Subgroups — #13269120

Order 1792.

358 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	C₂×C₇×#364	C₂	1	1	yes	maximal, characteristic
2	Dic₇×M₃₂	C₂	1	2	yes	maximal
3	#11727503	C₂	1	1	yes	maximal, characteristic
4	C₂×#89258	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×#364	C₄	1	4	yes	complemented
16	#89297	—	2	2	no	—
17	C₁₆×Dic₇	—	2	2	no	—
18	#89258	C₄	1	4	yes	complemented
19	#12656	—	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	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₇×M₃₂	—	2	2	no	—
28	C₇×M₃₂	C₂×C₄	1	4	yes	—
29	C₂×C₇×C₁₆	—	2	2	no	—
30	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
32	C₇⋊#95	—	2	2	no	—
33	#6389	C₈	1	4	yes	—
34	#6389	C₈	1	4	yes	—
35	C₈×Dic₇	—	2	1	no	—
36	C₈×Dic₇	—	2	1	no	—
37	C₈×Dic₇	D₄	1	2	yes	—
38	C₈×Dic₇	Q₈	1	2	yes	—
39	#6386	—	2	2	no	—
40	#6386	C₂×C₄	1	4	yes	—
41	C₂×C₇⋊C₁₆	—	2	2	no	—
42	#6377	—	2	2	no	—
43	C₂×#364	—	7	1	no	—
44	C₂×#364	D₇	1	1	yes	characteristic
45	C₄×M₃₂	—	7	2	no	—
46	#1726	—	7	1	no	—
47	#1376	—	7	2	no	—
48	C₂²×C₄×C₇	—	2	1	no	—
49	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₇	—	2	1	no	—
51	C₂²×Dic₇	M₁₆	1	2	yes	—
52	C₂×C₇×C₈	C₄²	1	2	yes	—
53	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
54	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
55	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₇×C₈	#41	1	1	yes	characteristic
60	C₂×C₇×C₈	#41	1	1	yes	characteristic
61	C₄²×C₇	—	2	1	no	—
62	C₄²×C₇	C₄²	1	2	yes	—
63	C₄×Dic₇	—	2	1	no	—
64	C₄×Dic₇	C₄⋊C₄	1	2	yes	—
65	C₄×Dic₇	C₂×C₈	1	2	yes	—
66	C₄×Dic₇	C₂×C₈	1	2	yes	—
67	C₂×C₇⋊C₈	—	2	1	no	—
68	C₂×C₇⋊C₈	—	2	1	no	—
69	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
71	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
72	C₂×C₇⋊C₈	M₁₆	1	2	yes	—
73	C₇×C₁₆	—	2	4	no	—
74	C₇⋊C₁₆	—	2	4	no	—
75	C₂×M₃₂	—	7	2	no	—
76	C₂×M₃₂	C₂×D₇	1	2	yes	—
77	C₂×#97	—	7	1	no	—
78	C₂×#97	—	7	1	no	—
79	C₂×#97	—	7	1	no	—
80	C₂×C₄×C₈	—	7	1	no	—
81	C₂×C₄×C₈	—	7	1	no	—
82	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
83	#376	—	7	1	no	—
84	#364	—	7	4	no	—
85	#364	Dic₇	1	4	yes	complemented
86	#346	—	14	2	no	—
87	C₄×C₁₆	—	14	2	no	—
88	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	4	2	no	—
91	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇	#117	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
97	C₂×Dic₇	—	2	2	no	—
98	C₂×Dic₇	—	4	2	no	—
99	C₂×Dic₇	—	4	2	no	—
100	C₂×Dic₇	C₄⋊C₈	1	4	yes	—
101	C₇×C₈	C₄×C₈	1	4	yes	—
102	C₇×C₈	C₄×D₄	1	1	yes	characteristic
103	C₇×C₈	C₄×D₄	1	1	yes	characteristic
104	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
105	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
106	C₇⋊C₈	—	2	2	no	—
107	C₇⋊C₈	C₄⋊C₈	1	4	yes	—
108	C₂²×C₈	—	7	1	no	—
109	C₂²×C₈	—	7	1	no	—
110	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
112	C₂×C₄²	—	7	1	no	—
113	C₂×C₄²	—	7	1	no	—
114	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	—	7	4	no	—
117	M₃₂	—	14	2	no	—
118	M₃₂	C₂×Dic₇	1	4	yes	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	14	2	no	—
121	#97	—	7	4	no	—
122	#97	—	7	4	no	—
123	#97	—	14	2	no	—
124	C₄×C₈	—	7	2	no	—
125	C₄×C₈	—	7	2	no	—
126	C₄×C₈	—	14	1	no	—
127	C₄×C₈	—	14	1	no	—
128	C₄×C₈	C₄×D₇	1	2	yes	—
129	C₄×C₈	C₂×Dic₇	1	2	yes	—
130	#95	—	14	2	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	C₄×M₁₆	1	1	yes	characteristic
133	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
134	C₂²×C₇	#431	1	1	yes	characteristic
135	C₄×C₇	—	4	2	no	—
136	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₄×C₇	#431	1	1	yes	characteristic
140	Dic₇	—	2	4	no	—
141	Dic₇	—	4	2	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	7	2	no	—
144	C₂²×C₄	—	14	1	no	—
145	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
146	C₂×C₈	—	7	2	no	—
147	C₂×C₈	—	7	2	no	—
148	C₂×C₈	—	7	2	no	—
149	C₂×C₈	—	7	2	no	—
150	C₂×C₈	—	14	1	no	—
151	C₂×C₈	—	14	1	no	—
152	C₂×C₈	C₈×D₇	1	2	yes	—
153	C₂×C₈	C₈×D₇	1	2	yes	—
154	C₂×C₈	C₄×Dic₇	1	2	yes	—
155	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
156	C₂×C₈	D₄×D₇	1	1	yes	characteristic
157	C₂×C₈	#1159	1	1	yes	characteristic
158	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
159	C₂×C₈	#1161	1	1	yes	characteristic
160	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
161	C₄²	—	2	1	no	—
162	C₄²	—	7	2	no	—
163	C₄²	—	7	2	no	—
164	C₄²	—	7	2	no	—
165	C₄²	—	14	1	no	—
166	C₄²	C₄×Dic₇	1	2	yes	—
167	C₁₆	—	2	4	no	—
168	C₁₆	—	14	4	no	—
169	C₂×C₇	—	2	1	no	—
170	C₂×C₇	—	2	1	no	—
171	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
172	C₂×C₇	#2153	1	2	yes	—
173	C₂³	D₇×M₁₆	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	7	4	no	—
177	C₂×C₄	—	14	2	no	—
178	C₂×C₄	—	28	2	no	—
179	C₂×C₄	—	28	2	no	—
180	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄	#6455	1	1	yes	characteristic
182	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
183	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
184	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
186	C₈	—	7	4	no	—
187	C₈	—	14	2	no	—
188	C₈	C₈×Dic₇	1	4	yes	—
189	C₈	#6450	1	1	yes	characteristic
190	C₈	#6456	1	1	yes	characteristic
191	C₈	D₄×Dic₇	1	1	yes	characteristic
192	C₈	Q₈×Dic₇	1	1	yes	characteristic
193	C₇	#12656	1	1	yes	characteristic, complemented, Sylow
194	C₂²	—	2	2	no	—
195	C₂²	D₇×C₄⋊C₈	1	1	yes	characteristic
196	C₂²	#89605	1	1	yes	characteristic
197	C₂²	M₁₆×Dic₇	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	14	4	no	—
200	C₄	—	28	2	no	—
201	C₄	D₇×C₄⋊C₈	1	1	yes	characteristic
202	C₄	#89605	1	1	yes	characteristic
203	C₄	C₄⋊C₄×Dic₇	1	1	yes	characteristic
204	C₄	C₂×C₈×Dic₇	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#11727057	1	2	yes	—
208	C₂	Dic₇×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#364	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₂×M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#364	D₇
hypocenter	C₇	#12656
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
15. C₇×#364	C₄	semidirect
18. #89258	C₄	semidirect
85. #364	Dic₇	semidirect
193. C₇	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#364	2
Dic₇×M₃₂	2
#11727503	2
C₂×#89258	2
#11725616	2
#12656	7