Subgroups — #34286239

Order 20224.

1166 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×#19098521	C₂	1	2	yes	maximal
2	C₂²×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23765807	C₂	1	1	yes	maximal, characteristic
4	#23763270	C₂	1	4	yes	maximal, complemented
5	#23763265	C₂	1	4	yes	maximal, complemented
6	#23763098	C₂	1	2	yes	maximal
7	C₂×#423×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×#14579120	C₂²	1	2	yes	—
11	#19098521	—	2	8	no	—
12	C₂×#14579071	C₂²	1	4	yes	—
13	C₂×C₄×C₇₉⋊C₈	C₄	1	8	yes	complemented
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇₉	C₂²	1	4	yes	—
17	#19098433	C₄	1	2	yes	—
18	#19098429	C₄	1	2	yes	—
19	#19098423	C₄	1	4	yes	—
20	C₄×C₄⋊Dic₇₉	C₂²	1	4	yes	—
21	#19098317	C₂²	1	8	yes	—
22	C₂²×M₁₆×C₇₉	C₂²	1	2	yes	—
23	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
24	#423×C₇₉	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₇₉	C₂²	1	4	yes	—
26	C₂²×C₇₉⋊C₈	—	2	2	no	—
27	C₂²×C₇₉⋊C₈	—	2	2	no	—
28	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
31	C₂²×C₇₉⋊C₈	Q₈	1	4	yes	—
32	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
35	#14579120	—	2	4	no	—
36	#14579120	—	2	4	no	—
37	C₂×C₄⋊Dic₇₉	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊Dic₇₉	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₇₉	C₂³	1	4	yes	—
41	#14579071	—	2	8	no	—
42	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
43	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
44	C₈×Dic₇₉	—	2	8	no	—
45	C₇₉⋊#95	C₂×C₄	1	8	yes	—
46	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
47	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₇₉	—	2	4	no	—
49	C₂×M₁₆×C₇₉	—	2	4	no	—
50	C₂×M₁₆×C₇₉	C₂×C₄	1	8	yes	—
51	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
52	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
57	C₄⋊C₈×C₇₉	C₂×C₄	1	16	yes	—
58	C₂²×Dic₇₉	C₂²×C₄	1	8	yes	—
59	C₂×C₇₉⋊C₈	—	2	8	no	—
60	C₂×C₇₉⋊C₈	—	2	2	no	—
61	C₂×C₇₉⋊C₈	—	2	2	no	—
62	C₂×C₇₉⋊C₈	C₄⋊C₄	1	16	yes	complemented
63	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
64	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
65	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
66	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
67	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
68	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
69	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
70	C₄⋊Dic₇₉	—	2	8	no	—
71	C₄⋊Dic₇₉	—	2	16	no	—
72	C₂²⋊Dic₇₉	—	2	8	no	—
73	C₄×Dic₇₉	—	2	4	no	—
74	C₄×Dic₇₉	—	2	4	no	—
75	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	—	2	8	no	—
77	C₂²×C₄×C₇₉	C₄²	1	8	yes	—
78	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₇₉	#41	1	2	yes	—
86	C₂²×C₄×C₇₉	#41	1	2	yes	—
87	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
88	M₁₆×C₇₉	—	2	16	no	—
89	C₂×C₈×C₇₉	—	2	4	no	—
90	C₂×C₈×C₇₉	—	2	4	no	—
91	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
92	C₄²×C₇₉	C₄²	1	4	yes	—
93	C₄²×C₇₉	C₄²	1	4	yes	—
94	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
95	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
96	C₂³×C₇₉	—	2	4	no	—
97	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
98	C₂³×C₇₉	C₄×D₄	1	2	yes	—
99	C₂³×C₇₉	C₄×Q₈	1	2	yes	—
100	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	—	2	4	no	—
102	C₂×C₄×C₇₉	—	2	16	no	—
103	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₇₉	#117	1	4	yes	—
108	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
110	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₇₉	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₇₉	#131	1	8	yes	—
113	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₇₉	—	2	8	no	—
120	C₂×Dic₇₉	—	2	16	no	—
121	C₈×C₇₉	—	2	8	no	—
122	C₇₉⋊C₈	—	2	4	no	—
123	C₇₉⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
124	C₇₉⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
125	C₂²×C₇₉	—	2	4	no	—
126	C₂²×C₇₉	—	2	4	no	—
127	C₂²×C₇₉	—	2	4	no	—
128	C₂²×C₇₉	C₄×C₄⋊C₄	1	4	yes	—
129	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
130	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₂²×C₇₉	C₂×#131	1	4	yes	—
133	C₄×C₇₉	—	2	4	no	—
134	C₄×C₇₉	C₄×C₄⋊C₄	1	4	yes	—
135	C₄×C₇₉	#405	1	8	yes	complemented
136	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
138	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
140	Dic₇₉	—	4	8	no	—
141	#20437	—	79	1	no	maximal, Sylow
142	C₂×C₇₉	—	2	4	no	—
143	C₂×C₇₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₇₉	C₂×#405	1	2	yes	—
145	C₂×C₇₉	#2931	1	2	yes	—
146	C₂×C₇₉	#2960	1	2	yes	—
147	C₂×#423	D₇₉	1	1	yes	characteristic
148	C₂×#405	—	79	2	no	—
149	C₂²×C₄×C₈	—	79	1	no	—
150	#2285	—	79	1	no	—
151	#1770	—	79	4	no	—
152	C₈×C₄⋊C₄	—	79	4	no	—
153	#1729	—	79	2	no	—
154	C₇₉	#20437	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	C₂×D₇₉	1	2	yes	—
156	C₂³×C₈	—	79	1	no	—
157	C₂³×C₈	—	79	1	no	—
158	C₂×#117	—	79	2	no	—
159	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
160	#423	Dic₇₉	1	8	yes	complemented
161	C₂×C₄⋊C₈	C₂×D₇₉	1	4	yes	—
162	#405	—	158	8	no	—
163	C₂×#97	—	79	4	no	—
164	C₂×C₄×C₈	—	79	2	no	—
165	C₂×C₄×C₈	—	79	2	no	—
166	C₂×C₄×C₈	—	79	8	no	—
167	C₂×C₄×C₈	—	79	4	no	—
168	#384	—	79	2	no	—
169	#382	—	79	2	no	—
170	#380	—	79	4	no	—
171	C₄×C₄⋊C₄	—	79	4	no	—
172	#336	—	79	8	no	—
173	C₂³×C₄	C₄×D₇₉	1	2	yes	—
174	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	C₂×Dic₇₉	1	8	yes	—
178	C₂²×C₈	—	79	4	no	—
179	C₂²×C₈	—	79	4	no	—
180	C₂²×C₈	—	79	4	no	—
181	C₂²×C₈	—	79	1	no	—
182	C₂²×C₈	—	79	1	no	—
183	C₂²×C₈	—	79	4	no	—
184	C₂²×C₈	—	79	2	no	—
185	C₂²×C₈	—	158	2	no	—
186	C₂²×C₈	—	158	2	no	—
187	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
188	#117	—	158	4	no	—
189	#117	—	158	4	no	—
190	C₂×C₄⋊C₄	—	79	4	no	—
191	C₂×C₄⋊C₄	—	79	8	no	—
192	C₂×C₂²⋊C₄	—	79	4	no	—
193	C₂×C₄²	—	79	4	no	—
194	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₇₉	1	4	yes	—
197	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
198	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
199	C₄⋊C₈	C₂×Dic₇₉	1	16	yes	—
200	#97	—	158	8	no	—
201	C₄×C₈	—	79	8	no	—
202	C₄×C₈	—	79	8	no	—
203	C₄×C₈	—	158	8	no	—
204	#95	—	79	8	no	—
205	C₂⁴	C₂×C₄×D₇₉	1	1	yes	characteristic
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	79	8	no	—
208	C₂²×C₄	C₄×Dic₇₉	1	8	yes	—
209	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
210	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
213	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
214	C₂²×C₄	D₄×D₇₉	1	2	yes	—
215	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
216	C₂²×C₄	#11929623	1	2	yes	—
217	C₂²×C₄	#11929625	1	2	yes	—
218	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
219	M₁₆	—	2	16	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	79	2	no	—
223	C₂×C₈	—	79	2	no	—
224	C₂×C₈	—	79	16	no	—
225	C₂×C₈	—	79	2	no	—
226	C₂×C₈	—	79	2	no	—
227	C₂×C₈	—	79	2	no	—
228	C₂×C₈	—	79	2	no	—
229	C₂×C₈	—	79	4	no	—
230	C₂×C₈	—	158	2	no	—
231	C₂×C₈	—	158	2	no	—
232	C₂×C₈	—	158	8	no	—
233	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
234	C₄⋊C₄	—	158	8	no	—
235	C₄⋊C₄	—	158	16	no	—
236	C₂²⋊C₄	—	158	8	no	—
237	C₄²	—	158	4	no	—
238	C₄²	—	158	4	no	—
239	C₄²	C₄×Dic₇₉	1	4	yes	—
240	C₄²	C₄×Dic₇₉	1	4	yes	—
241	C₄²	C₂×C₄×D₇₉	1	4	yes	—
242	C₄²	C₂²×Dic₇₉	1	4	yes	—
243	C₂³	—	2	4	no	—
244	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
245	C₂³	#14579123	1	2	yes	—
246	C₂³	#14579126	1	2	yes	—
247	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	158	8	no	—
251	C₂×C₄	—	158	16	no	—
252	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
253	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
255	C₂×C₄	C₄⋊C₄×D₇₉	1	4	yes	—
256	C₂×C₄	#14579119	1	4	yes	—
257	C₂×C₄	#14579123	1	2	yes	—
258	C₂×C₄	D₄×Dic₇₉	1	4	yes	—
259	C₂×C₄	#14579126	1	2	yes	—
260	C₂×C₄	Q₈×Dic₇₉	1	4	yes	—
261	C₂×C₄	#14579169	1	8	yes	—
262	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
263	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
267	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	79	4	no	—
270	C₈	—	79	4	no	—
271	C₈	—	158	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₄⋊C₄×Dic₇₉	1	4	yes	—
276	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
277	C₂²	C₂×#14579123	1	1	yes	characteristic
278	C₂²	C₂×#14579126	1	1	yes	characteristic
279	C₂²	C₂×#14579169	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	316	8	no	—
282	C₄	C₄⋊C₄×Dic₇₉	1	4	yes	—
283	C₄	#19098521	1	8	yes	complemented
284	C₄	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
285	C₄	C₂×#14579119	1	1	yes	characteristic
286	C₄	C₂×D₄×Dic₇₉	1	1	yes	characteristic
287	C₄	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄⋊C₄×Dic₇₉	1	1	yes	characteristic
290	C₂	C₂×#19098521	1	2	yes	—
291	C₂	#23772111	1	2	yes	—
292	C₂	#23772330	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Fitting	C₂×#423×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#423	D₇₉
hypocenter	C₇₉	#20437
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #23763270	C₂	semidirect
5. #23763265	C₂	semidirect
13. C₂×C₄×C₇₉⋊C₈	C₄	semidirect
24. #423×C₇₉	C₄	semidirect
62. C₂×C₇₉⋊C₈	C₄⋊C₄	semidirect
135. C₄×C₇₉	#405	semidirect
154. C₇₉	#20437	semidirect
160. #423	Dic₇₉	semidirect
283. C₄	#19098521	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098521	2
C₂²×C₄×C₇₉⋊C₈	2
#23765807	2
#23763270	2
#23763265	2
#23763098	2
C₂×#423×C₇₉	2
#20437	79