Subgroups — C₂×#23765207 #34441344

Order 20224.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765207	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	1	2	yes	maximal
4	C₂×#19098339	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#405×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	8	yes	complemented
10	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	8	yes	—
11	#19098339	C₂²	1	4	yes	complemented
12	C₄×C₇₉⋊C₁₆	C₂²	1	4	yes	complemented
13	C₂²×M₁₆×C₇₉	C₄	1	1	yes	characteristic
14	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₇₉	C₄	1	1	yes	characteristic
16	#405×C₇₉	C₄	1	4	yes	—
17	#405×C₇₉	C₂²	1	4	yes	—
18	C₂×#97×C₇₉	C₄	1	1	yes	characteristic
19	C₂×#97×C₇₉	C₄	1	1	yes	characteristic
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₇₉⋊C₁₆	—	2	4	no	—
23	C₂×C₇₉⋊C₁₆	—	2	4	no	—
24	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
25	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
26	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₇₉	—	2	4	no	—
28	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₇₉	—	2	4	no	—
30	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
36	#117×C₇₉	C₈	1	4	yes	—
37	#117×C₇₉	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₇₉	C₈	1	2	yes	—
39	C₂×C₂²⋊C₄×C₇₉	C₈	1	2	yes	—
40	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
42	#97×C₇₉	C₂×C₄	1	4	yes	—
43	#97×C₇₉	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₇₉	C₂³	1	2	yes	—
47	C₄×C₈×C₇₉	C₂³	1	2	yes	—
48	C₇₉⋊C₁₆	—	2	4	no	—
49	C₇₉⋊C₁₆	—	2	4	no	—
50	C₂⁴×C₇₉	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₇₉	—	2	2	no	—
52	C₂²×C₄×C₇₉	—	2	2	no	—
53	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₇₉	—	2	8	no	—
60	C₂×C₈×C₇₉	—	2	4	no	—
61	C₂×C₈×C₇₉	—	2	4	no	—
62	C₂×C₈×C₇₉	—	2	2	no	—
63	C₂×C₈×C₇₉	—	2	2	no	—
64	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
65	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₇₉	#41	1	4	yes	—
70	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₇₉	C₂×C₈	1	8	yes	—
72	C₂²⋊C₄×C₇₉	C₂×C₈	1	8	yes	—
73	C₄²×C₇₉	C₂×C₈	1	2	yes	—
74	C₄²×C₇₉	C₂×C₈	1	2	yes	—
75	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
77	C₂³×C₇₉	—	2	2	no	—
78	C₂³×C₇₉	—	2	4	no	—
79	C₂³×C₇₉	C₂²×C₈	1	2	yes	—
80	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	2	4	no	—
83	C₂×C₄×C₇₉	—	2	4	no	—
84	C₂×C₄×C₇₉	—	2	2	no	—
85	C₂×C₄×C₇₉	—	2	2	no	—
86	C₂×C₄×C₇₉	—	2	4	no	—
87	C₂×C₄×C₇₉	#117	1	2	yes	—
88	C₂×C₄×C₇₉	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₇₉	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
94	C₈×C₇₉	—	2	2	no	—
95	C₈×C₇₉	—	2	2	no	—
96	C₈×C₇₉	#117	1	2	yes	—
97	C₈×C₇₉	C₂×M₁₆	1	4	yes	—
98	C₈×C₇₉	C₂×#41	1	2	yes	—
99	C₂²×C₇₉	—	2	4	no	—
100	C₂²×C₇₉	—	2	2	no	—
101	C₂²×C₇₉	—	4	4	no	—
102	C₂²×C₇₉	#431	1	2	yes	—
103	C₂²×C₇₉	#504	1	4	yes	—
104	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
105	C₄×C₇₉	—	2	4	no	—
106	C₄×C₇₉	—	2	4	no	—
107	C₄×C₇₉	—	2	2	no	—
108	C₄×C₇₉	#431	1	2	yes	—
109	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
110	C₄×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
111	C₂×#2164	—	79	1	no	maximal, Sylow
112	C₂×C₇₉	—	4	2	no	—
113	C₂×C₇₉	#2164	1	4	yes	complemented
114	C₂×C₇₉	C₂×#431	1	1	yes	characteristic
115	C₂×C₇₉	C₂×#504	1	2	yes	—
116	C₂×#405	D₇₉	1	1	yes	characteristic
117	#2164	—	79	8	no	—
118	C₂×C₄⋊C₁₆	—	79	2	no	—
119	C₂×C₂²⋊C₁₆	—	79	2	no	—
120	C₂×#346	—	79	1	no	—
121	C₂×C₄×C₁₆	—	79	1	no	—
122	C₇₉	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
123	C₂²×M₁₆	Dic₇₉	1	1	yes	characteristic
124	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
125	C₂×#117	Dic₇₉	1	1	yes	characteristic
126	C₂²×C₁₆	—	79	2	no	—
127	C₂²×C₁₆	—	79	2	no	—
128	#405	Dic₇₉	1	4	yes	—
129	#405	C₂×D₇₉	1	4	yes	—
130	C₂×#97	Dic₇₉	1	1	yes	characteristic
131	C₂×#97	Dic₇₉	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
134	C₄⋊C₁₆	—	79	8	no	—
135	C₂²⋊C₁₆	—	79	8	no	—
136	#346	—	79	4	no	—
137	C₄×C₁₆	—	79	4	no	—
138	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	C₂×Dic₇₉	1	4	yes	—
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
144	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
147	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
148	#117	C₇₉⋊C₈	1	4	yes	—
149	#117	C₂×Dic₇₉	1	4	yes	—
150	C₂×C₄⋊C₄	C₇₉⋊C₈	1	2	yes	—
151	C₂×C₂²⋊C₄	C₇₉⋊C₈	1	2	yes	—
152	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
154	C₂×C₁₆	—	79	4	no	—
155	C₂×C₁₆	—	79	4	no	—
156	C₂×C₁₆	—	158	4	no	—
157	C₂×C₁₆	—	158	4	no	—
158	#97	C₂×Dic₇₉	1	4	yes	—
159	#97	C₂×Dic₇₉	1	4	yes	—
160	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
161	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
162	C₄×C₈	C₂²×D₇₉	1	2	yes	—
163	C₄×C₈	C₂²×D₇₉	1	2	yes	—
164	C₂⁴	C₂×C₇₉⋊C₈	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
168	C₂²×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
169	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
173	M₁₆	—	2	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	#11929608	1	4	yes	—
179	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
180	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
181	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
182	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
183	C₂×C₈	#11929624	1	4	yes	—
184	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
185	C₄⋊C₄	C₂×C₇₉⋊C₈	1	8	yes	—
186	C₂²⋊C₄	C₂×C₇₉⋊C₈	1	8	yes	—
187	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
188	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
189	C₄²	C₂²×Dic₇₉	1	2	yes	—
190	C₄²	C₂²×Dic₇₉	1	2	yes	—
191	C₁₆	—	158	4	no	—
192	C₁₆	—	158	4	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂²×C₇₉⋊C₈	1	2	yes	—
196	C₂³	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	#14579120	1	2	yes	—
204	C₂×C₄	C₂²×C₇₉⋊C₈	1	4	yes	—
205	C₂×C₄	C₂²×C₇₉⋊C₈	1	4	yes	—
206	C₂×C₄	C₂²×C₇₉⋊C₈	1	2	yes	—
207	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
208	C₂×C₄	C₂×#11929608	1	2	yes	—
209	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	#14579120	1	2	yes	—
213	C₈	C₂×#11929608	1	4	yes	—
214	C₈	C₂×#11929624	1	2	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	4	no	—
218	C₂²	#19098616	1	2	yes	—
219	C₂²	#19098980	1	4	yes	—
220	C₂²	C₂³×C₇₉⋊C₈	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	2	4	no	—
224	C₄	#19098616	1	2	yes	—
225	C₄	C₂×#14579120	1	1	yes	characteristic
226	C₄	C₂²×#11929608	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	#23765207	1	4	yes	direct product factor
229	C₂	C₂×#19098616	1	1	yes	characteristic
230	C₂	C₂×#19098980	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#405×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×C₈
prime radical	C₂⁴×C₇₉	C₂×C₈
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#405	D₇₉
hypocenter	C₇₉	C₂×#2164
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #23765207	C₂	direct (N × H)
2. C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
4. C₂×#19098339	C₂	semidirect
5. C₂×C₄×C₇₉⋊C₁₆	C₂	semidirect
9. C₄⋊(C₇₉⋊C₁₆)	C₂²	semidirect
11. #19098339	C₂²	semidirect
12. C₄×C₇₉⋊C₁₆	C₂²	semidirect
113. C₂×C₇₉	#2164	semidirect
122. C₇₉	C₂×#2164	semidirect
228. C₂	#23765207	direct (N × H)

Maximal subgroups

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

subgroup	index
#23765207	2
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₂×C₂²⋊(C₇₉⋊C₁₆)	2
C₂×#19098339	2
C₂×C₄×C₇₉⋊C₁₆	2
C₂×#405×C₇₉	2
C₂×#2164	79

Subgroups — C2×#23765207 #34441344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765207 #34441344