Subgroups — #34204199

Order 20224.

454 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098517	C₂	1	1	yes	maximal, characteristic
2	#23765156	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098344	C₂	1	2	yes	maximal
4	#23761483	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11929607	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₇₉	C₄	1	1	yes	characteristic
8	C₂×#14579118	C₄	1	1	yes	characteristic
9	C₂×#14579101	C₂²	1	2	yes	—
10	#19098517	—	2	2	no	—
11	#19098517	—	2	2	no	—
12	C₂×#14579072	C₄	1	1	yes	characteristic
13	C₂×#14579071	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₉	C₄	1	1	yes	characteristic
15	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
16	#19098344	—	2	4	no	—
17	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₇₉	—	2	2	no	—
20	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂×#11929607	—	2	1	no	—
22	C₂×#11929607	—	2	1	no	—
23	C₂×#11929607	—	2	1	no	—
24	C₂×#11929607	—	2	1	no	—
25	C₂×#11929607	D₄	1	2	yes	—
26	C₂×#11929607	Q₈	1	2	yes	—
27	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₇₉	—	2	2	no	—
30	C₂×C₈×D₇₉	—	2	2	no	—
31	C₂×C₈×D₇₉	—	2	2	no	—
32	#14579118	—	2	2	no	—
33	#14579118	—	2	2	no	—
34	C₂×Dic₇₉⋊C₄	C₈	1	2	yes	—
35	C₂×(C₂×D₇₉)⋊C₄	C₈	1	2	yes	—
36	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
37	#14579101	—	2	2	no	—
38	#14579101	—	2	2	no	—
39	C₂×C₇₉⋊C₁₆	—	2	2	no	—
40	#14579072	—	2	2	no	—
41	#14579071	—	2	1	no	—
42	#14579071	—	2	1	no	—
43	C₈×Dic₇₉	—	2	2	no	—
44	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₇₉	—	2	2	no	—
48	M₃₂×C₇₉	—	2	2	no	—
49	C₂×C₁₆×C₇₉	—	2	2	no	—
50	C₄×C₈×C₇₉	D₄	1	2	yes	—
51	C₄×C₈×C₇₉	D₄	1	2	yes	—
52	C₂³×D₇₉	C₂×C₈	1	1	yes	characteristic
53	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₇₉	—	2	1	no	—
56	C₂×C₄×D₇₉	—	2	1	no	—
57	C₂×C₄×D₇₉	—	2	1	no	—
58	C₂×C₄×D₇₉	—	2	1	no	—
59	C₂×C₄×D₇₉	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×D₇₉	M₁₆	1	2	yes	—
61	#11929607	—	2	2	no	—
62	#11929607	—	2	2	no	—
63	#11929607	—	2	2	no	—
64	#11929607	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	2	1	no	—
66	C₂×C₇₉⋊C₈	—	2	1	no	—
67	C₂×C₇₉⋊C₈	—	2	1	no	—
68	C₂×C₇₉⋊C₈	—	2	1	no	—
69	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₇₉⋊C₈	M₁₆	1	2	yes	—
71	C₈×D₇₉	—	2	8	no	—
72	Dic₇₉⋊C₄	—	2	4	no	—
73	(C₂×D₇₉)⋊C₄	—	2	4	no	—
74	C₄×Dic₇₉	—	2	1	no	—
75	C₄×Dic₇₉	—	2	1	no	—
76	C₇₉⋊C₁₆	—	4	2	no	—
77	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	—	2	1	no	—
80	C₂×C₈×C₇₉	—	2	1	no	—
81	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
88	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₇₉	M₁₆	1	2	yes	—
90	C₁₆×C₇₉	—	4	2	no	—
91	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
92	C₂²×D₇₉	—	2	1	no	—
93	C₂²×D₇₉	—	2	1	no	—
94	C₂²×D₇₉	—	4	2	no	—
95	C₂²×D₇₉	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₇₉	—	4	2	no	—
97	C₂×C₄×C₇₉	C₂²⋊C₈	1	4	yes	—
98	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×Dic₇₉	—	2	1	no	—
105	C₂×Dic₇₉	—	2	1	no	—
106	C₂×Dic₇₉	—	2	2	no	—
107	C₂×Dic₇₉	—	4	2	no	—
108	C₂×Dic₇₉	C₄⋊C₈	1	2	yes	—
109	C₄×D₇₉	—	2	2	no	—
110	C₄×D₇₉	—	2	2	no	—
111	C₄×D₇₉	—	2	2	no	—
112	C₄×D₇₉	—	2	2	no	—
113	C₈×C₇₉	—	2	2	no	—
114	C₈×C₇₉	#121	1	1	yes	characteristic
115	C₈×C₇₉	#121	1	1	yes	characteristic
116	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₇₉⋊C₈	—	2	2	no	—
119	C₇₉⋊C₈	—	2	2	no	—
120	C₂²×C₇₉	—	2	2	no	—
121	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂²×C₇₉	#431	1	1	yes	characteristic
124	C₂×D₇₉	—	2	2	no	—
125	C₂×D₇₉	—	4	2	no	—
126	C₂×D₇₉	—	4	2	no	—
127	C₂×D₇₉	—	4	2	no	—
128	C₄×C₇₉	—	2	4	no	—
129	C₄×C₇₉	#380	1	1	yes	characteristic
130	C₄×C₇₉	#407	1	1	yes	characteristic
131	C₄×C₇₉	#423	1	1	yes	characteristic
132	C₄×C₇₉	#432	1	1	yes	characteristic
133	Dic₇₉	—	2	2	no	—
134	Dic₇₉	—	4	2	no	—
135	#12786	—	79	1	no	maximal, Sylow
136	C₂×C₇₉	—	2	1	no	—
137	C₂×C₇₉	—	2	1	no	—
138	C₂×C₇₉	#1797	1	1	yes	characteristic
139	C₂×C₇₉	#2115	1	2	yes	—
140	D₇₉	—	4	2	no	—
141	C₂×#405	—	79	1	no	—
142	#2150	—	79	1	no	—
143	#2150	D₇₉	1	1	yes	characteristic, complemented
144	C₂×#349	—	79	2	no	—
145	#1377	—	79	2	no	—
146	C₇₉	#12786	1	1	yes	characteristic, complemented, Sylow
147	C₂²×M₁₆	—	79	1	no	—
148	C₂³×C₈	—	79	1	no	—
149	C₂×#117	—	79	1	no	—
150	C₂×M₃₂	—	79	2	no	—
151	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
152	#405	—	158	2	no	—
153	#405	—	158	2	no	—
154	C₂×#97	—	79	1	no	—
155	C₂×#97	—	79	1	no	—
156	C₂×C₄×C₈	—	79	1	no	—
157	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
158	C₄⋊C₁₆	—	2	2	no	—
159	C₄⋊C₁₆	—	158	2	no	—
160	#349	—	158	4	no	—
161	C₂³×C₄	—	79	1	no	—
162	C₂×M₁₆	—	79	2	no	—
163	C₂×M₁₆	—	79	2	no	—
164	C₂×M₁₆	—	158	1	no	—
165	C₂×M₁₆	—	158	1	no	—
166	C₂×M₁₆	—	158	1	no	—
167	C₂×M₁₆	—	158	1	no	—
168	C₂²×C₈	—	79	1	no	—
169	C₂²×C₈	—	79	1	no	—
170	C₂²×C₈	—	158	2	no	—
171	C₂²×C₈	—	158	2	no	—
172	C₂²×C₈	—	158	2	no	—
173	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
175	#117	—	158	2	no	—
176	#117	—	158	2	no	—
177	C₂×C₄⋊C₄	—	79	2	no	—
178	C₂×C₂²⋊C₄	—	79	2	no	—
179	C₂×C₄²	—	79	1	no	—
180	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	—	158	2	no	—
184	M₃₂	—	158	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	—	158	2	no	—
187	#97	—	158	1	no	—
188	#97	—	158	1	no	—
189	#97	—	158	2	no	—
190	C₄×C₈	—	158	2	no	—
191	C₄×C₈	D₃₁₆	1	2	yes	—
192	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
193	C₂⁴	—	79	1	no	—
194	C₂²×C₄	—	79	2	no	—
195	C₂²×C₄	—	79	1	no	—
196	C₂²×C₄	—	79	2	no	—
197	C₂²×C₄	—	79	2	no	—
198	C₂²×C₄	—	158	1	no	—
199	C₂²×C₄	—	158	1	no	—
200	C₂²×C₄	—	158	1	no	—
201	C₂²×C₄	—	158	1	no	—
202	C₂²×C₄	C₈×D₇₉	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
204	M₁₆	—	158	2	no	—
205	M₁₆	—	158	2	no	—
206	M₁₆	—	158	2	no	—
207	M₁₆	—	158	2	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	79	2	no	—
211	C₂×C₈	—	79	2	no	—
212	C₂×C₈	—	158	8	no	—
213	C₂×C₈	—	158	1	no	—
214	C₂×C₈	—	158	1	no	—
215	C₂×C₈	—	158	1	no	—
216	C₂×C₈	—	158	1	no	—
217	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
218	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
220	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
221	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
222	C₂×C₈	#11929623	1	1	yes	characteristic
223	C₂×C₈	#11929625	1	1	yes	characteristic
224	C₄⋊C₄	—	158	4	no	—
225	C₂²⋊C₄	—	158	4	no	—
226	C₄²	—	158	1	no	—
227	C₄²	—	158	1	no	—
228	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
229	C₄²	#11929607	1	2	yes	—
230	C₁₆	—	4	2	no	—
231	C₁₆	—	316	2	no	—
232	C₂³	—	316	2	no	—
233	C₂³	—	79	2	no	—
234	C₂³	—	158	1	no	—
235	C₂³	—	158	1	no	—
236	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	316	2	no	—
239	C₂×C₄	—	79	2	no	—
240	C₂×C₄	—	158	2	no	—
241	C₂×C₄	—	158	1	no	—
242	C₂×C₄	—	158	1	no	—
243	C₂×C₄	—	158	2	no	—
244	C₂×C₄	—	158	2	no	—
245	C₂×C₄	—	158	2	no	—
246	C₂×C₄	—	158	2	no	—
247	C₂×C₄	(C₂×D₇₉)⋊C₈	1	4	yes	—
248	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
250	C₂×C₄	#14579119	1	1	yes	characteristic
251	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
252	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
253	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	—	158	2	no	—
256	C₈	—	158	2	no	—
257	C₈	#14579132	1	1	yes	characteristic
258	C₈	#14579136	1	1	yes	characteristic
259	C₈	#14579137	1	1	yes	characteristic
260	C₈	#14579141	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	316	2	no	—
263	C₂²	—	316	2	no	—
264	C₂²	—	316	2	no	—
265	C₂²	—	158	2	no	—
266	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
267	C₂²	C₄⋊C₈×D₇₉	1	1	yes	characteristic
268	C₂²	#19098615	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	316	2	no	—
271	C₄	—	158	2	no	—
272	C₄	#19098420	1	1	yes	characteristic
273	C₄	#19098526	1	1	yes	characteristic
274	C₄	#19098585	1	1	yes	characteristic
275	C₄	#19098619	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	316	2	no	—
279	C₂	#23763377	1	1	yes	characteristic
280	C₂	#23765020	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂³×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2150	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂³×D₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2150	D₇₉
hypocenter	C₇₉	#12786
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #23765156	C₂	semidirect
4. #23761483	C₂	semidirect
5. C₇₉×#2150	C₂	semidirect
143. #2150	D₇₉	semidirect
146. C₇₉	#12786	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098517	2
#23765156	2
C₂×#19098344	2
#23761483	2
C₇₉×#2150	2
#12786	79