Subgroups — #34203380

Order 20224.

418 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098980	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098347	C₂	1	1	yes	maximal, characteristic
3	#23764997	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098339	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761489	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761482	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1764	C₂	1	1	yes	maximal, characteristic
8	#19098980	—	2	2	no	—
9	#19098980	C₄	1	4	yes	complemented
10	C₂×#14579101	—	2	1	no	—
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	C₂×#14579101	C₂²	1	1	yes	characteristic
13	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
14	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#19098347	C₄	1	4	yes	complemented
17	#19098339	—	2	1	no	—
18	#19098339	—	2	2	no	—
19	C₄×C₇₉⋊C₁₆	—	2	1	no	—
20	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#104×C₇₉	C₄	1	1	yes	characteristic
22	#416×C₇₉	C₄	1	1	yes	characteristic
23	#405×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	#341×C₇₉	C₄	1	1	yes	characteristic
26	#339×C₇₉	C₄	1	1	yes	characteristic
27	#14579101	—	2	4	no	—
28	#14579101	—	2	1	no	—
29	#14579101	—	2	1	no	—
30	#14579101	C₂×C₄	1	2	yes	—
31	#14579101	C₂×C₄	1	2	yes	—
32	C₂×C₇₉⋊C₁₆	—	2	4	no	—
33	C₂×C₇₉⋊C₁₆	—	2	1	no	—
34	C₂×C₇₉⋊C₁₆	—	2	1	no	—
35	C₂×C₇₉⋊C₁₆	—	2	1	no	—
36	C₂×C₇₉⋊C₁₆	—	2	1	no	—
37	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
42	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
43	#131×C₇₉	—	2	1	no	—
44	#131×C₇₉	—	2	1	no	—
45	#131×C₇₉	C₂×C₄	1	2	yes	—
46	#131×C₇₉	C₂×C₄	1	2	yes	—
47	C₂×M₁₆×C₇₉	—	2	1	no	—
48	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₉	—	2	1	no	—
51	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
55	#104×C₇₉	C₈	1	4	yes	—
56	#103×C₇₉	C₈	1	2	yes	—
57	#102×C₇₉	C₈	1	2	yes	—
58	#97×C₇₉	—	2	1	no	—
59	C₄×C₈×C₇₉	—	2	1	no	—
60	C₄×C₈×C₇₉	—	2	1	no	—
61	C₄×C₈×C₇₉	—	2	1	no	—
62	C₇₉⋊C₁₆	—	2	2	no	—
63	C₇₉⋊C₁₆	—	2	2	no	—
64	C₇₉⋊C₁₆	—	2	2	no	—
65	C₇₉⋊C₁₆	C₂²⋊C₄	1	2	yes	—
66	C₇₉⋊C₁₆	C₂²⋊C₄	1	2	yes	—
67	#41×C₇₉	—	2	1	no	—
68	#41×C₇₉	—	2	1	no	—
69	#41×C₇₉	C₄²	1	2	yes	—
70	#41×C₇₉	C₂×C₈	1	2	yes	—
71	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
72	C₂×D₄×C₇₉	—	2	1	no	—
73	C₂×D₄×C₇₉	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
77	M₁₆×C₇₉	—	2	2	no	—
78	M₁₆×C₇₉	—	2	2	no	—
79	M₁₆×C₇₉	—	2	1	no	—
80	M₁₆×C₇₉	—	2	1	no	—
81	M₁₆×C₇₉	C₄²	1	2	yes	—
82	M₁₆×C₇₉	C₂×C₈	1	2	yes	—
83	C₂×C₈×C₇₉	—	2	2	no	—
84	C₂×C₈×C₇₉	—	2	2	no	—
85	C₂×C₈×C₇₉	—	2	1	no	—
86	C₂×C₈×C₇₉	—	2	1	no	—
87	C₂×C₈×C₇₉	—	2	1	no	—
88	C₂×C₈×C₇₉	—	2	1	no	—
89	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
90	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₇₉	C₂×C₈	1	2	yes	—
97	C₂²⋊C₄×C₇₉	—	2	1	no	—
98	C₄²×C₇₉	—	2	1	no	—
99	C₄²×C₇₉	—	2	1	no	—
100	C₄²×C₇₉	C₂×C₈	1	2	yes	—
101	C₂³×C₇₉	—	2	1	no	—
102	C₂³×C₇₉	#131	1	1	yes	characteristic
103	Q₈×C₇₉	—	2	1	no	—
104	Q₈×C₇₉	C₄×C₈	1	2	yes	—
105	D₄×C₇₉	—	2	2	no	—
106	D₄×C₇₉	—	2	2	no	—
107	D₄×C₇₉	—	2	1	no	—
108	D₄×C₇₉	C₄×C₈	1	2	yes	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	2	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	2	2	no	—
113	C₂×C₄×C₇₉	—	2	2	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
121	C₈×C₇₉	—	2	2	no	—
122	C₈×C₇₉	—	2	2	no	—
123	C₈×C₇₉	—	2	2	no	—
124	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₇₉	#117	1	1	yes	characteristic
126	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
127	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
128	C₂²×C₇₉	—	2	2	no	—
129	C₂²×C₇₉	—	2	2	no	—
130	C₂²×C₇₉	—	4	2	no	—
131	C₂²×C₇₉	#405	1	1	yes	characteristic
132	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
133	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
134	C₄×C₇₉	—	2	2	no	—
135	C₄×C₇₉	—	4	2	no	—
136	C₄×C₇₉	—	4	2	no	—
137	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
139	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
140	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
141	#12615	—	79	1	no	maximal, Sylow
142	C₂×C₇₉	—	2	1	no	—
143	C₂×C₇₉	—	2	1	no	—
144	C₂×C₇₉	—	4	2	no	—
145	C₂×C₇₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₇₉	#2171	1	2	yes	—
147	C₂×#504	—	79	1	no	—
148	C₂×#350	—	79	1	no	—
149	#2108	—	79	1	no	—
150	C₂×#346	—	79	1	no	—
151	#1764	D₇₉	1	1	yes	characteristic
152	#1379	—	79	1	no	—
153	#1376	—	79	1	no	—
154	C₇₉	#12615	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
156	#504	—	79	4	no	—
157	#504	—	158	2	no	—
158	C₂×M₃₂	—	79	1	no	—
159	C₂×M₃₂	—	79	1	no	—
160	C₂×M₃₂	—	158	1	no	—
161	C₂²×C₁₆	—	79	1	no	—
162	C₂²×C₁₆	—	79	1	no	—
163	C₂²×C₁₆	—	158	1	no	—
164	C₂×#104	Dic₇₉	1	1	yes	characteristic
165	#416	Dic₇₉	1	1	yes	characteristic
166	#405	C₂×D₇₉	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
168	#350	—	79	4	no	—
169	#346	—	158	2	no	—
170	#346	—	158	1	no	—
171	C₄×C₁₆	—	158	1	no	—
172	#341	Dic₇₉	1	1	yes	characteristic
173	#339	Dic₇₉	1	1	yes	characteristic
174	C₂×#41	C₂×Dic₇₉	1	1	yes	characteristic
175	#131	—	2	1	no	—
176	#131	—	2	1	no	—
177	#131	C₄×D₇₉	1	2	yes	—
178	#131	C₄×D₇₉	1	2	yes	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
181	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
185	#117	C₂×Dic₇₉	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
187	M₃₂	—	79	2	no	—
188	M₃₂	—	79	2	no	—
189	M₃₂	—	158	4	no	—
190	M₃₂	—	158	1	no	—
191	M₃₂	—	158	1	no	—
192	C₂×C₁₆	—	79	2	no	—
193	C₂×C₁₆	—	79	2	no	—
194	C₂×C₁₆	—	79	1	no	—
195	C₂×C₁₆	—	79	1	no	—
196	C₂×C₁₆	—	79	2	no	—
197	C₂×C₁₆	—	158	4	no	—
198	C₂×C₁₆	—	158	1	no	—
199	C₂×C₁₆	—	158	1	no	—
200	C₂×C₁₆	—	158	1	no	—
201	C₂×C₁₆	—	158	1	no	—
202	#104	C₇₉⋊C₈	1	4	yes	—
203	#103	C₇₉⋊C₈	1	2	yes	—
204	#102	C₇₉⋊C₈	1	2	yes	—
205	#97	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	—	2	1	no	—
211	#41	C₄×Dic₇₉	1	2	yes	—
212	#41	C₂×C₇₉⋊C₈	1	2	yes	—
213	C₂×Q₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	C₄×Dic₇₉	1	2	yes	—
224	M₁₆	C₂×C₇₉⋊C₈	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂×C₇₉⋊C₈	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
234	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
235	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
236	C₂×C₈	#11929623	1	1	yes	characteristic
237	C₂×C₈	#11929623	1	1	yes	characteristic
238	C₄⋊C₄	C₂×C₇₉⋊C₈	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
243	C₁₆	—	79	2	no	—
244	C₁₆	—	79	2	no	—
245	C₁₆	—	158	2	no	—
246	C₁₆	—	158	2	no	—
247	C₁₆	—	158	2	no	—
248	C₂³	—	2	1	no	—
249	C₂³	#14579168	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×C₇₉⋊C₈	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×C₇₉⋊C₈	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
263	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
264	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
267	C₂×C₄	#14579168	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
272	C₈	#14579117	1	1	yes	characteristic
273	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
274	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	#19098519	1	1	yes	characteristic
279	C₂²	D₄×C₇₉⋊C₈	1	1	yes	characteristic
280	C₂²	D₄×C₇₉⋊C₈	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	C₂²⋊C₄×Dic₇₉	1	1	yes	characteristic
285	C₄	C₂×C₄×C₇₉⋊C₈	1	1	yes	characteristic
286	C₄	D₄×C₇₉⋊C₈	1	1	yes	characteristic
287	C₄	D₄×C₇₉⋊C₈	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	4	2	no	—
291	C₂	C₂²⋊C₄×C₇₉⋊C₈	1	1	yes	characteristic
292	C₂	#23765242	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1764	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1764	D₇₉
hypocenter	C₇₉	#12615
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1764

Complements

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

N	H ≅ G/N	type
3. #23764997	C₂	semidirect
4. C₂×#19098339	C₂	semidirect
5. #23761489	C₂	semidirect
6. #23761482	C₂	semidirect
9. #19098980	C₄	semidirect
16. #19098347	C₄	semidirect
154. C₇₉	#12615	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098980	2
C₂×#19098347	2
#23764997	2
C₂×#19098339	2
#23761489	2
#23761482	2
C₇₉×#1764	2
#12615	79