Subgroups — #13253556

Order 1792.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	#11729541	C₂	1	2	yes	maximal
3	#11729539	C₂	1	2	yes	maximal
4	#11727519	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₇×#500	—	2	2	no	—
8	C₇×#500	C₄	1	4	yes	complemented
9	C₂×C₇×#107	—	2	2	no	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×#6409	—	2	4	no	—
12	C₂×#6392	—	2	2	no	—
13	#89707	C₂²	1	2	yes	—
14	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	#89324	—	2	4	no	—
17	C₈×C₇⋊C₈	—	2	2	no	—
18	#9823	—	7	1	no	maximal, Sylow
19	C₂²×C₇×Q₈	—	2	2	no	—
20	C₂²×C₇×C₈	D₄	1	2	yes	—
21	C₇×C₄⋊Q₈	—	2	2	no	—
22	C₇×C₄⋊Q₈	—	2	2	no	—
23	C₇×C₄⋊Q₈	—	2	2	no	—
24	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₇×C₄⋊C₄	—	2	2	no	—
26	C₂×C₇×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₇	—	2	2	no	—
29	C₂×C₄⋊Dic₇	—	2	2	no	—
30	C₂×C₄⋊Dic₇	—	2	2	no	—
31	C₂²×C₇⋊C₈	D₄	1	2	yes	—
32	C₂²×C₇⋊C₈	D₄	1	2	yes	—
33	C₇×#107	—	2	8	no	—
34	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₇×C₈	D₄	1	2	yes	—
36	#6409	—	4	8	no	—
37	#6392	—	4	4	no	—
38	C₄×C₇⋊C₈	—	2	1	no	—
39	C₄×C₇⋊C₈	—	2	1	no	—
40	C₄×C₇⋊C₈	—	2	1	no	—
41	C₄×C₇⋊C₈	—	2	1	no	—
42	C₂×#500	D₇	1	1	yes	characteristic
43	#1990	—	7	2	no	—
44	#1956	—	7	2	no	—
45	#1845	—	7	1	no	—
46	C₂×C₈²	—	7	1	no	—
47	C₂×C₇×Q₈	—	2	4	no	—
48	C₂×C₇×Q₈	—	4	2	no	—
49	C₂×C₇×Q₈	—	4	4	no	—
50	C₂×C₇×Q₈	—	4	4	no	—
51	C₂²×C₄×C₇	—	4	2	no	—
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₇	—	4	2	no	—
56	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
57	C₂×C₇×C₈	SD₁₆	1	4	yes	—
58	C₂×C₇×C₈	Q₁₆	1	4	yes	—
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₄⋊C₄	—	2	4	no	—
61	C₇×C₄⋊C₄	—	4	2	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇	#41	1	1	yes	characteristic
66	C₄²×C₇	#41	1	1	yes	characteristic
67	C₄⋊Dic₇	—	4	4	no	—
68	C₄⋊Dic₇	—	4	4	no	—
69	C₄⋊Dic₇	—	4	4	no	—
70	C₂×C₇⋊C₈	—	2	2	no	—
71	C₂×C₇⋊C₈	—	2	2	no	—
72	C₂×C₇⋊C₈	—	2	2	no	—
73	C₂×C₇⋊C₈	—	2	2	no	—
74	C₂×C₇⋊C₈	D₈	1	4	yes	—
75	C₂×C₇⋊C₈	Q₁₆	1	4	yes	—
76	C₂×C₄⋊Q₈	C₂×D₇	1	2	yes	—
77	#500	—	2	2	no	—
78	#500	Dic₇	1	4	yes	complemented
79	C₂×#107	—	2	2	no	—
80	C₂×#107	—	14	2	no	—
81	C₂×#103	—	14	4	no	—
82	C₂×C₄×C₈	—	7	1	no	—
83	C₂×C₄×C₈	—	7	1	no	—
84	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
85	#384	—	7	2	no	—
86	#340	—	14	4	no	—
87	C₈²	—	14	2	no	—
88	C₂³×C₇	#127	1	1	yes	characteristic
89	C₇×Q₈	—	4	8	no	—
90	C₇×Q₈	—	8	4	no	—
91	C₂×C₄×C₇	—	4	4	no	—
92	C₂×C₄×C₇	—	8	2	no	—
93	C₂×C₄×C₇	—	8	2	no	—
94	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₇	#124	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₇	#135	1	2	yes	—
105	C₂×C₄×C₇	#135	1	2	yes	—
106	C₂×Dic₇	—	8	2	no	—
107	C₂×Dic₇	—	8	2	no	—
108	C₂×Dic₇	—	8	2	no	—
109	C₇×C₈	#103	1	8	yes	complemented
110	C₇⋊C₈	—	2	4	no	—
111	C₇⋊C₈	—	2	4	no	—
112	C₂²×Q₈	—	2	2	no	—
113	C₂²×C₈	—	7	2	no	—
114	C₂²×C₈	—	7	2	no	—
115	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	C₂×Dic₇	1	4	yes	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	14	2	no	—
123	C₂×C₄⋊C₄	—	14	2	no	—
124	C₂×C₄⋊C₄	—	14	2	no	—
125	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
126	#107	—	2	8	no	—
127	#107	—	28	4	no	—
128	#103	—	28	8	no	—
129	C₄×C₈	—	14	1	no	—
130	C₄×C₈	—	14	1	no	—
131	C₄×C₈	—	14	1	no	—
132	C₄×C₈	—	14	1	no	—
133	C₄×C₈	C₂×Dic₇	1	2	yes	—
134	C₄×C₈	C₂²⋊D₇	1	2	yes	—
135	C₂²×C₇	#390	1	1	yes	characteristic
136	C₂²×C₇	#492	1	1	yes	characteristic
137	C₂²×C₇	#493	1	1	yes	characteristic
138	C₂²×C₇	#494	1	1	yes	characteristic
139	C₂²×C₇	#494	1	1	yes	characteristic
140	C₂²×C₇	#495	1	1	yes	characteristic
141	C₂²×C₇	#495	1	1	yes	characteristic
142	C₄×C₇	—	8	4	no	—
143	C₄×C₇	C₂×#103	1	2	yes	—
144	C₄×C₇	C₄×D₈	1	2	yes	—
145	C₄×C₇	C₄×Q₁₆	1	2	yes	—
146	C₄×C₇	#466	1	2	yes	—
147	C₄×C₇	#467	1	2	yes	—
148	C₄×C₇	#487	1	2	yes	—
149	Dic₇	—	16	2	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	28	2	no	—
156	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
159	C₂×C₈	—	7	4	no	—
160	C₂×C₈	—	7	4	no	—
161	C₂×C₈	—	14	2	no	—
162	C₂×C₈	—	14	2	no	—
163	C₂×C₈	—	14	2	no	—
164	C₂×C₈	—	14	2	no	—
165	C₂×C₈	#1143	1	4	yes	—
166	C₂×C₈	#1144	1	4	yes	—
167	C₂×C₈	C₂²⋊Dic₇	1	4	yes	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	28	4	no	—
173	C₄⋊C₄	—	28	4	no	—
174	C₄⋊C₄	—	28	4	no	—
175	C₄²	#1159	1	1	yes	characteristic
176	C₄²	#1159	1	1	yes	characteristic
177	C₄²	C₂²×Dic₇	1	1	yes	characteristic
178	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
179	C₂×C₇	#1706	1	1	yes	characteristic
180	C₂×C₇	#1707	1	1	yes	characteristic
181	C₂×C₇	#1710	1	1	yes	characteristic
182	C₂×C₇	#1713	1	1	yes	characteristic
183	C₂×C₇	#1956	1	1	yes	characteristic
184	C₂×C₇	#1969	1	1	yes	characteristic
185	C₂×C₇	#1970	1	1	yes	characteristic
186	C₂³	#6503	1	1	yes	characteristic
187	Q₈	—	4	8	no	—
188	Q₈	—	8	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	56	2	no	—
193	C₂×C₄	—	56	2	no	—
194	C₂×C₄	—	56	2	no	—
195	C₂×C₄	D₇×D₈	1	2	yes	—
196	C₂×C₄	#6475	1	2	yes	—
197	C₂×C₄	D₇×Q₁₆	1	2	yes	—
198	C₂×C₄	#6482	1	2	yes	—
199	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
200	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
201	C₂×C₄	#6499	1	1	yes	characteristic
202	C₂×C₄	#6501	1	1	yes	characteristic
203	C₂×C₄	#6501	1	1	yes	characteristic
204	C₂×C₄	C₂×#1143	1	2	yes	—
205	C₂×C₄	C₂×#1144	1	2	yes	—
206	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
207	C₈	—	14	4	no	—
208	C₈	—	14	4	no	—
209	C₈	#6409	1	8	yes	complemented
210	C₇	#9823	1	1	yes	characteristic, complemented, Sylow
211	C₂²	#89857	1	1	yes	characteristic
212	C₂²	#89864	1	1	yes	characteristic
213	C₂²	#89923	1	1	yes	characteristic
214	C₂²	#89927	1	1	yes	characteristic
215	C₂²	#89953	1	1	yes	characteristic
216	C₂²	#89963	1	1	yes	characteristic
217	C₂²	#89993	1	1	yes	characteristic
218	C₄	—	8	4	no	—
219	C₄	—	112	2	no	—
220	C₄	#89853	1	2	yes	—
221	C₄	D₈×Dic₇	1	2	yes	—
222	C₄	#89929	1	2	yes	—
223	C₄	Q₁₆×Dic₇	1	2	yes	—
224	C₄	#89960	1	2	yes	—
225	C₄	C₂×#6409	1	2	yes	—
226	C₂	#11729113	1	1	yes	characteristic
227	C₂	#11729114	1	1	yes	characteristic
228	C₂	#11729115	1	1	yes	characteristic
229	C₂	#11729116	1	1	yes	characteristic
230	C₂	#11729539	1	1	yes	characteristic
231	C₂	#11729792	1	1	yes	characteristic
232	C₂	#11729820	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6503
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×#500	C₂
socle, prime radical	C₂³×C₇	#127
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×#500	D₇
hypocenter	C₇	#9823
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
8. C₇×#500	C₄	semidirect
78. #500	Dic₇	semidirect
109. C₇×C₈	#103	semidirect
209. C₈	#6409	semidirect
210. C₇	#9823	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#500	2
#11729541	2
#11729539	2
#11727519	2
C₂×C₈×C₇⋊C₈	2
#9823	7