Subgroups — #34443624

Order 20224.

578 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773911	C₂	1	1	yes	maximal, characteristic
2	#494×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23765644	C₂	1	4	yes	maximal, complemented
4	#23765629	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765628	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765627	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765118	C₂	1	4	yes	maximal
8	C₄×#14579100	C₂	1	1	yes	maximal, characteristic
9	C₇₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₇₉	—	2	1	no	—
11	C₂×Q₁₆×D₇₉	C₂²	1	2	yes	—
12	#19099118	—	2	1	no	—
13	C₄⋊Q₈×D₇₉	—	2	1	no	—
14	C₂×#14579100	C₂²	1	2	yes	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098964	—	2	1	no	—
17	#19098964	C₂²	1	2	yes	—
18	#19098959	—	2	1	no	—
19	#19098943	C₂²	1	1	yes	characteristic
20	#19098942	—	2	1	no	—
21	#19098941	—	2	1	no	—
22	#19098941	C₂²	1	2	yes	—
23	#19098600	—	2	1	no	—
24	#19098600	C₂²	1	2	yes	—
25	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
26	#19098368	C₂²	1	4	yes	—
27	#19098367	C₂²	1	4	yes	—
28	#19098366	C₂²	1	4	yes	—
29	#19098338	C₂²	1	2	yes	—
30	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
31	#500×C₇₉	C₂²	1	1	yes	characteristic
32	#494×C₇₉	C₂²	1	1	yes	characteristic
33	#358×C₇₉	C₂²	1	4	yes	—
34	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₇₉	—	2	2	no	—
36	C₂×Dic₃₁₆	—	2	1	no	—
37	C₂×Dic₃₁₆	C₂³	1	2	yes	—
38	C₂×#11929614	—	2	2	no	—
39	C₂×#11929614	—	2	2	no	—
40	Q₁₆×D₇₉	—	2	8	no	—
41	Q₁₆×D₇₉	—	4	4	no	—
42	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
43	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
44	#14579161	—	2	1	no	—
45	#14579161	—	2	1	no	—
46	#14579160	—	4	1	no	—
47	#14579160	—	4	1	no	—
48	#14579159	—	2	2	no	—
49	#14579152	—	2	2	no	—
50	#14579141	—	4	1	no	—
51	#14579137	—	4	1	no	—
52	Q₈×Dic₇₉	—	4	1	no	—
53	#14579126	—	4	1	no	—
54	#14579119	—	2	2	no	—
55	C₄⋊C₄×D₇₉	—	4	1	no	—
56	C₄⋊C₄×D₇₉	—	4	1	no	—
57	C₄²×D₇₉	D₄	1	1	yes	characteristic
58	#14579100	—	2	4	no	—
59	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
60	#14579095	—	2	1	no	—
61	#14579095	C₂³	1	2	yes	—
62	#14579092	—	2	2	no	—
63	#14579092	—	2	2	no	—
64	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₇₉	D₄	1	1	yes	characteristic
66	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₇₉	—	2	1	no	—
68	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₇₉	—	2	1	no	—
70	C₄⋊Q₈×C₇₉	—	2	1	no	—
71	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
72	#107×C₇₉	—	2	1	no	—
73	#107×C₇₉	C₂³	1	2	yes	—
74	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₅₈	—	2	2	no	—
76	C₂×Dic₁₅₈	—	4	2	no	—
77	C₂×Dic₁₅₈	—	4	1	no	—
78	Q₈×D₇₉	—	4	4	no	—
79	Q₈×D₇₉	—	4	4	no	—
80	C₂×C₄×D₇₉	—	4	2	no	—
81	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
82	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
83	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
84	Dic₃₁₆	—	2	4	no	—
85	Dic₃₁₆	—	4	2	no	—
86	#11929614	—	4	4	no	—
87	#11929614	—	4	4	no	—
88	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₈×D₇₉	SD₁₆	1	4	yes	—
91	C₈×D₇₉	#41	1	4	yes	—
92	C₄⋊Dic₇₉	—	2	2	no	—
93	C₄⋊Dic₇₉	—	4	2	no	—
94	C₄⋊Dic₇₉	—	4	1	no	—
95	C₄⋊Dic₇₉	—	4	1	no	—
96	C₄⋊Dic₇₉	—	4	1	no	—
97	Dic₇₉⋊C₄	—	4	2	no	—
98	Dic₇₉⋊C₄	—	4	2	no	—
99	Dic₇₉⋊C₄	—	4	2	no	—
100	(C₂×D₇₉)⋊C₄	—	4	2	no	—
101	C₄×Dic₇₉	—	4	2	no	—
102	C₄×Dic₇₉	D₈	1	1	yes	characteristic
103	C₄×Dic₇₉	D₈	1	1	yes	characteristic
104	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
105	C₇₉⋊C₁₆	—	2	2	no	—
106	C₂×Q₈×C₇₉	—	2	2	no	—
107	C₂×Q₈×C₇₉	—	4	1	no	—
108	Q₁₆×C₇₉	—	2	4	no	—
109	Q₁₆×C₇₉	—	4	2	no	—
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	C₄⋊C₄×C₇₉	—	4	1	no	—
114	C₄⋊C₄×C₇₉	—	4	1	no	—
115	C₄⋊C₄×C₇₉	—	4	1	no	—
116	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₉	—	2	2	no	—
118	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₇₉	—	4	2	no	—
120	Q₈×C₇₉	—	4	2	no	—
121	Q₈×C₇₉	—	8	2	no	—
122	C₂×C₄×C₇₉	—	4	2	no	—
123	C₂×C₄×C₇₉	—	4	2	no	—
124	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₇₉	—	4	2	no	—
128	C₂×Dic₇₉	—	4	2	no	—
129	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
132	C₄×D₇₉	—	2	2	no	—
133	C₄×D₇₉	—	2	2	no	—
134	C₄×D₇₉	—	8	4	no	—
135	C₄×D₇₉	#124	1	2	yes	—
136	C₄×D₇₉	C₂×SD₁₆	1	2	yes	—
137	Dic₁₅₈	—	4	2	no	—
138	Dic₁₅₈	—	4	2	no	—
139	Dic₁₅₈	—	8	2	no	—
140	Dic₁₅₈	—	8	4	no	—
141	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
142	C₈×C₇₉	C₂×#41	1	2	yes	—
143	C₇₉⋊C₈	#124	1	2	yes	—
144	C₇₉⋊C₈	C₂×SD₁₆	1	2	yes	—
145	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₇₉	—	2	1	no	—
147	C₂×D₇₉	—	2	1	no	—
148	C₂×D₇₉	#486	1	2	yes	—
149	C₄×C₇₉	—	8	2	no	—
150	C₄×C₇₉	—	8	2	no	—
151	C₄×C₇₉	#510	1	2	yes	—
152	C₄×C₇₉	#510	1	2	yes	—
153	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
154	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
155	Dic₇₉	—	2	1	no	—
156	Dic₇₉	—	2	1	no	—
157	Dic₇₉	—	8	2	no	—
158	Dic₇₉	—	8	2	no	—
159	Dic₇₉	#486	1	2	yes	—
160	#33705	—	79	1	no	maximal, Sylow
161	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
162	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
163	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
164	D₇₉	—	2	2	no	—
165	#3160	—	79	1	no	—
166	C₂×#494	—	79	1	no	—
167	#2245	—	79	4	no	—
168	#2241	—	79	1	no	—
169	#2241	—	79	1	no	—
170	#2241	—	79	1	no	—
171	#2241	D₇₉	1	1	yes	characteristic, complemented
172	#2141	—	79	4	no	—
173	C₄×M₃₂	—	79	1	no	—
174	C₇₉	#33705	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₁₆	—	79	2	no	—
176	C₂²×Q₁₆	—	158	1	no	—
177	#533	—	158	1	no	—
178	C₂×C₄⋊Q₈	—	158	1	no	—
179	C₂×M₃₂	—	79	2	no	—
180	#500	—	79	1	no	—
181	#500	C₂×D₇₉	1	1	yes	characteristic
182	#499	—	79	2	no	—
183	#499	—	158	1	no	—
184	#498	—	158	1	no	—
185	#494	—	79	2	no	—
186	#494	—	79	1	no	—
187	#494	—	158	1	no	—
188	#494	—	158	1	no	—
189	#494	C₂×D₇₉	1	1	yes	characteristic
190	#427	—	79	2	no	—
191	#427	—	158	1	no	—
192	C₂×C₄×C₈	—	79	1	no	—
193	#358	—	79	4	no	—
194	#358	—	79	4	no	—
195	#358	—	79	4	no	—
196	#358	C₂×D₇₉	1	4	yes	—
197	#346	—	79	2	no	—
198	C₄×C₁₆	—	79	1	no	—
199	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
200	C₂²×Q₈	—	158	2	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	316	4	no	—
203	C₂×Q₁₆	—	79	2	no	—
204	C₂×Q₁₆	—	158	8	no	—
205	C₂×Q₁₆	—	158	2	no	—
206	C₂×Q₁₆	—	158	2	no	—
207	C₂×Q₁₆	—	158	1	no	—
208	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
209	C₂²×C₈	—	79	1	no	—
210	C₂²×C₈	—	79	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	316	1	no	—
214	C₄⋊Q₈	—	316	1	no	—
215	C₄⋊Q₈	—	158	2	no	—
216	C₄⋊Q₈	—	158	1	no	—
217	C₄⋊Q₈	—	158	1	no	—
218	#125	—	158	2	no	—
219	C₂²⋊Q₈	—	316	1	no	—
220	C₂²⋊Q₈	—	316	1	no	—
221	C₄×Q₈	—	316	1	no	—
222	C₄×Q₈	—	316	1	no	—
223	#117	—	158	2	no	—
224	C₂×C₄⋊C₄	—	316	1	no	—
225	C₂×C₄⋊C₄	—	316	1	no	—
226	C₂×C₄²	—	79	1	no	—
227	M₃₂	—	158	4	no	—
228	C₂×C₁₆	—	79	2	no	—
229	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
230	#107	—	2	1	no	—
231	#107	—	79	2	no	—
232	#107	—	158	1	no	—
233	#107	C₂²×D₇₉	1	2	yes	—
234	#106	—	158	2	no	—
235	#106	—	158	2	no	—
236	C₄×C₈	—	79	1	no	—
237	C₄×C₈	—	79	1	no	—
238	C₄×C₈	—	79	1	no	—
239	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	316	2	no	—
243	C₂×Q₈	—	316	1	no	—
244	C₂×Q₈	—	316	4	no	—
245	C₂×Q₈	—	316	4	no	—
246	C₂×Q₈	—	158	2	no	—
247	C₂²×C₄	—	316	2	no	—
248	C₂²×C₄	—	79	1	no	—
249	C₂²×C₄	—	79	1	no	—
250	C₂²×C₄	—	79	1	no	—
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	316	4	no	—
254	Q₁₆	—	316	4	no	—
255	Q₁₆	—	316	2	no	—
256	Q₁₆	—	158	4	no	—
257	C₂×C₈	—	79	1	no	—
258	C₂×C₈	—	79	1	no	—
259	C₂×C₈	—	79	4	no	—
260	C₂×C₈	—	79	4	no	—
261	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	316	2	no	—
268	C₄⋊C₄	—	316	2	no	—
269	C₄⋊C₄	—	316	2	no	—
270	C₄⋊C₄	—	316	2	no	—
271	C₄⋊C₄	—	316	1	no	—
272	C₄⋊C₄	—	316	1	no	—
273	C₄⋊C₄	—	316	1	no	—
274	C₄⋊C₄	—	158	2	no	—
275	C₂²⋊C₄	—	316	2	no	—
276	C₄²	—	316	2	no	—
277	C₄²	—	79	1	no	—
278	C₄²	—	79	1	no	—
279	C₄²	—	79	1	no	—
280	C₄²	D₄×D₇₉	1	1	yes	characteristic
281	C₁₆	—	2	2	no	—
282	C₁₆	—	158	2	no	—
283	C₂³	—	79	1	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	316	2	no	—
288	Q₈	—	316	2	no	—
289	Q₈	—	632	4	no	—
290	Q₈	—	632	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	316	2	no	—
294	C₂×C₄	—	316	2	no	—
295	C₂×C₄	—	79	1	no	—
296	C₂×C₄	—	79	2	no	—
297	C₂×C₄	—	79	2	no	—
298	C₂×C₄	—	79	1	no	—
299	C₂×C₄	—	79	1	no	—
300	C₂×C₄	—	632	4	no	—
301	C₂×C₄	—	158	2	no	—
302	C₂×C₄	—	158	2	no	—
303	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
304	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
306	C₈	—	79	2	no	—
307	C₈	—	79	2	no	—
308	C₈	SD₁₆×D₇₉	1	2	yes	—
309	C₈	#41×D₇₉	1	2	yes	—
310	C₂²	—	79	2	no	—
311	C₂²	—	158	1	no	—
312	C₂²	—	158	1	no	—
313	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	79	2	no	—
317	C₄	—	632	2	no	—
318	C₄	—	632	2	no	—
319	C₄	—	158	1	no	—
320	C₄	—	158	1	no	—
321	C₄	#19099002	1	2	yes	—
322	C₄	#19099005	1	2	yes	—
323	C₄	#124×D₇₉	1	1	yes	characteristic
324	C₄	C₂×SD₁₆×D₇₉	1	1	yes	characteristic
325	C₂	—	158	2	no	—
326	C₂	#486×D₇₉	1	1	yes	characteristic
327	C₂	C₂×#19099002	1	1	yes	characteristic
328	C₂	C₂×#19099005	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2241	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂²×D₇₉	C₂×D₈
hypercenter	#2241	D₇₉
hypocenter	C₇₉	#33705
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
3. #23765644	C₂	semidirect
4. #23765629	C₂	semidirect
5. #23765628	C₂	semidirect
6. #23765627	C₂	semidirect
9. C₇₉×#2241	C₂	semidirect
171. #2241	D₇₉	semidirect
174. C₇₉	#33705	semidirect

Maximal subgroups

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

subgroup	index
#23773911	2
#494×D₇₉	2
#23765644	2
#23765629	2
#23765628	2
#23765627	2
#23765118	2
C₄×#14579100	2
C₇₉×#2241	2
#33705	79