Subgroups — #13269170

Order 1792.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 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	#11729179	C₂	1	2	yes	maximal
3	#11727533	C₂	1	1	yes	maximal, characteristic
4	C₂×#89258	C₂	1	1	yes	maximal, characteristic
5	#11725619	C₂	1	2	yes	maximal
6	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×#6393	C₄	1	1	yes	characteristic
10	#89878	C₄	1	1	yes	characteristic
11	#89873	C₂²	1	2	yes	—
12	C₂×#6386	C₂²	1	2	yes	—
13	C₇×#346	C₄	1	2	yes	complemented
14	C₄×C₇×C₁₆	C₄	1	2	yes	complemented
15	#89318	C₄	1	2	yes	—
16	#89307	—	2	4	no	—
17	#89258	C₄	1	2	yes	complemented
18	#89258	C₄	1	2	yes	complemented
19	#12667	—	7	1	no	maximal, Sylow
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	#6489	C₂×C₄	1	2	yes	—
24	C₂×#1136	C₂×C₄	1	2	yes	—
25	C₇×M₃₂	—	2	1	no	—
26	C₇×M₃₂	—	2	1	no	—
27	C₇×M₃₂	D₄	1	2	yes	—
28	C₇×M₃₂	Q₈	1	2	yes	—
29	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
31	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
35	#6393	—	2	1	no	—
36	#6393	—	2	1	no	—
37	#6392	—	2	1	no	—
38	#6391	—	2	1	no	—
39	#6389	—	2	2	no	—
40	C₈×Dic₇	—	2	2	no	—
41	#6386	—	2	2	no	—
42	#6386	C₂×C₄	1	4	yes	—
43	C₂×C₇⋊C₁₆	—	2	2	no	—
44	C₂×#364	—	7	1	no	—
45	#2110	—	7	2	no	—
46	C₄×M₃₂	D₇	1	1	yes	characteristic
47	#1772	—	7	1	no	—
48	#1378	—	7	2	no	—
49	C₂²×C₄×C₇	—	2	1	no	—
50	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
52	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
53	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
54	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
55	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
56	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
62	C₂×C₇×C₈	#41	1	1	yes	characteristic
63	C₄²×C₇	—	2	1	no	—
64	C₄²×C₇	C₄²	1	1	yes	characteristic
65	C₄²×C₇	C₄²	1	1	yes	characteristic
66	C₂²⋊Dic₇	—	2	2	no	—
67	C₄⋊Dic₇	—	2	2	no	—
68	C₄×Dic₇	—	2	2	no	—
69	#1136	—	2	2	no	—
70	#1136	—	2	2	no	—
71	C₂×C₇⋊C₈	—	2	2	no	—
72	C₇×C₁₆	—	2	2	no	—
73	C₇×C₁₆	C₄⋊C₄	1	4	yes	—
74	C₇⋊C₁₆	—	2	4	no	—
75	C₂×M₃₂	—	7	2	no	—
76	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
77	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
78	C₂×#108	—	7	1	no	—
79	#427	—	7	1	no	—
80	#405	—	7	2	no	—
81	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
82	#364	—	7	2	no	—
83	#364	—	7	2	no	—
84	#347	—	14	4	no	—
85	#346	Dic₇	1	2	yes	complemented
86	C₄×C₁₆	Dic₇	1	2	yes	complemented
87	#339	—	7	2	no	—
88	C₂³×C₇	#131	1	1	yes	characteristic
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	4	2	no	—
92	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₇	#131	1	1	yes	characteristic
98	C₂×Dic₇	—	4	2	no	—
99	C₇×C₈	#97	1	1	yes	characteristic
100	C₇×C₈	#97	1	1	yes	characteristic
101	C₇×C₈	#97	1	1	yes	characteristic
102	C₇×C₈	#97	1	1	yes	characteristic
103	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₇×C₈	#117	1	1	yes	characteristic
105	C₇×C₈	C₄×D₄	1	1	yes	characteristic
106	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
107	C₇⋊C₈	—	4	2	no	—
108	C₂×M₁₆	—	7	2	no	—
109	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
111	#117	—	7	2	no	—
112	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
113	M₃₂	—	2	1	no	—
114	M₃₂	—	2	1	no	—
115	M₃₂	—	7	4	no	—
116	M₃₂	—	14	2	no	—
117	M₃₂	Dic₁₄	1	2	yes	—
118	M₃₂	D₂₈	1	2	yes	—
119	C₂×C₁₆	—	14	2	no	—
120	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
121	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
122	#108	—	14	1	no	—
123	#108	—	14	1	no	—
124	#107	—	14	1	no	—
125	#106	—	14	1	no	—
126	#97	—	14	2	no	—
127	C₄×C₈	—	14	2	no	—
128	C₄×C₈	C₄×D₇	1	1	yes	characteristic
129	C₄×C₈	C₄×D₇	1	1	yes	characteristic
130	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
131	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
132	C₂²×C₇	—	4	1	no	—
133	C₂²×C₇	#405	1	1	yes	characteristic
134	C₂²×C₇	#436	1	1	yes	characteristic
135	C₂²×C₇	#446	1	1	yes	characteristic
136	C₄×C₇	—	2	2	no	—
137	C₄×C₇	—	4	1	no	—
138	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₇	C₂×#97	1	1	yes	characteristic
140	C₄×C₇	#436	1	1	yes	characteristic
141	C₄×C₇	#446	1	1	yes	characteristic
142	Dic₇	—	8	2	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
145	M₁₆	—	14	2	no	—
146	M₁₆	—	14	2	no	—
147	C₂×C₈	—	14	2	no	—
148	C₂×C₈	#1131	1	1	yes	characteristic
149	C₂×C₈	#1131	1	1	yes	characteristic
150	C₂×C₈	#1131	1	1	yes	characteristic
151	C₂×C₈	#1131	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
154	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
155	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
156	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
157	C₂×C₈	#1157	1	1	yes	characteristic
158	C₂×C₈	#1157	1	1	yes	characteristic
159	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
160	C₄⋊C₄	—	14	2	no	—
161	C₂²⋊C₄	—	14	2	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	14	2	no	—
164	C₄²	C₄×Dic₇	1	1	yes	characteristic
165	C₄²	C₄×Dic₇	1	1	yes	characteristic
166	C₁₆	—	2	2	no	—
167	C₁₆	—	14	4	no	—
168	C₁₆	C₄⋊Dic₇	1	4	yes	—
169	C₂×C₇	—	2	1	no	—
170	C₂×C₇	—	2	1	no	—
171	C₂×C₇	—	2	1	no	—
172	C₂×C₇	#1770	1	1	yes	characteristic
173	C₂³	#6464	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	28	2	no	—
178	C₂×C₄	C₄×Dic₁₄	1	1	yes	characteristic
179	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
180	C₂×C₄	C₂×#1131	1	1	yes	characteristic
181	C₂×C₄	C₂×#1131	1	1	yes	characteristic
182	C₂×C₄	#6464	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
184	C₈	—	28	2	no	—
185	C₈	#6389	1	1	yes	characteristic
186	C₈	#6389	1	1	yes	characteristic
187	C₈	#6389	1	1	yes	characteristic
188	C₈	#6389	1	1	yes	characteristic
189	C₈	C₄×Dic₁₄	1	1	yes	characteristic
190	C₈	C₄×D₂₈	1	1	yes	characteristic
191	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
192	C₈	#6489	1	1	yes	characteristic
193	C₇	#12667	1	1	yes	characteristic, complemented, Sylow
194	C₂²	—	4	1	no	—
195	C₂²	#89451	1	1	yes	characteristic
196	C₂²	#89460	1	1	yes	characteristic
197	C₂²	#89873	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	56	2	no	—
201	C₄	#89451	1	1	yes	characteristic
202	C₄	#89460	1	1	yes	characteristic
203	C₄	C₄×C₄⋊Dic₇	1	1	yes	characteristic
204	C₄	C₂×#6389	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#11727511	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#6489
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₄×C₇×M₃₂	C₂
socle	C₂×C₇	#1770
prime radical	C₂³×C₇	#131
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₄×M₃₂	D₇
hypocenter	C₇	#12667
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. C₇×#346	C₄	semidirect
14. C₄×C₇×C₁₆	C₄	semidirect
17. #89258	C₄	semidirect
18. #89258	C₄	semidirect
85. #346	Dic₇	semidirect
86. C₄×C₁₆	Dic₇	semidirect
193. C₇	#12667	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
#11729179	2
#11727533	2
C₂×#89258	2
#11725619	2
#12667	7