Subgroups — #34441358

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₇₉	C₂	1	1	yes	maximal, characteristic
2	#23765204	C₂	1	4	yes	maximal, complemented
3	#23765196	C₂	1	4	yes	maximal, complemented
4	#23765160	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765158	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765014	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#2113	C₂	1	1	yes	maximal, characteristic
10	C₂×#14579168	C₄	1	1	yes	characteristic
11	#41×Dic₇₉	C₄	1	1	yes	characteristic
12	C₂×#14579101	C₂²	1	1	yes	characteristic
13	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	#19098521	C₄	1	2	yes	—
15	#19098521	C₄	1	1	yes	characteristic
16	#19098518	C₂²	1	1	yes	characteristic
17	#19098518	C₂²	1	2	yes	—
18	M₁₆×Dic₇₉	C₄	1	1	yes	characteristic
19	M₁₆×Dic₇₉	C₄	1	2	yes	—
20	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
22	Dic₇₉⋊C₁₆	C₂²	1	4	yes	—
23	Dic₇₉⋊C₁₆	C₂²	1	4	yes	—
24	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	4	yes	—
25	#19098338	C₂²	1	2	yes	—
26	C₁₆×Dic₇₉	C₂²	1	2	yes	—
27	C₁₆×Dic₇₉	C₂²	1	2	yes	—
28	C₁₆×Dic₇₉	C₂²	1	2	yes	—
29	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
31	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
32	C₂²⋊C₁₆×C₇₉	C₂²	1	4	yes	—
33	#14579168	—	2	4	no	—
34	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
35	C₂×#11929608	C₂×C₄	1	2	yes	—
36	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
38	Q₈×Dic₇₉	C₈	1	2	yes	—
39	D₄×Dic₇₉	C₈	1	4	yes	—
40	D₄×Dic₇₉	C₈	1	2	yes	—
41	#14579120	C₂×C₄	1	2	yes	—
42	#14579120	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
45	#14579101	—	2	2	no	—
46	C₂×C₇₉⋊C₁₆	—	2	2	no	—
47	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
49	#14579071	C₂×C₄	1	2	yes	—
50	#14579071	C₂×C₄	1	2	yes	—
51	#14579071	C₂×C₄	1	2	yes	—
52	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
53	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
55	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₇₉	C₂³	1	2	yes	—
57	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
58	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
59	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
60	#131×C₇₉	—	2	2	no	—
61	#131×C₇₉	—	2	2	no	—
62	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
63	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
64	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
65	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	M₃₂×C₇₉	—	2	2	no	—
67	C₂×C₁₆×C₇₉	—	2	2	no	—
68	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
69	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
70	C₂²×Dic₇₉	C₂×C₈	1	4	yes	—
71	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
72	#11929608	—	2	4	no	—
73	#11929608	—	2	2	no	—
74	C₂×C₇₉⋊C₈	—	2	4	no	—
75	C₂×C₇₉⋊C₈	M₁₆	1	4	yes	—
76	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊Dic₇₉	C₂×C₈	1	4	yes	—
80	C₄⋊Dic₇₉	C₂×C₈	1	2	yes	—
81	C₂²⋊Dic₇₉	C₂×C₈	1	4	yes	—
82	C₂²⋊Dic₇₉	C₂×C₈	1	2	yes	—
83	C₄×Dic₇₉	C₂×C₈	1	4	yes	—
84	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
85	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
86	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
87	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
88	C₇₉⋊C₁₆	—	2	2	no	—
89	C₇₉⋊C₁₆	—	2	2	no	—
90	#41×C₇₉	—	2	2	no	—
91	#41×C₇₉	—	2	2	no	—
92	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
93	C₂×D₄×C₇₉	C₂×C₈	1	2	yes	—
94	C₂×D₄×C₇₉	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	M₁₆×C₇₉	—	2	4	no	—
98	M₁₆×C₇₉	—	2	2	no	—
99	C₂×C₈×C₇₉	—	2	2	no	—
100	C₂×C₈×C₇₉	—	2	2	no	—
101	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₇₉	#41	1	2	yes	—
104	C₂×C₈×C₇₉	#41	1	2	yes	—
105	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
106	C₁₆×C₇₉	—	2	2	no	—
107	C₁₆×C₇₉	—	2	2	no	—
108	C₂³×C₇₉	C₂²×C₈	1	2	yes	—
109	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
110	Q₈×C₇₉	—	2	2	no	—
111	D₄×C₇₉	—	2	4	no	—
112	D₄×C₇₉	—	2	2	no	—
113	C₂×C₄×C₇₉	—	2	2	no	—
114	C₂×C₄×C₇₉	—	2	2	no	—
115	C₂×C₄×C₇₉	#117	1	2	yes	—
116	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₇₉	—	2	4	no	—
121	C₂×Dic₇₉	—	2	8	no	—
122	C₂×Dic₇₉	C₂²×C₈	1	4	yes	—
123	C₂×Dic₇₉	C₂²×C₈	1	2	yes	—
124	C₂×Dic₇₉	C₂²×C₈	1	2	yes	—
125	C₈×C₇₉	—	2	2	no	—
126	C₈×C₇₉	#117	1	2	yes	—
127	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
128	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
129	C₇₉⋊C₈	—	2	2	no	—
130	C₇₉⋊C₈	C₂×M₁₆	1	2	yes	—
131	C₇₉⋊C₈	C₂×M₁₆	1	2	yes	—
132	C₂²×C₇₉	—	2	2	no	—
133	C₂²×C₇₉	—	2	4	no	—
134	C₂²×C₇₉	—	2	2	no	—
135	C₂²×C₇₉	#431	1	2	yes	—
136	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
137	C₄×C₇₉	—	2	2	no	—
138	C₄×C₇₉	#431	1	2	yes	—
139	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
140	C₄×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
141	Dic₇₉	—	2	2	no	—
142	Dic₇₉	—	2	4	no	—
143	Dic₇₉	#504	1	4	yes	—
144	#33420	—	79	1	no	maximal, Sylow
145	C₂×C₇₉	—	2	2	no	—
146	C₂×C₇₉	—	4	2	no	—
147	C₂×C₇₉	C₂×#431	1	1	yes	characteristic
148	C₂×C₇₉	C₂×#504	1	2	yes	—
149	C₄×#131	—	79	1	no	—
150	#2164	—	79	4	no	—
151	#2162	—	79	4	no	—
152	#2151	—	79	1	no	—
153	#2151	—	79	1	no	—
154	#2113	—	79	1	no	—
155	#2113	D₇₉	1	1	yes	characteristic
156	C₄×M₃₂	—	79	1	no	—
157	C₂×C₄×C₁₆	—	79	1	no	—
158	C₇₉	#33420	1	1	yes	characteristic, complemented, Sylow
159	C₂×#131	—	79	1	no	—
160	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
161	C₄×#41	—	79	1	no	—
162	C₂×M₃₂	—	79	1	no	—
163	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
164	C₂²×C₁₆	—	79	1	no	—
165	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
166	#405	—	79	1	no	—
167	#405	—	79	2	no	—
168	#405	—	79	1	no	—
169	#405	—	79	2	no	—
170	C₄×M₁₆	—	79	2	no	—
171	C₄×M₁₆	—	79	1	no	—
172	C₂×C₄×C₈	—	79	1	no	—
173	C₂×C₄×C₈	—	79	2	no	—
174	C₄⋊C₁₆	—	79	4	no	—
175	C₄⋊C₁₆	—	79	4	no	—
176	C₂²⋊C₁₆	—	79	4	no	—
177	C₂²⋊C₁₆	C₂×D₇₉	1	4	yes	—
178	#346	—	79	2	no	—
179	C₄×C₁₆	—	79	2	no	—
180	C₄×C₁₆	—	79	2	no	—
181	C₄×C₁₆	—	79	2	no	—
182	C₂×#41	C₄×D₇₉	1	1	yes	characteristic
183	#131	—	2	2	no	—
184	#131	—	2	2	no	—
185	#131	—	158	4	no	—
186	C₂×M₁₆	—	79	2	no	—
187	C₂×M₁₆	—	79	1	no	—
188	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
189	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
190	C₂²×C₈	—	79	1	no	—
191	C₂²×C₈	—	79	2	no	—
192	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
193	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
194	C₄×Q₈	—	79	2	no	—
195	C₄×D₄	—	79	4	no	—
196	C₄×D₄	—	79	2	no	—
197	#117	—	79	2	no	—
198	#117	—	79	1	no	—
199	C₂×C₄²	—	79	2	no	—
200	C₂×C₄²	—	79	1	no	—
201	M₃₂	—	2	2	no	—
202	M₃₂	—	158	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	79	2	no	—
205	C₂×C₁₆	—	79	2	no	—
206	C₂×C₁₆	—	158	2	no	—
207	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
208	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
209	#97	—	79	2	no	—
210	#97	—	79	2	no	—
211	#97	—	79	2	no	—
212	C₄×C₈	—	79	1	no	—
213	C₄×C₈	—	79	2	no	—
214	C₄×C₈	—	79	2	no	—
215	C₄×C₈	—	79	1	no	—
216	C₄×C₈	—	79	1	no	—
217	C₄×C₈	—	79	1	no	—
218	C₄×C₈	—	79	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	C₂×Q₈	C₈×D₇₉	1	1	yes	characteristic
222	C₂×D₄	C₈×D₇₉	1	2	yes	—
223	C₂×D₄	C₈×D₇₉	1	1	yes	characteristic
224	C₂²×C₄	—	79	4	no	—
225	C₂²×C₄	—	79	2	no	—
226	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	158	4	no	—
231	M₁₆	—	158	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	79	4	no	—
235	C₂×C₈	—	79	1	no	—
236	C₂×C₈	—	79	1	no	—
237	C₂×C₈	—	79	2	no	—
238	C₂×C₈	—	158	4	no	—
239	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
241	C₂×C₈	#11929623	1	2	yes	—
242	C₂×C₈	#11929623	1	2	yes	—
243	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
244	C₄⋊C₄	—	79	4	no	—
245	C₄⋊C₄	—	79	2	no	—
246	C₂²⋊C₄	—	79	4	no	—
247	C₂²⋊C₄	—	79	2	no	—
248	C₄²	—	79	4	no	—
249	C₄²	—	79	2	no	—
250	C₄²	—	79	1	no	—
251	C₄²	—	79	1	no	—
252	C₄²	—	79	2	no	—
253	C₁₆	—	2	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	158	2	no	—
256	C₁₆	—	158	2	no	—
257	C₂³	C₂×C₈×D₇₉	1	2	yes	—
258	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	79	2	no	—
265	C₂×C₄	—	79	4	no	—
266	C₂×C₄	—	79	2	no	—
267	C₂×C₄	—	158	4	no	—
268	C₂×C₄	—	158	8	no	—
269	C₂×C₄	#14579117	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	79	2	no	—
276	C₈	—	79	2	no	—
277	C₈	—	158	2	no	—
278	C₈	#14579117	1	2	yes	—
279	C₈	C₂×#11929623	1	1	yes	characteristic
280	C₈	C₂×#11929623	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	#19098613	1	2	yes	—
285	C₂²	C₂²×C₈×D₇₉	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	79	4	no	—
288	C₄	—	158	2	no	—
289	C₄	—	158	4	no	—
290	C₄	#19098613	1	2	yes	—
291	C₄	C₂×#14579117	1	1	yes	characteristic
292	C₄	C₂×M₁₆×D₇₉	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	C₂×#19098613	1	1	yes	characteristic
296	C₂	#504×D₇₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂³×C₈
Fitting	C₇₉×#2113	C₂
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	#2113	D₇₉
hypocenter	C₇₉	#33420
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #23765204	C₂	semidirect
3. #23765196	C₂	semidirect
4. #23765160	C₂	semidirect
5. #23765158	C₂	semidirect
7. M₃₂×Dic₇₉	C₂	semidirect
8. C₂×C₁₆×Dic₇₉	C₂	semidirect
158. C₇₉	#33420	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₇₉	2
#23765204	2
#23765196	2
#23765160	2
#23765158	2
#23765014	2
M₃₂×Dic₇₉	2
C₂×C₁₆×Dic₇₉	2
C₇₉×#2113	2
#33420	79