Subgroups — #34181701

Order 20224.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763380	C₂	1	1	yes	maximal, characteristic
2	#23763380	C₂	1	1	yes	maximal, characteristic
3	#23761283	C₂	1	2	yes	maximal, complemented
4	#23761283	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2285	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
9	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
10	#19098318	—	2	1	no	—
11	#19098318	—	2	1	no	—
12	C₂×#117×C₇₉	—	2	1	no	—
13	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
14	#384×C₇₉	C₂²	1	1	yes	characteristic
15	#384×C₇₉	C₂²	1	1	yes	characteristic
16	#382×C₇₉	—	2	1	no	—
17	#380×C₇₉	—	2	1	no	—
18	#380×C₇₉	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
20	C₄×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₂²×C₇₉⋊C₈	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
26	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
27	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
30	#117×C₇₉	—	4	4	no	—
31	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
38	C₂×C₄²×C₇₉	—	2	4	no	—
39	C₂×C₄²×C₇₉	—	2	2	no	—
40	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	#95×C₇₉	—	2	2	no	—
45	#95×C₇₉	—	2	2	no	—
46	C₂×C₇₉⋊C₈	—	4	2	no	—
47	C₂×C₇₉⋊C₈	—	4	2	no	—
48	C₂×C₇₉⋊C₈	—	4	2	no	—
49	C₂×C₇₉⋊C₈	—	4	2	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₂×C₇₉⋊C₈	—	4	2	no	—
52	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	—	2	2	no	—
54	C₂²×C₄×C₇₉	—	2	2	no	—
55	C₂²×C₄×C₇₉	—	2	2	no	—
56	C₂²×C₄×C₇₉	—	2	2	no	—
57	C₂²×C₄×C₇₉	—	2	2	no	—
58	C₂²×C₄×C₇₉	—	2	1	no	—
59	C₂²×C₄×C₇₉	—	2	1	no	—
60	C₂²×C₄×C₇₉	—	2	2	no	—
61	C₂²×C₄×C₇₉	—	2	2	no	—
62	C₂²×C₄×C₇₉	—	4	1	no	—
63	C₂²×C₄×C₇₉	—	4	1	no	—
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₇₉	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₇₉	—	4	2	no	—
73	C₄⋊C₄×C₇₉	—	4	2	no	—
74	C₄⋊C₄×C₇₉	—	4	2	no	—
75	C₄⋊C₄×C₇₉	—	4	1	no	—
76	C₄⋊C₄×C₇₉	—	4	2	no	—
77	C₄⋊C₄×C₇₉	—	4	2	no	—
78	C₄⋊C₄×C₇₉	—	4	1	no	—
79	C₂²⋊C₄×C₇₉	—	4	4	no	—
80	C₄²×C₇₉	—	2	2	no	—
81	C₄²×C₇₉	—	2	2	no	—
82	C₄²×C₇₉	—	2	2	no	—
83	C₄²×C₇₉	—	4	4	no	—
84	C₄²×C₇₉	#41	1	4	yes	—
85	C₂³×C₇₉	—	2	1	no	—
86	C₂³×C₇₉	—	2	1	no	—
87	C₂³×C₇₉	—	2	1	no	—
88	C₂³×C₇₉	—	2	1	no	—
89	C₂³×C₇₉	—	2	1	no	—
90	C₂³×C₇₉	—	2	1	no	—
91	C₂³×C₇₉	#103	1	2	yes	—
92	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₉	—	2	1	no	—
94	C₂×C₄×C₇₉	—	2	1	no	—
95	C₂×C₄×C₇₉	—	2	1	no	—
96	C₂×C₄×C₇₉	—	2	1	no	—
97	C₂×C₄×C₇₉	—	2	4	no	—
98	C₂×C₄×C₇₉	—	2	4	no	—
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	2	1	no	—
101	C₂×C₄×C₇₉	—	2	1	no	—
102	C₂×C₄×C₇₉	—	2	1	no	—
103	C₂×C₄×C₇₉	—	2	1	no	—
104	C₂×C₄×C₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	—	4	4	no	—
106	C₂×C₄×C₇₉	—	4	2	no	—
107	C₂×C₄×C₇₉	—	4	2	no	—
108	C₂×C₄×C₇₉	—	4	2	no	—
109	C₂×C₄×C₇₉	—	4	2	no	—
110	C₂×C₄×C₇₉	—	4	2	no	—
111	C₂×C₄×C₇₉	—	4	2	no	—
112	C₂×C₄×C₇₉	—	4	2	no	—
113	C₂×C₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	#103	1	2	yes	—
115	C₂×C₄×C₇₉	#117	1	2	yes	—
116	C₂×C₄×C₇₉	#123	1	2	yes	—
117	C₂×C₄×C₇₉	#123	1	2	yes	—
118	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#135	1	2	yes	—
121	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
124	C₇₉⋊C₈	—	8	2	no	—
125	C₇₉⋊C₈	—	8	2	no	—
126	C₂²×C₇₉	—	2	2	no	—
127	C₂²×C₇₉	—	2	2	no	—
128	C₂²×C₇₉	—	2	2	no	—
129	C₂²×C₇₉	—	4	1	no	—
130	C₂²×C₇₉	—	4	1	no	—
131	C₂²×C₇₉	—	4	1	no	—
132	C₂²×C₇₉	—	4	1	no	—
133	C₂²×C₇₉	#381	1	1	yes	characteristic
134	C₂²×C₇₉	#413	1	1	yes	characteristic
135	C₂²×C₇₉	#413	1	1	yes	characteristic
136	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
137	C₂²×C₇₉	#416	1	1	yes	characteristic
138	C₂²×C₇₉	#417	1	1	yes	characteristic
139	C₂²×C₇₉	#418	1	1	yes	characteristic
140	C₄×C₇₉	—	2	2	no	—
141	C₄×C₇₉	—	2	2	no	—
142	C₄×C₇₉	—	4	1	no	—
143	C₄×C₇₉	—	4	1	no	—
144	C₄×C₇₉	—	8	2	no	—
145	C₄×C₇₉	—	8	2	no	—
146	C₄×C₇₉	#481	1	2	yes	—
147	C₄×C₇₉	#482	1	2	yes	—
148	C₄×C₇₉	#484	1	2	yes	—
149	C₄×C₇₉	#485	1	2	yes	—
150	#8727	—	79	1	no	maximal, Sylow
151	C₂×C₇₉	—	4	2	no	—
152	C₂×C₇₉	#1542	1	1	yes	characteristic
153	C₂×C₇₉	#1550	1	1	yes	characteristic
154	C₂×C₇₉	#1553	1	1	yes	characteristic
155	C₂×C₇₉	#1557	1	1	yes	characteristic
156	C₂×C₇₉	#1822	1	1	yes	characteristic
157	C₂×C₇₉	#1824	1	1	yes	characteristic
158	C₂×C₇₉	#1825	1	1	yes	characteristic
159	#2285	D₇₉	1	1	yes	characteristic
160	#1797	—	79	1	no	—
161	#1797	—	79	1	no	—
162	#1296	—	79	2	no	—
163	#1296	—	79	2	no	—
164	C₇₉	#8727	1	1	yes	characteristic, complemented, Sylow
165	C₂×#117	—	2	1	no	—
166	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
167	C₂×C₄⋊C₈	—	79	2	no	—
168	C₂×C₄⋊C₈	—	79	2	no	—
169	C₂×C₂²⋊C₈	—	158	1	no	—
170	C₂×C₂²⋊C₈	—	158	1	no	—
171	#384	C₂×D₇₉	1	1	yes	characteristic
172	#384	C₂×D₇₉	1	1	yes	characteristic
173	#382	—	2	1	no	—
174	#380	—	2	1	no	—
175	#380	Dic₇₉	1	2	yes	—
176	C₄×C₄⋊C₄	—	2	1	no	—
177	C₄×C₄⋊C₄	Dic₇₉	1	2	yes	—
178	#336	—	158	1	no	—
179	#336	—	158	1	no	—
180	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
181	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
182	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
183	C₂²×C₈	—	158	2	no	—
184	C₂²×C₈	—	158	2	no	—
185	#117	—	4	4	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
191	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₄²	—	2	4	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
199	C₄⋊C₈	—	158	4	no	—
200	C₄⋊C₈	—	158	4	no	—
201	C₂²⋊C₈	—	316	2	no	—
202	C₂²⋊C₈	—	316	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
220	C₂²×C₄	#11929611	1	2	yes	—
221	C₂²×C₄	#11929614	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
225	C₂×C₈	—	316	2	no	—
226	C₂×C₈	—	316	2	no	—
227	C₂×C₈	—	316	2	no	—
228	C₂×C₈	—	316	2	no	—
229	C₂×C₈	—	316	2	no	—
230	C₂×C₈	—	316	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	4	no	—
243	C₄²	#11929624	1	4	yes	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	#14579085	1	2	yes	—
251	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	#14579085	1	2	yes	—
274	C₂×C₄	#14579120	1	2	yes	—
275	C₂×C₄	#14579144	1	2	yes	—
276	C₂×C₄	#14579144	1	2	yes	—
277	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
278	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
279	C₂×C₄	#14579181	1	2	yes	—
280	C₂×C₄	#14579186	1	1	yes	characteristic
281	C₂×C₄	#14579187	1	2	yes	—
282	C₂×C₄	#14579191	1	1	yes	characteristic
283	C₈	—	632	2	no	—
284	C₈	—	632	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	#19098426	1	1	yes	characteristic
293	C₂²	#19098547	1	1	yes	characteristic
294	C₂²	#19098547	1	1	yes	characteristic
295	C₂²	C₂×#14579085	1	1	yes	characteristic
296	C₂²	#19098559	1	1	yes	characteristic
297	C₂²	#19098564	1	1	yes	characteristic
298	C₂²	#19098568	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	#19098881	1	2	yes	—
306	C₄	#19098887	1	2	yes	—
307	C₄	#19098898	1	2	yes	—
308	C₄	#19098904	1	2	yes	—
309	C₂	—	4	2	no	—
310	C₂	#23762057	1	1	yes	characteristic
311	C₂	#23762090	1	1	yes	characteristic
312	C₂	#23762105	1	1	yes	characteristic
313	C₂	#23762122	1	1	yes	characteristic
314	C₂	#23763483	1	1	yes	characteristic
315	C₂	#23763490	1	1	yes	characteristic
316	C₂	#23763495	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2285	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2285	D₇₉
hypocenter	C₇₉	#8727
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #23761283	C₂	semidirect
4. #23761283	C₂	semidirect
164. C₇₉	#8727	semidirect

Maximal subgroups

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

subgroup	index
#23763380	2
#23761283	2
C₇₉×#2285	2
#8727	79