Subgroups — #34178900

Order 20224.

542 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765874	C₂	1	1	yes	maximal, characteristic
2	#23763255	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098295	C₂	1	1	yes	maximal, characteristic
5	C₂×#19098293	C₂	1	1	yes	maximal, characteristic
6	#23761241	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14579071	—	2	1	no	—
11	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₉	—	2	2	no	—
13	#19098468	C₄	1	2	yes	—
14	#19098459	C₄	1	2	yes	—
15	#19098445	—	2	2	no	—
16	C₄×Dic₇₉⋊C₄	—	2	1	no	—
17	#19098409	—	2	1	no	—
18	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
19	#19098317	—	2	1	no	—
20	#19098295	C₄	1	4	yes	complemented
21	#19098293	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇₉⋊C₈	—	2	1	no	—
27	C₂²×C₇₉⋊C₈	—	2	1	no	—
28	C₄×Dic₁₅₈	—	2	1	no	—
29	C₄×Dic₁₅₈	—	2	1	no	—
30	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
31	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₇₉⋊C₄	—	2	1	no	—
34	C₂×Dic₇₉⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₇₉	—	2	1	no	—
36	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
39	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
40	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
41	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
42	#14579071	—	4	2	no	—
43	C₄×C₇₉⋊C₈	—	2	1	no	—
44	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
45	C₈×Dic₇₉	—	2	8	no	—
46	C₇₉⋊#95	—	2	2	no	—
47	C₇₉⋊#95	—	2	2	no	—
48	C₇₉⋊#95	—	2	2	no	—
49	C₂²×C₈×C₇₉	—	2	1	no	—
50	C₂²×C₈×C₇₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₇₉	—	2	1	no	—
53	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₇₉	—	2	1	no	—
55	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₇₉	—	2	1	no	—
57	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
58	#95×C₇₉	—	2	1	no	—
59	C₂×Dic₁₅₈	—	2	1	no	—
60	C₂×Dic₁₅₈	—	2	2	no	—
61	C₂×Dic₁₅₈	—	2	2	no	—
62	C₂×Dic₁₅₈	C₄²	1	2	yes	—
63	C₂²×Dic₇₉	—	2	1	no	—
64	C₂²×Dic₇₉	—	2	1	no	—
65	C₂²×Dic₇₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×Dic₇₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₇₉⋊C₈	—	2	4	no	—
68	C₂×C₇₉⋊C₈	—	2	2	no	—
69	C₂×C₇₉⋊C₈	—	2	2	no	—
70	C₂×C₇₉⋊C₈	—	4	1	no	—
71	C₂×C₇₉⋊C₈	—	4	1	no	—
72	C₄⋊Dic₇₉	—	2	1	no	—
73	C₄⋊Dic₇₉	C₄²	1	2	yes	—
74	Dic₇₉⋊C₄	—	2	2	no	—
75	Dic₇₉⋊C₄	—	2	2	no	—
76	Dic₇₉⋊C₄	—	4	4	no	—
77	C₄×Dic₇₉	—	2	4	no	—
78	C₄×Dic₇₉	—	2	2	no	—
79	C₄×Dic₇₉	—	2	2	no	—
80	C₄×Dic₇₉	—	2	2	no	—
81	C₄×Dic₇₉	—	2	2	no	—
82	C₄×Dic₇₉	—	2	1	no	—
83	C₄×Dic₇₉	—	2	1	no	—
84	C₄×Dic₇₉	SD₁₆	1	2	yes	—
85	C₄×Dic₇₉	Q₁₆	1	2	yes	—
86	C₂²×C₄×C₇₉	—	2	2	no	—
87	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₇₉	—	2	4	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	4	1	no	—
94	C₂×C₈×C₇₉	—	4	1	no	—
95	C₄⋊C₄×C₇₉	—	2	2	no	—
96	C₄⋊C₄×C₇₉	—	2	2	no	—
97	C₄²×C₇₉	—	4	1	no	—
98	C₄²×C₇₉	—	4	1	no	—
99	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₉	#41	1	1	yes	characteristic
102	C₄²×C₇₉	#41	1	1	yes	characteristic
103	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	—	4	2	no	—
106	C₂×C₄×C₇₉	—	4	2	no	—
107	C₂×C₄×C₇₉	—	4	2	no	—
108	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	#131	1	2	yes	—
115	C₂×C₄×C₇₉	#131	1	2	yes	—
116	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
119	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
120	C₂×Dic₇₉	—	2	2	no	—
121	C₂×Dic₇₉	—	2	2	no	—
122	C₂×Dic₇₉	—	2	2	no	—
123	C₂×Dic₇₉	—	2	2	no	—
124	C₂×Dic₇₉	—	2	2	no	—
125	C₂×Dic₇₉	—	2	2	no	—
126	C₂×Dic₇₉	—	4	2	no	—
127	C₂×Dic₇₉	—	4	2	no	—
128	C₂×Dic₇₉	#103	1	4	yes	—
129	C₂×Dic₇₉	#104	1	4	yes	—
130	Dic₁₅₈	—	2	2	no	—
131	Dic₁₅₈	—	2	2	no	—
132	Dic₁₅₈	—	4	2	no	—
133	C₈×C₇₉	—	2	4	no	—
134	C₈×C₇₉	—	4	2	no	—
135	C₇₉⋊C₈	—	2	4	no	—
136	C₇₉⋊C₈	—	4	2	no	—
137	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₇₉	#408	1	2	yes	—
139	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
140	C₂²×C₇₉	#416	1	1	yes	characteristic
141	C₂²×C₇₉	C₂×#104	1	2	yes	—
142	C₄×C₇₉	—	2	1	no	—
143	C₄×C₇₉	—	2	1	no	—
144	C₄×C₇₉	—	8	2	no	—
145	C₄×C₇₉	#405	1	1	yes	characteristic
146	C₄×C₇₉	#405	1	1	yes	characteristic
147	C₄×C₇₉	#433	1	2	yes	—
148	C₄×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
149	C₄×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
150	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
151	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
152	Dic₇₉	—	2	4	no	—
153	Dic₇₉	—	2	4	no	—
154	Dic₇₉	—	4	2	no	—
155	Dic₇₉	—	4	2	no	—
156	#8176	—	79	1	no	maximal, Sylow
157	C₂×C₇₉	#1493	1	2	yes	—
158	C₂×C₇₉	#1494	1	2	yes	—
159	C₂×C₇₉	#1750	1	1	yes	characteristic
160	C₂×C₇₉	C₄×#104	1	1	yes	characteristic
161	C₂×C₇₉	C₄×#103	1	1	yes	characteristic
162	#2295	—	79	1	no	—
163	#1767	—	79	1	no	—
164	C₄²×C₈	—	79	1	no	—
165	C₂×Q₈⋊C₈	—	79	1	no	—
166	C₂×Q₈⋊C₈	—	79	1	no	—
167	#1279	—	79	1	no	—
168	#1279	D₇₉	1	1	yes	characteristic
169	C₇₉	#8176	1	1	yes	characteristic, complemented, Sylow
170	C₂×C₄×Q₈	—	79	1	no	—
171	C₂×C₄⋊C₈	—	79	1	no	—
172	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
173	C₂×#97	—	158	1	no	—
174	C₂×C₄×C₈	—	79	1	no	—
175	C₂×C₄×C₈	—	158	2	no	—
176	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
177	#391	—	79	2	no	—
178	#389	—	79	2	no	—
179	#387	—	158	2	no	—
180	C₄×C₄⋊C₄	—	158	1	no	—
181	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
182	#376	—	158	1	no	—
183	C₄³	—	79	1	no	—
184	#336	—	158	1	no	—
185	Q₈⋊C₈	—	79	4	no	—
186	Q₈⋊C₈	—	79	4	no	—
187	C₂²×Q₈	—	79	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	158	1	no	—
191	C₂²×C₈	—	158	1	no	—
192	C₄×Q₈	—	79	2	no	—
193	C₄×Q₈	—	79	2	no	—
194	C₄×Q₈	—	158	1	no	—
195	C₄×Q₈	—	158	1	no	—
196	C₂×C₄⋊C₄	—	79	1	no	—
197	C₂×C₄⋊C₄	—	158	2	no	—
198	C₂×C₄⋊C₄	—	158	1	no	—
199	C₂×C₄⋊C₄	C₄×D₇₉	1	2	yes	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	79	1	no	—
202	C₂×C₄²	—	79	2	no	—
203	C₂×C₄²	—	79	2	no	—
204	C₂×C₄²	—	79	1	no	—
205	C₂×C₄²	—	158	1	no	—
206	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	79	2	no	—
209	C₄⋊C₈	—	158	1	no	—
210	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
211	#97	—	316	2	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	79	2	no	—
214	C₄×C₈	—	158	8	no	—
215	C₄×C₈	—	158	1	no	—
216	C₄×C₈	C₄×D₇₉	1	2	yes	—
217	#95	—	2	1	no	—
218	#95	—	158	2	no	—
219	#95	—	158	2	no	—
220	#95	—	158	2	no	—
221	C₂×Q₈	—	79	2	no	—
222	C₂×Q₈	—	158	2	no	—
223	C₂×Q₈	—	158	2	no	—
224	C₂×Q₈	—	158	1	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	79	2	no	—
227	C₂²×C₄	—	79	2	no	—
228	C₂²×C₄	—	158	1	no	—
229	C₂²×C₄	—	158	1	no	—
230	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	316	1	no	—
239	C₂×C₈	—	316	1	no	—
240	C₂×C₈	—	158	2	no	—
241	C₂×C₈	—	158	4	no	—
242	C₂×C₈	—	158	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	316	4	no	—
246	C₄⋊C₄	—	79	2	no	—
247	C₄⋊C₄	—	158	1	no	—
248	C₄⋊C₄	—	158	2	no	—
249	C₄⋊C₄	—	158	2	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	79	2	no	—
253	C₄²	—	79	2	no	—
254	C₄²	—	158	4	no	—
255	C₄²	—	158	2	no	—
256	C₄²	—	158	2	no	—
257	C₄²	—	158	2	no	—
258	C₄²	—	158	2	no	—
259	C₄²	—	158	1	no	—
260	C₄²	—	158	1	no	—
261	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
263	C₄²	#11929623	1	1	yes	characteristic
264	C₄²	#11929625	1	1	yes	characteristic
265	C₂³	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
266	Q₈	—	316	2	no	—
267	Q₈	—	158	2	no	—
268	Q₈	—	158	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	316	2	no	—
274	C₂×C₄	—	316	2	no	—
275	C₂×C₄	—	79	4	no	—
276	C₂×C₄	—	79	4	no	—
277	C₂×C₄	—	158	2	no	—
278	C₂×C₄	—	158	2	no	—
279	C₂×C₄	—	158	2	no	—
280	C₂×C₄	—	158	2	no	—
281	C₂×C₄	—	158	2	no	—
282	C₂×C₄	—	158	2	no	—
283	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
284	C₂×C₄	#14579119	1	1	yes	characteristic
285	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	#14579123	1	1	yes	characteristic
288	C₂×C₄	#14579123	1	1	yes	characteristic
289	C₂×C₄	#14579167	1	2	yes	—
290	C₂×C₄	#14579169	1	2	yes	—
291	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
292	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
293	C₂×C₄	#14579180	1	1	yes	characteristic
294	C₂×C₄	#14579182	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	4	2	no	—
297	C₈	—	316	2	no	—
298	C₈	—	158	4	no	—
299	C₂²	#19098410	1	1	yes	characteristic
300	C₂²	#19098528	1	2	yes	—
301	C₂²	#103×D₇₉	1	1	yes	characteristic
302	C₂²	#19098556	1	1	yes	characteristic
303	C₂²	#104×D₇₉	1	2	yes	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	316	2	no	—
308	C₄	—	316	2	no	—
309	C₄	—	158	4	no	—
310	C₄	—	158	4	no	—
311	C₄	#19098517	1	1	yes	characteristic
312	C₄	#19098520	1	1	yes	characteristic
313	C₄	#19098623	1	2	yes	—
314	C₄	#19098650	1	1	yes	characteristic
315	C₄	#19098654	1	1	yes	characteristic
316	C₄	#19098655	1	1	yes	characteristic
317	C₄	#19098657	1	1	yes	characteristic
318	C₂	#23761853	1	2	yes	—
319	C₂	#23761857	1	2	yes	—
320	C₂	#23763165	1	1	yes	characteristic
321	C₂	#23763204	1	1	yes	characteristic
322	C₂	#23763219	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1279	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1279	D₇₉
hypocenter	C₇₉	#8176
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
20. #19098295	C₄	semidirect
21. #19098293	C₄	semidirect
169. C₇₉	#8176	semidirect

Maximal subgroups

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

subgroup	index
#23765874	2
#23763255	2
C₄×C₈×Dic₇₉	2
C₂×#19098295	2
C₂×#19098293	2
#23761241	2
C₇₉×#1279	2
#8176	79