Subgroups — #13506371

Order 1792.

1006 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89646	C₂	1	4	yes	maximal
2	#11731801	C₂	1	2	yes	maximal
3	C₇×#1845	C₂	1	1	yes	maximal, characteristic
4	#11728360	C₂	1	4	yes	maximal
5	#11727519	C₂	1	1	yes	maximal, characteristic
6	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
7	#11727331	C₂	1	2	yes	maximal
8	C₂×#6451	—	2	4	no	—
9	C₂×#6451	C₂²	1	4	yes	—
10	C₂×C₇×#107	C₂²	1	4	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#384	C₂²	1	2	yes	—
13	C₂×#6392	C₂²	1	4	yes	—
14	C₂×C₈×Dic₇	C₂²	1	2	yes	—
15	C₂×C₈×Dic₇	C₂²	1	2	yes	—
16	#89748	—	2	4	no	—
17	#89748	C₂²	1	4	yes	—
18	C₂×#6381	C₂²	1	8	yes	—
19	#89707	C₂²	1	2	yes	—
20	C₄²×Dic₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
22	#89646	—	2	16	no	—
23	#33312	—	7	1	no	maximal, Sylow
24	C₂²×Dic₁₄	—	2	4	no	—
25	C₂²×C₇×C₈	C₂³	1	2	yes	—
26	C₂×C₇×C₄⋊C₄	—	2	4	no	—
27	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
28	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₇	—	2	4	no	—
30	C₂×C₄⋊Dic₇	C₂³	1	4	yes	—
31	C₂×Dic₇⋊C₄	—	2	8	no	—
32	C₂×Dic₇⋊C₄	—	2	4	no	—
33	C₂×C₄×Dic₇	D₄	1	2	yes	—
34	C₂×C₄×Dic₇	D₄	1	2	yes	—
35	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
37	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
38	#6451	—	2	16	no	—
39	#6451	—	4	16	no	—
40	C₇×#107	—	2	8	no	—
41	C₄×C₇×C₈	Q₈	1	4	yes	complemented
42	#6392	—	2	8	no	—
43	C₈×Dic₇	D₄	1	8	yes	—
44	C₈×Dic₇	Q₈	1	8	yes	complemented
45	#6381	—	2	16	no	—
46	C₄×C₇⋊C₈	Q₈	1	4	yes	complemented
47	C₂×#500	—	7	4	no	—
48	#2868	—	7	2	no	—
49	#1845	—	7	4	no	—
50	#1845	—	7	1	no	—
51	#1845	—	7	2	no	—
52	#1845	D₇	1	1	yes	characteristic
53	C₄²×C₈	—	7	1	no	—
54	C₂²×C₄×C₇	—	2	4	no	—
55	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₇	—	2	4	no	—
58	C₂²×Dic₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₇	C₂×D₄	1	2	yes	—
61	C₂×Dic₁₄	—	4	8	no	—
62	C₂×Dic₁₄	—	4	16	no	—
63	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
64	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
65	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
66	C₇×C₄⋊C₄	—	2	8	no	—
67	C₇×C₄⋊C₄	—	4	8	no	—
68	C₄²×C₇	C₂×Q₈	1	2	yes	—
69	C₄²×C₇	C₂×Q₈	1	2	yes	—
70	C₄⋊Dic₇	—	2	8	no	—
71	C₄⋊Dic₇	—	4	8	no	—
72	Dic₇⋊C₄	—	4	8	no	—
73	Dic₇⋊C₄	—	4	16	no	—
74	C₄×Dic₇	D₈	1	8	yes	—
75	C₄×Dic₇	Q₁₆	1	8	yes	—
76	C₄×Dic₇	C₂×D₄	1	4	yes	—
77	C₄×Dic₇	C₂×Q₈	1	4	yes	—
78	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
79	C₂×C₇⋊C₈	C₂×Q₈	1	4	yes	—
80	C₂×C₇⋊C₈	C₂×Q₈	1	4	yes	—
81	C₂×C₄⋊Q₈	—	7	4	no	—
82	C₂×C₄⋊Q₈	—	14	4	no	—
83	#500	—	14	16	no	—
84	C₂×#107	—	7	4	no	—
85	C₂×#107	—	7	8	no	—
86	C₂×#107	C₂×D₇	1	4	yes	—
87	C₂×C₄×C₈	—	7	2	no	—
88	C₂×C₄×C₈	—	7	2	no	—
89	C₂×C₄×C₈	—	7	1	no	—
90	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
91	#384	—	7	4	no	—
92	#384	—	7	2	no	—
93	#384	—	14	4	no	—
94	#384	C₂×D₇	1	2	yes	—
95	C₄³	—	7	1	no	—
96	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇	—	4	4	no	—
98	C₂×C₄×C₇	—	4	8	no	—
99	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
100	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
101	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
102	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
103	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
106	C₂×Dic₇	—	4	4	no	—
107	C₂×Dic₇	—	4	8	no	—
108	C₂×Dic₇	#127	1	2	yes	—
109	C₂×Dic₇	#127	1	2	yes	—
110	C₂×Dic₇	C₄⋊Q₈	1	4	yes	—
111	C₂×Dic₇	C₂×D₈	1	4	yes	—
112	C₂×Dic₇	C₂×D₈	1	4	yes	—
113	C₂×Dic₇	C₂×Q₁₆	1	4	yes	—
114	C₂×Dic₇	C₂×Q₁₆	1	4	yes	—
115	Dic₁₄	—	8	16	no	—
116	C₇×C₈	C₄⋊Q₈	1	8	yes	complemented
117	C₇⋊C₈	C₄⋊Q₈	1	8	yes	complemented
118	C₂²×Q₈	—	14	4	no	—
119	C₂²×C₈	—	7	2	no	—
120	C₂²×C₈	C₂²×D₇	1	2	yes	—
121	C₄⋊Q₈	—	14	16	no	—
122	C₄⋊Q₈	—	28	16	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	7	4	no	—
125	C₂×C₄⋊C₄	—	14	4	no	—
126	C₂×C₄⋊C₄	—	14	8	no	—
127	C₂×C₄⋊C₄	—	14	4	no	—
128	C₂×C₄⋊C₄	C₂²×D₇	1	4	yes	—
129	C₂×C₄²	—	7	2	no	—
130	C₂×C₄²	—	7	2	no	—
131	C₂×C₄²	—	7	1	no	—
132	C₂×C₄²	—	7	1	no	—
133	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
134	#107	—	2	8	no	—
135	#107	—	14	16	no	—
136	#107	—	14	8	no	—
137	C₄×C₈	—	7	8	no	—
138	C₄×C₈	—	7	8	no	—
139	C₄×C₈	—	7	4	no	—
140	C₄×C₈	Dic₁₄	1	4	yes	complemented
141	C₂²×C₇	C₂×#127	1	1	yes	characteristic
142	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₂²×C₇	C₂²×D₈	1	2	yes	—
145	C₂²×C₇	C₂²×Q₁₆	1	2	yes	—
146	C₄×C₇	—	8	4	no	—
147	C₄×C₇	#500	1	8	yes	complemented
148	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
149	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
150	Dic₇	—	8	4	no	—
151	Dic₇	#493	1	4	yes	—
152	Dic₇	#494	1	4	yes	—
153	Dic₇	#500	1	8	yes	complemented
154	C₂×Q₈	—	28	8	no	—
155	C₂×Q₈	—	28	16	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	7	1	no	—
158	C₂²×C₄	—	7	2	no	—
159	C₂²×C₄	—	7	1	no	—
160	C₂²×C₄	—	14	4	no	—
161	C₂²×C₄	D₄×D₇	1	2	yes	—
162	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
163	C₂×C₈	—	7	4	no	—
164	C₂×C₈	—	7	4	no	—
165	C₂×C₈	—	7	4	no	—
166	C₂×C₈	C₂×Dic₁₄	1	4	yes	—
167	C₂×C₈	D₄×D₇	1	4	yes	—
168	C₂×C₈	Q₈×D₇	1	4	yes	—
169	C₄⋊C₄	—	2	8	no	—
170	C₄⋊C₄	—	4	8	no	—
171	C₄⋊C₄	—	14	8	no	—
172	C₄⋊C₄	—	28	8	no	—
173	C₄⋊C₄	—	28	16	no	—
174	C₄⋊C₄	—	28	8	no	—
175	C₄²	—	7	8	no	—
176	C₄²	—	7	8	no	—
177	C₄²	—	7	4	no	—
178	C₄²	—	7	4	no	—
179	C₄²	C₂×Dic₁₄	1	2	yes	—
180	C₄²	Q₈×D₇	1	2	yes	—
181	C₂×C₇	#2868	1	1	yes	characteristic
182	C₂×C₇	C₂×#493	1	1	yes	characteristic
183	C₂×C₇	C₂×#494	1	1	yes	characteristic
184	C₂×C₇	C₂×#500	1	2	yes	—
185	C₂×C₇	C₂×#500	1	2	yes	—
186	C₂³	C₂×D₄×D₇	1	1	yes	characteristic
187	Q₈	—	56	16	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	—	7	2	no	—
191	C₂×C₄	—	7	2	no	—
192	C₂×C₄	—	7	4	no	—
193	C₂×C₄	—	7	4	no	—
194	C₂×C₄	—	7	4	no	—
195	C₂×C₄	—	7	4	no	—
196	C₂×C₄	—	7	4	no	—
197	C₂×C₄	—	28	4	no	—
198	C₂×C₄	—	28	8	no	—
199	C₂×C₄	#6451	1	4	yes	—
200	C₂×C₄	D₇×D₈	1	4	yes	—
201	C₂×C₄	D₇×Q₁₆	1	4	yes	—
202	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×D₇	1	2	yes	—
204	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
205	C₂×C₄	C₂×Q₈×D₇	1	2	yes	—
206	C₈	—	7	8	no	—
207	C₈	#6451	1	8	yes	complemented
208	C₇	#33312	1	1	yes	characteristic, complemented, Sylow
209	C₂²	C₂×#6451	1	1	yes	characteristic
210	C₂²	D₇×#127	1	1	yes	characteristic
211	C₂²	D₇×C₄⋊Q₈	1	1	yes	characteristic
212	C₂²	C₂×D₇×D₈	1	2	yes	—
213	C₂²	C₂×D₇×Q₁₆	1	2	yes	—
214	C₄	—	7	4	no	—
215	C₄	—	7	4	no	—
216	C₄	—	7	8	no	—
217	C₄	—	8	4	no	—
218	C₄	—	56	4	no	—
219	C₄	#89646	1	8	yes	complemented
220	C₄	C₂×#6451	1	2	yes	—
221	C₄	D₇×C₄⋊Q₈	1	2	yes	—
222	C₂	#11731801	1	1	yes	characteristic
223	C₂	C₂×#89646	1	2	yes	—
224	C₂	D₇×#493	1	1	yes	characteristic
225	C₂	D₇×#494	1	1	yes	characteristic
226	C₂	D₇×#500	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇
commutator, cosocle	C₂²×C₄×C₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₇×#1845	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
hypercenter	#1845	D₇
hypocenter	C₇	#33312
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
41. C₄×C₇×C₈	Q₈	semidirect
44. C₈×Dic₇	Q₈	semidirect
46. C₄×C₇⋊C₈	Q₈	semidirect
116. C₇×C₈	C₄⋊Q₈	semidirect
117. C₇⋊C₈	C₄⋊Q₈	semidirect
140. C₄×C₈	Dic₁₄	semidirect
147. C₄×C₇	#500	semidirect
153. Dic₇	#500	semidirect
207. C₈	#6451	semidirect
208. C₇	#33312	semidirect
219. C₄	#89646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89646	2
#11731801	2
C₇×#1845	2
#11728360	2
#11727519	2
C₄×C₈×Dic₇	2
#11727331	2
#33312	7