Subgroups — #34275952

Order 20224.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773986	C₂	1	1	yes	maximal, characteristic, complemented
2	#23773955	C₂	1	2	yes	maximal, complemented
3	#23773722	C₂	1	2	yes	maximal, complemented
4	#23763011	C₂	1	2	yes	maximal, complemented
5	#23763009	C₂	1	1	yes	maximal, characteristic, complemented
6	#23763005	C₂	1	1	yes	maximal, characteristic, complemented
7	#23763000	C₂	1	2	yes	maximal, complemented
8	#23762998	C₂	1	1	yes	maximal, characteristic, complemented
9	#23762989	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#3169	C₂	1	1	yes	maximal, characteristic
12	#19099120	C₂²	1	2	yes	—
13	#19098968	C₂²	1	1	yes	characteristic
14	#19098959	C₂²	1	2	yes	—
15	#19098943	C₂²	1	1	yes	characteristic
16	#19098935	C₂²	1	2	yes	—
17	#19098917	—	2	4	no	—
18	#19098915	C₂²	1	2	yes	—
19	#19098914	C₂²	1	2	yes	—
20	#19098913	C₂²	1	2	yes	—
21	#19098913	C₂²	1	2	yes	—
22	#19098909	C₂²	1	2	yes	—
23	#19098909	C₂²	1	2	yes	—
24	#19098909	C₂²	1	2	yes	—
25	#19098804	—	2	2	no	—
26	#19098790	—	2	4	no	—
27	C₄×Dic₃₁₆	—	2	2	no	—
28	C₄×#11929612	—	2	4	no	—
29	#19098559	—	2	4	no	—
30	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
32	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
33	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
36	#532×C₇₉	C₂²	1	2	yes	—
37	#500×C₇₉	C₂²	1	1	yes	characteristic
38	#493×C₇₉	C₂²	1	1	yes	characteristic
39	#486×C₇₉	C₂²	1	2	yes	—
40	#466×C₇₉	—	2	2	no	—
41	C₄×D₈×C₇₉	—	2	2	no	—
42	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
43	C₂×Dic₃₁₆	—	2	2	no	—
44	C₂×#11929612	—	2	4	no	—
45	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
46	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
47	#14579161	C₂³	1	2	yes	—
48	#14579161	C₂³	1	2	yes	—
49	#14579153	—	2	2	no	—
50	#14579140	—	2	2	no	—
51	#14579140	—	2	2	no	—
52	C₄×Dic₁₅₈	—	2	2	no	—
53	C₄×Dic₁₅₈	—	2	2	no	—
54	#14579120	—	2	2	no	—
55	#14579095	—	2	2	no	—
56	#14579092	—	2	4	no	—
57	#14579086	—	2	4	no	—
58	#14579085	—	2	4	no	—
59	#14579085	—	2	4	no	—
60	#14579084	—	2	4	no	—
61	#14579082	—	2	4	no	—
62	#14579082	—	2	4	no	—
63	#14579082	—	2	4	no	—
64	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
65	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
66	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×#41×C₇₉	—	2	2	no	—
71	C₂×D₈×C₇₉	—	2	2	no	—
72	C₂²×C₈×C₇₉	D₄	1	2	yes	—
73	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
74	#127×C₇₉	C₂³	1	2	yes	—
75	#124×C₇₉	—	2	2	no	—
76	#121×C₇₉	—	2	2	no	—
77	#121×C₇₉	—	2	2	no	—
78	C₄×D₄×C₇₉	—	2	2	no	—
79	C₄×D₄×C₇₉	—	2	2	no	—
80	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
81	#107×C₇₉	—	2	2	no	—
82	#102×C₇₉	—	2	4	no	—
83	C₄×C₈×C₇₉	D₄	1	2	yes	—
84	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₂×Dic₁₅₈	—	2	2	no	—
87	C₂×Dic₁₅₈	—	2	2	no	—
88	Dic₃₁₆	—	4	4	no	—
89	#11929612	—	4	8	no	—
90	C₂×C₇₉⋊C₈	—	2	4	no	—
91	C₂×C₇₉⋊C₈	—	2	4	no	—
92	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
93	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
94	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
95	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
96	C₄⋊Dic₇₉	—	2	2	no	—
97	C₄⋊Dic₇₉	—	2	2	no	—
98	C₄⋊Dic₇₉	—	2	2	no	—
99	C₄⋊Dic₇₉	—	2	2	no	—
100	Dic₇₉⋊C₄	—	4	2	no	—
101	Dic₇₉⋊C₄	—	4	2	no	—
102	C₂²⋊Dic₇₉	—	4	2	no	—
103	C₄×Dic₇₉	—	4	2	no	—
104	#41×C₇₉	—	4	8	no	—
105	C₂×Q₈×C₇₉	—	2	2	no	—
106	C₂×D₄×C₇₉	—	2	2	no	—
107	C₂×D₄×C₇₉	—	2	2	no	—
108	C₂×D₄×C₇₉	—	2	2	no	—
109	C₂×D₄×C₇₉	—	4	2	no	—
110	C₂²×C₄×C₇₉	—	4	2	no	—
111	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
114	D₈×C₇₉	—	4	4	no	—
115	C₂×C₈×C₇₉	—	2	4	no	—
116	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
118	C₄⋊C₄×C₇₉	—	2	2	no	—
119	C₄⋊C₄×C₇₉	—	2	2	no	—
120	C₂²⋊C₄×C₇₉	—	4	2	no	—
121	C₂²⋊C₄×C₇₉	—	4	2	no	—
122	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
126	C₂³×C₇₉	—	4	2	no	—
127	C₂³×C₇₉	#127	1	1	yes	characteristic
128	Q₈×C₇₉	—	4	4	no	—
129	D₄×C₇₉	—	4	4	no	—
130	D₄×C₇₉	—	4	4	no	—
131	D₄×C₇₉	—	4	4	no	—
132	D₄×C₇₉	—	8	4	no	—
133	C₂×C₄×C₇₉	—	2	2	no	—
134	C₂×C₄×C₇₉	—	2	2	no	—
135	C₂×C₄×C₇₉	—	4	2	no	—
136	C₂×C₄×C₇₉	—	8	4	no	—
137	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
138	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
139	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
140	C₂×C₄×C₇₉	#135	1	4	yes	—
141	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₇₉	—	4	2	no	—
145	C₂×Dic₇₉	—	4	2	no	—
146	Dic₁₅₈	—	4	4	no	—
147	Dic₁₅₈	—	4	4	no	—
148	C₈×C₇₉	—	2	2	no	—
149	C₈×C₇₉	#135	1	4	yes	—
150	C₇₉⋊C₈	—	2	2	no	—
151	C₇₉⋊C₈	—	2	2	no	—
152	C₇₉⋊C₈	—	2	2	no	—
153	C₇₉⋊C₈	#135	1	4	yes	—
154	C₂²×C₇₉	—	2	1	no	—
155	C₂²×C₇₉	—	2	1	no	—
156	C₂²×C₇₉	—	8	2	no	—
157	C₂²×C₇₉	—	8	2	no	—
158	C₂²×C₇₉	—	8	2	no	—
159	C₂²×C₇₉	#495	1	2	yes	—
160	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
161	C₄×C₇₉	—	2	1	no	—
162	C₄×C₇₉	—	2	1	no	—
163	C₄×C₇₉	—	8	2	no	—
164	C₄×C₇₉	#495	1	2	yes	—
165	C₄×C₇₉	C₂×#135	1	2	yes	—
166	C₄×C₇₉	C₂×#135	1	2	yes	—
167	C₄×C₇₉	C₂×#135	1	2	yes	—
168	Dic₇₉	—	8	2	no	—
169	Dic₇₉	—	8	2	no	—
170	#19600	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	—	2	2	no	—
172	C₂×C₇₉	—	16	2	no	—
173	C₂×C₇₉	C₂×#495	1	1	yes	characteristic
174	C₂×C₇₉	#3147	1	1	yes	characteristic
175	C₂×C₇₉	#3147	1	1	yes	characteristic
176	#3170	—	79	1	no	—
177	#3169	D₇₉	1	1	yes	characteristic
178	#3167	—	79	2	no	—
179	#3131	—	79	2	no	—
180	#1711	—	79	2	no	—
181	#1710	—	79	1	no	—
182	#1709	—	79	1	no	—
183	#1708	—	79	2	no	—
184	#1707	—	79	1	no	—
185	#1706	—	79	1	no	—
186	C₂×C₈²	—	79	1	no	—
187	C₇₉	#19600	1	1	yes	characteristic, complemented, Sylow
188	#533	—	79	2	no	—
189	#532	C₂×D₇₉	1	2	yes	—
190	#500	—	79	1	no	—
191	#500	C₂×D₇₉	1	1	yes	characteristic
192	#498	—	79	2	no	—
193	#494	—	79	1	no	—
194	#493	C₂×D₇₉	1	1	yes	characteristic
195	#492	—	79	2	no	—
196	#488	—	158	4	no	—
197	#487	—	79	2	no	—
198	#487	—	79	2	no	—
199	#487	—	79	2	no	—
200	#487	—	79	2	no	—
201	#486	—	79	2	no	—
202	#486	—	79	2	no	—
203	#486	—	79	2	no	—
204	#486	C₂×D₇₉	1	2	yes	—
205	#467	—	158	2	no	—
206	#466	—	2	2	no	—
207	#465	—	158	4	no	—
208	C₄×Q₁₆	—	158	2	no	—
209	C₄×SD₁₆	—	158	4	no	—
210	C₄×D₈	—	2	2	no	—
211	#416	—	158	4	no	—
212	C₂×C₄×C₈	—	79	1	no	—
213	C₂×C₄×C₈	—	79	1	no	—
214	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
215	C₈²	—	79	1	no	—
216	C₈²	—	79	1	no	—
217	C₈²	—	79	1	no	—
218	C₈²	—	79	1	no	—
219	C₂×#41	—	2	2	no	—
220	C₂×Q₁₆	—	158	2	no	—
221	C₂×SD₁₆	—	158	4	no	—
222	C₂×D₈	—	2	2	no	—
223	C₂²×C₈	—	79	2	no	—
224	C₂²×C₈	—	79	2	no	—
225	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
226	C₄⋊Q₈	—	79	2	no	—
227	C₄⋊Q₈	—	79	2	no	—
228	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
229	#127	C₂²×D₇₉	1	2	yes	—
230	#125	—	158	2	no	—
231	#124	—	2	2	no	—
232	C₂²⋊Q₈	—	158	2	no	—
233	C₂²⋊Q₈	—	158	2	no	—
234	#121	—	2	2	no	—
235	#121	—	2	2	no	—
236	C₄×Q₈	—	158	2	no	—
237	C₄×Q₈	—	158	2	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	—	2	2	no	—
240	#117	—	158	2	no	—
241	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
242	#107	—	2	2	no	—
243	#107	—	158	2	no	—
244	#106	—	158	4	no	—
245	#103	—	158	4	no	—
246	#103	—	158	4	no	—
247	#103	—	158	4	no	—
248	#103	—	158	4	no	—
249	#102	—	2	4	no	—
250	#102	—	158	4	no	—
251	#102	—	158	4	no	—
252	#102	—	158	4	no	—
253	C₄×C₈	—	79	2	no	—
254	C₄×C₈	—	79	2	no	—
255	C₄×C₈	—	79	1	no	—
256	C₄×C₈	—	79	1	no	—
257	C₄×C₈	—	79	1	no	—
258	C₄×C₈	—	79	1	no	—
259	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
260	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
262	#41	—	4	8	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	158	2	no	—
265	C₂×Q₈	—	158	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
274	Q₁₆	—	316	4	no	—
275	SD₁₆	—	316	8	no	—
276	D₈	—	4	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	79	2	no	—
279	C₂×C₈	—	79	2	no	—
280	C₂×C₈	—	79	2	no	—
281	C₂×C₈	—	79	2	no	—
282	C₂×C₈	—	158	4	no	—
283	C₂×C₈	—	158	4	no	—
284	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
285	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	316	2	no	—
289	C₄⋊C₄	—	316	2	no	—
290	C₄⋊C₄	—	158	2	no	—
291	C₄⋊C₄	—	158	2	no	—
292	C₄⋊C₄	—	158	2	no	—
293	C₄⋊C₄	—	158	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	316	2	no	—
297	C₄²	—	316	2	no	—
298	C₄²	D₄×D₇₉	1	1	yes	characteristic
299	C₄²	D₄×D₇₉	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
301	C₄²	C₂³×D₇₉	1	1	yes	characteristic
302	C₂³	—	4	2	no	—
303	C₂³	#14579157	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	316	4	no	—
306	Q₈	—	316	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	8	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	—	316	2	no	—
316	C₂×C₄	—	316	2	no	—
317	C₂×C₄	#14579157	1	1	yes	characteristic
318	C₂×C₄	#14579157	1	1	yes	characteristic
319	C₂×C₄	#14579157	1	1	yes	characteristic
320	C₂×C₄	#14579180	1	4	yes	—
321	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	79	4	no	—
326	C₈	—	158	2	no	—
327	C₈	—	158	2	no	—
328	C₈	—	158	2	no	—
329	C₈	#14579181	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	#19098946	1	2	yes	—
336	C₂²	C₂×#14579157	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	632	2	no	—
341	C₄	—	632	2	no	—
342	C₄	#19098946	1	2	yes	—
343	C₄	#135×D₇₉	1	2	yes	—
344	C₄	C₂×#14579180	1	2	yes	—
345	C₄	C₂×#14579181	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₂×#19098946	1	1	yes	characteristic
349	C₂	#23773824	1	1	yes	characteristic
350	C₂	#23773826	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579157
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting, prime radical	C₇₉×#3169	C₂
socle	C₂²×C₇₉	C₂×#127
hypercenter	#3169	D₇₉
hypocenter	C₇₉	#19600
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #23773986	C₂	semidirect
2. #23773955	C₂	semidirect
3. #23773722	C₂	semidirect
4. #23763011	C₂	semidirect
5. #23763009	C₂	semidirect
6. #23763005	C₂	semidirect
7. #23763000	C₂	semidirect
8. #23762998	C₂	semidirect
9. #23762989	C₂	semidirect
10. C₂×C₈×C₇₉⋊C₈	C₂	semidirect
187. C₇₉	#19600	semidirect

Maximal subgroups

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

subgroup	index
#23773986	2
#23773955	2
#23773722	2
#23763011	2
#23763009	2
#23763005	2
#23763000	2
#23762998	2
#23762989	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#3169	2
#19600	79