no	—
164	C₄²	—	7	2	no	—
165	C₄²	—	14	1	no	—
166	C₄²	—	14	1	no	—
167	C₄²	C₈×D₇	1	2	yes	—
168	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
169	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
170	C₁₆	—	2	4	no	—
171	C₁₆	—	14	4	no	—
172	C₂×C₇	—	2	1	no	—
173	C₂×C₇	—	2	1	no	—
174	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
175	C₂×C₇	#2153	1	2	yes	—
176	C₂³	C₂×#1131	1	1	yes	characteristic
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	14	2	no	—
179	C₂×C₄	—	14	2	no	—
180	C₂×C₄	—	28	2	no	—
181	C₂×C₄	—	28	2	no	—
182	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
183	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
184	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
185	C₂×C₄	#6435	1	1	yes	characteristic
186	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
188	C₂×C₄	C₂×#1131	1	1	yes	characteristic
189	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
190	C₈	—	7	4	no	—
191	C₈	—	14	2	no	—
192	C₈	Dic₇⋊C₈	1	2	yes	—
193	C₈	Dic₇⋊C₈	1	2	yes	—
194	C₈	C₄×Dic₁₄	1	2	yes	—
195	C₈	C₄×C₂²⋊D₇	1	2	yes	—
196	C₇	#12656	1	1	yes	characteristic, complemented, Sylow
197	C₂²	—	2	2	no	—
198	C₂²	#89457	1	1	yes	characteristic
199	C₂²	C₄×#1131	1	1	yes	characteristic
200	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	28	2	no	—
204	C₄	—	28	2	no	—
205	C₄	C₄×C₈×D₇	1	1	yes	characteristic
206	C₄	#89457	1	1	yes	characteristic
207	C₄	C₄×Dic₇⋊C₄	1	1	yes	characteristic
208	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	C₄×Dic₇⋊C₈	1	1	yes	characteristic
212	C₂	#11729177	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₄×C₇×M₃₂	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₂×M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₄×M₃₂	D₇
hypocenter	C₇	#12656
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
17. #89306	C₄	semidirect
196. C₇	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×M₃₂	2
C₂×#89306	2
#11727503	2
C₄×#6386	2
#11725616	2
#12656	7