Subgroups — #34441659

Order 20224.

674 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098996	C₂	1	1	yes	maximal, characteristic
2	#23772275	C₂	1	2	yes	maximal, complemented
3	#23765277	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765275	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765271	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14579104	C₂	1	2	yes	maximal, complemented
7	#23765249	C₂	1	2	yes	maximal, complemented
8	#367×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765111	C₂	1	2	yes	maximal, complemented
10	#23764998	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579184	C₂²	1	2	yes	—
13	#19099045	—	2	2	no	—
14	#19098996	C₄	1	8	yes	complemented
15	C₂×#14579104	C₂²	1	2	yes	—
16	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
17	C₂×#14579100	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
19	#19098666	—	2	2	no	—
20	#19098661	—	2	2	no	—
21	Q₁₆×Dic₇₉	C₂²	1	2	yes	—
22	C₄×D₆₃₂	C₂²	1	2	yes	—
23	#19098645	C₂²	1	2	yes	—
24	#107×D₇₉	C₂²	1	2	yes	—
25	#19098563	—	2	2	no	—
26	#19098517	C₂²	1	1	yes	characteristic
27	#19098391	C₂²	1	1	yes	characteristic
28	#19098390	C₂²	1	2	yes	—
29	#19098367	C₂²	1	2	yes	—
30	#19098365	C₂²	1	2	yes	—
31	#19098363	C₂²	1	2	yes	—
32	#19098338	C₂²	1	1	yes	characteristic
33	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₇₉	C₂²	1	2	yes	—
37	#367×C₇₉	C₂²	1	1	yes	characteristic
38	#358×C₇₉	C₂²	1	2	yes	—
39	#346×C₇₉	C₂²	1	1	yes	characteristic
40	C₂×#11929625	—	2	2	no	—
41	#14579184	—	2	4	no	—
42	#14579184	C₂×C₄	1	8	yes	complemented
43	C₂×#11929611	—	2	2	no	—
44	C₂×D₆₃₂	C₂³	1	2	yes	—
45	C₂×#11929607	D₄	1	1	yes	characteristic
46	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
47	Q₈×Dic₇₉	—	2	2	no	—
48	C₄×D₃₁₆	—	2	2	no	—
49	C₄×D₃₁₆	—	4	2	no	—
50	#14579123	—	2	2	no	—
51	#14579123	—	4	2	no	—
52	#14579118	—	4	2	no	—
53	#14579118	D₄	1	1	yes	characteristic
54	C₄⋊C₄×D₇₉	—	2	2	no	—
55	C₄⋊C₄×D₇₉	—	4	2	no	—
56	#14579104	C₂×C₄	1	8	yes	complemented
57	D₁₂₆₄	C₂×C₄	1	4	yes	complemented
58	#14579100	—	2	2	no	—
59	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₇₉	C₂×C₄	1	4	yes	complemented
62	#14579095	C₂³	1	2	yes	—
63	#14579094	—	2	2	no	—
64	#14579085	—	2	2	no	—
65	#14579083	—	2	2	no	—
66	#14579081	—	2	2	no	—
67	#14579072	D₄	1	1	yes	characteristic
68	#14579071	D₄	1	1	yes	characteristic
69	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
71	C₄×Q₈×C₇₉	—	2	2	no	—
72	Q₃₂×C₇₉	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
75	#107×C₇₉	C₂³	1	2	yes	—
76	#103×C₇₉	—	2	2	no	—
77	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	#11929625	—	2	8	no	—
79	#11929625	—	4	4	no	—
80	C₂×D₃₁₆	—	2	2	no	—
81	C₂×D₃₁₆	—	4	2	no	—
82	C₂×C₄×D₇₉	—	4	2	no	—
83	C₂×C₄×D₇₉	—	4	2	no	—
84	C₂×C₄×D₇₉	—	4	2	no	—
85	C₂×C₄×D₇₉	—	4	2	no	—
86	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
87	#11929611	—	2	8	no	—
88	D₆₃₂	—	2	2	no	—
89	D₆₃₂	C₂²×C₄	1	4	yes	—
90	#11929607	—	2	2	no	—
91	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₈×D₇₉	C₂²×C₄	1	2	yes	—
94	C₈×D₇₉	#41	1	2	yes	—
95	C₄⋊Dic₇₉	—	2	2	no	—
96	C₄⋊Dic₇₉	—	4	2	no	—
97	Dic₇₉⋊C₄	—	4	2	no	—
98	Dic₇₉⋊C₄	D₈	1	2	yes	—
99	(C₂×D₇₉)⋊C₄	—	4	2	no	—
100	(C₂×D₇₉)⋊C₄	—	4	2	no	—
101	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
102	C₄×Dic₇₉	—	4	2	no	—
103	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
104	C₇₉⋊C₁₆	—	2	1	no	—
105	C₇₉⋊C₁₆	C₂²×C₄	1	2	yes	—
106	C₂×Q₈×C₇₉	—	2	2	no	—
107	Q₁₆×C₇₉	—	2	2	no	—
108	Q₁₆×C₇₉	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₇₉	—	2	2	no	—
112	C₄⋊C₄×C₇₉	—	4	2	no	—
113	C₄²×C₇₉	—	4	2	no	—
114	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₇₉	—	2	1	no	—
116	C₁₆×C₇₉	C₂²×C₄	1	2	yes	—
117	C₂²×D₇₉	—	4	2	no	—
118	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₇₉	—	2	4	no	—
120	Q₈×C₇₉	—	4	2	no	—
121	C₂×C₄×C₇₉	—	4	2	no	—
122	C₂×C₄×C₇₉	—	4	2	no	—
123	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₇₉	—	4	2	no	—
126	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
127	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
128	D₃₁₆	—	2	4	no	—
129	D₃₁₆	—	4	8	no	—
130	D₃₁₆	—	4	2	no	—
131	C₄×D₇₉	—	4	8	no	—
132	C₄×D₇₉	—	16	2	no	—
133	C₄×D₇₉	—	16	2	no	—
134	C₄×D₇₉	—	16	2	no	—
135	C₄×D₇₉	C₄×D₄	1	2	yes	—
136	C₄×D₇₉	#135	1	2	yes	—
137	C₈×C₇₉	—	2	1	no	—
138	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
139	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
140	C₇₉⋊C₈	—	2	1	no	—
141	C₇₉⋊C₈	C₄×D₄	1	2	yes	—
142	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
143	C₂×D₇₉	—	2	2	no	—
144	C₂×D₇₉	—	4	4	no	—
145	C₂×D₇₉	—	8	2	no	—
146	C₂×D₇₉	—	8	2	no	—
147	C₂×D₇₉	C₄×D₈	1	2	yes	—
148	C₄×C₇₉	—	2	2	no	—
149	C₄×C₇₉	—	4	4	no	—
150	C₄×C₇₉	—	8	2	no	—
151	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
153	Dic₇₉	—	2	2	no	—
154	Dic₇₉	—	8	2	no	—
155	Dic₇₉	C₄×D₈	1	2	yes	—
156	#33460	—	79	1	no	maximal, Sylow
157	C₂×C₇₉	C₂×C₄×D₈	1	1	yes	characteristic
158	C₂×C₇₉	#3416	1	2	yes	—
159	D₇₉	—	2	2	no	—
160	D₇₉	—	8	4	no	—
161	C₂×#508	—	79	1	no	—
162	#2951	—	79	2	no	—
163	#2177	—	79	1	no	—
164	#2177	—	79	1	no	—
165	#2176	—	79	1	no	—
166	#2176	D₇₉	1	1	yes	characteristic, complemented
167	C₄×SD₃₂	—	79	2	no	—
168	C₄×SD₃₂	—	79	2	no	—
169	C₂×#367	—	79	1	no	—
170	#2139	—	79	2	no	—
171	#2108	—	79	1	no	—
172	C₇₉	#33460	1	1	yes	characteristic, complemented, Sylow
173	C₂×#135	—	79	2	no	—
174	#520	—	158	2	no	—
175	#508	—	79	8	no	—
176	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
177	C₂×SD₃₂	—	79	2	no	—
178	C₂×D₁₆	—	79	1	no	—
179	C₂×M₃₂	—	79	1	no	—
180	C₂²×C₁₆	—	79	1	no	—
181	#440	—	158	2	no	—
182	#440	—	158	2	no	—
183	C₄×Q₁₆	—	79	2	no	—
184	C₄×Q₁₆	C₂×D₇₉	1	2	yes	—
185	C₄×D₈	—	79	2	no	—
186	C₄×D₈	—	79	2	no	—
187	C₂×#107	—	79	2	no	—
188	#417	—	158	2	no	—
189	#405	—	79	1	no	—
190	#367	—	79	1	no	—
191	#367	—	79	2	no	—
192	#367	C₂×D₇₉	1	1	yes	characteristic
193	#358	—	79	2	no	—
194	#358	C₂×D₇₉	1	2	yes	—
195	#357	—	79	2	no	—
196	#357	—	79	2	no	—
197	#346	—	79	1	no	—
198	#346	C₂×D₇₉	1	1	yes	characteristic
199	C₄×C₁₆	—	79	1	no	—
200	C₄×C₁₆	—	79	1	no	—
201	C₂×#41	—	158	2	no	—
202	#135	—	79	8	no	—
203	#135	—	158	4	no	—
204	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
205	C₂×SD₁₆	—	158	2	no	—
206	C₂×D₈	—	79	2	no	—
207	C₂×M₁₆	—	79	1	no	—
208	C₂²×C₈	—	79	1	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	158	2	no	—
211	C₄×D₄	—	316	2	no	—
212	C₄×D₄	—	316	2	no	—
213	C₄×D₄	—	158	2	no	—
214	C₄×D₄	—	158	2	no	—
215	#117	—	316	2	no	—
216	#117	—	79	1	no	—
217	C₂×C₄⋊C₄	—	316	2	no	—
218	C₂×C₄⋊C₄	—	158	2	no	—
219	Q₃₂	C₄×D₇₉	1	4	yes	complemented
220	SD₃₂	—	79	8	no	—
221	D₁₆	—	79	4	no	—
222	M₃₂	—	158	2	no	—
223	C₂×C₁₆	—	79	1	no	—
224	C₂×C₁₆	—	79	4	no	—
225	C₂×C₁₆	—	79	1	no	—
226	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
228	#107	—	79	2	no	—
229	#107	—	158	2	no	—
230	#107	C₂²×D₇₉	1	2	yes	—
231	#103	—	2	2	no	—
232	#103	—	158	2	no	—
233	#102	—	158	2	no	—
234	#102	—	158	2	no	—
235	#97	—	79	1	no	—
236	#97	—	79	1	no	—
237	C₄×C₈	—	79	1	no	—
238	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
239	#41	—	316	4	no	—
240	#41	—	158	8	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×D₄	—	316	2	no	—
243	C₂×D₄	—	158	2	no	—
244	C₂²×C₄	—	316	2	no	—
245	C₂²×C₄	—	316	2	no	—
246	C₂²×C₄	—	316	2	no	—
247	C₂²×C₄	—	316	2	no	—
248	C₂²×C₄	—	79	1	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	C₂×C₄×D₇₉	1	4	yes	—
251	SD₁₆	—	158	8	no	—
252	D₈	—	79	4	no	—
253	D₈	—	158	2	no	—
254	M₁₆	—	158	2	no	—
255	C₂×C₈	—	79	1	no	—
256	C₂×C₈	—	79	2	no	—
257	C₂×C₈	—	79	2	no	—
258	C₂×C₈	—	79	1	no	—
259	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
260	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	316	2	no	—
264	C₄⋊C₄	—	316	2	no	—
265	C₄⋊C₄	—	79	2	no	—
266	C₄⋊C₄	—	158	2	no	—
267	C₂²⋊C₄	—	316	2	no	—
268	C₂²⋊C₄	—	316	2	no	—
269	C₂²⋊C₄	—	79	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	316	2	no	—
272	C₄²	—	79	1	no	—
273	C₄²	D₄×D₇₉	1	1	yes	characteristic
274	C₁₆	—	2	1	no	—
275	C₁₆	—	79	2	no	—
276	C₁₆	—	158	1	no	—
277	C₁₆	C₂×C₄×D₇₉	1	2	yes	—
278	C₂³	—	316	2	no	—
279	C₂³	—	79	1	no	—
280	Q₈	—	2	4	no	—
281	Q₈	—	4	2	no	—
282	D₄	—	316	8	no	—
283	D₄	—	316	2	no	—
284	D₄	—	158	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	316	8	no	—
288	C₂×C₄	—	316	2	no	—
289	C₂×C₄	—	79	1	no	—
290	C₂×C₄	—	79	2	no	—
291	C₂×C₄	—	79	2	no	—
292	C₂×C₄	—	79	2	no	—
293	C₂×C₄	—	1264	2	no	—
294	C₂×C₄	—	1264	2	no	—
295	C₂×C₄	—	1264	2	no	—
296	C₂×C₄	D₈×D₇₉	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	79	2	no	—
300	C₈	—	158	1	no	—
301	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
302	C₈	#41×D₇₉	1	1	yes	characteristic
303	C₂²	—	316	4	no	—
304	C₂²	—	79	2	no	—
305	C₂²	—	632	2	no	—
306	C₂²	—	632	2	no	—
307	C₂²	—	158	2	no	—
308	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	8	2	no	—
312	C₄	—	79	2	no	—
313	C₄	—	632	2	no	—
314	C₄	—	158	2	no	—
315	C₄	C₄×D₄×D₇₉	1	1	yes	characteristic
316	C₄	#135×D₇₉	1	1	yes	characteristic
317	C₂	—	632	4	no	—
318	C₂	—	158	2	no	—
319	C₂	C₄×D₈×D₇₉	1	1	yes	characteristic
320	C₂	#23776545	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2176	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×#19098996	C₂
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	#2176	D₇₉
hypocenter	C₇₉	#33460
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
2. #23772275	C₂	semidirect
3. #23765277	C₂	semidirect
4. #23765275	C₂	semidirect
5. #23765271	C₂	semidirect
6. C₄×#14579104	C₂	semidirect
7. #23765249	C₂	semidirect
8. #367×D₇₉	C₂	semidirect
9. #23765111	C₂	semidirect
10. #23764998	C₂	semidirect
11. C₇₉×#2176	C₂	semidirect
14. #19098996	C₄	semidirect
42. #14579184	C₂×C₄	semidirect
56. #14579104	C₂×C₄	semidirect
57. D₁₂₆₄	C₂×C₄	semidirect
61. C₁₆×D₇₉	C₂×C₄	semidirect
72. Q₃₂×C₇₉	C₂×C₄	semidirect
166. #2176	D₇₉	semidirect
172. C₇₉	#33460	semidirect
219. Q₃₂	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098996	2
#23772275	2
#23765277	2
#23765275	2
#23765271	2
C₄×#14579104	2
#23765249	2
#367×D₇₉	2
#23765111	2
#23764998	2
C₇₉×#2176	2
#33460	79