Subgroups — #34196542

Order 20224.

634 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579095	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098331	C₂	1	2	yes	maximal
3	C₂×#19098328	C₂	1	1	yes	maximal, characteristic
4	#23763121	C₂	1	2	yes	maximal
5	C₇₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₇₉	C₂²	1	2	yes	—
8	C₂×#14579095	—	2	4	no	—
9	C₂×#14579095	—	2	2	no	—
10	#19098331	—	2	4	no	—
11	#19098328	C₄	1	4	yes	complemented
12	#19098325	—	2	4	no	—
13	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
16	#336×C₇₉	—	2	2	no	—
17	C₂³×Dic₇₉	—	2	2	no	—
18	C₂×#11929608	—	2	2	no	—
19	C₂×#11929608	—	2	2	no	—
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929608	—	2	2	no	—
23	C₂×#11929608	C₂×C₄	1	4	yes	—
24	C₂²×C₇₉⋊C₈	—	2	2	no	—
25	C₂×C₄⋊Dic₇₉	—	2	2	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	2	2	no	—
28	C₂×C₄⋊Dic₇₉	—	2	2	no	—
29	C₂×C₄⋊Dic₇₉	—	2	2	no	—
30	C₂×C₄⋊Dic₇₉	D₄	1	4	yes	—
31	#14579095	—	2	8	no	—
32	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₇₉	—	2	1	no	—
34	C₂×M₁₆×C₇₉	—	2	2	no	—
35	C₂×M₁₆×C₇₉	—	2	2	no	—
36	C₂×M₁₆×C₇₉	Q₈	1	2	yes	—
37	C₂²×C₈×C₇₉	—	2	2	no	—
38	C₂²×C₈×C₇₉	—	2	2	no	—
39	C₂²×C₈×C₇₉	—	2	1	no	—
40	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₇₉	D₄	1	2	yes	—
42	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
43	#117×C₇₉	—	2	2	no	—
44	#117×C₇₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
47	C₂×C₄²×C₇₉	—	2	1	no	—
48	C₂²×Dic₇₉	—	2	4	no	—
49	C₂²×Dic₇₉	—	4	2	no	—
50	C₂²×Dic₇₉	—	4	2	no	—
51	C₂²×Dic₇₉	—	8	4	no	—
52	#11929608	—	2	4	no	—
53	#11929608	—	2	4	no	—
54	#11929608	—	4	2	no	—
55	#11929608	—	4	2	no	—
56	C₂×C₇₉⋊C₈	—	2	4	no	—
57	C₂×C₇₉⋊C₈	—	4	2	no	—
58	C₂×C₇₉⋊C₈	—	4	2	no	—
59	C₄⋊Dic₇₉	—	2	4	no	—
60	C₄⋊Dic₇₉	—	2	4	no	—
61	C₄⋊Dic₇₉	—	4	2	no	—
62	C₄⋊Dic₇₉	—	4	2	no	—
63	C₂⁴×C₇₉	#41	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	—	2	2	no	—
65	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	#41	1	2	yes	—
72	C₂²×C₄×C₇₉	#41	1	2	yes	—
73	C₂²×C₄×C₇₉	#41	1	2	yes	—
74	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
75	M₁₆×C₇₉	—	2	4	no	—
76	M₁₆×C₇₉	—	4	2	no	—
77	C₂×C₈×C₇₉	—	2	4	no	—
78	C₂×C₈×C₇₉	—	2	2	no	—
79	C₂×C₈×C₇₉	—	2	4	no	—
80	C₂×C₈×C₇₉	—	2	2	no	—
81	C₂×C₈×C₇₉	—	2	2	no	—
82	C₂×C₈×C₇₉	—	4	1	no	—
83	C₂×C₈×C₇₉	—	4	1	no	—
84	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
85	C₄⋊C₄×C₇₉	—	2	4	no	—
86	C₂²⋊C₄×C₇₉	—	4	2	no	—
87	C₄²×C₇₉	—	4	1	no	—
88	C₄²×C₇₉	—	4	1	no	—
89	C₂³×C₇₉	—	2	2	no	—
90	C₂³×C₇₉	—	2	2	no	—
91	C₂³×C₇₉	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₇₉	#121	1	1	yes	characteristic
93	C₂³×C₇₉	#123	1	2	yes	—
94	C₂³×C₇₉	#125	1	1	yes	characteristic
95	C₂³×C₇₉	#126	1	2	yes	—
96	C₂×C₄×C₇₉	—	4	2	no	—
97	C₂×C₄×C₇₉	—	4	2	no	—
98	C₂×C₄×C₇₉	—	4	2	no	—
99	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#117	1	2	yes	—
101	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
104	C₂×C₄×C₇₉	#121	1	2	yes	—
105	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#123	1	2	yes	—
110	C₂×C₄×C₇₉	#124	1	2	yes	—
111	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#125	1	2	yes	—
114	C₂×C₄×C₇₉	#125	1	2	yes	—
115	C₂×C₄×C₇₉	#126	1	2	yes	—
116	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Dic₇₉	—	4	4	no	—
118	C₂×Dic₇₉	—	8	4	no	—
119	C₂×Dic₇₉	—	8	4	no	—
120	C₂×Dic₇₉	—	8	4	no	—
121	C₈×C₇₉	—	2	4	no	—
122	C₈×C₇₉	—	4	2	no	—
123	C₇₉⋊C₈	—	4	4	no	—
124	C₂²×C₇₉	—	2	4	no	—
125	C₂²×C₇₉	—	2	2	no	—
126	C₂²×C₇₉	—	2	2	no	—
127	C₂²×C₇₉	—	2	2	no	—
128	C₂²×C₇₉	—	2	2	no	—
129	C₂²×C₇₉	#387	1	2	yes	—
130	C₂²×C₇₉	#389	1	1	yes	characteristic
131	C₂²×C₇₉	#394	1	1	yes	characteristic
132	C₂²×C₇₉	#397	1	1	yes	characteristic
133	C₂²×C₇₉	#400	1	2	yes	—
134	C₂²×C₇₉	#472	1	2	yes	—
135	C₂²×C₇₉	#473	1	2	yes	—
136	C₄×C₇₉	—	8	2	no	—
137	C₄×C₇₉	#382	1	1	yes	characteristic
138	C₄×C₇₉	#385	1	1	yes	characteristic
139	C₄×C₇₉	#388	1	2	yes	—
140	C₄×C₇₉	#393	1	1	yes	characteristic
141	C₄×C₇₉	#398	1	2	yes	—
142	C₄×C₇₉	#400	1	1	yes	characteristic
143	Dic₇₉	—	8	4	no	—
144	#11270	—	79	1	no	maximal, Sylow
145	C₂×C₇₉	—	2	2	no	—
146	C₂×C₇₉	—	2	2	no	—
147	C₂×C₇₉	#1442	1	1	yes	characteristic
148	C₂×C₇₉	#1952	1	2	yes	—
149	C₂×C₇₉	#2045	1	1	yes	characteristic
150	C₂×C₇₉	#2047	1	1	yes	characteristic
151	C₂×C₇₉	#2078	1	2	yes	—
152	C₂²×#107	—	79	1	no	—
153	C₂×#343	—	79	2	no	—
154	C₂×#341	—	79	1	no	—
155	#1736	—	79	2	no	—
156	#1729	D₇₉	1	1	yes	characteristic
157	C₇₉	#11270	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	79	2	no	—
159	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
160	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
161	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
162	C₂²×C₄⋊C₄	—	79	2	no	—
163	C₂×#107	—	158	4	no	—
164	C₂×#107	—	158	2	no	—
165	#343	—	158	4	no	—
166	#341	—	79	4	no	—
167	#340	—	158	4	no	—
168	#336	—	2	2	no	—
169	C₂³×C₄	—	158	2	no	—
170	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	79	4	no	—
175	C₂×M₁₆	—	158	2	no	—
176	C₂×M₁₆	—	158	2	no	—
177	C₂×M₁₆	—	158	2	no	—
178	C₂×M₁₆	—	158	2	no	—
179	C₂×M₁₆	—	158	2	no	—
180	C₂×M₁₆	Dic₁₅₈	1	2	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	158	2	no	—
185	C₂²×C₈	Dic₁₅₈	1	2	yes	—
186	C₂²×C₈	C₄×D₇₉	1	2	yes	—
187	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	C₂×C₄⋊C₄	—	79	4	no	—
191	C₂×C₄⋊C₄	—	158	2	no	—
192	C₂×C₄⋊C₄	—	158	2	no	—
193	C₂×C₄⋊C₄	—	158	2	no	—
194	C₂×C₄⋊C₄	—	158	2	no	—
195	C₂×C₄⋊C₄	—	158	2	no	—
196	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	C₂×C₄²	—	2	1	no	—
199	#107	—	158	8	no	—
200	C₂⁴	#11929624	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	316	2	no	—
203	C₂²×C₄	—	316	2	no	—
204	C₂²×C₄	—	632	4	no	—
205	C₂²×C₄	—	158	4	no	—
206	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₇₉	1	2	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
210	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
211	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
212	C₂²×C₄	#11929623	1	2	yes	—
213	C₂²×C₄	#11929623	1	2	yes	—
214	C₂²×C₄	#11929624	1	1	yes	characteristic
215	C₂²×C₄	#11929625	1	2	yes	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	316	2	no	—
219	M₁₆	—	316	2	no	—
220	M₁₆	—	158	4	no	—
221	M₁₆	—	158	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	316	2	no	—
230	C₂×C₈	—	316	2	no	—
231	C₂×C₈	—	158	4	no	—
232	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	316	2	no	—
235	C₄⋊C₄	—	316	2	no	—
236	C₄⋊C₄	—	158	4	no	—
237	C₄⋊C₄	—	158	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₄×Dic₁₅₈	1	1	yes	characteristic
244	C₂³	#14579135	1	1	yes	characteristic
245	C₂³	#14579146	1	2	yes	—
246	C₂³	#14579153	1	1	yes	characteristic
247	C₂³	#14579154	1	2	yes	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	316	4	no	—
252	C₂×C₄	—	632	4	no	—
253	C₂×C₄	—	632	4	no	—
254	C₂×C₄	—	632	4	no	—
255	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
256	C₂×C₄	#14579119	1	2	yes	—
257	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
258	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
259	C₂×C₄	#14579129	1	1	yes	characteristic
260	C₂×C₄	#14579136	1	2	yes	—
261	C₂×C₄	#14579138	1	2	yes	—
262	C₂×C₄	#14579138	1	2	yes	—
263	C₂×C₄	#14579140	1	1	yes	characteristic
264	C₂×C₄	#14579142	1	1	yes	characteristic
265	C₂×C₄	#14579145	1	2	yes	—
266	C₂×C₄	#14579147	1	2	yes	—
267	C₂×C₄	#14579148	1	1	yes	characteristic
268	C₂×C₄	#14579150	1	1	yes	characteristic
269	C₂×C₄	#14579152	1	2	yes	—
270	C₂×C₄	#14579152	1	2	yes	—
271	C₂×C₄	#14579156	1	2	yes	—
272	C₂×C₄	#14579161	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	4	2	no	—
275	C₈	—	316	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	#19098450	1	2	yes	—
282	C₂²	#19098461	1	1	yes	characteristic
283	C₂²	#19098480	1	1	yes	characteristic
284	C₂²	#19098488	1	1	yes	characteristic
285	C₂²	#19098505	1	2	yes	—
286	C₂²	#19098833	1	2	yes	—
287	C₂²	#19098838	1	2	yes	—
288	C₄	—	8	2	no	—
289	C₄	—	632	4	no	—
290	C₄	#19098428	1	1	yes	characteristic
291	C₄	#19098435	1	1	yes	characteristic
292	C₄	#19098455	1	2	yes	—
293	C₄	#19098475	1	1	yes	characteristic
294	C₄	#19098495	1	2	yes	—
295	C₄	#19098502	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	#23761656	1	1	yes	characteristic
299	C₂	#23764185	1	2	yes	—
300	C₂	#23764656	1	1	yes	characteristic
301	C₂	#23764668	1	1	yes	characteristic
302	C₂	#23764840	1	2	yes	—

	subgroup	G/H
center	C₂³	#14579135
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1729	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂⁴×C₇₉	#41
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1729	D₇₉
hypocenter	C₇₉	#11270
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
11. #19098328	C₄	semidirect
157. C₇₉	#11270	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579095	2
C₂×#19098331	2
C₂×#19098328	2
#23763121	2
C₇₉×#1729	2
#11270	79