Subgroups — #13509287

Order 1792.

490 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11739361	C₂	1	2	yes	maximal
2	C₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729108	C₂	1	2	yes	maximal, complemented
4	#11727099	C₂	1	2	yes	maximal
5	#11727094	C₂	1	2	yes	maximal, complemented
6	#11727079	C₂	1	2	yes	maximal
7	#11727074	C₂	1	2	yes	maximal, complemented
8	C₄×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic
9	#11726799	C₂	1	1	yes	maximal, characteristic, complemented
10	#90379	—	2	2	no	—
11	C₇×#499	C₂²	1	2	yes	—
12	#89836	—	2	2	no	—
13	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
15	#89652	C₂²	1	4	yes	—
16	#89646	C₂²	1	4	yes	—
17	#89632	—	2	2	no	—
18	#89627	—	2	2	no	—
19	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
20	#89455	C₂²	1	2	yes	—
21	C₇×#367	C₂²	1	2	yes	—
22	C₇×#366	C₂²	1	2	yes	—
23	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
24	#89300	C₂²	1	2	yes	—
25	#89299	C₂²	1	2	yes	—
26	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
27	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
28	#89285	C₂²	1	4	yes	—
29	#89284	C₂²	1	4	yes	—
30	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
31	#33726	—	7	1	no	maximal, Sylow
32	C₇×#125	—	2	2	no	—
33	C₂×C₈×D₇	D₄	1	1	yes	characteristic
34	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
35	#6459	—	4	4	no	—
36	#6455	—	2	4	no	—
37	#6451	—	2	4	no	—
38	#6450	—	4	4	no	—
39	C₄²×D₇	D₄	1	1	yes	characteristic
40	#6433	—	2	2	no	—
41	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
43	C₇×#107	C₂³	1	4	yes	—
44	C₇×#106	—	2	2	no	—
45	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
46	#6392	C₂³	1	4	yes	—
47	#6391	—	2	2	no	—
48	C₈×Dic₇	D₄	1	1	yes	characteristic
49	C₈×Dic₇	C₂³	1	1	yes	characteristic
50	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
52	#6382	—	2	4	no	—
53	#6381	—	2	4	no	—
54	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
55	D₇×C₁₆	Q₈	1	4	yes	complemented
56	D₇×C₁₆	Q₈	1	4	yes	complemented
57	#3160	—	7	2	no	—
58	#2254	—	7	2	no	—
59	#2253	—	7	2	no	—
60	#2253	—	7	1	no	—
61	#2253	D₇	1	1	yes	characteristic, complemented
62	#2252	—	7	2	no	—
63	#2158	—	7	2	no	—
64	#2158	—	7	2	no	—
65	C₂×C₄×C₁₆	—	7	1	no	—
66	C₂×C₄×D₇	D₈	1	2	yes	—
67	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₁₄	—	4	4	no	—
69	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
71	C₇×C₄⋊C₄	—	2	4	no	—
72	C₇×C₄⋊C₄	—	4	4	no	—
73	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
74	(C₂×D₇)⋊C₄	—	4	4	no	—
75	C₄⋊Dic₇	—	2	4	no	—
76	Dic₇⋊C₄	—	4	4	no	—
77	Dic₇⋊C₄	—	4	4	no	—
78	C₄×Dic₇	—	4	4	no	—
79	C₄×Dic₇	D₈	1	2	yes	—
80	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
83	C₈×D₇	Q₁₆	1	4	yes	—
84	C₈×D₇	C₂×Q₈	1	2	yes	—
85	C₈×D₇	C₂×Q₈	1	2	yes	—
86	C₇×C₁₆	C₂×Q₈	1	2	yes	—
87	C₇×C₁₆	C₂×Q₈	1	2	yes	—
88	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
89	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
90	#533	—	14	2	no	—
91	C₂²×C₁₆	—	7	1	no	—
92	C₂²×C₁₆	—	7	1	no	—
93	#500	—	7	4	no	—
94	#499	—	7	2	no	—
95	#499	—	14	2	no	—
96	#499	C₂×D₇	1	2	yes	—
97	#498	—	14	2	no	—
98	#427	—	7	4	no	—
99	#427	—	14	2	no	—
100	C₂×C₄×C₈	—	7	1	no	—
101	#367	—	7	2	no	—
102	#367	—	7	4	no	—
103	#367	C₂×D₇	1	2	yes	—
104	#366	—	7	2	no	—
105	#366	—	7	4	no	—
106	#366	C₂×D₇	1	2	yes	—
107	C₄×C₁₆	—	7	1	no	—
108	C₄×C₁₆	—	7	1	no	—
109	C₄×C₁₆	—	7	1	no	—
110	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
111	C₂²×D₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇	—	4	4	no	—
113	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₇	—	4	4	no	—
116	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
117	C₂×Dic₇	C₂×D₈	1	2	yes	—
118	C₄×D₇	—	2	4	no	—
119	C₄×D₇	C₄⋊Q₈	1	2	yes	—
120	C₄×D₇	C₂×Q₁₆	1	2	yes	—
121	Dic₁₄	—	8	8	no	—
122	C₇×C₈	C₂×Q₁₆	1	2	yes	—
123	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
124	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
125	C₇⋊C₈	C₄⋊Q₈	1	2	yes	—
126	C₇⋊C₈	C₂×Q₁₆	1	2	yes	—
127	C₂²×C₈	—	7	1	no	—
128	C₂²×C₈	—	7	1	no	—
129	C₄⋊Q₈	—	14	4	no	—
130	#125	—	2	2	no	—
131	#125	—	14	2	no	—
132	C₂²⋊Q₈	—	28	4	no	—
133	C₄×Q₈	—	28	4	no	—
134	#117	—	14	4	no	—
135	C₂×C₄²	—	7	1	no	—
136	C₂×C₁₆	—	7	1	no	—
137	C₂×C₁₆	—	7	1	no	—
138	C₂×C₁₆	—	7	4	no	—
139	C₂×C₁₆	—	7	4	no	—
140	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
141	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
142	#107	—	7	4	no	—
143	#107	—	14	4	no	—
144	#107	C₂²×D₇	1	4	yes	—
145	#106	—	2	2	no	—
146	#106	—	14	4	no	—
147	#106	—	14	2	no	—
148	C₄×C₈	—	7	1	no	—
149	C₄×C₈	—	7	1	no	—
150	C₄×C₈	—	7	1	no	—
151	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
152	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₇	—	2	2	no	—
154	C₂×D₇	#500	1	2	yes	—
155	C₄×C₇	—	2	2	no	—
156	C₄×C₇	—	8	4	no	—
157	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
159	Dic₇	—	8	4	no	—
160	Dic₇	#500	1	2	yes	—
161	Dic₇	#508	1	4	yes	—
162	C₂×Q₈	—	28	4	no	—
163	C₂²×C₄	—	7	1	no	—
164	C₂²×C₄	—	7	2	no	—
165	C₂×C₈	—	7	1	no	—
166	C₂×C₈	—	7	1	no	—
167	C₂×C₈	—	7	4	no	—
168	C₂×C₈	—	7	2	no	—
169	C₂×C₈	—	7	2	no	—
170	C₂×C₈	D₄×D₇	1	1	yes	characteristic
171	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	14	4	no	—
175	C₄⋊C₄	—	28	4	no	—
176	C₄⋊C₄	—	28	4	no	—
177	C₂²⋊C₄	—	28	4	no	—
178	C₄²	—	7	2	no	—
179	C₄²	—	7	1	no	—
180	C₄²	—	28	4	no	—
181	C₄²	D₄×D₇	1	1	yes	characteristic
182	C₁₆	—	7	2	no	—
183	C₁₆	—	7	2	no	—
184	C₁₆	Q₈×D₇	1	2	yes	—
185	C₁₆	Q₈×D₇	1	2	yes	—
186	C₂×C₇	C₂×#500	1	1	yes	characteristic
187	C₂×C₇	C₂×#508	1	2	yes	—
188	D₇	—	2	2	no	—
189	C₂³	—	7	1	no	—
190	Q₈	—	56	8	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	7	1	no	—
193	C₂×C₄	—	7	2	no	—
194	C₂×C₄	—	7	2	no	—
195	C₂×C₄	—	7	2	no	—
196	C₂×C₄	—	14	4	no	—
197	C₂×C₄	—	28	4	no	—
198	C₂×C₄	D₇×D₈	1	2	yes	—
199	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
200	C₈	—	7	2	no	—
201	C₈	—	7	2	no	—
202	C₈	D₇×Q₁₆	1	2	yes	—
203	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
204	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
205	C₇	#33726	1	1	yes	characteristic, complemented, Sylow
206	C₂²	—	7	2	no	—
207	C₂²	—	14	2	no	—
208	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	7	4	no	—
211	C₄	—	7	2	no	—
212	C₄	—	8	4	no	—
213	C₄	—	56	4	no	—
214	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
215	C₄	C₂×D₇×Q₁₆	1	1	yes	characteristic
216	C₂	—	14	2	no	—
217	C₂	D₇×#500	1	1	yes	characteristic
218	C₂	D₇×#508	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2253	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂²×D₇	C₂×D₈
hypercenter	#2253	D₇
hypocenter	C₇	#33726
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. C₇×#2253	C₂	semidirect
3. #11729108	C₂	semidirect
5. #11727094	C₂	semidirect
7. #11727074	C₂	semidirect
9. #11726799	C₂	semidirect
55. D₇×C₁₆	Q₈	semidirect
56. D₇×C₁₆	Q₈	semidirect
61. #2253	D₇	semidirect
205. C₇	#33726	semidirect

Maximal subgroups

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

subgroup	index
#11739361	2
C₇×#2253	2
#11729108	2
#11727099	2
#11727094	2
#11727079	2
#11727074	2
C₄×D₇×C₁₆	2
#11726799	2
#33726	7