Subgroups — C₂×#23765281 #34441669

Order 20224.

662 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098680	C₂	1	2	yes	maximal, complemented
2	#23765281	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098393	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098370	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#508×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579168	—	2	2	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	#19098680	—	2	4	no	—
10	#19098680	C₂²	1	8	yes	complemented
11	C₂×#14579097	C₂²	1	2	yes	—
12	C₂×#14579088	—	2	2	no	—
13	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
14	#19098393	C₂²	1	4	yes	complemented
15	#19098370	C₂²	1	8	yes	complemented
16	C₁₆×Dic₇₉	C₂²	1	4	yes	complemented
17	C₂×#135×C₇₉	C₂²	1	2	yes	—
18	#508×C₇₉	C₄	1	4	yes	complemented
19	#508×C₇₉	C₂²	1	4	yes	—
20	C₂×Q₃₂×C₇₉	C₄	1	1	yes	characteristic, complemented
21	C₂×SD₃₂×C₇₉	C₄	1	2	yes	complemented
22	C₂×D₁₆×C₇₉	C₄	1	1	yes	characteristic, complemented
23	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	#14579168	—	2	8	no	—
25	#14579168	—	4	4	no	—
26	C₂×#11929608	—	2	2	no	—
27	C₂×#11929608	—	4	2	no	—
28	C₂²×C₇₉⋊C₈	—	4	2	no	—
29	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
31	C₂×C₇₉⋊C₁₆	—	2	1	no	—
32	C₂×C₇₉⋊C₁₆	—	2	1	no	—
33	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
34	#14579097	—	2	2	no	—
35	#14579097	C₂³	1	4	yes	—
36	#14579088	—	2	8	no	—
37	C₈×Dic₇₉	D₄	1	2	yes	—
38	C₈×Dic₇₉	C₂³	1	2	yes	—
39	C₂×#41×C₇₉	—	2	2	no	—
40	#135×C₇₉	—	2	2	no	—
41	#135×C₇₉	—	2	2	no	—
42	#135×C₇₉	C₂×C₄	1	4	yes	—
43	#135×C₇₉	C₂³	1	4	yes	—
44	C₂×Q₁₆×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×SD₁₆×C₇₉	—	2	2	no	—
46	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
48	Q₃₂×C₇₉	C₂×C₄	1	4	yes	complemented
49	SD₃₂×C₇₉	C₂×C₄	1	8	yes	complemented
50	D₁₆×C₇₉	C₂×C₄	1	4	yes	complemented
51	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
54	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
55	C₂²×Dic₇₉	—	2	1	no	—
56	#11929608	—	2	4	no	—
57	#11929608	—	4	8	no	—
58	#11929608	—	4	2	no	—
59	C₂×C₇₉⋊C₈	—	4	8	no	—
60	C₂×C₇₉⋊C₈	—	4	2	no	—
61	C₂×C₇₉⋊C₈	D₈	1	1	yes	characteristic
62	C₂×C₇₉⋊C₈	D₈	1	1	yes	characteristic
63	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
64	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
65	C₄×Dic₇₉	—	2	1	no	—
66	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
67	C₇₉⋊C₁₆	—	2	2	no	—
68	#41×C₇₉	—	2	4	no	—
69	#41×C₇₉	—	2	4	no	—
70	#41×C₇₉	—	4	2	no	—
71	#41×C₇₉	—	4	2	no	—
72	C₂×Q₈×C₇₉	—	2	2	no	—
73	C₂×D₄×C₇₉	—	2	2	no	—
74	C₂×D₄×C₇₉	—	4	2	no	—
75	C₂²×C₄×C₇₉	—	4	2	no	—
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₇₉	—	2	2	no	—
78	Q₁₆×C₇₉	C₂²×C₄	1	4	yes	—
79	SD₁₆×C₇₉	—	2	4	no	—
80	SD₁₆×C₇₉	—	2	4	no	—
81	D₈×C₇₉	—	2	2	no	—
82	D₈×C₇₉	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
88	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
89	C₁₆×C₇₉	C₂²×C₄	1	2	yes	—
90	C₁₆×C₇₉	C₂²×C₄	1	2	yes	—
91	C₂³×C₇₉	—	4	2	no	—
92	C₂³×C₇₉	#135	1	1	yes	characteristic
93	Q₈×C₇₉	—	2	4	no	—
94	Q₈×C₇₉	—	4	2	no	—
95	D₄×C₇₉	—	2	4	no	—
96	D₄×C₇₉	—	4	4	no	—
97	D₄×C₇₉	—	4	4	no	—
98	D₄×C₇₉	—	4	2	no	—
99	C₂×C₄×C₇₉	—	4	2	no	—
100	C₂×C₄×C₇₉	—	4	4	no	—
101	C₂×C₄×C₇₉	—	4	4	no	—
102	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₇₉	—	2	4	no	—
109	C₂×Dic₇₉	—	2	2	no	—
110	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
111	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
112	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
114	C₇₉⋊C₈	—	4	4	no	—
115	C₇₉⋊C₈	C₂×D₈	1	2	yes	—
116	C₇₉⋊C₈	C₂×D₈	1	2	yes	—
117	C₂²×C₇₉	—	2	2	no	—
118	C₂²×C₇₉	—	4	4	no	—
119	C₂²×C₇₉	—	8	4	no	—
120	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
121	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
122	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
123	C₄×C₇₉	—	4	4	no	—
124	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
125	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
126	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
128	Dic₇₉	—	2	4	no	—
129	C₂×#2178	—	79	1	no	maximal, Sylow
130	C₂×C₇₉	—	2	1	no	—
131	C₂×C₇₉	—	2	1	no	—
132	C₂×C₇₉	—	8	4	no	—
133	C₂×C₇₉	#2178	1	2	yes	complemented
134	C₂×C₇₉	C₂×C₄×D₈	1	1	yes	characteristic
135	C₂×#508	D₇₉	1	1	yes	characteristic
136	C₂×#443	—	79	2	no	—
137	#2178	—	79	8	no	—
138	C₂×#368	—	79	1	no	—
139	C₂×#359	—	79	2	no	—
140	C₂×C₄×C₁₆	—	79	1	no	—
141	C₇₉	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	C₂×D₇₉	1	2	yes	—
143	C₂×#131	—	158	2	no	—
144	#508	Dic₇₉	1	4	yes	complemented
145	#508	C₂×D₇₉	1	4	yes	—
146	C₂×Q₃₂	Dic₇₉	1	1	yes	characteristic, complemented
147	C₂×SD₃₂	Dic₇₉	1	2	yes	complemented
148	C₂×D₁₆	Dic₇₉	1	1	yes	characteristic, complemented
149	C₂²×C₁₆	—	79	1	no	—
150	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
151	#443	—	79	8	no	—
152	#443	—	158	4	no	—
153	C₂×#108	—	79	2	no	—
154	C₂×#104	—	158	2	no	—
155	C₂×C₄×C₈	—	79	1	no	—
156	#368	—	79	4	no	—
157	#359	—	79	8	no	—
158	C₄×C₁₆	—	79	4	no	—
159	C₂×#41	—	2	2	no	—
160	#135	—	2	2	no	—
161	#135	—	2	2	no	—
162	#135	C₂×Dic₇₉	1	4	yes	—
163	#135	C₂²×D₇₉	1	4	yes	—
164	C₂×Q₁₆	C₂×Dic₇₉	1	2	yes	—
165	C₂×SD₁₆	—	2	2	no	—
166	C₂×D₈	C₂×Dic₇₉	1	2	yes	—
167	#131	—	316	4	no	—
168	#131	—	158	8	no	—
169	C₂×M₁₆	—	316	2	no	—
170	C₂×M₁₆	—	158	2	no	—
171	C₂²×C₈	—	316	2	no	—
172	C₂²×C₈	—	79	1	no	—
173	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
174	C₂×C₄²	—	79	1	no	—
175	Q₃₂	C₂×Dic₇₉	1	4	yes	complemented
176	SD₃₂	C₂×Dic₇₉	1	8	yes	complemented
177	D₁₆	C₂×Dic₇₉	1	4	yes	complemented
178	C₂×C₁₆	—	79	2	no	—
179	C₂×C₁₆	—	158	1	no	—
180	C₂×C₁₆	—	158	1	no	—
181	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
182	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
183	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
184	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
185	#108	—	79	4	no	—
186	#108	—	158	2	no	—
187	#104	—	158	8	no	—
188	C₄×C₈	—	79	2	no	—
189	C₄×C₈	—	79	2	no	—
190	#41	—	2	4	no	—
191	#41	—	2	4	no	—
192	#41	—	4	2	no	—
193	#41	—	4	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	158	1	no	—
199	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
200	Q₁₆	—	2	2	no	—
201	Q₁₆	C₂²×Dic₇₉	1	4	yes	—
202	SD₁₆	—	2	4	no	—
203	SD₁₆	—	2	4	no	—
204	D₈	—	2	2	no	—
205	D₈	C₂²×Dic₇₉	1	4	yes	—
206	M₁₆	—	316	2	no	—
207	M₁₆	—	316	8	no	—
208	M₁₆	—	158	4	no	—
209	C₂×C₈	—	316	2	no	—
210	C₂×C₈	—	316	8	no	—
211	C₂×C₈	—	79	2	no	—
212	C₂×C₈	—	79	1	no	—
213	C₂×C₈	—	79	1	no	—
214	C₂×C₈	—	79	2	no	—
215	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
216	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
217	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
218	C₂×C₈	#11929623	1	1	yes	characteristic
219	C₂×C₈	#11929623	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
221	C₄²	—	79	2	no	—
222	C₄²	—	158	1	no	—
223	C₁₆	—	158	2	no	—
224	C₁₆	C₂²×Dic₇₉	1	2	yes	—
225	C₁₆	C₂²×Dic₇₉	1	2	yes	—
226	C₂³	—	4	2	no	—
227	C₂³	#14579180	1	1	yes	characteristic
228	Q₈	—	2	4	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	158	4	no	—
238	C₂×C₄	—	158	2	no	—
239	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
240	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
241	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
242	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
243	C₂×C₄	#14579180	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
245	C₈	—	316	4	no	—
246	C₈	—	79	2	no	—
247	C₈	—	79	2	no	—
248	C₈	D₄×Dic₇₉	1	1	yes	characteristic
249	C₈	D₄×Dic₇₉	1	1	yes	characteristic
250	C₈	C₂³×Dic₇₉	1	1	yes	characteristic
251	C₈	C₂×#11929623	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	8	4	no	—
255	C₂²	D₈×Dic₇₉	1	1	yes	characteristic
256	C₂²	D₈×Dic₇₉	1	1	yes	characteristic
257	C₂²	C₂×#14579180	1	1	yes	characteristic
258	C₄	—	4	4	no	—
259	C₄	—	158	4	no	—
260	C₄	D₈×Dic₇₉	1	1	yes	characteristic
261	C₄	D₈×Dic₇₉	1	1	yes	characteristic
262	C₄	C₂×D₄×Dic₇₉	1	1	yes	characteristic
263	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	8	4	no	—
267	C₂	#23765281	1	2	yes	direct product factor
268	C₂	C₂×D₈×Dic₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₉
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting, prime radical	C₂×#508×C₇₉	C₂
socle	C₂²×C₇₉	C₄×D₈
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#508	D₇₉
hypocenter	C₇₉	C₂×#2178
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#19098680	C₂	semidirect
2. #23765281	C₂	direct (N × H)
3. C₂×#19098393	C₂	semidirect
4. C₂×#19098370	C₂	semidirect
5. C₂×C₁₆×Dic₇₉	C₂	semidirect
10. #19098680	C₂²	semidirect
14. #19098393	C₂²	semidirect
15. #19098370	C₂²	semidirect
16. C₁₆×Dic₇₉	C₂²	semidirect
18. #508×C₇₉	C₄	semidirect
20. C₂×Q₃₂×C₇₉	C₄	semidirect
21. C₂×SD₃₂×C₇₉	C₄	semidirect
22. C₂×D₁₆×C₇₉	C₄	semidirect
48. Q₃₂×C₇₉	C₂×C₄	semidirect
49. SD₃₂×C₇₉	C₂×C₄	semidirect
50. D₁₆×C₇₉	C₂×C₄	semidirect
133. C₂×C₇₉	#2178	semidirect
141. C₇₉	C₂×#2178	semidirect
144. #508	Dic₇₉	semidirect
146. C₂×Q₃₂	Dic₇₉	semidirect
147. C₂×SD₃₂	Dic₇₉	semidirect
148. C₂×D₁₆	Dic₇₉	semidirect
175. Q₃₂	C₂×Dic₇₉	semidirect
176. SD₃₂	C₂×Dic₇₉	semidirect
177. D₁₆	C₂×Dic₇₉	semidirect
267. C₂	#23765281	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×#19098680	2
#23765281	2
C₂×#19098393	2
C₂×#19098370	2
C₂×C₁₆×Dic₇₉	2
C₂×#508×C₇₉	2
C₂×#2178	79

Subgroups — C2×#23765281 #34441669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765281 #34441669