Subgroups — C₂×#23765249 #34441533

Order 20224.

742 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579104	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098645	C₂	1	1	yes	maximal, characteristic
4	#23765249	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19098390	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19098365	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#358×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
10	C₂×Q₈×Dic₇₉	—	2	1	no	—
11	C₂×#14579123	—	2	1	no	—
12	C₂×#14579104	C₄	1	8	yes	complemented
13	C₂×#14579104	C₂²	1	4	yes	—
14	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	Q₁₆×Dic₇₉	—	2	2	no	—
16	Q₁₆×Dic₇₉	C₂²	1	4	yes	complemented
17	#19098645	—	2	2	no	—
18	#19098645	C₂²	1	4	yes	—
19	C₂×#14579095	C₂²	1	1	yes	characteristic
20	C₂×#14579085	—	2	1	no	—
21	C₂×#14579081	—	2	1	no	—
22	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
23	#19098390	C₂²	1	4	yes	complemented
24	#19098365	C₂²	1	4	yes	—
25	C₁₆×Dic₇₉	C₂²	1	4	yes	complemented
26	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
29	#358×C₇₉	C₂²	1	4	yes	complemented
30	C₂²×D₃₁₆	—	2	1	no	—
31	C₂²×C₄×D₇₉	—	4	1	no	—
32	C₂×D₆₃₂	—	2	1	no	—
33	C₂×D₆₃₂	—	2	2	no	—
34	C₂×D₆₃₂	C₂×C₄	1	4	yes	—
35	C₂×D₆₃₂	C₂³	1	2	yes	—
36	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
37	Q₈×Dic₇₉	—	2	4	no	—
38	Q₈×Dic₇₉	—	4	2	no	—
39	#14579123	—	2	4	no	—
40	#14579123	—	4	2	no	—
41	C₂×C₄⋊Dic₇₉	—	2	1	no	—
42	C₂×C₄⋊Dic₇₉	—	4	1	no	—
43	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
44	C₂×C₄×Dic₇₉	—	4	1	no	—
45	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
46	#14579104	C₂×C₄	1	16	yes	complemented
47	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	complemented
48	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
49	#14579095	—	2	1	no	—
50	#14579095	C₂³	1	2	yes	—
51	#14579085	—	2	4	no	—
52	#14579081	—	2	4	no	—
53	C₈×Dic₇₉	D₄	1	2	yes	—
54	C₈×Dic₇₉	C₂³	1	2	yes	—
55	C₂²×Q₈×C₇₉	—	2	1	no	—
56	C₂×Q₁₆×C₇₉	—	2	1	no	—
57	C₂×Q₁₆×C₇₉	—	2	2	no	—
58	C₂×Q₁₆×C₇₉	C₂×C₄	1	4	yes	complemented
59	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
60	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
62	C₂×C₁₆×C₇₉	—	2	2	no	—
63	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
64	#107×C₇₉	—	2	1	no	—
65	#107×C₇₉	C₂³	1	2	yes	—
66	C₂³×D₇₉	—	4	1	no	—
67	C₂×D₃₁₆	—	2	4	no	—
68	C₂×D₃₁₆	—	2	2	no	—
69	C₂×D₃₁₆	—	4	1	no	—
70	C₂×D₃₁₆	—	4	2	no	—
71	C₂²×Dic₇₉	—	4	1	no	—
72	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
73	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
74	C₂×C₄×D₇₉	—	4	4	no	—
75	C₂×C₄×D₇₉	—	8	4	no	—
76	D₆₃₂	—	2	4	no	—
77	D₆₃₂	—	2	2	no	—
78	D₆₃₂	C₂²×C₄	1	4	yes	—
79	C₂×C₇₉⋊C₈	—	2	2	no	—
80	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
81	C₄⋊Dic₇₉	—	2	2	no	—
82	C₄⋊Dic₇₉	—	4	4	no	—
83	C₄⋊Dic₇₉	—	4	1	no	—
84	(C₂×D₇₉)⋊C₄	—	4	4	no	—
85	C₄×Dic₇₉	—	4	4	no	—
86	C₄×Dic₇₉	D₈	1	2	yes	—
87	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
88	C₇₉⋊C₁₆	C₂²×C₄	1	4	yes	complemented
89	C₂×Q₈×C₇₉	—	2	4	no	—
90	C₂×Q₈×C₇₉	—	2	2	no	—
91	C₂×Q₈×C₇₉	—	4	1	no	—
92	C₂×Q₈×C₇₉	—	4	2	no	—
93	C₂²×C₄×C₇₉	—	4	1	no	—
94	C₂²×C₄×C₇₉	—	4	1	no	—
95	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
96	Q₁₆×C₇₉	—	2	4	no	—
97	Q₁₆×C₇₉	—	2	2	no	—
98	Q₁₆×C₇₉	C₂²×C₄	1	4	yes	complemented
99	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₇₉	#41	1	2	yes	—
102	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₇₉	—	2	2	no	—
104	C₄⋊C₄×C₇₉	—	4	1	no	—
105	C₁₆×C₇₉	—	2	2	no	—
106	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₇₉	—	4	2	no	—
108	C₂²×D₇₉	—	4	4	no	—
109	C₂²×D₇₉	—	8	2	no	—
110	C₂²×D₇₉	—	8	2	no	—
111	Q₈×C₇₉	—	2	4	no	—
112	Q₈×C₇₉	—	4	2	no	—
113	Q₈×C₇₉	—	4	4	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	—	4	4	no	—
116	C₂×C₄×C₇₉	—	4	2	no	—
117	C₂×C₄×C₇₉	—	8	2	no	—
118	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#135	1	2	yes	—
121	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₇₉	—	2	2	no	—
123	C₂×Dic₇₉	—	4	2	no	—
124	C₂×Dic₇₉	—	8	2	no	—
125	C₂×Dic₇₉	SD₃₂	1	4	yes	complemented
126	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
127	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
128	D₃₁₆	—	2	4	no	—
129	D₃₁₆	—	4	2	no	—
130	D₃₁₆	—	4	4	no	—
131	C₄×D₇₉	—	8	8	no	—
132	C₈×C₇₉	C₄×D₄	1	2	yes	—
133	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
134	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
135	C₇₉⋊C₈	—	2	2	no	—
136	C₂²×C₇₉	C₄×D₈	1	2	yes	—
137	C₂²×C₇₉	C₂×SD₃₂	1	2	yes	—
138	C₂²×C₇₉	#508	1	2	yes	—
139	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₇₉	—	4	4	no	—
141	C₂×D₇₉	—	8	2	no	—
142	C₂×D₇₉	—	8	2	no	—
143	C₂×D₇₉	—	8	8	no	—
144	C₄×C₇₉	—	4	4	no	—
145	C₄×C₇₉	—	8	2	no	—
146	C₄×C₇₉	C₄×D₈	1	2	yes	—
147	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
149	Dic₇₉	—	2	2	no	—
150	Dic₇₉	—	8	2	no	—
151	Dic₇₉	C₂×SD₃₂	1	4	yes	complemented
152	C₂×C₄×SD₃₂	—	79	1	no	maximal, Sylow
153	C₂×C₇₉	C₄×SD₃₂	1	4	yes	complemented
154	C₂×C₇₉	C₂×C₄×D₈	1	1	yes	characteristic
155	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
156	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
157	D₇₉	—	8	4	no	—
158	C₂²×SD₃₂	—	79	1	no	—
159	C₂×C₄×Q₁₆	—	79	1	no	—
160	C₂×C₄×D₈	—	79	1	no	—
161	C₄×SD₃₂	—	79	8	no	—
162	C₂×#367	—	79	1	no	—
163	C₂×#358	D₇₉	1	1	yes	characteristic, complemented
164	C₂×#357	—	79	1	no	—
165	C₂×C₄×C₁₆	—	79	1	no	—
166	C₇₉	C₂×C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
168	C₂²×D₈	—	79	1	no	—
169	C₂×C₄×Q₈	—	158	1	no	—
170	C₂×C₄×D₄	—	158	1	no	—
171	C₂×SD₃₂	—	79	4	no	—
172	C₂×SD₃₂	—	79	8	no	—
173	C₂²×C₁₆	—	79	1	no	—
174	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
175	C₄×Q₁₆	—	79	4	no	—
176	C₄×Q₁₆	—	158	2	no	—
177	C₄×D₈	—	79	4	no	—
178	C₄×D₈	—	158	2	no	—
179	C₂×#107	—	79	1	no	—
180	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
181	C₂×#103	—	158	1	no	—
182	C₂×#102	—	158	1	no	—
183	C₂×C₄×C₈	—	79	1	no	—
184	#367	—	79	4	no	—
185	#358	C₂×D₇₉	1	4	yes	complemented
186	#357	—	79	4	no	—
187	C₄×C₁₆	—	79	4	no	—
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×D₄	—	158	1	no	—
190	C₂³×C₄	—	316	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	C₄×D₇₉	1	4	yes	complemented
194	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
195	C₂×D₈	—	79	4	no	—
196	C₂×D₈	—	79	2	no	—
197	C₂×D₈	—	158	1	no	—
198	C₂×D₈	—	158	2	no	—
199	C₂²×C₈	—	79	1	no	—
200	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
201	C₄×Q₈	—	316	2	no	—
202	C₄×Q₈	—	158	4	no	—
203	C₄×D₄	—	316	2	no	—
204	C₄×D₄	—	158	4	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	316	1	no	—
207	C₂×C₄⋊C₄	—	158	1	no	—
208	C₂×C₂²⋊C₄	—	316	1	no	—
209	C₂×C₄²	—	316	1	no	—
210	C₂×C₄²	—	79	1	no	—
211	SD₃₂	—	79	16	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	—	79	4	no	—
214	C₂×C₁₆	—	79	2	no	—
215	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
216	#107	—	2	1	no	—
217	#107	—	79	2	no	—
218	#107	—	158	1	no	—
219	#107	C₂²×D₇₉	1	2	yes	—
220	#103	—	158	4	no	—
221	#102	—	158	4	no	—
222	C₄×C₈	—	79	2	no	—
223	C₄×C₈	—	79	2	no	—
224	C₂⁴	—	316	1	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×D₄	—	316	1	no	—
230	C₂×D₄	—	316	2	no	—
231	C₂×D₄	—	158	4	no	—
232	C₂×D₄	—	158	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	316	4	no	—
236	C₂²×C₄	—	316	1	no	—
237	C₂²×C₄	—	79	1	no	—
238	C₂²×C₄	—	79	1	no	—
239	C₂²×C₄	—	632	4	no	—
240	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	2	4	no	—
243	Q₁₆	C₂×C₄×D₇₉	1	4	yes	complemented
244	D₈	—	79	4	no	—
245	D₈	—	158	2	no	—
246	D₈	—	158	4	no	—
247	C₂×C₈	—	79	2	no	—
248	C₂×C₈	—	158	2	no	—
249	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
250	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
251	C₂×C₈	#11929623	1	2	yes	—
252	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	316	4	no	—
256	C₄⋊C₄	—	316	1	no	—
257	C₄⋊C₄	—	158	2	no	—
258	C₂²⋊C₄	—	316	4	no	—
259	C₄²	—	316	4	no	—
260	C₄²	—	79	2	no	—
261	C₄²	—	79	2	no	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	79	4	no	—
264	C₂³	—	316	4	no	—
265	C₂³	—	316	2	no	—
266	C₂³	—	632	2	no	—
267	C₂³	—	632	2	no	—
268	C₂³	D₈×D₇₉	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	4	2	no	—
272	D₄	—	316	4	no	—
273	D₄	—	316	2	no	—
274	D₄	—	158	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	316	2	no	—
280	C₂×C₄	—	79	4	no	—
281	C₂×C₄	—	79	2	no	—
282	C₂×C₄	—	79	2	no	—
283	C₂×C₄	—	632	8	no	—
284	C₂×C₄	—	632	2	no	—
285	C₂×C₄	—	158	2	no	—
286	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
287	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
288	C₂×C₄	#14579182	1	2	yes	—
289	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
290	C₈	—	158	2	no	—
291	C₈	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
292	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
293	C₈	C₂×#11929623	1	1	yes	characteristic
294	C₂²	—	316	4	no	—
295	C₂²	—	632	8	no	—
296	C₂²	—	632	2	no	—
297	C₂²	—	632	2	no	—
298	C₂²	#19098643	1	2	yes	—
299	C₂²	SD₃₂×D₇₉	1	2	yes	—
300	C₂²	#19098996	1	2	yes	—
301	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
302	C₄	—	4	4	no	—
303	C₄	—	8	2	no	—
304	C₄	—	79	4	no	—
305	C₄	—	632	2	no	—
306	C₄	—	158	2	no	—
307	C₄	#19098643	1	2	yes	—
308	C₄	C₂×(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
309	C₄	C₂×#14579182	1	1	yes	characteristic
310	C₂	—	632	4	no	—
311	C₂	#23765249	1	4	yes	direct product factor
312	C₂	C₂×#19098643	1	1	yes	characteristic
313	C₂	C₂×SD₃₂×D₇₉	1	1	yes	characteristic
314	C₂	C₂×#19098996	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#358×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₆₃₂	C₂²
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#358	D₇₉
hypocenter	C₇₉	C₂×C₄×SD₃₂
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×Dic₇₉	C₂	semidirect
4. #23765249	C₂	direct (N × H)
5. C₂×#19098390	C₂	semidirect
7. C₂×C₁₆×Dic₇₉	C₂	semidirect
8. C₂×#358×C₇₉	C₂	semidirect
12. C₂×#14579104	C₄	semidirect
16. Q₁₆×Dic₇₉	C₂²	semidirect
23. #19098390	C₂²	semidirect
25. C₁₆×Dic₇₉	C₂²	semidirect
29. #358×C₇₉	C₂²	semidirect
46. #14579104	C₂×C₄	semidirect
47. C₂×C₇₉⋊C₁₆	C₂×C₄	semidirect
58. C₂×Q₁₆×C₇₉	C₂×C₄	semidirect
88. C₇₉⋊C₁₆	C₂²×C₄	semidirect
98. Q₁₆×C₇₉	C₂²×C₄	semidirect
125. C₂×Dic₇₉	SD₃₂	semidirect
151. Dic₇₉	C₂×SD₃₂	semidirect
153. C₂×C₇₉	C₄×SD₃₂	semidirect
163. C₂×#358	D₇₉	semidirect
166. C₇₉	C₂×C₄×SD₃₂	semidirect
185. #358	C₂×D₇₉	semidirect
193. C₂×Q₁₆	C₄×D₇₉	semidirect
243. Q₁₆	C₂×C₄×D₇₉	semidirect
311. C₂	#23765249	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14579104	2
C₂×Q₁₆×Dic₇₉	2
C₂×#19098645	2
#23765249	2
C₂×#19098390	2
C₂×#19098365	2
C₂×C₁₆×Dic₇₉	2
C₂×#358×C₇₉	2
C₂×C₄×SD₃₂	79

Subgroups — C2×#23765249 #34441533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765249 #34441533