Subgroups — #34441628

Order 20224.

718 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic
2	#23772247	C₂	1	2	yes	maximal, complemented
3	#23765277	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765276	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14579102	C₂	1	2	yes	maximal, complemented
7	#23765244	C₂	1	2	yes	maximal, complemented
8	#23765183	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765114	C₂	1	2	yes	maximal, complemented
10	#23764998	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₇₉	C₂²	1	2	yes	—
13	#19099035	—	2	2	no	—
14	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
15	C₂×#14579102	C₂²	1	2	yes	—
16	D₁₆×D₇₉	C₄	1	8	yes	complemented
17	C₂×#14579100	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
19	#19098673	—	2	2	no	—
20	#19098672	—	2	2	no	—
21	C₄×D₆₃₂	C₂²	1	2	yes	complemented
22	D₈×Dic₇₉	C₂²	1	2	yes	complemented
23	#19098645	C₂²	1	2	yes	complemented
24	#19098600	C₂²	1	2	yes	—
25	#19098566	—	2	2	no	—
26	#19098517	C₂²	1	1	yes	characteristic
27	#19098391	C₂²	1	1	yes	characteristic, complemented
28	#19098388	C₂²	1	2	yes	complemented
29	#19098365	C₂²	1	2	yes	complemented
30	#19098364	C₂²	1	2	yes	complemented
31	#19098363	C₂²	1	2	yes	complemented
32	#19098338	C₂²	1	1	yes	characteristic, complemented
33	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
34	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₇₉	C₂²	1	2	yes	complemented
37	#367×C₇₉	C₂²	1	1	yes	characteristic, complemented
38	#357×C₇₉	C₂²	1	2	yes	complemented
39	#346×C₇₉	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₇₉	—	2	2	no	—
41	C₂×D₆₃₂	C₂³	1	2	yes	—
42	C₂×#11929609	—	2	2	no	—
43	D₈×D₇₉	—	2	4	no	—
44	D₈×D₇₉	C₂×C₄	1	8	yes	complemented
45	C₂×#11929607	D₄	1	1	yes	characteristic
46	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
47	C₄×D₃₁₆	—	2	2	no	—
48	D₄×Dic₇₉	—	2	2	no	—
49	#14579123	—	2	2	no	—
50	C₄×C₂²⋊D₇₉	—	4	2	no	—
51	(C₂²⋊D₇₉)⋊C₄	—	4	2	no	—
52	#14579119	—	2	2	no	—
53	#14579118	D₄	1	1	yes	characteristic
54	C₂×(C₂×D₇₉)⋊C₄	—	4	2	no	—
55	C₂²⋊C₄×D₇₉	—	4	2	no	—
56	D₁₂₆₄	C₂×C₄	1	4	yes	complemented
57	#14579102	C₂×C₄	1	8	yes	complemented
58	#14579100	—	2	2	no	—
59	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₇₉	C₂×C₄	1	4	yes	complemented
62	#14579095	C₂³	1	2	yes	—
63	#14579092	—	2	2	no	—
64	#14579083	—	2	2	no	—
65	#14579082	—	2	2	no	—
66	#14579081	—	2	2	no	—
67	#14579072	D₄	1	1	yes	characteristic
68	#14579071	D₄	1	1	yes	characteristic
69	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₇₉	C₂³	1	2	yes	—
71	C₄×D₄×C₇₉	—	2	2	no	—
72	D₁₆×C₇₉	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
75	#107×C₇₉	C₂³	1	2	yes	—
76	#102×C₇₉	—	2	2	no	—
77	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	C₂³×D₇₉	—	4	2	no	—
79	C₂×D₃₁₆	—	2	2	no	—
80	C₂×C₂²⋊D₇₉	—	4	2	no	—
81	D₄×D₇₉	—	2	8	no	—
82	D₄×D₇₉	—	4	4	no	—
83	C₂²×Dic₇₉	—	4	2	no	—
84	C₂×C₄×D₇₉	—	4	2	no	—
85	C₂×C₄×D₇₉	—	4	2	no	—
86	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
87	D₆₃₂	—	2	2	no	—
88	D₆₃₂	C₂²×C₄	1	4	yes	—
89	#11929609	—	2	8	no	—
90	#11929607	—	2	2	no	—
91	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₈×D₇₉	C₂²×C₄	1	2	yes	—
94	C₈×D₇₉	#41	1	2	yes	—
95	C₄⋊Dic₇₉	—	2	2	no	—
96	Dic₇₉⋊C₄	D₈	1	2	yes	—
97	C₂²⋊Dic₇₉	—	4	2	no	—
98	(C₂×D₇₉)⋊C₄	—	4	2	no	—
99	(C₂×D₇₉)⋊C₄	—	4	2	no	—
100	(C₂×D₇₉)⋊C₄	—	4	2	no	—
101	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
102	C₄×Dic₇₉	—	4	2	no	—
103	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
104	C₇₉⋊C₁₆	—	2	1	no	—
105	C₇₉⋊C₁₆	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₇₉	—	2	2	no	—
107	C₂²×C₄×C₇₉	—	4	2	no	—
108	D₈×C₇₉	—	2	2	no	—
109	D₈×C₇₉	C₂²×C₄	1	4	yes	—
110	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₇₉	—	2	2	no	—
113	C₂²⋊C₄×C₇₉	—	4	2	no	—
114	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₇₉	—	2	1	no	—
116	C₁₆×C₇₉	C₂²×C₄	1	2	yes	—
117	C₂³×C₇₉	—	4	2	no	—
118	C₂²×D₇₉	—	4	2	no	—
119	C₂²×D₇₉	—	4	8	no	—
120	C₂²×D₇₉	—	16	2	no	—
121	C₂²×D₇₉	—	16	2	no	—
122	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
123	D₄×C₇₉	—	2	4	no	—
124	D₄×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	—	4	2	no	—
126	C₂×C₄×C₇₉	—	16	2	no	—
127	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₇₉	—	4	8	no	—
130	C₂×Dic₇₉	—	4	2	no	—
131	C₂×Dic₇₉	—	16	2	no	—
132	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
133	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
134	D₃₁₆	—	2	4	no	—
135	D₃₁₆	—	4	2	no	—
136	C₄×D₇₉	—	16	2	no	—
137	C₄×D₇₉	—	16	2	no	—
138	C₄×D₇₉	C₄×D₄	1	2	yes	—
139	C₄×D₇₉	#135	1	2	yes	—
140	C₈×C₇₉	—	2	1	no	—
141	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
142	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
143	C₇₉⋊C₈	—	2	1	no	—
144	C₇₉⋊C₈	C₄×D₄	1	2	yes	—
145	C₂²×C₇₉	—	4	4	no	—
146	C₂²×C₇₉	—	8	2	no	—
147	C₂²×C₇₉	—	8	2	no	—
148	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₇₉	—	2	2	no	—
150	C₂×D₇₉	—	4	4	no	—
151	C₂×D₇₉	—	8	2	no	—
152	C₂×D₇₉	—	8	2	no	—
153	C₂×D₇₉	—	16	8	no	—
154	C₂×D₇₉	C₄×D₈	1	2	yes	—
155	C₄×C₇₉	—	2	2	no	—
156	C₄×C₇₉	—	8	2	no	—
157	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
159	Dic₇₉	—	2	2	no	—
160	Dic₇₉	—	8	2	no	—
161	Dic₇₉	C₄×D₈	1	2	yes	—
162	#33456	—	79	1	no	maximal, Sylow
163	C₂×C₇₉	—	8	4	no	—
164	C₂×C₇₉	C₂×C₄×D₈	1	1	yes	characteristic
165	C₂×C₇₉	#3415	1	2	yes	—
166	D₇₉	—	2	2	no	—
167	D₇₉	—	8	4	no	—
168	C₂²×D₁₆	—	79	1	no	—
169	#2948	—	79	2	no	—
170	#2177	—	79	1	no	—
171	#2177	—	79	1	no	—
172	#2177	—	79	1	no	—
173	#2177	D₇₉	1	1	yes	characteristic, complemented
174	C₄×D₁₆	—	79	2	no	—
175	C₄×D₁₆	—	79	2	no	—
176	#2158	—	79	1	no	—
177	#2140	—	79	2	no	—
178	#2108	—	79	1	no	—
179	C₇₉	#33456	1	1	yes	characteristic, complemented, Sylow
180	C₂²×D₈	—	79	2	no	—
181	#518	—	158	2	no	—
182	C₂×D₁₆	—	79	2	no	—
183	C₂×D₁₆	—	79	8	no	—
184	C₂×D₁₆	—	79	1	no	—
185	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
186	C₂×M₃₂	—	79	1	no	—
187	C₂²×C₁₆	—	79	1	no	—
188	#442	—	158	2	no	—
189	#442	—	158	2	no	—
190	C₄×D₈	—	79	2	no	—
191	C₄×D₈	—	79	2	no	—
192	C₄×D₈	—	79	2	no	—
193	C₄×D₈	C₂×D₇₉	1	2	yes	complemented
194	#427	—	79	2	no	—
195	#418	—	158	2	no	—
196	#405	—	79	1	no	—
197	#367	—	79	1	no	—
198	#367	C₂×D₇₉	1	1	yes	characteristic, complemented
199	#366	—	79	2	no	—
200	#357	—	79	2	no	—
201	#357	—	79	2	no	—
202	#357	—	79	2	no	—
203	#357	C₂×D₇₉	1	2	yes	complemented
204	#346	—	79	1	no	—
205	#346	C₂×D₇₉	1	1	yes	characteristic, complemented
206	C₄×C₁₆	—	79	1	no	—
207	C₄×C₁₆	—	79	1	no	—
208	C₂²×D₄	—	158	2	no	—
209	C₂×D₈	—	79	8	no	—
210	C₂×D₈	—	79	2	no	—
211	C₂×D₈	—	158	4	no	—
212	C₂×D₈	—	158	2	no	—
213	C₂×D₈	C₂²×D₇₉	1	2	yes	—
214	C₂×M₁₆	—	79	1	no	—
215	C₂²×C₈	—	79	1	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	316	2	no	—
218	C₄×D₄	—	316	2	no	—
219	C₄×D₄	—	158	2	no	—
220	C₄×D₄	—	158	2	no	—
221	C₄×D₄	—	158	2	no	—
222	#117	—	79	1	no	—
223	#117	—	158	2	no	—
224	C₂×C₂²⋊C₄	—	316	2	no	—
225	C₂×C₂²⋊C₄	—	316	2	no	—
226	D₁₆	—	79	8	no	—
227	D₁₆	—	79	4	no	—
228	D₁₆	C₄×D₇₉	1	4	yes	complemented
229	M₃₂	—	158	2	no	—
230	C₂×C₁₆	—	79	1	no	—
231	C₂×C₁₆	—	79	4	no	—
232	C₂×C₁₆	—	79	1	no	—
233	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
235	#107	—	79	2	no	—
236	#107	C₂²×D₇₉	1	2	yes	—
237	#106	—	158	2	no	—
238	#102	—	2	2	no	—
239	#102	—	158	2	no	—
240	#102	—	158	2	no	—
241	#102	—	158	2	no	—
242	#97	—	79	1	no	—
243	#97	—	79	1	no	—
244	C₄×C₈	—	79	1	no	—
245	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
246	C₂⁴	—	316	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	316	4	no	—
249	C₂×D₄	—	316	2	no	—
250	C₂×D₄	—	158	8	no	—
251	C₂×D₄	—	158	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	316	2	no	—
254	C₂²×C₄	—	316	2	no	—
255	C₂²×C₄	—	316	2	no	—
256	C₂²×C₄	—	79	1	no	—
257	D₈	—	2	2	no	—
258	D₈	—	79	4	no	—
259	D₈	—	158	8	no	—
260	D₈	—	158	2	no	—
261	D₈	C₂×C₄×D₇₉	1	4	yes	—
262	M₁₆	—	158	2	no	—
263	C₂×C₈	—	79	1	no	—
264	C₂×C₈	—	79	2	no	—
265	C₂×C₈	—	79	2	no	—
266	C₂×C₈	—	79	1	no	—
267	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	79	2	no	—
271	C₄⋊C₄	—	158	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	316	2	no	—
274	C₂²⋊C₄	—	316	2	no	—
275	C₂²⋊C₄	—	316	2	no	—
276	C₂²⋊C₄	—	316	2	no	—
277	C₂²⋊C₄	—	79	2	no	—
278	C₄²	—	316	2	no	—
279	C₄²	—	79	1	no	—
280	C₄²	D₄×D₇₉	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	79	2	no	—
283	C₁₆	—	158	1	no	—
284	C₁₆	C₂×C₄×D₇₉	1	2	yes	—
285	C₂³	—	4	2	no	—
286	C₂³	—	316	8	no	—
287	C₂³	—	316	2	no	—
288	C₂³	—	79	1	no	—
289	C₂³	—	1264	2	no	—
290	C₂³	—	1264	2	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	316	8	no	—
294	D₄	—	316	2	no	—
295	D₄	—	158	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	—	316	2	no	—
299	C₂×C₄	—	79	1	no	—
300	C₂×C₄	—	79	2	no	—
301	C₂×C₄	—	79	2	no	—
302	C₂×C₄	—	79	2	no	—
303	C₂×C₄	—	1264	2	no	—
304	C₂×C₄	—	1264	2	no	—
305	C₂×C₄	—	1264	2	no	—
306	C₂×C₄	D₈×D₇₉	1	2	yes	—
307	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	79	2	no	—
310	C₈	—	158	1	no	—
311	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
312	C₈	#41×D₇₉	1	1	yes	characteristic
313	C₂²	—	4	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	316	4	no	—
317	C₂²	—	79	2	no	—
318	C₂²	—	632	2	no	—
319	C₂²	—	632	2	no	—
320	C₂²	—	158	2	no	—
321	C₂²	—	1264	8	no	—
322	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	79	2	no	—
326	C₄	—	632	2	no	—
327	C₄	—	158	2	no	—
328	C₄	C₄×D₄×D₇₉	1	1	yes	characteristic
329	C₄	#135×D₇₉	1	1	yes	characteristic
330	C₂	—	8	4	no	—
331	C₂	—	632	4	no	—
332	C₂	—	158	2	no	—
333	C₂	C₄×D₈×D₇₉	1	1	yes	characteristic
334	C₂	#23776540	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2177	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×D₁₆×D₇₉	C₂
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	#2177	D₇₉
hypocenter	C₇₉	#33456
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. #23772247	C₂	semidirect
3. #23765277	C₂	semidirect
4. #23765276	C₂	semidirect
5. #23765275	C₂	semidirect
6. C₄×#14579102	C₂	semidirect
7. #23765244	C₂	semidirect
8. #23765183	C₂	semidirect
9. #23765114	C₂	semidirect
10. #23764998	C₂	semidirect
11. C₇₉×#2177	C₂	semidirect
16. D₁₆×D₇₉	C₄	semidirect
21. C₄×D₆₃₂	C₂²	semidirect
22. D₈×Dic₇₉	C₂²	semidirect
23. #19098645	C₂²	semidirect
27. #19098391	C₂²	semidirect
28. #19098388	C₂²	semidirect
29. #19098365	C₂²	semidirect
30. #19098364	C₂²	semidirect
31. #19098363	C₂²	semidirect
32. #19098338	C₂²	semidirect
33. C₄×C₇₉⋊C₁₆	C₂²	semidirect
34. C₁₆×Dic₇₉	C₂²	semidirect
36. C₄×D₈×C₇₉	C₂²	semidirect
37. #367×C₇₉	C₂²	semidirect
38. #357×C₇₉	C₂²	semidirect
39. #346×C₇₉	C₂²	semidirect
44. D₈×D₇₉	C₂×C₄	semidirect
56. D₁₂₆₄	C₂×C₄	semidirect
57. #14579102	C₂×C₄	semidirect
61. C₁₆×D₇₉	C₂×C₄	semidirect
72. D₁₆×C₇₉	C₂×C₄	semidirect
173. #2177	D₇₉	semidirect
179. C₇₉	#33456	semidirect
193. C₄×D₈	C₂×D₇₉	semidirect
198. #367	C₂×D₇₉	semidirect
203. #357	C₂×D₇₉	semidirect
205. #346	C₂×D₇₉	semidirect
228. D₁₆	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₇₉	2
#23772247	2
#23765277	2
#23765276	2
#23765275	2
C₄×#14579102	2
#23765244	2
#23765183	2
#23765114	2
#23764998	2
C₇₉×#2177	2
#33456	79