Subgroups — C₂×#23765636 #34443575

Order 20224.

634 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098968	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765636	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098367	C₂	1	2	yes	maximal, complemented
4	C₂×#19098364	C₂	1	2	yes	maximal
5	C₂×#19098339	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#495×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579161	—	2	1	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	#19098968	—	2	2	no	—
10	#19098968	C₂²	1	4	yes	complemented
11	C₂×#14579095	—	2	1	no	—
12	C₂×#14579095	C₂²	1	2	yes	—
13	#19098367	C₂²	1	8	yes	complemented
14	#19098364	C₂²	1	8	yes	—
15	#19098339	C₂²	1	4	yes	complemented
16	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₇₉	—	2	1	no	—
18	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#124×C₇₉	—	2	1	no	—
20	#495×C₇₉	—	2	2	no	—
21	#495×C₇₉	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₅₈	—	4	1	no	—
24	#14579161	—	2	4	no	—
25	#14579161	—	4	2	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	4	1	no	—
28	C₂×C₇₉⋊C₁₆	—	2	4	no	—
29	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
30	#14579095	—	2	2	no	—
31	#14579095	—	2	4	no	—
32	#14579095	C₂³	1	4	yes	—
33	C₂²×Q₈×C₇₉	—	2	1	no	—
34	C₂²×D₄×C₇₉	—	2	1	no	—
35	C₂×Q₁₆×C₇₉	—	2	1	no	—
36	C₂×Q₁₆×C₇₉	—	2	4	no	—
37	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
38	C₂×SD₁₆×C₇₉	—	2	4	no	—
39	C₂×SD₁₆×C₇₉	—	4	4	no	—
40	C₂×D₈×C₇₉	—	2	1	no	—
41	C₂×D₈×C₇₉	—	2	4	no	—
42	C₂×D₈×C₇₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	#124×C₇₉	—	2	4	no	—
46	#124×C₇₉	—	4	2	no	—
47	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
48	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
49	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₉	D₄	1	2	yes	—
51	C₄×C₈×C₇₉	C₂³	1	2	yes	—
52	C₂×Dic₁₅₈	—	4	4	no	—
53	C₂×Dic₁₅₈	—	8	4	no	—
54	C₂²×Dic₇₉	—	4	2	no	—
55	C₄⋊Dic₇₉	—	2	4	no	—
56	C₄⋊Dic₇₉	—	4	2	no	—
57	C₄⋊Dic₇₉	—	4	4	no	—
58	C₇₉⋊C₁₆	—	2	4	no	—
59	C₂⁴×C₇₉	—	4	1	no	—
60	C₂×Q₈×C₇₉	—	2	2	no	—
61	C₂×Q₈×C₇₉	—	4	1	no	—
62	C₂×Q₈×C₇₉	—	4	2	no	—
63	C₂×Q₈×C₇₉	—	4	2	no	—
64	C₂×D₄×C₇₉	—	2	2	no	—
65	C₂×D₄×C₇₉	—	4	2	no	—
66	C₂×D₄×C₇₉	—	4	1	no	—
67	C₂×D₄×C₇₉	—	4	2	no	—
68	C₂²×C₄×C₇₉	—	4	1	no	—
69	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₇₉	—	2	4	no	—
73	Q₁₆×C₇₉	—	4	2	no	—
74	SD₁₆×C₇₉	—	4	8	no	—
75	D₈×C₇₉	—	2	4	no	—
76	D₈×C₇₉	—	4	2	no	—
77	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
78	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₇₉	#41	1	4	yes	—
81	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
82	C₂²⋊C₄×C₇₉	—	4	4	no	—
83	C₂²⋊C₄×C₇₉	—	4	4	no	—
84	C₄²×C₇₉	D₈	1	2	yes	—
85	C₄²×C₇₉	C₂×D₄	1	2	yes	—
86	C₂³×C₇₉	—	4	2	no	—
87	C₂³×C₇₉	—	8	2	no	—
88	C₂³×C₇₉	—	8	2	no	—
89	C₂³×C₇₉	—	8	2	no	—
90	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₇₉	—	4	2	no	—
92	Q₈×C₇₉	—	4	2	no	—
93	Q₈×C₇₉	—	8	2	no	—
94	D₄×C₇₉	—	4	2	no	—
95	D₄×C₇₉	—	4	2	no	—
96	D₄×C₇₉	—	8	2	no	—
97	C₂×C₄×C₇₉	—	2	2	no	—
98	C₂×C₄×C₇₉	—	2	2	no	—
99	C₂×C₄×C₇₉	—	4	2	no	—
100	C₂×C₄×C₇₉	—	8	2	no	—
101	C₂×C₄×C₇₉	#124	1	2	yes	—
102	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×Dic₇₉	—	4	4	no	—
108	C₂×Dic₇₉	—	8	4	no	—
109	Dic₁₅₈	—	8	8	no	—
110	C₈×C₇₉	#124	1	2	yes	—
111	C₈×C₇₉	C₂×SD₁₆	1	4	yes	—
112	C₈×C₇₉	C₂×#41	1	2	yes	—
113	C₂²×C₇₉	—	8	2	no	—
114	C₂²×C₇₉	—	8	2	no	—
115	C₂²×C₇₉	—	8	2	no	—
116	C₂²×C₇₉	—	16	4	no	—
117	C₂²×C₇₉	#486	1	2	yes	—
118	C₂²×C₇₉	#509	1	2	yes	—
119	C₂²×C₇₉	#510	1	2	yes	—
120	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₇₉	—	2	2	no	—
122	C₄×C₇₉	—	2	2	no	—
123	C₄×C₇₉	—	8	2	no	—
124	C₄×C₇₉	#486	1	2	yes	—
125	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
126	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
127	Dic₇₉	—	8	4	no	—
128	C₂×#2243	—	79	1	no	maximal, Sylow
129	C₂×C₇₉	—	16	2	no	—
130	C₂×C₇₉	#2243	1	4	yes	complemented
131	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
132	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
133	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
134	C₂×#500	—	79	1	no	—
135	C₂×#495	D₇₉	1	1	yes	characteristic
136	#2243	—	79	8	no	—
137	C₂×#358	—	79	2	no	—
138	C₂×#357	—	79	2	no	—
139	C₂×#346	—	79	1	no	—
140	C₇₉	C₂×#2243	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
142	C₂²×SD₁₆	—	2	1	no	—
143	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	—	158	1	no	—
145	C₂×#124	—	2	1	no	—
146	C₂²×C₁₆	—	79	2	no	—
147	#500	—	79	4	no	—
148	#500	—	158	2	no	—
149	#495	—	2	2	no	—
150	#495	C₂×D₇₉	1	4	yes	—
151	C₂×#107	—	79	2	no	—
152	C₂×#107	—	158	1	no	—
153	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
154	#358	—	79	8	no	—
155	#357	—	79	8	no	—
156	#346	—	79	4	no	—
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×Q₈	—	316	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
163	C₂×SD₁₆	—	2	4	no	—
164	C₂×SD₁₆	—	4	4	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	—	2	4	no	—
167	C₂×D₈	C₂²×D₇₉	1	2	yes	—
168	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
170	C₄⋊Q₈	—	316	2	no	—
171	C₄⋊Q₈	—	158	4	no	—
172	#124	—	2	4	no	—
173	#124	—	4	2	no	—
174	C₂×C₄⋊C₄	—	316	1	no	—
175	C₂×C₄⋊C₄	—	158	2	no	—
176	C₂×C₂²⋊C₄	—	4	1	no	—
177	C₂×C₂²⋊C₄	—	4	1	no	—
178	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
179	C₂×C₁₆	—	79	4	no	—
180	C₂×C₁₆	—	158	4	no	—
181	#107	—	79	4	no	—
182	#107	—	158	2	no	—
183	#107	—	158	4	no	—
184	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
185	C₄×C₈	C₂²×D₇₉	1	2	yes	—
186	C₂⁴	—	4	1	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	316	4	no	—
192	C₂×Q₈	—	632	4	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	316	2	no	—
199	C₂²×C₄	#11929609	1	1	yes	characteristic
200	C₂²×C₄	#11929609	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
202	Q₁₆	—	2	4	no	—
203	Q₁₆	—	4	2	no	—
204	SD₁₆	—	4	8	no	—
205	D₈	—	2	4	no	—
206	D₈	—	4	2	no	—
207	C₂×C₈	#11929612	1	4	yes	—
208	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
210	C₂×C₈	#11929623	1	4	yes	—
211	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
212	C₄⋊C₄	—	316	4	no	—
213	C₄⋊C₄	—	316	2	no	—
214	C₄⋊C₄	—	158	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₄²	#11929609	1	2	yes	—
218	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
219	C₁₆	—	158	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	8	2	no	—
222	C₂³	—	8	2	no	—
223	C₂³	—	8	2	no	—
224	C₂³	C₂×#11929609	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	8	2	no	—
228	Q₈	—	632	8	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	8	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	316	4	no	—
237	C₂×C₄	—	632	4	no	—
238	C₂×C₄	#14579148	1	2	yes	—
239	C₂×C₄	C₂×#11929609	1	2	yes	—
240	C₂×C₄	C₂×#11929609	1	2	yes	—
241	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
242	C₂×C₄	C₂×#11929612	1	2	yes	—
243	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
244	C₈	#14579148	1	2	yes	—
245	C₈	C₂×#11929612	1	4	yes	—
246	C₈	C₂×#11929623	1	2	yes	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	16	4	no	—
251	C₂²	#19098909	1	2	yes	—
252	C₂²	#19098998	1	2	yes	—
253	C₂²	#19099003	1	2	yes	—
254	C₂²	C₂²×#11929609	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	8	2	no	—
258	C₄	—	632	4	no	—
259	C₄	#19098909	1	2	yes	—
260	C₄	C₂×#14579148	1	1	yes	characteristic
261	C₄	C₂²×#11929612	1	1	yes	characteristic
262	C₂	—	16	2	no	—
263	C₂	#23765636	1	4	yes	direct product factor
264	C₂	C₂×#19098909	1	1	yes	characteristic
265	C₂	C₂×#19098998	1	1	yes	characteristic
266	C₂	C₂×#19099003	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#495×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₈×C₇₉	C₂²
hypercenter	C₂×#495	D₇₉
hypocenter	C₇₉	C₂×#2243
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. C₂×#19098968	C₂	semidirect
2. #23765636	C₂	direct (N × H)
3. C₂×#19098367	C₂	semidirect
5. C₂×#19098339	C₂	semidirect
10. #19098968	C₂²	semidirect
13. #19098367	C₂²	semidirect
15. #19098339	C₂²	semidirect
130. C₂×C₇₉	#2243	semidirect
140. C₇₉	C₂×#2243	semidirect
263. C₂	#23765636	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098968	2
#23765636	2
C₂×#19098367	2
C₂×#19098364	2
C₂×#19098339	2
C₂×#495×C₇₉	2
C₂×#2243	79

Subgroups — C2×#23765636 #34443575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765636 #34443575