Subgroups — #126×C₇₉⋊C₈ #34296483

Order 20224.

786 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764008	C₂	1	3	yes	maximal
2	#23763959	C₂	1	3	yes	maximal, complemented
3	#23763543	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₄×C₇₉⋊C₈	C₂	1	3	yes	maximal, complemented
5	C₂²⋊C₄×C₇₉⋊C₈	C₂	1	3	yes	maximal
6	C₄²×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#126×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	3	yes	—
10	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	3	yes	—
11	C₂×C₄×C₇₉⋊C₈	C₂²	1	3	yes	—
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	3	yes	—
13	C₂×C₄×C₇₉⋊C₈	C₂²	1	3	yes	—
14	#19098318	C₂²	1	3	yes	—
15	#19098318	C₂²	1	3	yes	—
16	#19098318	C₂²	1	3	yes	—
17	#19098318	C₂²	1	3	yes	—
18	C₂×#126×C₇₉	C₄	1	1	yes	characteristic
19	#388×C₇₉	C₄	1	3	yes	—
20	#387×C₇₉	C₄	1	3	yes	—
21	#383×C₇₉	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₇₉	C₂²	1	3	yes	—
23	C₄×C₂²⋊C₄×C₇₉	C₂²	1	3	yes	—
24	C₄³×C₇₉	C₂²	1	1	yes	characteristic
25	C₂²×C₇₉⋊C₈	—	2	6	no	—
26	C₂²×C₇₉⋊C₈	C₂³	1	3	yes	—
27	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₇₉⋊C₈	C₂³	1	3	yes	—
29	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
30	C₄⋊(C₇₉⋊C₈)	—	2	6	no	—
31	C₂²⋊(C₇₉⋊C₈)	—	2	6	no	—
32	C₄×C₇₉⋊C₈	—	2	6	no	—
33	C₄×C₇₉⋊C₈	—	2	6	no	—
34	C₄×C₇₉⋊C₈	—	2	6	no	—
35	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	#126×C₇₉	C₈	1	8	yes	complemented
37	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
38	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
39	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
40	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
41	C₂×C₄²×C₇₉	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
44	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
45	#95×C₇₉	C₂×C₄	1	3	yes	—
46	#95×C₇₉	C₂×C₄	1	3	yes	—
47	#95×C₇₉	C₂×C₄	1	3	yes	—
48	#95×C₇₉	C₂×C₄	1	3	yes	—
49	C₂×C₇₉⋊C₈	—	2	3	no	—
50	C₂×C₇₉⋊C₈	—	2	3	no	—
51	C₂×C₇₉⋊C₈	—	2	3	no	—
52	C₂×C₇₉⋊C₈	—	2	3	no	—
53	C₂×C₇₉⋊C₈	—	2	3	no	—
54	C₂×C₇₉⋊C₈	—	2	3	no	—
55	C₂×C₇₉⋊C₈	—	4	4	no	—
56	C₂×C₇₉⋊C₈	#41	1	6	yes	—
57	C₂×C₇₉⋊C₈	#41	1	6	yes	—
58	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	—	2	3	no	—
60	C₂²×C₄×C₇₉	—	2	3	no	—
61	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₇₉	—	2	6	no	—
68	C₄⋊C₄×C₇₉	C₂×C₈	1	12	yes	complemented
69	C₂²⋊C₄×C₇₉	—	2	6	no	—
70	C₂²⋊C₄×C₇₉	C₂×C₈	1	12	yes	—
71	C₄²×C₇₉	—	2	6	no	—
72	C₄²×C₇₉	—	2	3	no	—
73	C₄²×C₇₉	—	2	3	no	—
74	C₄²×C₇₉	—	2	3	no	—
75	C₄²×C₇₉	—	2	3	no	—
76	C₄²×C₇₉	C₂×C₈	1	4	yes	complemented
77	C₂³×C₇₉	—	2	3	no	—
78	C₂³×C₇₉	—	2	3	no	—
79	C₂³×C₇₉	C₂²×C₈	1	2	yes	—
80	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	—	2	3	no	—
82	C₂×C₄×C₇₉	—	2	3	no	—
83	C₂×C₄×C₇₉	—	2	3	no	—
84	C₂×C₄×C₇₉	—	2	3	no	—
85	C₂×C₄×C₇₉	—	2	3	no	—
86	C₂×C₄×C₇₉	—	2	3	no	—
87	C₂×C₄×C₇₉	—	2	3	no	—
88	C₂×C₄×C₇₉	—	2	3	no	—
89	C₂×C₄×C₇₉	—	2	3	no	—
90	C₂×C₄×C₇₉	—	2	3	no	—
91	C₂×C₄×C₇₉	—	4	1	no	—
92	C₂×C₄×C₇₉	—	4	3	no	—
93	C₂×C₄×C₇₉	C₂²×C₈	1	6	yes	—
94	C₂×C₄×C₇₉	C₂²×C₈	1	6	yes	—
95	C₂×C₄×C₇₉	#131	1	6	yes	—
96	C₂×C₄×C₇₉	C₂×#41	1	3	yes	—
97	C₂×C₄×C₇₉	C₂×#41	1	3	yes	—
98	C₇₉⋊C₈	—	2	6	no	—
99	C₇₉⋊C₈	—	4	1	no	—
100	C₇₉⋊C₈	—	4	3	no	—
101	C₇₉⋊C₈	#126	1	4	yes	direct product factor
102	C₂²×C₇₉	—	2	6	no	—
103	C₂²×C₇₉	—	4	1	no	—
104	C₂²×C₇₉	—	4	3	no	—
105	C₂²×C₇₉	C₄×#41	1	3	yes	—
106	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
107	C₂²×C₇₉	C₂×#131	1	3	yes	—
108	C₄×C₇₉	—	2	6	no	—
109	C₄×C₇₉	—	2	6	no	—
110	C₄×C₇₉	—	2	6	no	—
111	C₄×C₇₉	—	4	1	no	—
112	C₄×C₇₉	—	4	3	no	—
113	C₄×C₇₉	C₂×#126	1	1	yes	characteristic
114	C₄×C₇₉	#529	1	3	yes	—
115	C₈×#126	—	79	1	no	maximal, Sylow
116	C₂×C₇₉	—	4	2	no	—
117	C₂×C₇₉	C₄×#126	1	1	yes	characteristic
118	C₂×C₇₉	C₈×#41	1	3	yes	—
119	C₂×C₇₉	#2966	1	3	yes	—
120	C₄×#126	D₇₉	1	1	yes	characteristic
121	#1923	—	79	3	no	—
122	#1916	—	79	3	no	—
123	#1839	—	79	1	no	—
124	C₈×C₄⋊C₄	—	79	3	no	—
125	C₈×C₂²⋊C₄	—	79	3	no	—
126	C₄²×C₈	—	79	1	no	—
127	C₇₉	C₈×#126	1	1	yes	characteristic, complemented, Sylow
128	C₂³×C₈	—	79	1	no	—
129	C₂×#126	Dic₇₉	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	79	3	no	—
131	C₂×C₂²⋊C₈	—	79	3	no	—
132	C₂×C₄×C₈	—	79	3	no	—
133	C₂×C₄×C₈	—	79	3	no	—
134	C₂×C₄×C₈	—	79	3	no	—
135	#388	Dic₇₉	1	3	yes	—
136	#387	Dic₇₉	1	3	yes	—
137	#383	Dic₇₉	1	1	yes	characteristic
138	C₄×C₄⋊C₄	C₂×D₇₉	1	3	yes	—
139	C₄×C₂²⋊C₄	C₂×D₇₉	1	3	yes	—
140	C₄³	C₂×D₇₉	1	1	yes	characteristic
141	#336	—	79	3	no	—
142	#336	—	79	3	no	—
143	#336	—	79	3	no	—
144	#336	—	79	3	no	—
145	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
146	C₂²×C₈	—	79	1	no	—
147	C₂²×C₈	—	79	3	no	—
148	C₂²×C₈	—	79	1	no	—
149	C₂²×C₈	—	79	3	no	—
150	C₂²×C₈	—	158	6	no	—
151	#126	C₇₉⋊C₈	1	8	yes	direct product factor
152	C₂×C₄⋊C₄	C₂×Dic₇₉	1	3	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₇₉	1	3	yes	—
154	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	3	yes	—
155	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	3	yes	—
156	C₂×C₄²	C₂×Dic₇₉	1	3	yes	—
157	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
159	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
160	C₄⋊C₈	—	158	6	no	—
161	C₂²⋊C₈	—	158	6	no	—
162	C₄×C₈	—	158	6	no	—
163	C₄×C₈	—	158	6	no	—
164	C₄×C₈	—	158	6	no	—
165	#95	C₂×Dic₇₉	1	3	yes	—
166	#95	C₂×Dic₇₉	1	3	yes	—
167	#95	C₂×Dic₇₉	1	3	yes	—
168	#95	C₂×Dic₇₉	1	3	yes	—
169	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
170	C₂²×C₄	—	2	3	no	—
171	C₂²×C₄	—	2	3	no	—
172	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
173	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
174	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
175	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
176	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
177	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
178	C₂×C₈	—	316	4	no	—
179	C₂×C₈	—	79	6	no	—
180	C₂×C₈	—	79	6	no	—
181	C₂×C₈	—	158	3	no	—
182	C₂×C₈	—	158	3	no	—
183	C₂×C₈	—	158	3	no	—
184	C₂×C₈	—	158	3	no	—
185	C₂×C₈	—	158	3	no	—
186	C₂×C₈	—	158	3	no	—
187	C₄⋊C₄	—	2	6	no	—
188	C₄⋊C₄	C₂×C₇₉⋊C₈	1	12	yes	complemented
189	C₂²⋊C₄	—	2	6	no	—
190	C₂²⋊C₄	C₂×C₇₉⋊C₈	1	12	yes	—
191	C₄²	—	2	3	no	—
192	C₄²	—	2	3	no	—
193	C₄²	—	2	3	no	—
194	C₄²	—	2	3	no	—
195	C₄²	—	2	6	no	—
196	C₄²	C₂×C₇₉⋊C₈	1	4	yes	complemented
197	C₂³	—	2	3	no	—
198	C₂³	—	2	3	no	—
199	C₂³	C₂²×C₇₉⋊C₈	1	2	yes	—
200	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	2	3	no	—
207	C₂×C₄	—	2	3	no	—
208	C₂×C₄	—	2	3	no	—
209	C₂×C₄	—	2	3	no	—
210	C₂×C₄	—	2	3	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	C₂²×C₇₉⋊C₈	1	6	yes	—
214	C₂×C₄	C₂²×C₇₉⋊C₈	1	6	yes	—
215	C₂×C₄	#14579168	1	6	yes	—
216	C₂×C₄	#41×D₇₉	1	3	yes	—
217	C₂×C₄	#41×D₇₉	1	3	yes	—
218	C₈	—	316	1	no	—
219	C₈	—	316	3	no	—
220	C₈	—	79	4	no	—
221	C₈	—	158	6	no	—
222	C₂²	—	2	6	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	3	no	—
225	C₂²	#41×Dic₇₉	1	3	yes	—
226	C₂²	C₂³×C₇₉⋊C₈	1	1	yes	characteristic
227	C₂²	C₂×#14579168	1	3	yes	—
228	C₄	—	2	6	no	—
229	C₄	—	2	6	no	—
230	C₄	—	2	6	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	3	no	—
233	C₄	#126×D₇₉	1	1	yes	characteristic
234	C₄	#19099096	1	3	yes	—
235	C₂	—	4	2	no	—
236	C₂	#126×Dic₇₉	1	1	yes	characteristic
237	C₂	#41×C₇₉⋊C₈	1	3	yes	—
238	C₂	#23772376	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂³×C₈
Fitting	C₄×#126×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₄×#126	D₇₉
hypocenter	C₇₉	C₈×#126
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#126

Complements

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

N	H ≅ G/N	type
2. #23763959	C₂	semidirect
3. #23763543	C₂	semidirect
4. C₄⋊C₄×C₇₉⋊C₈	C₂	semidirect
6. C₄²×C₇₉⋊C₈	C₂	semidirect
36. #126×C₇₉	C₈	semidirect
68. C₄⋊C₄×C₇₉	C₂×C₈	semidirect
76. C₄²×C₇₉	C₂×C₈	semidirect
101. C₇₉⋊C₈	#126	direct (N × H)
127. C₇₉	C₈×#126	semidirect
151. #126	C₇₉⋊C₈	direct (N × H)
188. C₄⋊C₄	C₂×C₇₉⋊C₈	semidirect
196. C₄²	C₂×C₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#23764008	2
#23763959	2
#23763543	2
C₄⋊C₄×C₇₉⋊C₈	2
C₂²⋊C₄×C₇₉⋊C₈	2
C₄²×C₇₉⋊C₈	2
C₄×#126×C₇₉	2
C₈×#126	79

Subgroups — #126×C79⋊C8 #34296483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #126×C₇₉⋊C₈ #34296483