Subgroups — C₂×#23765491 #34442693

Order 20224.

702 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098804	C₂	1	2	yes	maximal
3	#23765491	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098368	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
8	C₂×#14579140	—	2	2	no	—
9	C₂×Dic₆₃₂	—	2	4	no	—
10	C₂×Dic₆₃₂	C₂²	1	4	yes	complemented
11	#19098804	—	2	4	no	—
12	#19098804	C₂²	1	8	yes	—
13	C₂×#14579095	C₂²	1	2	yes	—
14	C₂×#14579086	—	2	2	no	—
15	#19098389	C₂²	1	4	yes	complemented
16	#19098368	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₇₉	—	2	4	no	—
19	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₇₉	C₂²	1	4	yes	—
21	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₅₈	—	2	2	no	—
23	C₂×Dic₃₁₆	—	2	4	no	—
24	C₂×Dic₃₁₆	—	2	4	no	—
25	C₂×Dic₃₁₆	—	2	2	no	—
26	C₂×Dic₃₁₆	C₂³	1	4	yes	—
27	#14579140	—	2	8	no	—
28	#14579140	—	4	4	no	—
29	C₂×C₄⋊Dic₇₉	—	2	2	no	—
30	C₂×Dic₇₉⋊C₄	—	4	2	no	—
31	C₂×C₂²⋊Dic₇₉	—	4	2	no	—
32	Dic₆₃₂	—	2	8	no	—
33	#14579095	—	2	2	no	—
34	#14579095	C₂³	1	4	yes	—
35	#14579086	—	2	8	no	—
36	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₉	—	2	2	no	—
38	C₂²×C₈×C₇₉	—	2	2	no	—
39	C₂²×C₈×C₇₉	D₄	1	2	yes	—
40	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
42	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₇₉	—	2	2	no	—
44	C₂×C₁₆×C₇₉	—	2	8	no	—
45	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
46	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
47	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
48	C₂×Dic₁₅₈	—	2	4	no	—
49	C₂×Dic₁₅₈	—	4	4	no	—
50	C₂×Dic₁₅₈	—	4	2	no	—
51	C₂×Dic₁₅₈	—	4	4	no	—
52	C₂²×Dic₇₉	—	4	2	no	—
53	C₂²×Dic₇₉	—	4	2	no	—
54	Dic₃₁₆	—	2	4	no	—
55	Dic₃₁₆	—	2	4	no	—
56	Dic₃₁₆	—	4	4	no	—
57	C₄⋊Dic₇₉	—	2	4	no	—
58	C₄⋊Dic₇₉	—	4	2	no	—
59	Dic₇₉⋊C₄	—	4	8	no	—
60	C₂²⋊Dic₇₉	—	4	8	no	—
61	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	—	2	2	no	—
63	C₂²×C₄×C₇₉	—	2	2	no	—
64	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	D₈	1	2	yes	—
66	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₉	—	2	4	no	—
69	C₂×C₈×C₇₉	—	2	2	no	—
70	C₂×C₈×C₇₉	—	2	2	no	—
71	C₂×C₈×C₇₉	—	2	2	no	—
72	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₉	#41	1	2	yes	—
76	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
77	C₁₆×C₇₉	—	2	2	no	—
78	C₁₆×C₇₉	C₂×D₄	1	4	yes	—
79	C₂³×C₇₉	—	2	2	no	—
80	C₂³×C₇₉	—	2	2	no	—
81	C₂³×C₇₉	Q₃₂	1	4	yes	complemented
82	C₂³×C₇₉	C₂×D₈	1	2	yes	—
83	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₇₉	—	2	2	no	—
85	C₂×C₄×C₇₉	—	2	2	no	—
86	C₂×C₄×C₇₉	—	2	2	no	—
87	C₂×C₄×C₇₉	—	2	4	no	—
88	C₂×C₄×C₇₉	#121	1	2	yes	—
89	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#135	1	2	yes	—
92	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
93	C₂×Dic₇₉	—	4	4	no	—
94	C₂×Dic₇₉	—	4	4	no	—
95	C₂×Dic₇₉	—	8	4	no	—
96	C₂×Dic₇₉	—	8	4	no	—
97	Dic₁₅₈	—	4	4	no	—
98	Dic₁₅₈	—	4	4	no	—
99	Dic₁₅₈	—	8	4	no	—
100	C₈×C₇₉	—	2	2	no	—
101	C₈×C₇₉	#121	1	2	yes	—
102	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
103	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
104	C₂²×C₇₉	—	2	2	no	—
105	C₂²×C₇₉	—	2	8	no	—
106	C₂²×C₇₉	—	2	2	no	—
107	C₂²×C₇₉	#466	1	2	yes	—
108	C₂²×C₇₉	C₂×Q₃₂	1	2	yes	—
109	C₂²×C₇₉	C₂×Q₃₂	1	4	yes	complemented
110	C₂²×C₇₉	#508	1	2	yes	—
111	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
112	C₄×C₇₉	—	2	2	no	—
113	C₄×C₇₉	#466	1	2	yes	—
114	C₄×C₇₉	C₂×#121	1	1	yes	characteristic
115	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
116	Dic₇₉	—	8	4	no	—
117	Dic₇₉	—	8	4	no	—
118	C₂×#2216	—	79	1	no	maximal, Sylow
119	C₂×C₇₉	—	2	4	no	—
120	C₂×C₇₉	#2216	1	4	yes	complemented
121	C₂×C₇₉	C₂×#466	1	1	yes	characteristic
122	C₂×C₇₉	C₂²×Q₃₂	1	1	yes	characteristic
123	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
124	C₂²×Q₃₂	—	79	1	no	—
125	C₂³×C₁₆	D₇₉	1	1	yes	characteristic
126	C₂×#467	—	79	2	no	—
127	#2216	—	79	8	no	—
128	C₂×#366	—	79	1	no	—
129	C₂×#358	—	79	2	no	—
130	C₇₉	C₂×#2216	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₁₆	—	79	2	no	—
132	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
133	C₂×C₂²⋊Q₈	—	158	2	no	—
134	C₂×Q₃₂	—	79	4	no	—
135	C₂×Q₃₂	—	158	4	no	—
136	C₂²×C₁₆	—	2	4	no	—
137	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
138	C₂²×C₁₆	C₂×D₇₉	1	4	yes	—
139	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
140	#467	—	79	8	no	—
141	#467	—	158	4	no	—
142	C₂×#107	—	79	2	no	—
143	C₂×#103	—	158	2	no	—
144	#366	—	79	4	no	—
145	#358	—	79	8	no	—
146	C₂²×Q₈	—	158	2	no	—
147	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
148	C₂×Q₁₆	—	79	4	no	—
149	C₂×Q₁₆	—	158	4	no	—
150	C₂×Q₁₆	—	158	2	no	—
151	C₂×Q₁₆	—	158	4	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
155	C₂²×C₈	D₃₁₆	1	2	yes	—
156	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
157	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
158	C₂²⋊Q₈	—	316	4	no	—
159	C₂²⋊Q₈	—	158	8	no	—
160	C₂×C₄⋊C₄	—	316	2	no	—
161	C₂×C₄⋊C₄	—	158	2	no	—
162	C₂×C₂²⋊C₄	—	316	2	no	—
163	Q₃₂	—	158	8	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	8	no	—
166	C₂×C₁₆	C₂²⋊D₇₉	1	4	yes	—
167	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
168	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
169	#107	—	79	4	no	—
170	#107	—	158	2	no	—
171	#103	—	158	8	no	—
172	C₂⁴	D₆₃₂	1	1	yes	characteristic
173	C₂×Q₈	—	316	2	no	—
174	C₂×Q₈	—	316	4	no	—
175	C₂×Q₈	—	316	4	no	—
176	C₂×Q₈	—	158	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	316	2	no	—
180	C₂²×C₄	—	316	2	no	—
181	C₂²×C₄	D₆₃₂	1	2	yes	—
182	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
184	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
185	Q₁₆	—	316	4	no	—
186	Q₁₆	—	158	4	no	—
187	Q₁₆	—	158	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
193	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
194	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
195	C₂×C₈	#11929624	1	2	yes	—
196	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
197	C₄⋊C₄	—	316	8	no	—
198	C₄⋊C₄	—	316	2	no	—
199	C₄⋊C₄	—	158	4	no	—
200	C₂²⋊C₄	—	316	8	no	—
201	C₁₆	—	2	2	no	—
202	C₁₆	C₂×C₂²⋊D₇₉	1	4	yes	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	Dic₆₃₂	1	4	yes	complemented
206	C₂³	C₂×D₆₃₂	1	2	yes	—
207	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
208	Q₈	—	316	4	no	—
209	Q₈	—	316	4	no	—
210	Q₈	—	632	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	316	4	no	—
216	C₂×C₄	—	316	4	no	—
217	C₂×C₄	—	632	4	no	—
218	C₂×C₄	—	632	4	no	—
219	C₂×C₄	#14579135	1	2	yes	—
220	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
221	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
222	C₂×C₄	#14579183	1	2	yes	—
223	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	#14579135	1	2	yes	—
226	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
227	C₈	C₂×#11929624	1	1	yes	characteristic
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	8	no	—
231	C₂²	#19098799	1	2	yes	—
232	C₂²	C₂×Dic₆₃₂	1	2	yes	—
233	C₂²	C₂×Dic₆₃₂	1	4	yes	complemented
234	C₂²	#19098995	1	2	yes	—
235	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	632	4	no	—
238	C₄	—	632	4	no	—
239	C₄	#19098799	1	2	yes	—
240	C₄	C₂×#14579135	1	1	yes	characteristic
241	C₄	C₂×#14579183	1	1	yes	characteristic
242	C₂	—	2	4	no	—
243	C₂	#23765491	1	4	yes	direct product factor
244	C₂	C₂×#19098799	1	1	yes	characteristic
245	C₂	C₂²×Dic₆₃₂	1	1	yes	characteristic
246	C₂	C₂×#19098995	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂³×C₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂⁴×C₇₉	D₈
hypercenter	C₂³×C₁₆	D₇₉
hypocenter	C₇₉	C₂×#2216
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₆₃₂	C₂	semidirect
3. #23765491	C₂	direct (N × H)
4. C₂×#19098389	C₂	semidirect
5. C₂×#19098368	C₂	semidirect
10. C₂×Dic₆₃₂	C₂²	semidirect
15. #19098389	C₂²	semidirect
16. #19098368	C₂²	semidirect
81. C₂³×C₇₉	Q₃₂	semidirect
109. C₂²×C₇₉	C₂×Q₃₂	semidirect
120. C₂×C₇₉	#2216	semidirect
130. C₇₉	C₂×#2216	semidirect
205. C₂³	Dic₆₃₂	semidirect
233. C₂²	C₂×Dic₆₃₂	semidirect
243. C₂	#23765491	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₃₂	2
C₂×#19098804	2
#23765491	2
C₂×#19098389	2
C₂×#19098368	2
C₂³×C₁₆×C₇₉	2
C₂×#2216	79

Subgroups — C2×#23765491 #34442693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765491 #34442693