Subgroups — C₂×#23762765 #34270504

Order 20224.

790 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098943	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098843	C₂	1	2	yes	maximal
3	#23762765	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098316	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098310	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₄×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₆	—	2	1	no	—
8	C₂²×Dic₃₁₆	—	2	1	no	—
9	C₂×#14579161	C₂²	1	2	yes	—
10	#19098943	—	2	2	no	—
11	#19098943	C₂²	1	4	yes	complemented
12	#19098843	—	2	4	no	—
13	#19098843	C₂²	1	8	yes	—
14	C₂×#14579094	—	2	2	no	—
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×#14579082	—	2	2	no	—
17	#19098316	C₂²	1	4	yes	complemented
18	#19098310	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₇₉	—	2	1	no	—
20	C₂×C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
21	C₈×D₄×C₇₉	—	2	2	no	—
22	C₈×D₄×C₇₉	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₇₉	—	2	1	no	—
25	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₅₈	—	2	2	no	—
27	C₂×Dic₃₁₆	—	2	4	no	—
28	C₂×Dic₃₁₆	—	2	4	no	—
29	C₂×Dic₃₁₆	—	4	4	no	—
30	C₂×Dic₃₁₆	—	4	4	no	—
31	C₂²×C₇₉⋊C₈	—	2	2	no	—
32	#14579161	—	2	2	no	—
33	#14579161	—	2	2	no	—
34	#14579161	—	2	2	no	—
35	#14579161	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₇₉	—	2	2	no	—
37	C₂×C₄⋊Dic₇₉	—	2	2	no	—
38	#14579094	—	2	8	no	—
39	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
40	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
41	#14579082	—	2	8	no	—
42	C₂²×D₄×C₇₉	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₇₉	—	2	1	no	—
44	C₂²×C₈×C₇₉	—	2	1	no	—
45	C₂²×C₈×C₇₉	—	2	4	no	—
46	C₂²×C₈×C₇₉	—	4	4	no	—
47	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
49	C₄×D₄×C₇₉	—	2	1	no	—
50	C₄×D₄×C₇₉	—	2	1	no	—
51	C₄×D₄×C₇₉	D₄	1	2	yes	—
52	C₄×D₄×C₇₉	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₉	—	2	1	no	—
57	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
58	C₂²⋊C₈×C₇₉	—	2	4	no	—
59	C₄×C₈×C₇₉	D₄	1	2	yes	—
60	C₄×C₈×C₇₉	C₂³	1	2	yes	—
61	C₂×Dic₁₅₈	—	2	4	no	—
62	C₂×Dic₁₅₈	—	4	2	no	—
63	C₂×Dic₁₅₈	—	4	4	no	—
64	C₂×Dic₁₅₈	—	4	4	no	—
65	C₂²×Dic₇₉	—	4	2	no	—
66	Dic₃₁₆	—	4	8	no	—
67	Dic₃₁₆	—	4	8	no	—
68	C₂×C₇₉⋊C₈	—	2	4	no	—
69	C₂×C₇₉⋊C₈	—	4	4	no	—
70	C₄⋊Dic₇₉	—	2	4	no	—
71	C₄⋊Dic₇₉	—	2	4	no	—
72	C₄⋊Dic₇₉	—	4	2	no	—
73	C₄⋊Dic₇₉	—	4	2	no	—
74	C₂⁴×C₇₉	—	2	1	no	—
75	C₂×D₄×C₇₉	—	2	1	no	—
76	C₂×D₄×C₇₉	—	2	2	no	—
77	C₂×D₄×C₇₉	Q₁₆	1	4	yes	complemented
78	C₂×D₄×C₇₉	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇₉	—	2	1	no	—
80	C₂²×C₄×C₇₉	—	2	2	no	—
81	C₂²×C₄×C₇₉	—	2	2	no	—
82	C₂²×C₄×C₇₉	—	4	2	no	—
83	C₂²×C₄×C₇₉	—	4	2	no	—
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	—	2	2	no	—
89	C₂×C₈×C₇₉	—	4	8	no	—
90	C₂×C₈×C₇₉	—	4	2	no	—
91	C₂×C₈×C₇₉	D₈	1	4	yes	—
92	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₇₉	—	2	1	no	—
95	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
96	C₂²⋊C₄×C₇₉	—	2	2	no	—
97	C₂²⋊C₄×C₇₉	—	2	2	no	—
98	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₉	#41	1	1	yes	characteristic
101	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
102	C₂³×C₇₉	—	2	2	no	—
103	C₂³×C₇₉	—	2	4	no	—
104	C₂³×C₇₉	—	4	2	no	—
105	C₂³×C₇₉	—	4	2	no	—
106	C₂³×C₇₉	#121	1	1	yes	characteristic
107	D₄×C₇₉	—	2	2	no	—
108	D₄×C₇₉	—	2	4	no	—
109	D₄×C₇₉	C₂×Q₁₆	1	4	yes	complemented
110	C₂×C₄×C₇₉	—	2	2	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	4	2	no	—
113	C₂×C₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	—	4	4	no	—
116	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
121	C₂×C₄×C₇₉	#135	1	2	yes	—
122	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₇₉	—	4	4	no	—
127	C₂×Dic₇₉	—	8	4	no	—
128	Dic₁₅₈	—	4	4	no	—
129	Dic₁₅₈	—	4	4	no	—
130	Dic₁₅₈	—	8	4	no	—
131	C₈×C₇₉	—	2	2	no	—
132	C₈×C₇₉	—	4	2	no	—
133	C₈×C₇₉	C₂×D₈	1	4	yes	—
134	C₇₉⋊C₈	—	4	4	no	—
135	C₂²×C₇₉	—	2	4	no	—
136	C₂²×C₇₉	—	4	8	no	—
137	C₂²×C₇₉	—	4	2	no	—
138	C₂²×C₇₉	—	4	2	no	—
139	C₂²×C₇₉	#459	1	2	yes	—
140	C₂²×C₇₉	#467	1	2	yes	—
141	C₂²×C₇₉	#473	1	2	yes	—
142	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
143	C₄×C₇₉	—	2	1	no	—
144	C₄×C₇₉	—	2	1	no	—
145	C₄×C₇₉	—	4	2	no	—
146	C₄×C₇₉	#459	1	2	yes	—
147	C₄×C₇₉	#467	1	2	yes	—
148	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
150	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
151	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	Dic₇₉	—	8	4	no	—
153	C₂×#1669	—	79	1	no	maximal, Sylow
154	C₂×C₇₉	—	4	4	no	—
155	C₂×C₇₉	#1669	1	4	yes	complemented
156	C₂×C₇₉	C₂×#459	1	1	yes	characteristic
157	C₂×C₇₉	C₂×#467	1	1	yes	characteristic
158	C₂×C₇₉	C₂×#473	1	1	yes	characteristic
159	C₂×#494	—	79	1	no	—
160	C₂×#474	—	79	2	no	—
161	C₂×C₈×D₄	D₇₉	1	1	yes	characteristic
162	#1669	—	79	8	no	—
163	C₂×#335	—	79	1	no	—
164	C₂×#332	—	79	2	no	—
165	C₇₉	C₂×#1669	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₁₆	—	158	1	no	—
167	C₂²×Q₁₆	—	158	1	no	—
168	C₂³×C₈	—	2	1	no	—
169	C₂×C₄⋊Q₈	—	79	2	no	—
170	C₂×C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
171	#494	—	79	4	no	—
172	#494	—	158	2	no	—
173	#474	—	79	8	no	—
174	#474	—	158	4	no	—
175	C₈×D₄	—	2	2	no	—
176	C₈×D₄	C₂×D₇₉	1	4	yes	—
177	C₂×#107	—	158	2	no	—
178	C₂×C₄⋊C₈	—	79	2	no	—
179	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
180	C₂×#102	—	158	2	no	—
181	C₂×C₂²⋊C₈	—	2	1	no	—
182	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
183	#335	—	79	4	no	—
184	#332	—	79	8	no	—
185	C₂²×Q₈	—	158	2	no	—
186	C₂²×D₄	D₃₁₆	1	1	yes	characteristic
187	C₂³×C₄	—	2	1	no	—
188	C₂×Q₁₆	—	316	4	no	—
189	C₂×Q₁₆	—	316	4	no	—
190	C₂×Q₁₆	—	158	4	no	—
191	C₂×Q₁₆	—	158	4	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	4	4	no	—
195	C₂²×C₈	—	158	2	no	—
196	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	79	4	no	—
199	C₄⋊Q₈	—	158	2	no	—
200	C₄⋊Q₈	—	158	2	no	—
201	C₄⋊Q₈	—	158	2	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	D₃₁₆	1	2	yes	—
205	C₄×D₄	C₂²×D₇₉	1	2	yes	—
206	C₂×C₄⋊C₄	—	158	2	no	—
207	C₂×C₄⋊C₄	—	158	2	no	—
208	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
211	#107	—	158	8	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	79	4	no	—
214	C₄⋊C₈	—	158	2	no	—
215	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
216	#102	—	158	8	no	—
217	C₂²⋊C₈	—	2	4	no	—
218	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
219	C₄×C₈	C₂²×D₇₉	1	2	yes	—
220	C₂⁴	—	2	1	no	—
221	C₂×Q₈	—	316	2	no	—
222	C₂×Q₈	—	316	4	no	—
223	C₂×Q₈	—	316	4	no	—
224	C₂×Q₈	—	158	4	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	Dic₃₁₆	1	4	yes	complemented
228	C₂×D₄	C₂×D₃₁₆	1	2	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	316	2	no	—
235	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
237	C₂²×C₄	#11929624	1	1	yes	characteristic
238	Q₁₆	—	316	8	no	—
239	Q₁₆	—	316	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	8	no	—
244	C₂×C₈	—	316	4	no	—
245	C₂×C₈	—	158	4	no	—
246	C₂×C₈	#11929609	1	4	yes	—
247	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	316	2	no	—
251	C₄⋊C₄	—	316	2	no	—
252	C₄⋊C₄	—	158	4	no	—
253	C₄⋊C₄	—	158	4	no	—
254	C₄⋊C₄	C₂×D₃₁₆	1	2	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
258	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
259	C₄²	#11929624	1	1	yes	characteristic
260	C₄²	C₂³×D₇₉	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	#14579135	1	1	yes	characteristic
266	Q₈	—	316	4	no	—
267	Q₈	—	316	4	no	—
268	Q₈	—	632	4	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	D₄	C₂×Dic₃₁₆	1	4	yes	complemented
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	316	4	no	—
279	C₂×C₄	—	632	4	no	—
280	C₂×C₄	#14579135	1	1	yes	characteristic
281	C₂×C₄	#14579135	1	1	yes	characteristic
282	C₂×C₄	#14579135	1	1	yes	characteristic
283	C₂×C₄	C₂×#11929609	1	2	yes	—
284	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
285	C₂×C₄	#14579183	1	2	yes	—
286	C₂×C₄	#14579186	1	2	yes	—
287	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
289	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	316	4	no	—
293	C₈	C₂×#11929609	1	4	yes	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	#19098758	1	2	yes	—
299	C₂²	#19098804	1	2	yes	—
300	C₂²	#19098838	1	2	yes	—
301	C₂²	C₂×#14579135	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	632	4	no	—
306	C₄	#19098758	1	2	yes	—
307	C₄	#19098804	1	2	yes	—
308	C₄	C₂²×#11929609	1	1	yes	characteristic
309	C₄	C₂²×Dic₃₁₆	1	1	yes	characteristic
310	C₄	C₂×#14579183	1	1	yes	characteristic
311	C₄	C₂×#14579186	1	1	yes	characteristic
312	C₂	—	4	4	no	—
313	C₂	#23762765	1	4	yes	direct product factor
314	C₂	C₂×#19098758	1	1	yes	characteristic
315	C₂	C₂×#19098804	1	1	yes	characteristic
316	C₂	C₂×#19098838	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579135
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×C₈×D₄×C₇₉	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂²×D₄×C₇₉	D₄
hypercenter	C₂×C₈×D₄	D₇₉
hypocenter	C₇₉	C₂×#1669
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#19098943	C₂	semidirect
3. #23762765	C₂	direct (N × H)
4. C₂×#19098316	C₂	semidirect
5. C₂×#19098310	C₂	semidirect
11. #19098943	C₂²	semidirect
17. #19098316	C₂²	semidirect
18. #19098310	C₂²	semidirect
77. C₂×D₄×C₇₉	Q₁₆	semidirect
109. D₄×C₇₉	C₂×Q₁₆	semidirect
155. C₂×C₇₉	#1669	semidirect
165. C₇₉	C₂×#1669	semidirect
227. C₂×D₄	Dic₃₁₆	semidirect
271. D₄	C₂×Dic₃₁₆	semidirect
313. C₂	#23762765	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×#19098943	2
C₂×#19098843	2
#23762765	2
C₂×#19098316	2
C₂×#19098310	2
C₂×C₈×D₄×C₇₉	2
C₂×#1669	79

Subgroups — C2×#23762765 #34270504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23762765 #34270504