Subgroups — #23764008

Order 10112.

292 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098318	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098318	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
13	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
14	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
15	C₄×C₇₉⋊C₈	—	2	2	no	—
16	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₇₉	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₇₉	C₄	1	1	yes	characteristic
19	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	#95×C₇₉	C₄	1	1	yes	characteristic
22	#95×C₇₉	C₄	1	1	yes	characteristic
23	C₂×C₇₉⋊C₈	—	2	1	no	—
24	C₂×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₇₉⋊C₈	—	2	1	no	—
26	C₂×C₇₉⋊C₈	—	2	1	no	—
27	C₂×C₇₉⋊C₈	—	2	1	no	—
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₂×C₇₉⋊C₈	—	2	1	no	—
30	C₂×C₇₉⋊C₈	—	2	1	no	—
31	C₂×C₇₉⋊C₈	—	2	1	no	—
32	C₂×C₇₉⋊C₈	—	2	1	no	—
33	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
34	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
35	C₂⁴×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	—	2	1	no	—
37	C₂²×C₄×C₇₉	—	2	1	no	—
38	C₂²×C₄×C₇₉	—	2	1	no	—
39	C₂²×C₄×C₇₉	—	2	1	no	—
40	C₂²×C₄×C₇₉	—	2	1	no	—
41	C₂²×C₄×C₇₉	—	2	1	no	—
42	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
48	C₂²⋊C₄×C₇₉	—	2	2	no	—
49	C₂²⋊C₄×C₇₉	C₈	1	4	yes	complemented
50	C₄²×C₇₉	—	2	1	no	—
51	C₄²×C₇₉	—	2	1	no	—
52	C₄²×C₇₉	—	2	1	no	—
53	C₄²×C₇₉	—	2	1	no	—
54	C₂³×C₇₉	—	2	1	no	—
55	C₂³×C₇₉	—	2	1	no	—
56	C₂³×C₇₉	—	2	1	no	—
57	C₂³×C₇₉	—	2	1	no	—
58	C₂³×C₇₉	—	2	1	no	—
59	C₂³×C₇₉	—	2	1	no	—
60	C₂³×C₇₉	C₂×C₈	1	2	yes	—
61	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	—	2	1	no	—
63	C₂×C₄×C₇₉	—	2	1	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	2	1	no	—
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	2	1	no	—
68	C₂×C₄×C₇₉	—	2	1	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	1	no	—
71	C₂×C₄×C₇₉	—	2	1	no	—
72	C₂×C₄×C₇₉	—	2	1	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	—	4	1	no	—
75	C₂×C₄×C₇₉	—	4	1	no	—
76	C₂×C₄×C₇₉	—	4	1	no	—
77	C₂×C₄×C₇₉	—	4	1	no	—
78	C₂×C₄×C₇₉	C₂×C₈	1	2	yes	—
79	C₂×C₄×C₇₉	C₂×C₈	1	2	yes	—
80	C₂×C₄×C₇₉	M₁₆	1	2	yes	—
81	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
84	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
85	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
86	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
87	C₇₉⋊C₈	—	2	2	no	—
88	C₇₉⋊C₈	—	2	2	no	—
89	C₇₉⋊C₈	—	4	1	no	—
90	C₇₉⋊C₈	—	4	1	no	—
91	C₂²×C₇₉	—	2	2	no	—
92	C₂²×C₇₉	—	2	2	no	—
93	C₂²×C₇₉	—	2	2	no	—
94	C₂²×C₇₉	—	4	1	no	—
95	C₂²×C₇₉	—	4	1	no	—
96	C₂²×C₇₉	—	4	1	no	—
97	C₂²×C₇₉	—	4	1	no	—
98	C₂²×C₇₉	#117	1	1	yes	characteristic
99	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
100	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
101	C₂²×C₇₉	C₂²×C₈	1	1	yes	characteristic
102	C₂²×C₇₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂²×C₇₉	#131	1	1	yes	characteristic
104	C₂²×C₇₉	#131	1	1	yes	characteristic
105	C₄×C₇₉	—	2	2	no	—
106	C₄×C₇₉	—	2	2	no	—
107	C₄×C₇₉	—	2	2	no	—
108	C₄×C₇₉	—	4	1	no	—
109	C₄×C₇₉	—	4	1	no	—
110	C₄×C₇₉	#121	1	1	yes	characteristic
111	C₄×C₇₉	#123	1	1	yes	characteristic
112	C₄×C₇₉	#124	1	1	yes	characteristic
113	C₄×C₇₉	#126	1	1	yes	characteristic
114	C₂×C₇₉	—	4	2	no	—
115	C₂×C₇₉	#388	1	1	yes	characteristic
116	C₂×C₇₉	#431	1	1	yes	characteristic
117	C₂×C₇₉	#433	1	1	yes	characteristic
118	C₂×C₇₉	C₈×D₄	1	1	yes	characteristic
119	C₂×C₇₉	C₈×D₄	1	1	yes	characteristic
120	C₂×C₇₉	#435	1	1	yes	characteristic
121	C₂×C₇₉	#436	1	1	yes	characteristic
122	#1923	—	79	1	no	maximal, Sylow
123	C₇₉	#1923	1	1	yes	characteristic, complemented, Sylow
124	C₂×C₄⋊C₈	—	79	1	no	—
125	C₂×C₂²⋊C₈	—	79	1	no	—
126	C₂×C₂²⋊C₈	—	79	1	no	—
127	C₂×C₄×C₈	—	79	1	no	—
128	C₄×C₂²⋊C₄	D₇₉	1	1	yes	characteristic
129	#336	—	79	1	no	—
130	#336	—	79	1	no	—
131	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
132	C₂²×C₈	—	79	1	no	—
133	C₂²×C₈	—	79	1	no	—
134	C₂²×C₈	—	79	1	no	—
135	C₂²×C₈	—	79	1	no	—
136	C₂×C₂²⋊C₄	Dic₇₉	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	Dic₇₉	1	1	yes	characteristic
138	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
139	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
140	C₄⋊C₈	—	158	2	no	—
141	C₂²⋊C₈	—	158	2	no	—
142	C₂²⋊C₈	—	158	2	no	—
143	C₄×C₈	—	158	2	no	—
144	#95	Dic₇₉	1	1	yes	characteristic
145	#95	Dic₇₉	1	1	yes	characteristic
146	C₂⁴	C₂×Dic₇₉	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	C₂×Dic₇₉	1	1	yes	characteristic
154	C₂²×C₄	C₂×Dic₇₉	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₇₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₇₉	1	1	yes	characteristic
157	C₂²×C₄	C₂×Dic₇₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
159	C₂×C₈	—	79	2	no	—
160	C₂×C₈	—	79	2	no	—
161	C₂×C₈	—	158	1	no	—
162	C₂×C₈	—	158	1	no	—
163	C₂×C₈	—	158	1	no	—
164	C₂×C₈	—	158	1	no	—
165	C₂×C₈	—	158	1	no	—
166	C₂×C₈	—	158	1	no	—
167	C₂×C₈	—	158	1	no	—
168	C₂×C₈	—	158	1	no	—
169	C₂×C₈	—	158	1	no	—
170	C₂×C₈	—	158	1	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	C₇₉⋊C₈	1	4	yes	complemented
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	1	no	—
182	C₂³	—	2	1	no	—
183	C₂³	C₂×C₇₉⋊C₈	1	2	yes	—
184	C₂³	C₂²×Dic₇₉	1	1	yes	characteristic
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
202	C₂×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
203	C₂×C₄	#11929608	1	2	yes	—
204	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
205	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
206	C₂×C₄	#11929623	1	1	yes	characteristic
207	C₂×C₄	#11929623	1	1	yes	characteristic
208	C₂×C₄	#11929624	1	1	yes	characteristic
209	C₂×C₄	#11929624	1	1	yes	characteristic
210	C₈	—	316	1	no	—
211	C₈	—	316	1	no	—
212	C₈	—	158	2	no	—
213	C₈	—	158	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	1	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	#14579120	1	1	yes	characteristic
222	C₂²	D₄×Dic₇₉	1	1	yes	characteristic
223	C₂²	D₄×Dic₇₉	1	1	yes	characteristic
224	C₂²	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
225	C₂²	C₂×#11929608	1	1	yes	characteristic
226	C₂²	#14579168	1	1	yes	characteristic
227	C₂²	#14579168	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	#14579134	1	1	yes	characteristic
234	C₄	#14579143	1	1	yes	characteristic
235	C₄	#14579147	1	1	yes	characteristic
236	C₄	#14579154	1	1	yes	characteristic
237	C₂	—	4	2	no	—
238	C₂	#19098456	1	1	yes	characteristic
239	C₂	#19098616	1	1	yes	characteristic
240	C₂	#19098624	1	1	yes	characteristic
241	C₂	D₄×C₇₉⋊C₈	1	1	yes	characteristic
242	C₂	D₄×C₇₉⋊C₈	1	1	yes	characteristic
243	C₂	#19098635	1	1	yes	characteristic
244	C₂	#19098641	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₈
Fitting	C₄×C₂²⋊C₄×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×C₄
prime radical	C₂⁴×C₇₉	C₂×C₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₄×C₂²⋊C₄	D₇₉
hypocenter	C₇₉	#1923
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₇₉⋊C₈)	C₂	semidirect
4. C₂×C₄×C₇₉⋊C₈	C₂	semidirect
5. #19098318	C₂	semidirect
6. #19098318	C₂	semidirect
49. C₂²⋊C₄×C₇₉	C₈	semidirect
123. C₇₉	#1923	semidirect
172. C₂²⋊C₄	C₇₉⋊C₈	semidirect