Subgroups — #34196696

Order 20224.

622 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771990	C₂	1	1	yes	maximal, characteristic
2	#23763138	C₂	1	2	yes	maximal
3	C₂×#19098323	C₂	1	2	yes	maximal
4	C₂×C₇₉⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	2	yes	—
7	C₂×#14579120	C₂²	1	2	yes	—
8	#19098605	—	2	4	no	—
9	#19098601	—	2	2	no	—
10	#19098331	—	2	4	no	—
11	#19098323	—	2	4	no	—
12	C₇₉⋊#338	C₄	1	4	yes	complemented
13	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	#336×C₇₉	—	2	2	no	—
17	C₂×#11929608	—	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	C₂×C₄	1	4	yes	—
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	#14579120	—	2	2	no	—
25	#14579120	—	2	2	no	—
26	#14579120	—	2	2	no	—
27	#14579120	—	2	2	no	—
28	C₂×C₄⋊Dic₇₉	D₄	1	4	yes	—
29	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
30	C₂×C₄×Dic₇₉	—	2	2	no	—
31	#14579095	—	2	4	no	—
32	#14579093	—	2	4	no	—
33	C₂³×C₄×C₇₉	—	2	1	no	—
34	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₇₉	—	2	2	no	—
36	C₂×M₁₆×C₇₉	—	2	2	no	—
37	C₂×M₁₆×C₇₉	—	2	1	no	—
38	C₂×M₁₆×C₇₉	—	2	2	no	—
39	C₂×M₁₆×C₇₉	—	2	2	no	—
40	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₇₉	Q₈	1	2	yes	—
42	C₂×M₁₆×C₇₉	Q₈	1	2	yes	—
43	C₂²×C₈×C₇₉	—	2	1	no	—
44	C₂²×C₈×C₇₉	D₄	1	2	yes	—
45	C₂×C₄²×C₇₉	—	2	2	no	—
46	C₂×C₄²×C₇₉	—	2	2	no	—
47	C₂×C₄²×C₇₉	—	2	1	no	—
48	C₂×C₄²×C₇₉	D₄	1	2	yes	—
49	C₂²×Dic₇₉	—	2	4	no	—
50	#11929608	—	2	4	no	—
51	#11929608	—	2	4	no	—
52	#11929608	—	4	2	no	—
53	#11929608	—	4	2	no	—
54	C₂×C₇₉⋊C₈	—	2	4	no	—
55	C₂×C₇₉⋊C₈	—	4	2	no	—
56	C₂×C₇₉⋊C₈	—	4	2	no	—
57	C₄⋊Dic₇₉	—	2	4	no	—
58	C₄⋊Dic₇₉	—	2	4	no	—
59	C₂²⋊Dic₇₉	—	4	2	no	—
60	C₂²⋊Dic₇₉	—	4	2	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	—	4	4	no	—
66	C₂²×C₄×C₇₉	—	4	2	no	—
67	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	#41	1	2	yes	—
74	C₂²×C₄×C₇₉	#41	1	2	yes	—
75	C₂²×C₄×C₇₉	#41	1	2	yes	—
76	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
77	M₁₆×C₇₉	—	2	4	no	—
78	M₁₆×C₇₉	—	2	4	no	—
79	M₁₆×C₇₉	—	2	4	no	—
80	M₁₆×C₇₉	—	4	2	no	—
81	C₂×C₈×C₇₉	—	2	2	no	—
82	C₂×C₈×C₇₉	—	2	2	no	—
83	C₂×C₈×C₇₉	—	2	2	no	—
84	C₂×C₈×C₇₉	—	4	1	no	—
85	C₂×C₈×C₇₉	—	4	1	no	—
86	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
87	C₄²×C₇₉	—	2	4	no	—
88	C₄²×C₇₉	—	4	1	no	—
89	C₄²×C₇₉	—	4	1	no	—
90	C₂³×C₇₉	—	4	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	—	2	4	no	—
97	C₂×C₄×C₇₉	—	4	2	no	—
98	C₂×C₄×C₇₉	—	4	8	no	—
99	C₂×C₄×C₇₉	—	4	2	no	—
100	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#117	1	2	yes	—
102	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
105	C₂×C₄×C₇₉	#121	1	2	yes	—
106	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
110	C₂×C₄×C₇₉	#123	1	2	yes	—
111	C₂×C₄×C₇₉	#124	1	2	yes	—
112	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
114	C₂×C₄×C₇₉	#125	1	2	yes	—
115	C₂×C₄×C₇₉	#125	1	2	yes	—
116	C₂×C₄×C₇₉	#126	1	2	yes	—
117	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₇₉	—	4	4	no	—
119	C₂×Dic₇₉	—	8	4	no	—
120	C₈×C₇₉	—	2	4	no	—
121	C₈×C₇₉	—	4	2	no	—
122	C₇₉⋊C₈	—	4	4	no	—
123	C₂²×C₇₉	—	2	2	no	—
124	C₂²×C₇₉	—	2	4	no	—
125	C₂²×C₇₉	—	4	2	no	—
126	C₂²×C₇₉	—	4	1	no	—
127	C₂²×C₇₉	—	4	1	no	—
128	C₂²×C₇₉	#387	1	2	yes	—
129	C₂²×C₇₉	#389	1	1	yes	characteristic
130	C₂²×C₇₉	#394	1	1	yes	characteristic
131	C₂²×C₇₉	#397	1	1	yes	characteristic
132	C₂²×C₇₉	#400	1	2	yes	—
133	C₄×C₇₉	—	4	4	no	—
134	C₄×C₇₉	#382	1	1	yes	characteristic
135	C₄×C₇₉	#385	1	1	yes	characteristic
136	C₄×C₇₉	#388	1	2	yes	—
137	C₄×C₇₉	#393	1	1	yes	characteristic
138	C₄×C₇₉	#398	1	2	yes	—
139	C₄×C₇₉	#400	1	1	yes	characteristic
140	Dic₇₉	—	8	4	no	—
141	#11294	—	79	1	no	maximal, Sylow
142	C₂×C₇₉	—	2	1	no	—
143	C₂×C₇₉	—	2	2	no	—
144	C₂×C₇₉	—	2	1	no	—
145	C₂×C₇₉	—	2	2	no	—
146	C₂×C₇₉	#1442	1	1	yes	characteristic
147	C₂×C₇₉	#2050	1	2	yes	—
148	#2911	—	79	1	no	—
149	#1742	—	79	2	no	—
150	C₂×#339	—	79	2	no	—
151	C₂×#338	—	79	1	no	—
152	#1728	D₇₉	1	1	yes	characteristic
153	C₇₉	#11294	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	79	2	no	—
155	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
156	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
157	C₂×#117	—	79	2	no	—
158	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
159	#428	—	158	4	no	—
160	#427	—	158	2	no	—
161	#343	—	158	4	no	—
162	#339	—	158	4	no	—
163	#338	—	79	4	no	—
164	#336	—	2	2	no	—
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	79	4	no	—
173	C₂×M₁₆	—	158	2	no	—
174	C₂×M₁₆	—	158	2	no	—
175	C₂×M₁₆	—	158	2	no	—
176	C₂×M₁₆	—	158	2	no	—
177	C₂×M₁₆	—	158	2	no	—
178	C₂×M₁₆	Dic₁₅₈	1	2	yes	—
179	C₂×M₁₆	Dic₁₅₈	1	2	yes	—
180	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	158	2	no	—
183	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
184	#117	—	158	2	no	—
185	#117	—	158	2	no	—
186	#117	—	158	2	no	—
187	#117	—	158	2	no	—
188	C₂×C₄⋊C₄	—	79	4	no	—
189	C₂×C₂²⋊C₄	—	158	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	158	2	no	—
194	C₂×C₄²	D₃₁₆	1	2	yes	—
195	#107	—	158	4	no	—
196	#106	—	158	4	no	—
197	C₂⁴	#11929624	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	158	4	no	—
202	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
203	C₂²×C₄	D₄×D₇₉	1	2	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
208	C₂²×C₄	#11929623	1	2	yes	—
209	C₂²×C₄	#11929623	1	2	yes	—
210	C₂²×C₄	#11929624	1	1	yes	characteristic
211	C₂²×C₄	#11929625	1	2	yes	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	316	2	no	—
217	M₁₆	—	316	2	no	—
218	M₁₆	—	158	4	no	—
219	M₁₆	—	158	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	316	2	no	—
226	C₂×C₈	—	316	2	no	—
227	C₂×C₈	—	158	4	no	—
228	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
229	C₄⋊C₄	—	158	4	no	—
230	C₄⋊C₄	—	158	4	no	—
231	C₂²⋊C₄	—	316	2	no	—
232	C₂²⋊C₄	—	316	2	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	316	2	no	—
237	C₄²	—	316	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₄×Dic₁₅₈	1	1	yes	characteristic
240	C₂³	#14579135	1	1	yes	characteristic
241	C₂³	#14579146	1	2	yes	—
242	C₂³	#14579153	1	1	yes	characteristic
243	C₂³	#14579154	1	2	yes	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	316	4	no	—
249	C₂×C₄	—	632	4	no	—
250	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
251	C₂×C₄	#14579119	1	2	yes	—
252	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
253	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
254	C₂×C₄	#14579129	1	1	yes	characteristic
255	C₂×C₄	#14579136	1	2	yes	—
256	C₂×C₄	#14579138	1	2	yes	—
257	C₂×C₄	#14579138	1	2	yes	—
258	C₂×C₄	#14579140	1	1	yes	characteristic
259	C₂×C₄	#14579142	1	1	yes	characteristic
260	C₂×C₄	#14579145	1	2	yes	—
261	C₂×C₄	#14579147	1	2	yes	—
262	C₂×C₄	#14579148	1	1	yes	characteristic
263	C₂×C₄	#14579150	1	1	yes	characteristic
264	C₂×C₄	#14579152	1	2	yes	—
265	C₂×C₄	#14579152	1	2	yes	—
266	C₂×C₄	#14579156	1	2	yes	—
267	C₂×C₄	#14579161	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	2	no	—
270	C₈	—	316	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#19098450	1	2	yes	—
277	C₂²	#19098461	1	1	yes	characteristic
278	C₂²	#19098480	1	1	yes	characteristic
279	C₂²	#19098488	1	1	yes	characteristic
280	C₂²	#19098505	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	632	4	no	—
283	C₄	#19098428	1	1	yes	characteristic
284	C₄	#19098435	1	1	yes	characteristic
285	C₄	#19098455	1	2	yes	—
286	C₄	#19098475	1	1	yes	characteristic
287	C₄	#19098495	1	2	yes	—
288	C₄	#19098502	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	#23761656	1	1	yes	characteristic
294	C₂	#23764687	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579150
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1728	C₂
socle	C₂²×C₇₉	#394
prime radical	C₂⁴×C₇₉	#41
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1728	D₇₉
hypocenter	C₇₉	#11294
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
12. C₇₉⋊#338	C₄	semidirect
153. C₇₉	#11294	semidirect

Maximal subgroups

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

subgroup	index
#23771990	2
#23763138	2
C₂×#19098323	2
C₂×C₇₉⋊#338	2
C₇₉×#1728	2
#11294	79