Subgroups — C₂×#23765242 #34441503

Order 20224.

654 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×#19098980	C₂	1	2	yes	maximal, complemented
2	#23765242	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098347	C₂	1	2	yes	maximal, complemented
5	C₂×#19098339	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#443×C₇₉	C₂	1	1	yes	maximal, characteristic
7	#19098980	—	2	4	no	—
8	#19098980	C₂²	1	8	yes	complemented
9	C₂×#14579101	—	2	2	no	—
10	C₂×#14579101	C₂²	1	2	yes	—
11	C₂²×C₇₉⋊C₁₆	—	2	2	no	—
12	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	#19098384	C₂²	1	4	yes	complemented
15	#19098347	C₂²	1	8	yes	complemented
16	#19098339	C₂²	1	4	yes	complemented
17	C₂×#135×C₇₉	C₄	1	1	yes	characteristic
18	C₂×#131×C₇₉	C₂²	1	2	yes	—
19	#443×C₇₉	C₄	1	4	yes	—
20	#443×C₇₉	C₂²	1	4	yes	—
21	C₂×#108×C₇₉	C₄	1	1	yes	characteristic
22	C₂×#104×C₇₉	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#14579101	—	2	8	no	—
25	#14579101	—	2	2	no	—
26	#14579101	C₂³	1	4	yes	—
27	C₂×C₇₉⋊C₁₆	—	2	8	no	—
28	C₂×C₇₉⋊C₁₆	—	2	2	no	—
29	C₂×C₇₉⋊C₁₆	—	2	1	no	—
30	C₂×C₇₉⋊C₁₆	—	2	1	no	—
31	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
32	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
33	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
34	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
35	C₂×#41×C₇₉	C₂×C₄	1	2	yes	—
36	#135×C₇₉	C₈	1	4	yes	—
37	#135×C₇₉	C₂×C₄	1	4	yes	—
38	C₂×Q₁₆×C₇₉	C₈	1	1	yes	characteristic
39	C₂×SD₁₆×C₇₉	C₈	1	2	yes	—
40	C₂×D₈×C₇₉	C₈	1	1	yes	characteristic
41	#131×C₇₉	—	2	2	no	—
42	#131×C₇₉	—	2	2	no	—
43	#131×C₇₉	C₂×C₄	1	4	yes	—
44	#131×C₇₉	C₂³	1	4	yes	—
45	C₂×M₁₆×C₇₉	—	2	2	no	—
46	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₇₉	—	2	2	no	—
48	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
51	#108×C₇₉	C₂×C₄	1	4	yes	—
52	#104×C₇₉	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₇₉	C₂³	1	2	yes	—
55	C₇₉⋊C₁₆	—	2	4	no	—
56	C₇₉⋊C₁₆	—	2	2	no	—
57	C₇₉⋊C₁₆	C₂×D₄	1	4	yes	—
58	#41×C₇₉	—	2	2	no	—
59	#41×C₇₉	—	2	2	no	—
60	#41×C₇₉	C₂×C₈	1	4	yes	—
61	#41×C₇₉	C₂²×C₄	1	4	yes	—
62	C₂×Q₈×C₇₉	C₂×C₈	1	2	yes	—
63	C₂×D₄×C₇₉	—	2	2	no	—
64	C₂×D₄×C₇₉	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₇₉	—	2	2	no	—
66	C₂²×C₄×C₇₉	—	2	1	no	—
67	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	Q₁₆×C₇₉	C₂×C₈	1	4	yes	—
69	SD₁₆×C₇₉	C₂×C₈	1	8	yes	—
70	D₈×C₇₉	C₂×C₈	1	4	yes	—
71	M₁₆×C₇₉	—	2	4	no	—
72	M₁₆×C₇₉	—	2	4	no	—
73	M₁₆×C₇₉	—	2	2	no	—
74	M₁₆×C₇₉	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₇₉	—	2	4	no	—
76	C₂×C₈×C₇₉	—	2	4	no	—
77	C₂×C₈×C₇₉	—	2	2	no	—
78	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
81	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
86	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
88	C₄²×C₇₉	—	2	1	no	—
89	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
90	C₂³×C₇₉	—	2	2	no	—
91	C₂³×C₇₉	#131	1	1	yes	characteristic
92	Q₈×C₇₉	—	2	2	no	—
93	Q₈×C₇₉	C₂²×C₈	1	4	yes	—
94	D₄×C₇₉	—	2	4	no	—
95	D₄×C₇₉	—	2	4	no	—
96	D₄×C₇₉	—	2	2	no	—
97	D₄×C₇₉	C₂²×C₈	1	4	yes	—
98	C₂×C₄×C₇₉	—	2	2	no	—
99	C₂×C₄×C₇₉	—	2	4	no	—
100	C₂×C₄×C₇₉	—	2	4	no	—
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	—	4	2	no	—
103	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₇₉	—	2	4	no	—
110	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
111	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
112	C₈×C₇₉	C₂²×C₈	1	2	yes	—
113	C₈×C₇₉	C₂²×C₈	1	2	yes	—
114	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
116	C₂²×C₇₉	—	2	2	no	—
117	C₂²×C₇₉	—	2	4	no	—
118	C₂²×C₇₉	—	4	4	no	—
119	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
120	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₇₉	C₂×#131	1	1	yes	characteristic
122	C₄×C₇₉	—	2	4	no	—
123	C₄×C₇₉	—	4	2	no	—
124	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
125	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
126	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₇₉	C₂³×C₈	1	1	yes	characteristic
128	C₂×#2171	—	79	1	no	maximal, Sylow
129	C₂×C₇₉	—	2	1	no	—
130	C₂×C₇₉	—	2	1	no	—
131	C₂×C₇₉	—	4	4	no	—
132	C₂×C₇₉	#2171	1	2	yes	complemented
133	C₂×C₇₉	C₂×C₈×D₄	1	1	yes	characteristic
134	C₂×#504	—	79	2	no	—
135	C₂×#443	D₇₉	1	1	yes	characteristic
136	#2171	—	79	8	no	—
137	C₂×#364	—	79	1	no	—
138	C₂×#350	—	79	2	no	—
139	C₂×#346	—	79	1	no	—
140	C₇₉	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	Dic₇₉	1	1	yes	characteristic
142	C₂×#131	C₂×D₇₉	1	2	yes	—
143	#504	—	79	8	no	—
144	#504	—	158	4	no	—
145	C₂×M₃₂	—	79	2	no	—
146	C₂×M₃₂	—	158	2	no	—
147	C₂²×C₁₆	—	79	1	no	—
148	C₂²×C₁₆	—	79	1	no	—
149	C₂²×C₁₆	—	158	2	no	—
150	#443	Dic₇₉	1	4	yes	—
151	#443	C₂×D₇₉	1	4	yes	—
152	C₂×#108	Dic₇₉	1	1	yes	characteristic
153	C₂×#104	Dic₇₉	1	2	yes	—
154	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
155	#364	—	79	4	no	—
156	#350	—	79	8	no	—
157	#346	—	79	4	no	—
158	C₂×#41	C₂×Dic₇₉	1	2	yes	—
159	#135	C₇₉⋊C₈	1	4	yes	—
160	#135	C₂×Dic₇₉	1	4	yes	—
161	C₂×Q₁₆	C₇₉⋊C₈	1	1	yes	characteristic
162	C₂×SD₁₆	C₇₉⋊C₈	1	2	yes	—
163	C₂×D₈	C₇₉⋊C₈	1	1	yes	characteristic
164	#131	—	2	2	no	—
165	#131	—	2	2	no	—
166	#131	C₂×Dic₇₉	1	4	yes	—
167	#131	C₂²×D₇₉	1	4	yes	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
174	M₃₂	—	79	4	no	—
175	M₃₂	—	158	8	no	—
176	M₃₂	—	158	2	no	—
177	C₂×C₁₆	—	79	2	no	—
178	C₂×C₁₆	—	79	2	no	—
179	C₂×C₁₆	—	79	2	no	—
180	C₂×C₁₆	—	79	2	no	—
181	C₂×C₁₆	—	158	8	no	—
182	C₂×C₁₆	—	158	2	no	—
183	C₂×C₁₆	—	158	1	no	—
184	C₂×C₁₆	—	158	1	no	—
185	#108	C₂×Dic₇₉	1	4	yes	—
186	#104	C₂×Dic₇₉	1	8	yes	—
187	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
188	C₄×C₈	C₂²×D₇₉	1	2	yes	—
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	C₂×C₇₉⋊C₈	1	4	yes	—
192	#41	C₂²×Dic₇₉	1	4	yes	—
193	C₂×Q₈	C₂×C₇₉⋊C₈	1	2	yes	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	C₂×C₇₉⋊C₈	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
199	Q₁₆	C₂×C₇₉⋊C₈	1	4	yes	—
200	SD₁₆	C₂×C₇₉⋊C₈	1	8	yes	—
201	D₈	C₂×C₇₉⋊C₈	1	4	yes	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	C₂²×Dic₇₉	1	4	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
210	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
211	C₂×C₈	C₂×C₇₉⋊C₈	1	2	yes	—
212	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
213	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
214	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
215	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
216	C₂×C₈	#11929623	1	1	yes	characteristic
217	C₂×C₈	#11929623	1	1	yes	characteristic
218	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂²×Dic₇₉	1	2	yes	—
221	C₁₆	—	79	4	no	—
222	C₁₆	—	158	2	no	—
223	C₁₆	—	158	4	no	—
224	C₂³	—	2	2	no	—
225	C₂³	#14579168	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	C₂²×C₇₉⋊C₈	1	4	yes	—
228	D₄	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	D₄	C₂²×C₇₉⋊C₈	1	4	yes	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
238	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
241	C₂×C₄	#14579168	1	1	yes	characteristic
242	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	D₄×Dic₇₉	1	1	yes	characteristic
245	C₈	D₄×Dic₇₉	1	1	yes	characteristic
246	C₈	C₂²×C₇₉⋊C₈	1	2	yes	—
247	C₈	C₂²×C₇₉⋊C₈	1	2	yes	—
248	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
249	C₈	C₂×#11929623	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	D₄×C₇₉⋊C₈	1	1	yes	characteristic
254	C₂²	D₄×C₇₉⋊C₈	1	1	yes	characteristic
255	C₂²	C₂×#14579168	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	D₄×C₇₉⋊C₈	1	1	yes	characteristic
259	C₄	D₄×C₇₉⋊C₈	1	1	yes	characteristic
260	C₄	C₂×D₄×Dic₇₉	1	1	yes	characteristic
261	C₄	C₂³×C₇₉⋊C₈	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	4	4	no	—
265	C₂	#23765242	1	2	yes	direct product factor
266	C₂	C₂×D₄×C₇₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₄×C₇₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#443×C₇₉	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×#135×C₇₉	C₄
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#443	D₇₉
hypocenter	C₇₉	C₂×#2171
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#19098980	C₂	semidirect
2. #23765242	C₂	direct (N × H)
3. C₂×#19098384	C₂	semidirect
4. C₂×#19098347	C₂	semidirect
5. C₂×#19098339	C₂	semidirect
8. #19098980	C₂²	semidirect
14. #19098384	C₂²	semidirect
15. #19098347	C₂²	semidirect
16. #19098339	C₂²	semidirect
132. C₂×C₇₉	#2171	semidirect
140. C₇₉	C₂×#2171	semidirect
265. C₂	#23765242	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×#19098980	2
#23765242	2
C₂×#19098384	2
C₂×#19098347	2
C₂×#19098339	2
C₂×#443×C₇₉	2
C₂×#2171	79

Subgroups — C2×#23765242 #34441503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765242 #34441503