Subgroups — C₂×#23765535 #34442870

Order 20224.

638 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098967	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23765535	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098388	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19098364	C₂	1	1	yes	maximal, characteristic
8	C₂×#357×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14579160	—	2	1	no	—
10	C₂×D₄×Dic₇₉	—	2	1	no	—
11	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#19098967	—	2	2	no	—
13	#19098967	C₂²	1	4	yes	complemented
14	D₈×Dic₇₉	—	2	2	no	—
15	D₈×Dic₇₉	C₂²	1	4	yes	—
16	C₂×#14579095	C₂²	1	1	yes	characteristic
17	C₂×#14579095	C₂²	1	1	yes	characteristic
18	C₂×#14579094	—	2	1	no	—
19	C₂×#14579082	—	2	1	no	—
20	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
21	#19098389	C₂²	1	4	yes	complemented
22	#19098388	C₂²	1	4	yes	complemented
23	Dic₇₉⋊C₁₆	C₂²	1	4	yes	complemented
24	#19098364	C₂²	1	4	yes	—
25	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
28	#357×C₇₉	C₂²	1	4	yes	—
29	C₂²×Dic₁₅₈	—	4	1	no	—
30	C₂³×Dic₇₉	—	4	1	no	—
31	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
32	#14579160	—	2	4	no	—
33	#14579160	—	4	2	no	—
34	D₄×Dic₇₉	—	2	4	no	—
35	D₄×Dic₇₉	—	4	2	no	—
36	C₂×C₄⋊Dic₇₉	—	2	1	no	—
37	C₂×C₄⋊Dic₇₉	—	2	1	no	—
38	C₂×Dic₇₉⋊C₄	—	4	1	no	—
39	C₂×C₂²⋊Dic₇₉	—	4	1	no	—
40	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
41	C₂×C₇₉⋊C₁₆	—	2	2	no	—
42	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
43	#14579095	—	2	1	no	—
44	#14579095	—	2	1	no	—
45	#14579095	C₂³	1	2	yes	—
46	#14579095	C₂³	1	2	yes	—
47	#14579094	—	2	4	no	—
48	#14579082	—	2	4	no	—
49	C₈×Dic₇₉	D₄	1	2	yes	—
50	C₈×Dic₇₉	C₂³	1	2	yes	—
51	C₂²×D₄×C₇₉	—	2	1	no	—
52	C₂×D₈×C₇₉	—	2	1	no	—
53	C₂×D₈×C₇₉	—	2	2	no	—
54	C₂×D₈×C₇₉	Q₈	1	4	yes	complemented
55	C₂×D₈×C₇₉	C₂³	1	2	yes	—
56	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
58	C₂×C₁₆×C₇₉	—	2	2	no	—
59	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
60	#107×C₇₉	—	2	1	no	—
61	#107×C₇₉	C₂³	1	2	yes	—
62	C₂×Dic₁₅₈	—	4	4	no	—
63	C₂×Dic₁₅₈	—	8	4	no	—
64	C₂²×Dic₇₉	—	4	4	no	—
65	C₂²×Dic₇₉	—	4	1	no	—
66	C₂²×Dic₇₉	—	4	1	no	—
67	C₂²×Dic₇₉	—	8	4	no	—
68	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
69	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
70	C₂×C₇₉⋊C₈	—	2	2	no	—
71	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
72	C₄⋊Dic₇₉	—	2	2	no	—
73	C₄⋊Dic₇₉	—	2	2	no	—
74	C₄⋊Dic₇₉	—	4	1	no	—
75	C₄⋊Dic₇₉	—	4	1	no	—
76	Dic₇₉⋊C₄	—	4	4	no	—
77	C₂²⋊Dic₇₉	—	4	4	no	—
78	C₄×Dic₇₉	D₈	1	2	yes	—
79	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
80	C₇₉⋊C₁₆	—	2	2	no	—
81	C₂⁴×C₇₉	—	4	1	no	—
82	C₂×D₄×C₇₉	—	2	4	no	—
83	C₂×D₄×C₇₉	—	2	2	no	—
84	C₂×D₄×C₇₉	—	4	1	no	—
85	C₂×D₄×C₇₉	—	4	2	no	—
86	C₂²×C₄×C₇₉	—	4	1	no	—
87	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
88	D₈×C₇₉	—	2	4	no	—
89	D₈×C₇₉	—	2	2	no	—
90	D₈×C₇₉	C₂×Q₈	1	4	yes	complemented
91	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₇₉	#41	1	2	yes	—
94	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₇₉	—	2	2	no	—
96	C₄⋊C₄×C₇₉	—	4	1	no	—
97	C₁₆×C₇₉	—	2	2	no	—
98	C₂³×C₇₉	—	4	2	no	—
99	C₂³×C₇₉	—	4	4	no	—
100	C₂³×C₇₉	—	8	2	no	—
101	C₂³×C₇₉	—	8	2	no	—
102	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
103	D₄×C₇₉	—	2	4	no	—
104	D₄×C₇₉	—	4	2	no	—
105	D₄×C₇₉	—	4	4	no	—
106	C₂×C₄×C₇₉	—	4	2	no	—
107	C₂×C₄×C₇₉	—	8	2	no	—
108	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₇₉	#137	1	2	yes	—
111	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₇₉	—	2	2	no	—
113	C₂×Dic₇₉	—	4	2	no	—
114	C₂×Dic₇₉	—	4	2	no	—
115	C₂×Dic₇₉	—	8	2	no	—
116	C₂×Dic₇₉	—	8	2	no	—
117	C₂×Dic₇₉	—	8	8	no	—
118	C₂×Dic₇₉	D₁₆	1	4	yes	—
119	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
120	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
121	Dic₁₅₈	—	8	8	no	—
122	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
123	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
125	C₇₉⋊C₈	—	2	2	no	—
126	C₂²×C₇₉	—	4	4	no	—
127	C₂²×C₇₉	—	8	2	no	—
128	C₂²×C₇₉	—	8	2	no	—
129	C₂²×C₇₉	—	8	8	no	—
130	C₂²×C₇₉	#474	1	2	yes	—
131	C₂²×C₇₉	C₂×D₁₆	1	2	yes	—
132	C₂²×C₇₉	#510	1	2	yes	—
133	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₇₉	—	8	2	no	—
135	C₄×C₇₉	#474	1	2	yes	—
136	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
138	Dic₇₉	—	2	2	no	—
139	Dic₇₉	—	8	2	no	—
140	Dic₇₉	—	8	2	no	—
141	Dic₇₉	C₂×D₁₆	1	4	yes	—
142	C₂×#2224	—	79	1	no	maximal, Sylow
143	C₂×C₇₉	—	8	4	no	—
144	C₂×C₇₉	#2224	1	4	yes	complemented
145	C₂×C₇₉	C₂×#474	1	1	yes	characteristic
146	C₂×C₇₉	C₂²×D₁₆	1	1	yes	characteristic
147	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
148	C₂×#500	—	79	1	no	—
149	C₂×C₄×D₈	—	79	1	no	—
150	#2224	—	79	8	no	—
151	C₂×#366	—	79	1	no	—
152	C₂×#366	—	79	1	no	—
153	C₂×C₄⋊C₁₆	—	79	1	no	—
154	C₂×#357	—	79	1	no	—
155	C₂×#357	D₇₉	1	1	yes	characteristic
156	C₇₉	C₂×#2224	1	1	yes	characteristic, complemented, Sylow
157	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
158	C₂×C₄⋊Q₈	—	158	1	no	—
159	C₂×C₄×D₄	—	158	1	no	—
160	C₂²×C₁₆	—	79	1	no	—
161	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
162	#500	—	79	4	no	—
163	#500	—	158	2	no	—
164	C₄×D₈	—	79	4	no	—
165	C₄×D₈	—	158	2	no	—
166	C₂×#107	—	79	1	no	—
167	C₂×#107	—	79	1	no	—
168	C₂×#107	—	158	1	no	—
169	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
170	C₂×#102	—	158	1	no	—
171	C₂×C₄×C₈	—	79	1	no	—
172	#366	—	79	4	no	—
173	#366	—	79	4	no	—
174	C₄⋊C₁₆	—	79	4	no	—
175	#357	—	79	4	no	—
176	#357	C₂×D₇₉	1	4	yes	—
177	C₂²×Q₈	—	316	1	no	—
178	C₂²×D₄	—	2	1	no	—
179	C₂³×C₄	—	316	1	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	Dic₁₅₈	1	4	yes	complemented
183	C₂×D₈	C₂²×D₇₉	1	2	yes	—
184	C₂²×C₈	—	79	1	no	—
185	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
186	C₄⋊Q₈	—	316	2	no	—
187	C₄⋊Q₈	—	158	4	no	—
188	C₄×D₄	—	316	2	no	—
189	C₄×D₄	—	158	4	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	316	1	no	—
192	C₂×C₄⋊C₄	—	158	1	no	—
193	C₂×C₄⋊C₄	—	158	1	no	—
194	C₂×C₂²⋊C₄	—	316	1	no	—
195	C₂×C₄²	—	79	1	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	—	79	2	no	—
198	C₂×C₁₆	—	158	2	no	—
199	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
200	#107	—	2	1	no	—
201	#107	—	79	2	no	—
202	#107	—	79	2	no	—
203	#107	—	158	4	no	—
204	#107	—	158	1	no	—
205	#107	—	158	1	no	—
206	#107	C₂²×D₇₉	1	2	yes	—
207	#102	—	158	4	no	—
208	C₄×C₈	—	79	2	no	—
209	C₄×C₈	—	79	2	no	—
210	C₂⁴	—	4	1	no	—
211	C₂×Q₈	—	316	4	no	—
212	C₂×Q₈	—	632	4	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	316	1	no	—
219	C₂²×C₄	—	316	1	no	—
220	C₂²×C₄	—	316	4	no	—
221	C₂²×C₄	—	79	1	no	—
222	C₂²×C₄	—	79	1	no	—
223	C₂²×C₄	—	632	4	no	—
224	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
225	D₈	—	2	2	no	—
226	D₈	—	2	4	no	—
227	D₈	C₂×Dic₁₅₈	1	4	yes	complemented
228	C₂×C₈	—	79	2	no	—
229	C₂×C₈	—	158	2	no	—
230	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
231	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
232	C₂×C₈	#11929623	1	2	yes	—
233	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	316	4	no	—
237	C₄⋊C₄	—	316	1	no	—
238	C₄⋊C₄	—	316	1	no	—
239	C₄⋊C₄	—	158	2	no	—
240	C₄⋊C₄	—	158	2	no	—
241	C₂²⋊C₄	—	316	4	no	—
242	C₄²	—	79	2	no	—
243	C₄²	—	79	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	158	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	8	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	D₈×D₇₉	1	1	yes	characteristic
251	Q₈	—	632	8	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	316	2	no	—
258	C₂×C₄	—	316	2	no	—
259	C₂×C₄	—	79	4	no	—
260	C₂×C₄	—	79	2	no	—
261	C₂×C₄	—	79	2	no	—
262	C₂×C₄	—	632	8	no	—
263	C₂×C₄	—	632	2	no	—
264	C₂×C₄	—	632	2	no	—
265	C₂×C₄	—	158	2	no	—
266	C₂×C₄	#14579138	1	2	yes	—
267	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
268	C₂×C₄	#14579190	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
270	C₈	—	158	2	no	—
271	C₈	#14579138	1	2	yes	—
272	C₈	C₂²×Dic₁₅₈	1	1	yes	characteristic
273	C₈	C₂×#11929623	1	1	yes	characteristic
274	C₂²	—	4	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	8	no	—
277	C₂²	—	8	2	no	—
278	C₂²	#19098841	1	2	yes	—
279	C₂²	D₁₆×D₇₉	1	2	yes	—
280	C₂²	#19099002	1	2	yes	—
281	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
282	C₄	—	8	2	no	—
283	C₄	—	79	4	no	—
284	C₄	—	632	2	no	—
285	C₄	—	632	2	no	—
286	C₄	—	158	2	no	—
287	C₄	#19098841	1	2	yes	—
288	C₄	C₂×#14579138	1	1	yes	characteristic
289	C₄	C₂×#14579190	1	1	yes	characteristic
290	C₂	—	8	4	no	—
291	C₂	#23765535	1	4	yes	direct product factor
292	C₂	C₂×#19098841	1	1	yes	characteristic
293	C₂	C₂×D₁₆×D₇₉	1	1	yes	characteristic
294	C₂	C₂×#19099002	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#357×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₈×C₇₉	C₂²
hypercenter	C₂×#357	D₇₉
hypocenter	C₇₉	C₂×#2224
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
1. C₂×#19098967	C₂	semidirect
3. #23765535	C₂	direct (N × H)
4. C₂×#19098389	C₂	semidirect
5. C₂×#19098388	C₂	semidirect
6. C₂×Dic₇₉⋊C₁₆	C₂	semidirect
13. #19098967	C₂²	semidirect
21. #19098389	C₂²	semidirect
22. #19098388	C₂²	semidirect
23. Dic₇₉⋊C₁₆	C₂²	semidirect
54. C₂×D₈×C₇₉	Q₈	semidirect
90. D₈×C₇₉	C₂×Q₈	semidirect
144. C₂×C₇₉	#2224	semidirect
156. C₇₉	C₂×#2224	semidirect
182. C₂×D₈	Dic₁₅₈	semidirect
227. D₈	C₂×Dic₁₅₈	semidirect
291. C₂	#23765535	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×#19098967	2
C₂×D₈×Dic₇₉	2
#23765535	2
C₂×#19098389	2
C₂×#19098388	2
C₂×Dic₇₉⋊C₁₆	2
C₂×#19098364	2
C₂×#357×C₇₉	2
C₂×#2224	79

Subgroups — C2×#23765535 #34442870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765535 #34442870