Subgroups — #13271827

Order 1792.

326 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×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×#89299	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic
4	#11728366	C₂	1	2	yes	maximal
5	#11725528	C₂	1	2	yes	maximal
6	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₇×#107	C₂²	1	2	yes	—
8	C₂×#6392	C₂²	1	2	yes	—
9	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	#89757	—	2	2	no	—
12	C₇×#366	C₄	1	4	yes	complemented
13	#89324	—	2	2	no	—
14	#89299	—	2	2	no	—
15	C₁₆×Dic₇	—	2	2	no	—
16	#13165	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄⋊Dic₇	—	2	2	no	—
20	C₂×C₄⋊Dic₇	—	4	2	no	—
21	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
22	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₇×C₁₆	D₄	1	2	yes	—
25	C₂×C₇×C₁₆	Q₈	1	2	yes	—
26	C₇×#107	—	2	2	no	—
27	C₇×#107	C₂×C₄	1	4	yes	—
28	C₇⋊#95	—	4	2	no	—
29	#6392	—	2	2	no	—
30	#6392	—	2	2	no	—
31	C₈×Dic₇	—	2	1	no	—
32	C₈×Dic₇	—	2	1	no	—
33	C₂×C₇⋊C₁₆	—	2	1	no	—
34	C₂×C₇⋊C₁₆	—	2	1	no	—
35	C₂×C₇⋊C₁₆	Q₈	1	2	yes	—
36	C₂×#366	—	7	1	no	—
37	C₂×#366	D₇	1	1	yes	characteristic
38	C₂×C₄×C₁₆	—	7	1	no	—
39	#1942	—	7	2	no	—
40	#1380	—	7	2	no	—
41	C₂²×C₄×C₇	—	4	2	no	—
42	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₇	—	4	2	no	—
44	C₂²×Dic₇	D₈	1	1	yes	characteristic
45	C₂²×Dic₇	D₈	1	1	yes	characteristic
46	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₇×C₈	#41	1	1	yes	characteristic
51	C₂×C₇×C₈	#41	1	1	yes	characteristic
52	C₇×C₄⋊C₄	—	2	4	no	—
53	C₇×C₄⋊C₄	—	4	2	no	—
54	C₄⋊Dic₇	—	4	2	no	—
55	C₄⋊Dic₇	—	4	2	no	—
56	C₄⋊Dic₇	—	8	4	no	—
57	C₄×Dic₇	—	2	1	no	—
58	C₄×Dic₇	—	2	1	no	—
59	C₂×C₇⋊C₈	—	2	1	no	—
60	C₂×C₇⋊C₈	—	2	1	no	—
61	C₂×C₇⋊C₈	Q₁₆	1	2	yes	—
62	C₇×C₁₆	C₄⋊C₄	1	4	yes	complemented
63	C₇⋊C₁₆	—	2	2	no	—
64	C₂²×C₁₆	—	7	1	no	—
65	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
66	C₂×#107	—	7	2	no	—
67	C₂×#107	C₂×D₇	1	2	yes	—
68	C₂×C₄×C₈	—	7	1	no	—
69	#389	—	14	2	no	—
70	#366	—	14	2	no	—
71	#366	Dic₇	1	4	yes	complemented
72	C₄×C₁₆	—	14	2	no	—
73	#340	—	14	2	no	—
74	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇	—	4	4	no	—
76	C₂×C₄×C₇	—	8	2	no	—
77	C₂×C₄×C₇	—	8	2	no	—
78	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇	#121	1	1	yes	characteristic
80	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇	#135	1	1	yes	characteristic
83	C₂×C₄×C₇	#135	1	1	yes	characteristic
84	C₂×Dic₇	—	2	1	no	—
85	C₂×Dic₇	—	2	1	no	—
86	C₂×Dic₇	—	2	1	no	—
87	C₂×Dic₇	—	2	1	no	—
88	C₂×Dic₇	—	8	2	no	—
89	C₂×Dic₇	—	8	2	no	—
90	C₂×Dic₇	—	8	2	no	—
91	C₂×Dic₇	D₁₆	1	2	yes	—
92	C₂×Dic₇	Q₃₂	1	2	yes	—
93	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
95	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
96	C₇×C₈	#125	1	1	yes	characteristic
97	C₇⋊C₈	—	2	2	no	—
98	C₂²×C₈	—	7	1	no	—
99	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	14	2	no	—
102	C₂×C₄⋊C₄	—	28	2	no	—
103	C₂×C₄²	—	7	1	no	—
104	C₂×C₁₆	—	7	2	no	—
105	C₂×C₁₆	—	14	1	no	—
106	C₂×C₁₆	—	14	1	no	—
107	C₂×C₁₆	Dic₁₄	1	2	yes	—
108	C₂×C₁₆	D₂₈	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
110	#107	—	2	2	no	—
111	#107	—	14	2	no	—
112	#107	—	14	2	no	—
113	#107	C₂×Dic₇	1	4	yes	—
114	C₄×C₈	—	14	1	no	—
115	C₄×C₈	—	14	1	no	—
116	#95	—	28	2	no	—
117	C₂²×C₇	C₄×D₈	1	1	yes	characteristic
118	C₂²×C₇	#466	1	1	yes	characteristic
119	C₂²×C₇	#500	1	1	yes	characteristic
120	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
121	C₂²×C₇	C₂×Q₃₂	1	1	yes	characteristic
122	C₂²×C₇	#508	1	1	yes	characteristic
123	C₂²×C₇	#508	1	1	yes	characteristic
124	C₄×C₇	—	8	4	no	—
125	C₄×C₇	#389	1	1	yes	characteristic
126	C₄×C₇	C₄×Q₁₆	1	1	yes	characteristic
127	C₄×C₇	#467	1	1	yes	characteristic
128	C₄×C₇	#499	1	1	yes	characteristic
129	Dic₇	—	2	2	no	—
130	Dic₇	—	2	2	no	—
131	Dic₇	—	16	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	7	1	no	—
134	C₂²×C₄	—	7	1	no	—
135	C₂²×C₄	—	28	2	no	—
136	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
137	C₂×C₈	—	7	2	no	—
138	C₂×C₈	—	14	1	no	—
139	C₂×C₈	—	14	1	no	—
140	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
141	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
142	C₂×C₈	#1159	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
144	C₂×C₈	#1161	1	1	yes	characteristic
145	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	28	2	no	—
149	C₄⋊C₄	—	28	2	no	—
150	C₄⋊C₄	—	56	4	no	—
151	C₄²	—	14	1	no	—
152	C₄²	—	14	1	no	—
153	C₁₆	—	14	2	no	—
154	C₁₆	C₄⋊Dic₇	1	4	yes	complemented
155	C₂×C₇	#1942	1	1	yes	characteristic
156	C₂×C₇	C₄×D₁₆	1	1	yes	characteristic
157	C₂×C₇	C₄×Q₃₂	1	1	yes	characteristic
158	C₂×C₇	#2215	1	1	yes	characteristic
159	C₂×C₇	#2216	1	1	yes	characteristic
160	C₂×C₇	#2252	1	1	yes	characteristic
161	C₂×C₇	#2253	1	1	yes	characteristic
162	C₂³	D₇×D₈	1	1	yes	characteristic
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	7	2	no	—
165	C₂×C₄	—	7	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	14	1	no	—
169	C₂×C₄	—	14	1	no	—
170	C₂×C₄	—	14	1	no	—
171	C₂×C₄	—	14	1	no	—
172	C₂×C₄	—	56	2	no	—
173	C₂×C₄	—	56	2	no	—
174	C₂×C₄	—	56	2	no	—
175	C₂×C₄	#6451	1	1	yes	characteristic
176	C₂×C₄	#6458	1	1	yes	characteristic
177	C₂×C₄	#6475	1	1	yes	characteristic
178	C₂×C₄	D₇×Q₁₆	1	1	yes	characteristic
179	C₂×C₄	#6482	1	1	yes	characteristic
180	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
181	C₈	—	14	2	no	—
182	C₈	#6453	1	1	yes	characteristic
183	C₈	#6460	1	1	yes	characteristic
184	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
185	C₈	Q₈×Dic₇	1	1	yes	characteristic
186	C₇	#13165	1	1	yes	characteristic, complemented, Sylow
187	C₂²	#89646	1	1	yes	characteristic
188	C₂²	#89655	1	1	yes	characteristic
189	C₂²	D₇×D₁₆	1	1	yes	characteristic
190	C₂²	#89684	1	1	yes	characteristic
191	C₂²	D₇×Q₃₂	1	1	yes	characteristic
192	C₂²	#89691	1	1	yes	characteristic
193	C₂²	D₈×Dic₇	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	—	14	2	no	—
196	C₄	—	14	2	no	—
197	C₄	—	112	2	no	—
198	C₄	#89650	1	1	yes	characteristic
199	C₄	#89659	1	1	yes	characteristic
200	C₄	#89757	1	1	yes	characteristic
201	C₄	Q₁₆×Dic₇	1	1	yes	characteristic
202	C₂	#11727074	1	1	yes	characteristic
203	C₂	#11727075	1	1	yes	characteristic
204	C₂	#11727081	1	1	yes	characteristic
205	C₂	#11727082	1	1	yes	characteristic
206	C₂	#11728366	1	1	yes	characteristic
207	C₂	Dic₇×D₁₆	1	1	yes	characteristic
208	C₂	Dic₇×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₇×D₈
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#366	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#366	D₇
hypocenter	C₇	#13165
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
12. C₇×#366	C₄	semidirect
62. C₇×C₁₆	C₄⋊C₄	semidirect
71. #366	Dic₇	semidirect
154. C₁₆	C₄⋊Dic₇	semidirect
186. C₇	#13165	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#366	2
C₂×#89299	2
C₂×C₁₆×Dic₇	2
#11728366	2
#11725528	2
#13165	7