Subgroups — #34443776

Order 20224.

474 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773937	C₂	1	2	yes	maximal
2	#23765687	C₂	1	2	yes	maximal, complemented
3	#23765685	C₂	1	2	yes	maximal, complemented
4	#23765677	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765675	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765674	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765187	C₂	1	2	yes	maximal
8	#23765181	C₂	1	1	yes	maximal, characteristic
9	#366×D₇₉	C₂	1	1	yes	maximal, characteristic
10	#23764995	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099193	—	2	2	no	—
13	C₂×#14579100	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
15	#19098973	—	2	2	no	—
16	#19098970	—	2	2	no	—
17	#19098968	C₂²	1	2	yes	—
18	#19098967	C₂²	1	2	yes	—
19	#19098964	C₂²	1	2	yes	—
20	#19098602	—	2	2	no	—
21	#19098600	C₂²	1	2	yes	—
22	#107×D₇₉	C₂²	1	2	yes	—
23	#19098517	C₂²	1	1	yes	characteristic
24	#19098390	C₂²	1	2	yes	—
25	#19098390	C₂²	1	2	yes	—
26	#19098389	C₂²	1	1	yes	characteristic
27	#19098389	C₂²	1	1	yes	characteristic
28	#19098389	C₂²	1	2	yes	—
29	#19098388	C₂²	1	2	yes	—
30	#19098388	C₂²	1	2	yes	—
31	#19098339	C₂²	1	1	yes	characteristic
32	#19098338	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
34	#499×C₇₉	C₂²	1	2	yes	—
35	#367×C₇₉	C₂²	1	2	yes	—
36	#366×C₇₉	C₂²	1	1	yes	characteristic
37	#366×C₇₉	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
39	C₂×#11929607	D₄	1	1	yes	characteristic
40	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
41	#14579161	—	2	2	no	—
42	#14579160	—	2	2	no	—
43	#14579160	—	4	2	no	—
44	#14579152	—	2	2	no	—
45	#14579151	—	4	2	no	—
46	#14579141	—	4	2	no	—
47	#14579139	—	4	2	no	—
48	#14579119	—	2	2	no	—
49	#14579118	D₄	1	1	yes	characteristic
50	C₄⋊C₄×D₇₉	—	2	2	no	—
51	#14579100	Q₈	1	4	yes	complemented
52	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₁₆×D₇₉	Q₈	1	4	yes	complemented
55	#14579095	—	2	2	no	—
56	#14579095	C₂³	1	2	yes	—
57	#14579095	C₂³	1	2	yes	—
58	#14579094	—	2	2	no	—
59	#14579094	—	2	2	no	—
60	#14579092	—	2	2	no	—
61	#14579092	—	2	2	no	—
62	#14579072	D₄	1	1	yes	characteristic
63	#14579071	D₄	1	1	yes	characteristic
64	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
65	#125×C₇₉	—	2	2	no	—
66	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
68	#107×C₇₉	C₂³	1	2	yes	—
69	#107×C₇₉	C₂³	1	2	yes	—
70	#106×C₇₉	—	2	2	no	—
71	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₅₈	—	4	2	no	—
73	C₂×Dic₁₅₈	—	4	2	no	—
74	C₂×C₄×D₇₉	—	4	2	no	—
75	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
76	#11929607	Q₁₆	1	4	yes	—
77	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
79	C₈×D₇₉	C₂×Q₈	1	2	yes	—
80	C₈×D₇₉	C₂×Q₈	1	2	yes	—
81	C₄⋊Dic₇₉	—	2	2	no	—
82	C₄⋊Dic₇₉	—	2	2	no	—
83	C₄⋊Dic₇₉	—	4	2	no	—
84	C₄⋊Dic₇₉	—	4	2	no	—
85	Dic₇₉⋊C₄	—	4	2	no	—
86	Dic₇₉⋊C₄	—	4	2	no	—
87	Dic₇₉⋊C₄	—	4	2	no	—
88	Dic₇₉⋊C₄	D₈	1	1	yes	characteristic
89	Dic₇₉⋊C₄	D₈	1	1	yes	characteristic
90	(C₂×D₇₉)⋊C₄	—	4	2	no	—
91	(C₂×D₇₉)⋊C₄	D₈	1	1	yes	characteristic
92	(C₂×D₇₉)⋊C₄	D₈	1	1	yes	characteristic
93	C₄×Dic₇₉	—	4	2	no	—
94	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
95	C₇₉⋊C₁₆	C₂×Q₈	1	2	yes	—
96	C₇₉⋊C₁₆	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₇₉	—	2	2	no	—
100	C₄⋊C₄×C₇₉	—	2	2	no	—
101	C₄⋊C₄×C₇₉	—	4	2	no	—
102	C₄⋊C₄×C₇₉	—	4	2	no	—
103	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₁₆×C₇₉	C₂×Q₈	1	2	yes	—
105	C₁₆×C₇₉	C₂×Q₈	1	2	yes	—
106	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₇₉	—	4	2	no	—
108	C₂×C₄×C₇₉	—	4	2	no	—
109	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₇₉	—	4	2	no	—
113	C₂×Dic₇₉	—	4	2	no	—
114	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
115	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
116	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
117	C₄×D₇₉	—	16	2	no	—
118	C₄×D₇₉	C₄⋊Q₈	1	2	yes	—
119	C₄×D₇₉	C₂×Q₁₆	1	2	yes	—
120	Dic₁₅₈	—	16	2	no	—
121	Dic₁₅₈	—	16	2	no	—
122	C₈×C₇₉	C₂×Q₁₆	1	2	yes	—
123	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
125	C₇₉⋊C₈	C₄⋊Q₈	1	2	yes	—
126	C₇₉⋊C₈	C₂×Q₁₆	1	2	yes	—
127	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
128	C₂×D₇₉	—	2	1	no	—
129	C₂×D₇₉	—	2	1	no	—
130	C₂×D₇₉	#500	1	2	yes	—
131	C₄×C₇₉	—	2	1	no	—
132	C₄×C₇₉	—	2	1	no	—
133	C₄×C₇₉	—	8	2	no	—
134	C₄×C₇₉	—	8	2	no	—
135	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
137	Dic₇₉	—	2	1	no	—
138	Dic₇₉	—	2	1	no	—
139	Dic₇₉	—	8	2	no	—
140	Dic₇₉	—	8	2	no	—
141	Dic₇₉	#500	1	2	yes	—
142	#33729	—	79	1	no	maximal, Sylow
143	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
144	C₂×C₇₉	#3415	1	1	yes	characteristic
145	C₂×C₇₉	#3417	1	1	yes	characteristic
146	D₇₉	—	2	2	no	—
147	#3164	—	79	2	no	—
148	#2255	—	79	2	no	—
149	#2255	—	79	2	no	—
150	#2253	—	79	1	no	—
151	#2253	D₇₉	1	1	yes	characteristic, complemented
152	#2252	—	79	1	no	—
153	#2252	—	79	1	no	—
154	#2159	—	79	2	no	—
155	#2158	—	79	1	no	—
156	C₂×#366	—	79	1	no	—
157	#2108	—	79	1	no	—
158	C₇₉	#33729	1	1	yes	characteristic, complemented, Sylow
159	#544	—	158	2	no	—
160	C₂×M₃₂	—	79	1	no	—
161	C₂²×C₁₆	—	79	1	no	—
162	#501	—	158	2	no	—
163	#501	—	158	2	no	—
164	#500	—	79	2	no	—
165	#500	—	79	2	no	—
166	#499	—	79	2	no	—
167	#499	C₂×D₇₉	1	2	yes	—
168	#428	—	158	2	no	—
169	#427	—	79	2	no	—
170	C₂×#107	—	79	2	no	—
171	#405	—	79	1	no	—
172	#367	—	79	2	no	—
173	#367	—	79	2	no	—
174	#367	C₂×D₇₉	1	2	yes	—
175	#366	—	79	1	no	—
176	#366	—	79	1	no	—
177	#366	—	79	2	no	—
178	#366	—	79	2	no	—
179	#366	—	79	2	no	—
180	#366	C₂×D₇₉	1	1	yes	characteristic
181	#366	C₂×D₇₉	1	1	yes	characteristic
182	#346	—	79	1	no	—
183	#346	—	79	1	no	—
184	C₄×C₁₆	—	79	1	no	—
185	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
186	C₂×M₁₆	—	79	1	no	—
187	C₂²×C₈	—	79	1	no	—
188	C₄⋊Q₈	—	316	2	no	—
189	C₄⋊Q₈	—	158	2	no	—
190	C₄⋊Q₈	—	158	2	no	—
191	#125	—	2	2	no	—
192	#125	—	316	2	no	—
193	#125	—	158	2	no	—
194	C₂²⋊Q₈	—	316	2	no	—
195	C₂²⋊Q₈	—	316	2	no	—
196	#117	—	79	1	no	—
197	#117	—	158	2	no	—
198	C₂×C₄⋊C₄	—	158	2	no	—
199	M₃₂	—	79	4	no	—
200	C₂×C₁₆	—	79	1	no	—
201	C₂×C₁₆	—	79	1	no	—
202	C₂×C₁₆	—	79	4	no	—
203	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
204	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
205	#107	—	79	2	no	—
206	#107	—	79	2	no	—
207	#107	—	158	2	no	—
208	#107	—	158	2	no	—
209	#107	—	158	2	no	—
210	#107	C₂²×D₇₉	1	2	yes	—
211	#107	C₂²×D₇₉	1	2	yes	—
212	#106	—	2	2	no	—
213	#106	—	158	2	no	—
214	#106	—	158	2	no	—
215	#97	—	79	1	no	—
216	#97	—	79	1	no	—
217	C₄×C₈	—	79	1	no	—
218	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
219	C₂×Q₈	—	316	2	no	—
220	C₂×Q₈	—	316	2	no	—
221	C₂²×C₄	—	316	2	no	—
222	C₂²×C₄	—	79	1	no	—
223	M₁₆	—	79	4	no	—
224	C₂×C₈	—	79	1	no	—
225	C₂×C₈	—	79	1	no	—
226	C₂×C₈	—	79	2	no	—
227	C₂×C₈	—	79	2	no	—
228	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	316	2	no	—
235	C₄⋊C₄	—	316	2	no	—
236	C₄⋊C₄	—	316	2	no	—
237	C₄⋊C₄	—	316	2	no	—
238	C₄⋊C₄	—	316	2	no	—
239	C₄⋊C₄	—	79	1	no	—
240	C₄⋊C₄	—	79	1	no	—
241	C₄⋊C₄	—	158	2	no	—
242	C₄⋊C₄	—	158	2	no	—
243	C₂²⋊C₄	—	316	2	no	—
244	C₂²⋊C₄	—	79	1	no	—
245	C₂²⋊C₄	—	79	1	no	—
246	C₄²	—	316	2	no	—
247	C₄²	—	79	1	no	—
248	C₄²	D₄×D₇₉	1	1	yes	characteristic
249	C₁₆	—	79	2	no	—
250	C₁₆	—	79	2	no	—
251	C₁₆	Q₈×D₇₉	1	2	yes	—
252	C₁₆	Q₈×D₇₉	1	2	yes	—
253	C₂³	—	79	1	no	—
254	Q₈	—	1264	2	no	—
255	Q₈	—	1264	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	316	2	no	—
259	C₂×C₄	—	316	2	no	—
260	C₂×C₄	—	79	1	no	—
261	C₂×C₄	—	79	2	no	—
262	C₂×C₄	—	79	2	no	—
263	C₂×C₄	—	79	1	no	—
264	C₂×C₄	—	79	1	no	—
265	C₂×C₄	—	1264	2	no	—
266	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
267	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
269	C₈	—	79	2	no	—
270	C₈	—	79	2	no	—
271	C₈	Q₁₆×D₇₉	1	2	yes	—
272	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
274	C₂²	—	79	2	no	—
275	C₂²	—	158	1	no	—
276	C₂²	—	158	1	no	—
277	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	79	2	no	—
283	C₄	—	632	2	no	—
284	C₄	—	632	2	no	—
285	C₄	—	158	1	no	—
286	C₄	—	158	1	no	—
287	C₄	C₄⋊Q₈×D₇₉	1	1	yes	characteristic
288	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
289	C₂	—	158	2	no	—
290	C₂	#500×D₇₉	1	1	yes	characteristic
291	C₂	#23776538	1	1	yes	characteristic
292	C₂	#23776550	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2253	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂²×D₇₉	C₂×D₈
hypercenter	#2253	D₇₉
hypocenter	C₇₉	#33729
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #23765687	C₂	semidirect
3. #23765685	C₂	semidirect
4. #23765677	C₂	semidirect
5. #23765675	C₂	semidirect
6. #23765674	C₂	semidirect
11. C₇₉×#2253	C₂	semidirect
51. #14579100	Q₈	semidirect
54. C₁₆×D₇₉	Q₈	semidirect
151. #2253	D₇₉	semidirect
158. C₇₉	#33729	semidirect

Maximal subgroups

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

subgroup	index
#23773937	2
#23765687	2
#23765685	2
#23765677	2
#23765675	2
#23765674	2
#23765187	2
#23765181	2
#366×D₇₉	2
#23764995	2
C₇₉×#2253	2
#33729	79