Subgroups — #34196951

Order 20224.

686 conjugacy classes of subgroups.

The subgroup lattice has 298 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	#23771989	C₂	1	2	yes	maximal
3	C₂×C₇₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	#23763096	C₂	1	2	yes	maximal
5	C₇₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	2	yes	—
7	C₂×#14579120	C₂²	1	2	yes	—
8	#19098606	—	2	4	no	—
9	#19098605	—	2	4	no	—
10	#19098603	—	2	4	no	—
11	#19098601	—	2	2	no	—
12	#19098599	—	2	4	no	—
13	C₇₉⋊#338	C₄	1	4	yes	complemented
14	#19098317	—	2	4	no	—
15	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
18	#336×C₇₉	—	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	C₂×#11929608	Q₈	1	4	yes	—
25	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
26	#14579120	—	2	2	no	—
27	#14579120	—	2	2	no	—
28	#14579120	—	2	2	no	—
29	#14579120	—	2	2	no	—
30	C₂×C₄⋊Dic₇₉	D₄	1	4	yes	—
31	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
32	C₂×C₄×Dic₇₉	—	2	2	no	—
33	#14579095	—	2	4	no	—
34	#14579094	—	2	8	no	—
35	#14579093	—	2	4	no	—
36	#14579092	—	2	8	no	—
37	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
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	C₂²×C₈×C₇₉	—	2	1	no	—
43	C₂²×C₈×C₇₉	—	2	2	no	—
44	C₂²×C₈×C₇₉	—	2	2	no	—
45	C₂²×C₈×C₇₉	—	2	1	no	—
46	C₂²×C₈×C₇₉	D₄	1	2	yes	—
47	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
48	#117×C₇₉	—	2	2	no	—
49	#117×C₇₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
52	C₂×C₄²×C₇₉	—	2	1	no	—
53	C₂²×Dic₇₉	—	2	4	no	—
54	#11929608	—	2	4	no	—
55	#11929608	—	2	4	no	—
56	#11929608	—	2	4	no	—
57	#11929608	—	2	4	no	—
58	C₂×C₇₉⋊C₈	—	2	4	no	—
59	C₂×C₇₉⋊C₈	—	2	4	no	—
60	C₂×C₇₉⋊C₈	C₄⋊C₄	1	8	yes	—
61	C₄⋊Dic₇₉	—	2	4	no	—
62	C₄⋊Dic₇₉	—	2	4	no	—
63	C₂²⋊Dic₇₉	—	4	2	no	—
64	C₂²⋊Dic₇₉	—	4	2	no	—
65	C₄×Dic₇₉	—	4	2	no	—
66	C₄×Dic₇₉	—	4	2	no	—
67	C₂⁴×C₇₉	#41	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	—	2	2	no	—
69	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	#41	1	2	yes	—
78	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
79	M₁₆×C₇₉	—	2	4	no	—
80	M₁₆×C₇₉	—	2	4	no	—
81	C₂×C₈×C₇₉	—	2	2	no	—
82	C₂×C₈×C₇₉	—	2	4	no	—
83	C₂×C₈×C₇₉	—	2	2	no	—
84	C₂×C₈×C₇₉	—	2	2	no	—
85	C₂×C₈×C₇₉	—	4	1	no	—
86	C₂×C₈×C₇₉	—	4	2	no	—
87	C₂×C₈×C₇₉	—	4	1	no	—
88	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
89	C₄⋊C₄×C₇₉	—	2	4	no	—
90	C₂²⋊C₄×C₇₉	—	4	2	no	—
91	C₄²×C₇₉	—	4	1	no	—
92	C₄²×C₇₉	—	4	1	no	—
93	C₂³×C₇₉	—	4	2	no	—
94	C₂³×C₇₉	C₄×Q₈	1	1	yes	characteristic
95	C₂³×C₇₉	#121	1	2	yes	—
96	C₂³×C₇₉	#121	1	1	yes	characteristic
97	C₂³×C₇₉	#125	1	2	yes	—
98	C₂³×C₇₉	#125	1	1	yes	characteristic
99	C₂×C₄×C₇₉	—	4	4	no	—
100	C₂×C₄×C₇₉	—	4	2	no	—
101	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
106	C₂×C₄×C₇₉	#120	1	2	yes	—
107	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#123	1	2	yes	—
114	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×Dic₇₉	—	4	8	no	—
120	C₂×Dic₇₉	—	4	4	no	—
121	C₈×C₇₉	—	2	4	no	—
122	C₈×C₇₉	—	4	2	no	—
123	C₇₉⋊C₈	—	2	8	no	—
124	C₂²×C₇₉	—	2	2	no	—
125	C₂²×C₇₉	—	2	4	no	—
126	C₂²×C₇₉	—	4	1	no	—
127	C₂²×C₇₉	—	4	2	no	—
128	C₂²×C₇₉	—	4	1	no	—
129	C₂²×C₇₉	#389	1	2	yes	—
130	C₂²×C₇₉	#389	1	1	yes	characteristic
131	C₂²×C₇₉	#392	1	1	yes	characteristic
132	C₂²×C₇₉	#398	1	1	yes	characteristic
133	C₂²×C₇₉	#398	1	2	yes	—
134	C₄×C₇₉	—	8	2	no	—
135	C₄×C₇₉	#384	1	1	yes	characteristic
136	C₄×C₇₉	#385	1	1	yes	characteristic
137	C₄×C₇₉	#385	1	2	yes	—
138	C₄×C₇₉	#395	1	1	yes	characteristic
139	C₄×C₇₉	#395	1	2	yes	—
140	C₄×C₇₉	#399	1	1	yes	characteristic
141	Dic₇₉	—	8	4	no	—
142	#11337	—	79	1	no	maximal, Sylow
143	C₂×C₇₉	—	2	2	no	—
144	C₂×C₇₉	—	2	1	no	—
145	C₂×C₇₉	—	2	1	no	—
146	C₂×C₇₉	—	2	2	no	—
147	C₂×C₇₉	#1444	1	1	yes	characteristic
148	C₂×C₇₉	#2017	1	2	yes	—
149	#2911	—	79	1	no	—
150	#2911	—	79	2	no	—
151	C₂×#338	—	79	1	no	—
152	#1729	—	79	2	no	—
153	#1729	D₇₉	1	1	yes	characteristic
154	C₇₉	#11337	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	79	2	no	—
156	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
157	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
158	C₂×#117	—	79	2	no	—
159	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
160	#428	—	158	4	no	—
161	#428	—	158	4	no	—
162	#428	—	158	4	no	—
163	#427	—	158	4	no	—
164	#427	—	158	2	no	—
165	#338	—	79	4	no	—
166	#336	—	2	2	no	—
167	#336	—	158	4	no	—
168	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	79	4	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	Dic₁₅₈	1	2	yes	—
178	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	79	4	no	—
184	C₂²×C₈	Dic₁₅₈	1	2	yes	—
185	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	158	2	no	—
189	#117	—	158	2	no	—
190	#117	—	158	2	no	—
191	#117	—	158	2	no	—
192	C₂×C₄⋊C₄	—	79	4	no	—
193	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
194	C₂×C₂²⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	—	158	2	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	—	158	2	no	—
198	#107	—	158	8	no	—
199	#107	—	158	4	no	—
200	#106	—	158	8	no	—
201	#106	—	158	4	no	—
202	C₂⁴	#11929624	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	158	4	no	—
205	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₇₉	1	2	yes	—
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	Q₈×D₇₉	1	2	yes	—
211	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
213	C₂²×C₄	#11929623	1	2	yes	—
214	C₂²×C₄	#11929624	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	158	4	no	—
218	M₁₆	—	158	4	no	—
219	M₁₆	—	158	4	no	—
220	M₁₆	—	158	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	79	8	no	—
229	C₂×C₈	—	158	4	no	—
230	C₂×C₈	—	158	4	no	—
231	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	158	4	no	—
234	C₄⋊C₄	—	158	4	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	316	2	no	—
237	C₂²⋊C₄	—	316	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	316	2	no	—
241	C₄²	—	316	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	C₄×Dic₁₅₈	1	1	yes	characteristic
244	C₂³	#14579134	1	2	yes	—
245	C₂³	#14579135	1	1	yes	characteristic
246	C₂³	#14579151	1	2	yes	—
247	C₂³	#14579153	1	1	yes	characteristic
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	316	8	no	—
251	C₂×C₄	—	316	4	no	—
252	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
253	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
254	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
255	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
256	C₂×C₄	#14579129	1	1	yes	characteristic
257	C₂×C₄	#14579131	1	2	yes	—
258	C₂×C₄	#14579138	1	2	yes	—
259	C₂×C₄	#14579138	1	2	yes	—
260	C₂×C₄	#14579139	1	2	yes	—
261	C₂×C₄	#14579140	1	1	yes	characteristic
262	C₂×C₄	#14579141	1	2	yes	—
263	C₂×C₄	#14579142	1	1	yes	characteristic
264	C₂×C₄	#14579143	1	2	yes	—
265	C₂×C₄	#14579157	1	1	yes	characteristic
266	C₂×C₄	#14579159	1	1	yes	characteristic
267	C₂×C₄	#14579160	1	2	yes	—
268	C₂×C₄	#14579160	1	2	yes	—
269	C₂×C₄	#14579161	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₈	—	158	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	#19098460	1	2	yes	—
279	C₂²	#19098461	1	1	yes	characteristic
280	C₂²	#19098472	1	1	yes	characteristic
281	C₂²	#19098493	1	1	yes	characteristic
282	C₂²	#19098497	1	2	yes	—
283	C₄	—	8	2	no	—
284	C₄	—	632	4	no	—
285	C₄	#19098432	1	1	yes	characteristic
286	C₄	#19098435	1	1	yes	characteristic
287	C₄	#19098437	1	2	yes	—
288	C₄	#19098483	1	1	yes	characteristic
289	C₄	#19098484	1	2	yes	—
290	C₄	#19098499	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	#23761664	1	1	yes	characteristic
296	C₂	#23764504	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579157
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1729	C₂
socle	C₂²×C₇₉	#392
prime radical	C₂⁴×C₇₉	#41
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1729	D₇₉
hypocenter	C₇₉	#11337
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
13. C₇₉⋊#338	C₄	semidirect
154. C₇₉	#11337	semidirect

Maximal subgroups

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

subgroup	index
#23771990	2
#23771989	2
C₂×C₇₉⋊#338	2
#23763096	2
C₇₉×#1729	2
#11337	79