Subgroups — #13253812

Order 1792.

458 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#125	C₂	1	1	yes	maximal, characteristic
2	#11727148	C₂	1	1	yes	maximal, characteristic
3	#11727142	C₂	1	1	yes	maximal, characteristic
4	#11725425	C₂	1	4	yes	maximal
5	C₂×C₇×#125	C₄	1	1	yes	characteristic
6	C₇×#389	C₄	1	2	yes	—
7	C₇×#387	—	2	2	no	—
8	C₇×#382	C₄	1	1	yes	characteristic
9	C₄×C₇×C₄⋊C₄	—	2	2	no	—
10	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₄³×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₇⋊C₈)	—	2	2	no	—
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
15	#89319	—	2	2	no	—
16	#9872	—	7	1	no	maximal, Sylow
17	C₇×#125	—	2	2	no	—
18	C₇×#125	C₈	1	4	yes	complemented
19	C₂×C₇×C₄⋊C₄	—	2	2	no	—
20	C₂×C₇×C₄⋊C₄	—	2	2	no	—
21	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
22	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇	—	2	2	no	—
24	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	D₄	1	2	yes	—
27	C₂×C₄²×C₇	D₄	1	2	yes	—
28	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
29	C₂²×C₇⋊C₈	—	2	2	no	—
30	C₂²×C₇⋊C₈	—	2	2	no	—
31	C₇×#95	—	2	2	no	—
32	C₇×#95	—	2	2	no	—
33	C₇×#95	—	2	2	no	—
34	C₄⋊(C₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₇⋊C₈)	—	4	4	no	—
36	C₄×C₇⋊C₈	—	2	2	no	—
37	C₄×C₇⋊C₈	Q₈	1	4	yes	—
38	C₄×#125	D₇	1	1	yes	characteristic
39	#1842	—	7	1	no	—
40	#1744	—	7	1	no	—
41	#1279	—	7	4	no	—
42	C₂²×C₄×C₇	—	2	2	no	—
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₇×C₄⋊C₄	—	2	8	no	—
50	C₇×C₄⋊C₄	—	2	2	no	—
51	C₇×C₄⋊C₄	—	2	4	no	—
52	C₇×C₄⋊C₄	—	4	4	no	—
53	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
54	C₄²×C₇	—	2	2	no	—
55	C₄²×C₇	—	2	2	no	—
56	C₄²×C₇	—	2	2	no	—
57	C₄²×C₇	—	4	2	no	—
58	C₄²×C₇	—	4	2	no	—
59	C₄²×C₇	—	4	2	no	—
60	C₄²×C₇	C₂×C₈	1	2	yes	—
61	C₄²×C₇	M₁₆	1	2	yes	—
62	C₄²×C₇	C₂×Q₈	1	2	yes	—
63	C₄²×C₇	#41	1	2	yes	—
64	C₂×C₇⋊C₈	—	2	4	no	—
65	C₂×C₇⋊C₈	—	4	4	no	—
66	C₂×C₇⋊C₈	—	4	2	no	—
67	C₂×C₇⋊C₈	—	4	4	no	—
68	C₂×#125	Dic₇	1	1	yes	characteristic
69	C₂×C₄⋊C₈	—	7	2	no	—
70	C₂×C₄⋊C₈	—	14	2	no	—
71	C₂×C₄×C₈	—	7	2	no	—
72	#389	Dic₇	1	2	yes	—
73	#387	—	2	2	no	—
74	#382	Dic₇	1	1	yes	characteristic
75	C₄×C₄⋊C₄	—	2	2	no	—
76	C₄×C₄⋊C₄	C₂×D₇	1	2	yes	—
77	C₄³	C₂×D₇	1	1	yes	characteristic
78	#336	—	14	2	no	—
79	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	4	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	4	2	no	—
88	C₂×C₄×C₇	—	4	2	no	—
89	C₂×C₄×C₇	—	4	2	no	—
90	C₂×C₄×C₇	—	4	2	no	—
91	C₂×C₄×C₇	—	4	2	no	—
92	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
93	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₇	#124	1	1	yes	characteristic
96	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇	#131	1	2	yes	—
100	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₄×C₇	#137	1	2	yes	—
102	C₇⋊C₈	—	4	4	no	—
103	C₇⋊C₈	—	8	2	no	—
104	C₂²×C₈	—	14	2	no	—
105	C₂²×C₈	—	14	2	no	—
106	#125	—	2	2	no	—
107	#125	C₇⋊C₈	1	4	yes	complemented
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
111	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
114	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
115	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
116	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
117	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
118	C₄⋊C₈	—	14	4	no	—
119	C₄⋊C₈	—	28	4	no	—
120	C₄×C₈	—	7	4	no	—
121	C₄×C₈	—	14	2	no	—
122	#95	—	2	2	no	—
123	#95	—	2	2	no	—
124	#95	—	2	2	no	—
125	C₂²×C₇	#391	1	1	yes	characteristic
126	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₇	#408	1	1	yes	characteristic
128	C₂²×C₇	#417	1	2	yes	—
129	C₂²×C₇	#421	1	2	yes	—
130	C₄×C₇	—	2	4	no	—
131	C₄×C₇	—	2	2	no	—
132	C₄×C₇	—	4	4	no	—
133	C₄×C₇	—	4	2	no	—
134	C₄×C₇	—	8	2	no	—
135	C₄×C₇	C₈×Q₈	1	2	yes	—
136	C₄×C₇	#446	1	2	yes	—
137	C₄×C₇	#490	1	1	yes	characteristic
138	C₄×C₇	#491	1	1	yes	characteristic
139	C₄×C₇	#501	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
143	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
147	C₂×C₈	—	14	4	no	—
148	C₂×C₈	—	28	4	no	—
149	C₂×C₈	—	28	2	no	—
150	C₂×C₈	—	28	4	no	—
151	C₄⋊C₄	—	2	8	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	—	4	2	no	—
162	C₄²	C₂×C₇⋊C₈	1	2	yes	—
163	C₄²	#1136	1	2	yes	—
164	C₄²	Q₈×D₇	1	2	yes	—
165	C₄²	#1161	1	2	yes	—
166	C₂×C₇	#1477	1	2	yes	—
167	C₂×C₇	#1495	1	2	yes	—
168	C₂×C₇	#1984	1	1	yes	characteristic
169	C₂×C₇	#1992	1	1	yes	characteristic
170	C₂×C₇	#1995	1	1	yes	characteristic
171	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	4	yes	—
185	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
186	C₂×C₄	C₂×#1136	1	1	yes	characteristic
187	C₂×C₄	#6507	1	1	yes	characteristic
188	C₂×C₄	Q₈×Dic₇	1	2	yes	—
189	C₂×C₄	#6509	1	2	yes	—
190	C₂×C₄	#6510	1	1	yes	characteristic
191	C₂×C₄	#6511	1	2	yes	—
192	C₂×C₄	#6512	1	2	yes	—
193	C₂×C₄	#6514	1	2	yes	—
194	C₈	—	28	4	no	—
195	C₈	—	56	2	no	—
196	C₇	#9872	1	1	yes	characteristic, complemented, Sylow
197	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
198	C₂²	#90002	1	1	yes	characteristic
199	C₂²	#90004	1	2	yes	—
200	C₂²	#90006	1	1	yes	characteristic
201	C₂²	#90008	1	2	yes	—
202	C₄	—	2	2	no	—
203	C₄	—	2	4	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	4	no	—
206	C₄	—	8	2	no	—
207	C₄	Q₈×C₇⋊C₈	1	2	yes	—
208	C₄	#89796	1	2	yes	—
209	C₄	#89837	1	2	yes	—
210	C₄	#89839	1	1	yes	characteristic
211	C₄	#89841	1	1	yes	characteristic
212	C₂	#11729388	1	1	yes	characteristic
213	C₂	#11729393	1	2	yes	—
214	C₂	#11729495	1	1	yes	characteristic
215	C₂	#11729507	1	2	yes	—
216	C₂	#11729594	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₄×C₇×#125	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₄×#125	D₇
hypocenter	C₇	#9872
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
18. C₇×#125	C₈	semidirect
107. #125	C₇⋊C₈	semidirect
196. C₇	#9872	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×#125	2
#11727148	2
#11727142	2
#11725425	2
#9872	7