Subgroups — #34288641

Order 20224.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771826	C₂	1	1	yes	maximal, characteristic
2	#23763547	C₂	1	2	yes	maximal, complemented
3	#23763523	C₂	1	2	yes	maximal, complemented
4	#23763501	C₂	1	2	yes	maximal
5	#23763450	C₂	1	1	yes	maximal, characteristic
6	#23763286	C₂	1	2	yes	maximal, complemented
7	#23763202	C₂	1	2	yes	maximal
8	#23763154	C₂	1	2	yes	maximal, complemented
9	C₇₉×#1812	C₂	1	1	yes	maximal, characteristic
10	C₂×#14579168	C₂²	1	1	yes	characteristic
11	#41×Dic₇₉	C₂²	1	1	yes	characteristic
12	C₂×#14579097	C₂²	1	2	yes	—
13	#19098547	C₂²	1	2	yes	—
14	#19098521	C₄	1	2	yes	complemented
15	#19098521	C₂²	1	1	yes	characteristic
16	#19098519	C₂²	1	4	yes	—
17	#19098518	C₄	1	2	yes	complemented
18	#19098518	C₂²	1	1	yes	characteristic
19	M₁₆×Dic₇₉	C₄	1	2	yes	complemented
20	M₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
21	C₂×#14579071	C₄	1	2	yes	complemented
22	C₂×#14579071	C₂²	1	1	yes	characteristic
23	#19098333	C₂²	1	4	yes	—
24	#19098322	C₂²	1	4	yes	—
25	C₇₉⋊#338	C₂²	1	2	yes	—
26	#19098319	C₂²	1	4	yes	—
27	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
28	#427×C₇₉	C₂²	1	2	yes	—
29	#410×C₇₉	C₂²	1	2	yes	—
30	H(3,ℤ/4)×C₇₉	C₂²	1	2	yes	—
31	#14579168	—	2	4	no	—
32	C₂×#11929608	C₂×C₄	1	2	yes	—
33	C₂×#11929608	C₂³	1	4	yes	—
34	C₂×#11929608	C₂³	1	1	yes	characteristic
35	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
37	Q₈×Dic₇₉	—	2	1	no	—
38	D₄×Dic₇₉	—	2	2	no	—
39	D₄×Dic₇₉	—	2	1	no	—
40	#14579120	C₂×C₄	1	2	yes	—
41	#14579120	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
44	#14579097	—	2	2	no	—
45	#14579097	—	2	2	no	—
46	#14579080	—	2	2	no	—
47	#14579078	—	2	2	no	—
48	#14579072	—	2	4	no	—
49	#14579071	C₂×C₄	1	1	yes	characteristic
50	#14579071	C₂×C₄	1	2	yes	—
51	#14579071	C₂×C₄	1	2	yes	—
52	#14579071	C₂×C₄	1	2	yes	—
53	#14579071	C₂×C₄	1	1	yes	characteristic
54	#14579071	C₂×C₄	1	1	yes	characteristic
55	#14579071	C₂×C₄	1	1	yes	characteristic
56	C₄×C₇₉⋊C₈	—	2	4	no	—
57	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
58	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
59	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
60	C₂×#41×C₇₉	C₂³	1	1	yes	characteristic
61	#131×C₇₉	—	2	2	no	—
62	#131×C₇₉	—	2	2	no	—
63	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
64	C₂×M₁₆×C₇₉	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	#117×C₇₉	C₂³	1	4	yes	—
68	#107×C₇₉	—	2	2	no	—
69	#106×C₇₉	—	2	2	no	—
70	#99×C₇₉	—	2	4	no	—
71	C₂²×Dic₇₉	—	2	2	no	—
72	C₂²×Dic₇₉	—	2	1	no	—
73	#11929608	—	2	4	no	—
74	#11929608	—	2	2	no	—
75	#11929608	—	2	4	no	—
76	#11929608	—	2	4	no	—
77	C₂×C₇₉⋊C₈	—	2	4	no	—
78	C₂×C₇₉⋊C₈	—	2	4	no	—
79	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₇₉⋊C₈	#41	1	4	yes	—
83	C₄⋊Dic₇₉	—	2	2	no	—
84	C₄⋊Dic₇₉	—	2	1	no	—
85	C₂²⋊Dic₇₉	—	2	2	no	—
86	C₂²⋊Dic₇₉	—	2	1	no	—
87	C₄×Dic₇₉	—	2	2	no	—
88	C₄×Dic₇₉	—	2	1	no	—
89	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
90	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
91	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
92	#41×C₇₉	—	2	2	no	—
93	#41×C₇₉	—	2	2	no	—
94	C₂×Q₈×C₇₉	#41	1	1	yes	characteristic
95	C₂×D₄×C₇₉	#41	1	2	yes	—
96	C₂×D₄×C₇₉	#41	1	1	yes	characteristic
97	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
99	M₁₆×C₇₉	—	2	4	no	—
100	M₁₆×C₇₉	—	2	2	no	—
101	C₂×C₈×C₇₉	—	2	2	no	—
102	C₂×C₈×C₇₉	—	2	2	no	—
103	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₇₉	#41	1	2	yes	—
107	C₂×C₈×C₇₉	#41	1	2	yes	—
108	C₄⋊C₄×C₇₉	—	2	4	no	—
109	C₂²⋊C₄×C₇₉	—	2	4	no	—
110	C₄²×C₇₉	—	2	4	no	—
111	C₂³×C₇₉	#117	1	2	yes	—
112	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
113	Q₈×C₇₉	—	2	2	no	—
114	D₄×C₇₉	—	2	4	no	—
115	D₄×C₇₉	—	2	2	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	4	4	no	—
119	C₂×C₄×C₇₉	#117	1	2	yes	—
120	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
122	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
123	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₇₉	—	2	1	no	—
125	C₂×Dic₇₉	—	2	2	no	—
126	C₂×Dic₇₉	—	2	1	no	—
127	C₂×Dic₇₉	—	8	1	no	—
128	C₂×Dic₇₉	—	8	2	no	—
129	C₈×C₇₉	—	2	2	no	—
130	C₈×C₇₉	#117	1	1	yes	characteristic
131	C₈×C₇₉	#117	1	1	yes	characteristic
132	C₈×C₇₉	#117	1	2	yes	—
133	C₇₉⋊C₈	—	2	2	no	—
134	C₇₉⋊C₈	—	4	4	no	—
135	C₇₉⋊C₈	#117	1	2	yes	—
136	C₇₉⋊C₈	#117	1	2	yes	—
137	C₂²×C₇₉	—	2	2	no	—
138	C₂²×C₇₉	—	4	1	no	—
139	C₂²×C₇₉	—	4	2	no	—
140	C₂²×C₇₉	C₂×#117	1	1	yes	characteristic
141	C₂²×C₇₉	#529	1	2	yes	—
142	C₄×C₇₉	—	2	2	no	—
143	C₄×C₇₉	—	8	4	no	—
144	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
145	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
146	C₄×C₇₉	#529	1	2	yes	—
147	Dic₇₉	—	4	2	no	—
148	Dic₇₉	—	4	1	no	—
149	Dic₇₉	—	4	1	no	—
150	#20788	—	79	1	no	maximal, Sylow
151	C₂×C₇₉	—	2	1	no	—
152	C₂×C₇₉	—	2	2	no	—
153	C₂×C₇₉	—	4	2	no	—
154	C₂×C₇₉	#2283	1	1	yes	characteristic
155	#2875	—	79	1	no	—
156	#1841	—	79	2	no	—
157	#1833	—	79	2	no	—
158	#1827	—	79	2	no	—
159	#1813	—	79	1	no	—
160	#1812	D₇₉	1	1	yes	characteristic
161	#1773	—	79	2	no	—
162	#1757	—	79	2	no	—
163	#1747	—	79	2	no	—
164	C₇₉	#20788	1	1	yes	characteristic, complemented, Sylow
165	C₂×#131	—	79	1	no	—
166	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
167	C₄×#41	—	79	1	no	—
168	C₂×#108	—	79	2	no	—
169	#427	C₂×D₇₉	1	2	yes	—
170	#413	—	79	2	no	—
171	#410	C₂×D₇₉	1	2	yes	—
172	#405	—	79	1	no	—
173	#405	—	79	2	no	—
174	#405	—	79	1	no	—
175	#405	—	79	2	no	—
176	#405	—	79	4	no	—
177	C₄×M₁₆	—	79	1	no	—
178	C₄×M₁₆	—	79	2	no	—
179	C₂×#97	—	79	1	no	—
180	C₂×#97	—	79	2	no	—
181	#344	—	79	4	no	—
182	#339	—	79	4	no	—
183	#338	—	79	2	no	—
184	H(3,ℤ/4)	—	79	4	no	—
185	H(3,ℤ/4)	C₂×D₇₉	1	2	yes	—
186	C₂×#41	C₂²×D₇₉	1	1	yes	characteristic
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	158	4	no	—
190	C₂×M₁₆	—	79	2	no	—
191	C₂×M₁₆	—	79	1	no	—
192	C₂×M₁₆	—	79	4	no	—
193	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
194	C₂×M₁₆	C₂²×D₇₉	1	1	yes	characteristic
195	C₂²×C₈	—	79	1	no	—
196	C₂²×C₈	—	79	2	no	—
197	C₂²×C₈	C₄×D₇₉	1	2	yes	—
198	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
199	C₄×Q₈	—	158	1	no	—
200	C₄×D₄	—	158	2	no	—
201	C₄×D₄	—	158	1	no	—
202	#117	—	79	2	no	—
203	#117	—	79	1	no	—
204	#117	C₂²×D₇₉	1	4	yes	—
205	C₂×C₄²	—	79	2	no	—
206	C₂×C₄²	—	79	1	no	—
207	#108	—	158	2	no	—
208	#108	—	158	2	no	—
209	#107	—	2	2	no	—
210	#106	—	2	2	no	—
211	#101	—	158	2	no	—
212	#100	—	158	2	no	—
213	#99	—	2	4	no	—
214	#97	—	79	1	no	—
215	#97	—	79	1	no	—
216	#97	—	79	2	no	—
217	#97	—	79	1	no	—
218	#97	—	79	2	no	—
219	#97	—	79	1	no	—
220	#97	—	79	2	no	—
221	#97	—	158	4	no	—
222	C₄×C₈	—	79	2	no	—
223	C₄×C₈	—	79	2	no	—
224	C₄×C₈	—	79	2	no	—
225	C₄×C₈	—	158	4	no	—
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	#11929625	1	1	yes	characteristic
229	C₂×D₄	#11929623	1	2	yes	—
230	C₂×D₄	#11929625	1	1	yes	characteristic
231	C₂²×C₄	—	158	1	no	—
232	C₂²×C₄	—	158	2	no	—
233	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
234	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	158	4	no	—
238	M₁₆	—	158	4	no	—
239	M₁₆	—	158	4	no	—
240	M₁₆	—	158	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	79	4	no	—
244	C₂×C₈	—	79	1	no	—
245	C₂×C₈	—	79	1	no	—
246	C₂×C₈	—	79	2	no	—
247	C₂×C₈	—	158	4	no	—
248	C₂×C₈	—	158	4	no	—
249	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
252	C₂×C₈	#11929623	1	2	yes	—
253	C₂×C₈	#11929623	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	158	1	no	—
256	C₄⋊C₄	—	158	2	no	—
257	C₂²⋊C₄	—	2	4	no	—
258	C₂²⋊C₄	—	158	1	no	—
259	C₂²⋊C₄	—	158	2	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	79	2	no	—
262	C₄²	—	79	1	no	—
263	C₄²	—	79	1	no	—
264	C₄²	—	158	1	no	—
265	C₄²	—	158	2	no	—
266	C₂³	#14579119	1	2	yes	—
267	C₂³	C₂×#11929623	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	632	1	no	—
275	C₂×C₄	—	632	2	no	—
276	C₂×C₄	—	158	1	no	—
277	C₂×C₄	—	158	1	no	—
278	C₂×C₄	—	158	2	no	—
279	C₂×C₄	#14579119	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
281	C₂×C₄	#41×D₇₉	1	2	yes	—
282	C₂×C₄	C₂×#11929623	1	2	yes	—
283	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	316	4	no	—
286	C₈	—	79	2	no	—
287	C₈	—	79	2	no	—
288	C₈	—	158	2	no	—
289	C₈	#14579117	1	1	yes	characteristic
290	C₈	#14579117	1	1	yes	characteristic
291	C₈	#14579117	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×#14579119	1	1	yes	characteristic
296	C₂²	#19099094	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	8	4	no	—
299	C₄	—	316	1	no	—
300	C₄	—	316	1	no	—
301	C₄	—	316	2	no	—
302	C₄	#117×D₇₉	1	1	yes	characteristic
303	C₄	C₂×#14579117	1	1	yes	characteristic
304	C₄	#19099099	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	4	2	no	—
308	C₂	#23765795	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1812	C₂
socle	C₂×C₇₉	#2283
prime radical	C₂×#41×C₇₉	C₂³
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1812	D₇₉
hypocenter	C₇₉	#20788
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1812

Complements

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

N	H ≅ G/N	type
2. #23763547	C₂	semidirect
3. #23763523	C₂	semidirect
6. #23763286	C₂	semidirect
8. #23763154	C₂	semidirect
14. #19098521	C₄	semidirect
17. #19098518	C₄	semidirect
19. M₁₆×Dic₇₉	C₄	semidirect
21. C₂×#14579071	C₄	semidirect
164. C₇₉	#20788	semidirect

Maximal subgroups

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

subgroup	index
#23771826	2
#23763547	2
#23763523	2
#23763501	2
#23763450	2
#23763286	2
#23763202	2
#23763154	2
C₇₉×#1812	2
#20788	79