Subgroups — #13258702

Order 1792.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1939	C₂	1	1	yes	maximal, characteristic
2	#11727519	C₂	1	1	yes	maximal, characteristic
3	#11727220	C₂	1	2	yes	maximal
4	#11725508	C₂	1	2	yes	maximal
5	#11725460	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊C₈	—	2	1	no	—
7	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×#6392	—	2	1	no	—
11	C₂×#6392	—	2	1	no	—
12	C₂×#6381	—	2	2	no	—
13	#89707	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
15	C₇×#336	—	2	1	no	—
16	#89324	—	2	2	no	—
17	#10649	—	7	1	no	maximal, Sylow
18	C₂²×C₇×C₈	—	2	1	no	—
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
22	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₇	—	2	1	no	—
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇	—	2	2	no	—
26	C₂×C₄⋊Dic₇	—	2	2	no	—
27	C₂×C₄⋊Dic₇	—	2	2	no	—
28	C₂²×C₇⋊C₈	—	2	2	no	—
29	C₇×C₄⋊C₈	—	2	2	no	—
30	C₇×C₄⋊C₈	—	4	2	no	—
31	C₄×C₇×C₈	—	2	2	no	—
32	C₇×#95	—	2	1	no	—
33	#6392	—	4	2	no	—
34	#6392	—	4	2	no	—
35	#6381	—	4	4	no	—
36	C₄⋊(C₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	—
38	#1947	—	7	2	no	—
39	#1939	D₇	1	1	yes	characteristic
40	#1845	—	7	1	no	—
41	#1296	—	7	2	no	—
42	#1277	—	7	1	no	—
43	C₂²×C₄×C₇	—	2	1	no	—
44	C₂²×C₄×C₇	—	2	1	no	—
45	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	#41	1	1	yes	characteristic
48	C₂²×Dic₇	—	4	2	no	—
49	C₂×C₇×C₈	—	2	1	no	—
50	C₂×C₇×C₈	—	2	1	no	—
51	C₂×C₇×C₈	—	2	1	no	—
52	C₂×C₇×C₈	—	2	1	no	—
53	C₂×C₇×C₈	—	4	1	no	—
54	C₂×C₇×C₈	—	4	1	no	—
55	C₂×C₇×C₈	—	4	1	no	—
56	C₂×C₇×C₈	D₈	1	2	yes	—
57	C₂×C₇×C₈	Q₁₆	1	2	yes	—
58	C₇×C₄⋊C₄	—	2	1	no	—
59	C₇×C₄⋊C₄	—	2	2	no	—
60	C₇×C₄⋊C₄	D₈	1	2	yes	—
61	C₄²×C₇	—	4	1	no	—
62	C₄²×C₇	—	4	1	no	—
63	C₄²×C₇	C₂×Q₈	1	2	yes	—
64	C₄²×C₇	#41	1	2	yes	—
65	C₄⋊Dic₇	—	4	2	no	—
66	C₄⋊Dic₇	—	4	2	no	—
67	C₄⋊Dic₇	—	4	2	no	—
68	C₄⋊Dic₇	—	4	2	no	—
69	C₄⋊Dic₇	—	8	2	no	—
70	C₂×C₇⋊C₈	—	2	4	no	—
71	C₂×C₇⋊C₈	—	4	2	no	—
72	C₂×C₇⋊C₈	—	4	2	no	—
73	C₂×#107	—	14	1	no	—
74	C₂×#107	—	14	1	no	—
75	C₂×#107	—	14	2	no	—
76	C₂×C₄⋊C₈	—	2	1	no	—
77	C₂×C₄⋊C₈	—	7	2	no	—
78	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
80	#384	—	7	2	no	—
81	C₄×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
82	#340	—	14	2	no	—
83	#336	—	2	1	no	—
84	C₂³×C₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	4	1	no	—
89	C₂×C₄×C₇	—	4	1	no	—
90	C₂×C₄×C₇	—	4	1	no	—
91	C₂×C₄×C₇	—	4	1	no	—
92	C₂×C₄×C₇	—	4	1	no	—
93	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₇	#123	1	1	yes	characteristic
95	C₂×C₄×C₇	#125	1	2	yes	—
96	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇	#135	1	1	yes	characteristic
101	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₇	#137	1	2	yes	—
103	C₂×C₄×C₇	#137	1	1	yes	characteristic
104	C₂×Dic₇	—	8	2	no	—
105	C₂×Dic₇	—	8	2	no	—
106	C₂×Dic₇	—	8	2	no	—
107	C₇×C₈	—	2	2	no	—
108	C₇×C₈	—	2	2	no	—
109	C₇×C₈	—	8	1	no	—
110	C₇⋊C₈	—	4	4	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	14	2	no	—
113	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	14	2	no	—
116	C₂×C₄⋊C₄	—	14	2	no	—
117	C₂×C₄⋊C₄	—	14	2	no	—
118	C₂×C₄⋊C₄	D₂₈	1	1	yes	characteristic
119	C₂×C₄⋊C₄	D₂₈	1	1	yes	characteristic
120	C₂×C₄²	—	2	1	no	—
121	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
122	#107	—	28	2	no	—
123	#107	—	28	2	no	—
124	#107	—	28	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	7	4	no	—
128	C₄⋊C₈	—	14	2	no	—
129	C₄×C₈	—	2	2	no	—
130	#95	—	2	1	no	—
131	C₂²×C₇	#398	1	1	yes	characteristic
132	C₂²×C₇	#459	1	1	yes	characteristic
133	C₂²×C₇	#462	1	1	yes	characteristic
134	C₂²×C₇	#466	1	1	yes	characteristic
135	C₂²×C₇	#470	1	1	yes	characteristic
136	C₂²×C₇	#472	1	1	yes	characteristic
137	C₂²×C₇	#473	1	1	yes	characteristic
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	—	4	2	no	—
140	C₄×C₇	—	8	1	no	—
141	C₄×C₇	#474	1	2	yes	—
142	C₄×C₇	#479	1	2	yes	—
143	C₄×C₇	#480	1	1	yes	characteristic
144	C₄×C₇	#484	1	1	yes	characteristic
145	C₄×C₇	#500	1	1	yes	characteristic
146	C₄×C₇	#501	1	1	yes	characteristic
147	Dic₇	—	16	2	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	28	2	no	—
151	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
152	C₂²×C₄	#1157	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	14	4	no	—
162	C₂×C₈	—	28	2	no	—
163	C₂×C₈	—	28	2	no	—
164	C₂×C₈	#1141	1	2	yes	—
165	C₂×C₈	#1144	1	2	yes	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	28	2	no	—
169	C₄⋊C₄	—	28	2	no	—
170	C₄⋊C₄	—	28	2	no	—
171	C₄⋊C₄	—	28	2	no	—
172	C₄⋊C₄	—	56	2	no	—
173	C₄⋊C₄	D₅₆	1	2	yes	—
174	C₄²	—	4	1	no	—
175	C₄²	—	4	1	no	—
176	C₄²	#1159	1	2	yes	—
177	C₄²	Q₈×D₇	1	2	yes	—
178	C₂×C₇	#1667	1	1	yes	characteristic
179	C₂×C₇	#1674	1	1	yes	characteristic
180	C₂×C₇	#1689	1	1	yes	characteristic
181	C₂×C₇	#1696	1	1	yes	characteristic
182	C₂×C₇	#2083	1	1	yes	characteristic
183	C₂×C₇	#2086	1	1	yes	characteristic
184	C₂×C₇	#2089	1	1	yes	characteristic
185	C₂³	#6493	1	1	yes	characteristic
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	56	2	no	—
195	C₂×C₄	—	56	2	no	—
196	C₂×C₄	—	56	2	no	—
197	C₂×C₄	#6452	1	2	yes	—
198	C₂×C₄	#6460	1	2	yes	—
199	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
200	C₂×C₄	#6467	1	1	yes	characteristic
201	C₂×C₄	#6472	1	2	yes	—
202	C₂×C₄	C₂×#1141	1	1	yes	characteristic
203	C₂×C₄	#6498	1	1	yes	characteristic
204	C₂×C₄	C₂×#1144	1	1	yes	characteristic
205	C₂×C₄	#6507	1	1	yes	characteristic
206	C₂×C₄	#6512	1	1	yes	characteristic
207	C₂×C₄	#6514	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	8	1	no	—
211	C₈	—	28	4	no	—
212	C₇	#10649	1	1	yes	characteristic, complemented, Sylow
213	C₂²	#89755	1	1	yes	characteristic
214	C₂²	#89787	1	1	yes	characteristic
215	C₂²	#89803	1	1	yes	characteristic
216	C₂²	#89887	1	1	yes	characteristic
217	C₂²	#89909	1	1	yes	characteristic
218	C₂²	#89914	1	1	yes	characteristic
219	C₂²	#89915	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	—	112	2	no	—
224	C₄	#89601	1	2	yes	—
225	C₄	#89618	1	2	yes	—
226	C₄	#89805	1	1	yes	characteristic
227	C₄	#89816	1	1	yes	characteristic
228	C₄	#89837	1	1	yes	characteristic
229	C₄	#89850	1	1	yes	characteristic
230	C₂	#11727361	1	1	yes	characteristic
231	C₂	#11728216	1	1	yes	characteristic
232	C₂	#11728286	1	1	yes	characteristic
233	C₂	#11728601	1	1	yes	characteristic
234	C₂	#11728625	1	1	yes	characteristic
235	C₂	#11728698	1	1	yes	characteristic
236	C₂	#11728719	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6493
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1939	C₂
socle, prime radical	C₂³×C₇	#121
hypercenter	#1939	D₇
hypocenter	C₇	#10649
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
212. C₇	#10649	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1939	2
#11727519	2
#11727220	2
#11725508	2
#11725460	2
#10649	7