Subgroups — #34443795

Order 20224.

482 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	#23773913	C₂	1	1	yes	maximal, characteristic
2	#499×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23765685	C₂	1	2	yes	maximal, complemented
4	#23765684	C₂	1	2	yes	maximal, complemented
5	#23765679	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765676	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765676	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765187	C₂	1	2	yes	maximal
9	#23765185	C₂	1	2	yes	maximal
10	C₄×#14579100	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099119	—	2	1	no	—
13	#125×D₇₉	—	2	1	no	—
14	C₂×#14579100	C₂²	1	1	yes	characteristic
15	C₂×#14579100	C₂²	1	1	yes	characteristic
16	#19098967	C₂²	1	2	yes	—
17	#19098965	C₂²	1	1	yes	characteristic
18	#19098965	C₂²	1	1	yes	characteristic
19	#19098964	C₂²	1	2	yes	—
20	#19098963	—	2	1	no	—
21	#19098962	—	2	1	no	—
22	#19098962	—	2	1	no	—
23	#19098960	—	2	1	no	—
24	#19098600	C₂²	1	2	yes	—
25	#19098598	—	2	1	no	—
26	#107×D₇₉	C₂²	1	2	yes	—
27	#106×D₇₉	—	2	1	no	—
28	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
29	#19098391	C₂²	1	2	yes	—
30	#19098390	C₂²	1	2	yes	—
31	#19098390	C₂²	1	2	yes	—
32	#19098389	C₂²	1	2	yes	—
33	#19098388	C₂²	1	2	yes	—
34	#19098388	C₂²	1	2	yes	—
35	#19098338	C₂²	1	1	yes	characteristic
36	#19098338	C₂²	1	1	yes	characteristic
37	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
38	#499×C₇₉	C₂²	1	1	yes	characteristic
39	#499×C₇₉	C₂²	1	1	yes	characteristic
40	#367×C₇₉	C₂²	1	2	yes	—
41	#366×C₇₉	C₂²	1	2	yes	—
42	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
43	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
44	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
45	#14579160	—	2	2	no	—
46	#14579153	—	2	1	no	—
47	#14579153	—	2	1	no	—
48	#14579152	—	2	2	no	—
49	#14579151	—	4	2	no	—
50	#14579139	—	4	2	no	—
51	#14579126	—	4	2	no	—
52	#14579119	—	2	2	no	—
53	C₄⋊C₄×D₇₉	—	2	2	no	—
54	C₄⋊C₄×D₇₉	—	4	2	no	—
55	C₄²×D₇₉	D₄	1	1	yes	characteristic
56	#14579100	Q₈	1	4	yes	complemented
57	#14579100	Q₈	1	4	yes	complemented
58	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
60	#14579095	C₂³	1	2	yes	—
61	#14579095	C₂³	1	2	yes	—
62	#14579094	—	2	2	no	—
63	#14579094	—	2	2	no	—
64	#14579093	—	2	1	no	—
65	#14579093	—	2	1	no	—
66	#14579092	—	2	2	no	—
67	#14579092	—	2	2	no	—
68	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
69	C₈×Dic₇₉	D₄	1	1	yes	characteristic
70	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
71	#125×C₇₉	—	2	1	no	—
72	#125×C₇₉	—	2	1	no	—
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	#107×C₇₉	C₂³	1	2	yes	—
77	#106×C₇₉	—	2	1	no	—
78	#106×C₇₉	—	2	1	no	—
79	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₅₈	—	4	2	no	—
81	C₂×C₄×D₇₉	—	4	2	no	—
82	C₂×C₄×D₇₉	D₈	1	2	yes	—
83	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
86	C₈×D₇₉	Q₁₆	1	4	yes	—
87	C₈×D₇₉	C₂×Q₈	1	2	yes	—
88	C₈×D₇₉	C₂×Q₈	1	2	yes	—
89	C₄⋊Dic₇₉	—	2	2	no	—
90	C₄⋊Dic₇₉	—	2	2	no	—
91	C₄⋊Dic₇₉	—	4	2	no	—
92	Dic₇₉⋊C₄	—	4	2	no	—
93	Dic₇₉⋊C₄	—	4	2	no	—
94	Dic₇₉⋊C₄	—	4	2	no	—
95	Dic₇₉⋊C₄	—	4	2	no	—
96	Dic₇₉⋊C₄	—	4	2	no	—
97	(C₂×D₇₉)⋊C₄	—	4	2	no	—
98	C₄×Dic₇₉	—	4	2	no	—
99	C₄×Dic₇₉	D₈	1	2	yes	—
100	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
101	C₇₉⋊C₁₆	C₂×Q₈	1	2	yes	—
102	C₇₉⋊C₁₆	C₂×Q₈	1	2	yes	—
103	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₇₉	—	2	2	no	—
106	C₄⋊C₄×C₇₉	—	2	2	no	—
107	C₄⋊C₄×C₇₉	—	4	2	no	—
108	C₄⋊C₄×C₇₉	—	4	2	no	—
109	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₇₉	C₂×Q₈	1	2	yes	—
111	C₁₆×C₇₉	C₂×Q₈	1	2	yes	—
112	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₇₉	—	4	2	no	—
118	C₂×Dic₇₉	—	4	2	no	—
119	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
120	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
121	C₄×D₇₉	—	4	2	no	—
122	C₄×D₇₉	—	8	4	no	—
123	C₄×D₇₉	C₄⋊Q₈	1	2	yes	—
124	C₄×D₇₉	C₂×Q₁₆	1	2	yes	—
125	Dic₁₅₈	—	8	4	no	—
126	C₈×C₇₉	C₂×Q₁₆	1	2	yes	—
127	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
128	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
129	C₇₉⋊C₈	C₄⋊Q₈	1	2	yes	—
130	C₇₉⋊C₈	C₂×Q₁₆	1	2	yes	—
131	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
132	C₂×D₇₉	—	2	2	no	—
133	C₂×D₇₉	#500	1	2	yes	—
134	C₄×C₇₉	—	2	2	no	—
135	C₄×C₇₉	—	8	2	no	—
136	C₄×C₇₉	—	8	2	no	—
137	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
139	Dic₇₉	—	2	2	no	—
140	Dic₇₉	—	8	2	no	—
141	Dic₇₉	—	8	2	no	—
142	Dic₇₉	#500	1	2	yes	—
143	#33732	—	79	1	no	maximal, Sylow
144	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
145	C₂×C₇₉	#3414	1	2	yes	—
146	D₇₉	—	2	2	no	—
147	#3160	—	79	1	no	—
148	C₂×#499	—	79	1	no	—
149	#2255	—	79	2	no	—
150	#2255	—	79	2	no	—
151	#2253	—	79	1	no	—
152	#2253	—	79	1	no	—
153	#2253	—	79	1	no	—
154	#2253	D₇₉	1	1	yes	characteristic, complemented
155	#2159	—	79	2	no	—
156	#2159	—	79	2	no	—
157	C₄×M₃₂	—	79	1	no	—
158	C₇₉	#33732	1	1	yes	characteristic, complemented, Sylow
159	#533	—	158	1	no	—
160	C₂×#125	—	158	1	no	—
161	C₂×M₃₂	—	79	1	no	—
162	C₂×M₃₂	—	79	1	no	—
163	#500	—	79	2	no	—
164	#499	—	79	2	no	—
165	#499	—	79	1	no	—
166	#499	—	79	1	no	—
167	#499	—	158	1	no	—
168	#499	—	158	1	no	—
169	#499	—	158	1	no	—
170	#499	C₂×D₇₉	1	1	yes	characteristic
171	#499	C₂×D₇₉	1	1	yes	characteristic
172	#498	—	158	1	no	—
173	#427	—	79	2	no	—
174	#427	—	158	1	no	—
175	C₂×#107	—	79	2	no	—
176	C₂×#106	—	158	1	no	—
177	C₂×C₄×C₈	—	79	1	no	—
178	#367	—	79	2	no	—
179	#367	—	79	2	no	—
180	#367	—	79	2	no	—
181	#367	C₂×D₇₉	1	2	yes	—
182	#366	—	79	2	no	—
183	#366	—	79	2	no	—
184	#366	—	79	2	no	—
185	#366	C₂×D₇₉	1	2	yes	—
186	#346	—	79	1	no	—
187	#346	—	79	1	no	—
188	C₄×C₁₆	—	79	1	no	—
189	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
190	C₂²×C₈	—	79	1	no	—
191	C₂²×C₈	—	79	1	no	—
192	C₄⋊Q₈	—	158	2	no	—
193	#125	—	2	1	no	—
194	#125	—	2	1	no	—
195	#125	—	316	2	no	—
196	#125	—	158	2	no	—
197	#125	—	158	1	no	—
198	#125	—	158	1	no	—
199	C₂²⋊Q₈	—	316	2	no	—
200	C₄×Q₈	—	316	2	no	—
201	#117	—	158	2	no	—
202	C₂×C₄⋊C₄	—	316	2	no	—
203	C₂×C₄⋊C₄	—	158	2	no	—
204	C₂×C₄²	—	79	1	no	—
205	M₃₂	—	79	4	no	—
206	M₃₂	—	79	4	no	—
207	C₂×C₁₆	—	79	1	no	—
208	C₂×C₁₆	—	79	1	no	—
209	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
210	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
211	#107	—	79	2	no	—
212	#107	—	79	2	no	—
213	#107	—	158	2	no	—
214	#107	—	158	2	no	—
215	#107	C₂²×D₇₉	1	2	yes	—
216	#107	C₂²×D₇₉	1	2	yes	—
217	#106	—	2	1	no	—
218	#106	—	2	1	no	—
219	#106	—	158	2	no	—
220	#106	—	158	2	no	—
221	#106	—	158	1	no	—
222	#106	—	158	1	no	—
223	C₄×C₈	—	79	1	no	—
224	C₄×C₈	—	79	1	no	—
225	C₄×C₈	—	79	1	no	—
226	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
227	C₂×Q₈	—	316	2	no	—
228	C₂²×C₄	—	316	2	no	—
229	C₂²×C₄	—	79	2	no	—
230	C₂²×C₄	—	79	1	no	—
231	C₂×C₈	—	79	1	no	—
232	C₂×C₈	—	79	1	no	—
233	C₂×C₈	—	79	2	no	—
234	C₂×C₈	—	79	2	no	—
235	C₂×C₈	—	79	4	no	—
236	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	316	2	no	—
243	C₄⋊C₄	—	316	2	no	—
244	C₄⋊C₄	—	316	2	no	—
245	C₄⋊C₄	—	316	2	no	—
246	C₄⋊C₄	—	316	2	no	—
247	C₄⋊C₄	—	316	2	no	—
248	C₄⋊C₄	—	158	2	no	—
249	C₄⋊C₄	—	158	2	no	—
250	C₂²⋊C₄	—	316	2	no	—
251	C₄²	—	316	2	no	—
252	C₄²	—	79	2	no	—
253	C₄²	—	79	1	no	—
254	C₄²	D₄×D₇₉	1	1	yes	characteristic
255	C₁₆	—	79	2	no	—
256	C₁₆	—	79	2	no	—
257	C₁₆	Q₈×D₇₉	1	2	yes	—
258	C₁₆	Q₈×D₇₉	1	2	yes	—
259	C₂³	—	79	1	no	—
260	Q₈	—	632	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	316	2	no	—
264	C₂×C₄	—	316	2	no	—
265	C₂×C₄	—	316	2	no	—
266	C₂×C₄	—	79	1	no	—
267	C₂×C₄	—	79	2	no	—
268	C₂×C₄	—	79	2	no	—
269	C₂×C₄	—	79	2	no	—
270	C₂×C₄	—	632	4	no	—
271	C₂×C₄	D₈×D₇₉	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
273	C₈	—	79	2	no	—
274	C₈	—	79	2	no	—
275	C₈	Q₁₆×D₇₉	1	2	yes	—
276	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
277	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
278	C₂²	—	79	2	no	—
279	C₂²	—	158	2	no	—
280	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	79	2	no	—
285	C₄	—	632	2	no	—
286	C₄	—	632	2	no	—
287	C₄	—	158	2	no	—
288	C₄	C₄⋊Q₈×D₇₉	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
290	C₂	—	158	2	no	—
291	C₂	#500×D₇₉	1	1	yes	characteristic
292	C₂	#23776533	1	2	yes	—

	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	#33732
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
3. #23765685	C₂	semidirect
4. #23765684	C₂	semidirect
5. #23765679	C₂	semidirect
6. #23765676	C₂	semidirect
7. #23765676	C₂	semidirect
11. C₇₉×#2253	C₂	semidirect
56. #14579100	Q₈	semidirect
57. #14579100	Q₈	semidirect
154. #2253	D₇₉	semidirect
158. C₇₉	#33732	semidirect

Maximal subgroups

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

subgroup	index
#23773913	2
#499×D₇₉	2
#23765685	2
#23765684	2
#23765679	2
#23765676	2
#23765187	2
#23765185	2
C₄×#14579100	2
C₇₉×#2253	2
#33732	79