Subgroups — #13380428

Order 1792.

632 conjugacy classes of subgroups.

The subgroup lattice has 222 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	#11727238	C₂	1	4	yes	maximal
3	#11727236	C₂	1	4	yes	maximal
4	#11727236	C₂	1	4	yes	maximal
5	#11727146	C₂	1	2	yes	maximal
6	C₂×C₇×#125	C₄	1	1	yes	characteristic
7	C₇×#389	C₄	1	2	yes	—
8	C₇×#387	C₄	1	4	yes	—
9	C₇×#382	C₄	1	1	yes	characteristic
10	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
12	C₄³×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
16	C₂×C₄×C₇⋊C₈	C₂²	1	4	yes	—
17	#89319	C₂²	1	4	yes	—
18	#89319	C₂²	1	4	yes	—
19	#89319	C₂²	1	4	yes	—
20	#89319	C₂²	1	4	yes	—
21	#23369	—	7	1	no	maximal, Sylow
22	C₇×#125	C₈	1	8	yes	complemented
23	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
24	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
25	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂³	1	4	yes	—
31	C₂×C₄²×C₇	C₂³	1	2	yes	—
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₂²×C₇⋊C₈	C₂³	1	4	yes	—
34	C₂²×C₇⋊C₈	C₂³	1	4	yes	—
35	C₇×#95	C₂×C₄	1	4	yes	—
36	C₇×#95	C₂×C₄	1	4	yes	—
37	C₇×#95	C₂×C₄	1	4	yes	—
38	C₄⋊(C₇⋊C₈)	—	4	2	no	—
39	C₄⋊(C₇⋊C₈)	—	4	4	no	—
40	C₄⋊(C₇⋊C₈)	—	4	2	no	—
41	C₄×C₇⋊C₈	—	4	4	no	—
42	C₄×#125	D₇	1	1	yes	characteristic
43	#1939	—	7	4	no	—
44	#1916	—	7	4	no	—
45	#1916	—	7	4	no	—
46	#1831	—	7	2	no	—
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
54	C₇×C₄⋊C₄	—	4	2	no	—
55	C₇×C₄⋊C₄	—	4	4	no	—
56	C₇×C₄⋊C₄	C₂×C₈	1	16	yes	—
57	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	—
58	C₄²×C₇	—	2	4	no	—
59	C₄²×C₇	—	2	2	no	—
60	C₄²×C₇	—	2	1	no	—
61	C₄²×C₇	—	2	1	no	—
62	C₄²×C₇	—	2	4	no	—
63	C₄²×C₇	—	2	2	no	—
64	C₄²×C₇	—	4	1	no	—
65	C₄²×C₇	—	4	2	no	—
66	C₄²×C₇	C₂×C₈	1	4	yes	—
67	C₂×C₇⋊C₈	—	2	4	no	—
68	C₂×C₇⋊C₈	—	2	4	no	—
69	C₂×C₇⋊C₈	—	2	4	no	—
70	C₂×C₇⋊C₈	—	2	4	no	—
71	C₂×C₇⋊C₈	—	2	4	no	—
72	C₂×C₇⋊C₈	—	2	4	no	—
73	C₂×#125	Dic₇	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	7	2	no	—
75	C₂×C₄⋊C₈	—	7	2	no	—
76	C₂×C₄⋊C₈	—	7	4	no	—
77	C₂×C₄×C₈	—	7	4	no	—
78	#389	Dic₇	1	2	yes	—
79	#387	Dic₇	1	4	yes	—
80	#382	Dic₇	1	1	yes	characteristic
81	C₄×C₄⋊C₄	C₂×D₇	1	2	yes	—
82	C₄×C₄⋊C₄	C₂×D₇	1	4	yes	—
83	C₄³	C₂×D₇	1	1	yes	characteristic
84	#336	—	7	4	no	—
85	#336	—	7	4	no	—
86	#336	—	7	4	no	—
87	#336	—	7	4	no	—
88	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	4	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	4	no	—
102	C₂×C₄×C₇	—	2	4	no	—
103	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
106	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
110	C₇⋊C₈	—	4	4	no	—
111	C₇⋊C₈	—	4	4	no	—
112	C₂²×C₈	—	7	4	no	—
113	C₂²×C₈	—	7	4	no	—
114	#125	C₇⋊C₈	1	8	yes	complemented
115	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
116	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
117	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
118	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
119	C₂×C₄²	C₂×Dic₇	1	2	yes	—
120	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
121	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₇	1	2	yes	—
124	C₂×C₄²	C₂²×D₇	1	4	yes	—
125	C₄⋊C₈	—	28	4	no	—
126	C₄⋊C₈	—	28	2	no	—
127	C₄⋊C₈	—	28	2	no	—
128	C₄×C₈	—	28	4	no	—
129	#95	C₂×Dic₇	1	4	yes	—
130	#95	C₂×Dic₇	1	4	yes	—
131	#95	C₂×Dic₇	1	4	yes	—
132	C₂²×C₇	#519	1	1	yes	characteristic
133	C₂²×C₇	#520	1	2	yes	—
134	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
135	C₂²×C₇	#568	1	1	yes	characteristic
136	C₂²×C₇	#568	1	2	yes	—
137	C₄×C₇	—	2	2	no	—
138	C₄×C₇	—	2	4	no	—
139	C₄×C₇	—	2	8	no	—
140	C₄×C₇	—	4	1	no	—
141	C₄×C₇	—	4	2	no	—
142	C₄×C₇	—	4	4	no	—
143	C₄×C₇	#562	1	1	yes	characteristic
144	C₄×C₇	#563	1	2	yes	—
145	C₄×C₇	#564	1	1	yes	characteristic
146	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
147	C₂²×C₄	C₂²×Dic₇	1	4	yes	—
148	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
150	C₂²×C₄	C₂²×Dic₇	1	4	yes	—
151	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
152	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
153	C₂×C₈	—	14	4	no	—
154	C₂×C₈	—	14	4	no	—
155	C₂×C₈	—	14	4	no	—
156	C₂×C₈	—	14	4	no	—
157	C₂×C₈	—	14	4	no	—
158	C₂×C₈	—	14	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	C₂×C₇⋊C₈	1	16	yes	—
162	C₄⋊C₄	C₂×C₇⋊C₈	1	8	yes	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	4	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	2	no	—
171	C₄²	C₂×C₇⋊C₈	1	4	yes	—
172	C₂×C₇	#2382	1	1	yes	characteristic
173	C₂×C₇	#2982	1	1	yes	characteristic
174	C₂×C₇	#2987	1	1	yes	characteristic
175	C₂×C₇	#2988	1	2	yes	—
176	C₂×C₇	#2995	1	2	yes	—
177	C₂³	C₂³×Dic₇	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
193	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
194	C₂×C₄	C₂²×C₇⋊C₈	1	8	yes	—
195	C₂×C₄	#6551	1	1	yes	characteristic
196	C₂×C₄	#6551	1	1	yes	characteristic
197	C₂×C₄	#6553	1	2	yes	—
198	C₂×C₄	#6554	1	2	yes	—
199	C₈	—	28	4	no	—
200	C₈	—	28	4	no	—
201	C₇	#23369	1	1	yes	characteristic, complemented, Sylow
202	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
203	C₂²	#90503	1	1	yes	characteristic
204	C₂²	#90510	1	2	yes	—
205	C₂²	#90510	1	1	yes	characteristic
206	C₂²	#90518	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	8	no	—
209	C₄	—	2	2	no	—
210	C₄	—	4	1	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	4	no	—
213	C₄	C₇⋊#564	1	1	yes	characteristic
214	C₄	#90390	1	2	yes	—
215	C₄	#90391	1	1	yes	characteristic
216	C₂	#11739880	1	1	yes	characteristic
217	C₂	#11739965	1	2	yes	—
218	C₂	#11740472	1	1	yes	characteristic
219	C₂	#11740578	1	2	yes	—
220	C₂	#11740778	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇
commutator	C₂²×C₇	C₂³×C₈
Fitting	C₄×C₇×#125	C₂
socle, prime radical	C₂³×C₇	C₂³×C₄
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₄×#125	D₇
hypocenter	C₇	#23369
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
22. C₇×#125	C₈	semidirect
114. #125	C₇⋊C₈	semidirect
201. C₇	#23369	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
#11727238	2
#11727236	2
#11727146	2
#23369	7