Subgroups — #34291637

Order 20224.

982 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579095	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098518	C₂	1	1	yes	maximal, characteristic
3	#23766023	C₂	1	2	yes	maximal
4	#23764144	C₂	1	4	yes	maximal, complemented
5	#23764134	C₂	1	4	yes	maximal, complemented
6	#23763121	C₂	1	2	yes	maximal
7	C₂×#427×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11929608	C₂²	1	1	yes	characteristic
9	C₂×#14579120	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₇₉	C₂²	1	2	yes	—
11	C₂×#14579095	—	2	4	no	—
12	C₂×#14579095	C₂²	1	2	yes	—
13	C₂×#14579095	C₂²	1	2	yes	—
14	#19098518	—	2	4	no	—
15	C₂×#14579071	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
17	#19098462	C₂²	1	8	yes	—
18	#19098438	—	2	4	no	—
19	#19098433	—	2	2	no	—
20	#19098429	—	2	2	no	—
21	#19098325	C₂²	1	8	yes	—
22	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×#117×C₇₉	C₂²	1	2	yes	—
24	#427×C₇₉	C₄	1	8	yes	complemented
25	C₂×#107×C₇₉	C₂²	1	2	yes	—
26	C₂×#106×C₇₉	C₂²	1	2	yes	—
27	C₂³×Dic₇₉	—	2	2	no	—
28	C₂×#11929608	—	2	2	no	—
29	C₂×#11929608	—	2	2	no	—
30	C₂×#11929608	Q₈	1	4	yes	—
31	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
32	#14579120	—	2	2	no	—
33	#14579120	—	2	2	no	—
34	C₂×C₄⋊Dic₇₉	—	2	4	no	—
35	C₂×C₄⋊Dic₇₉	—	2	4	no	—
36	C₂×C₄⋊Dic₇₉	—	2	2	no	—
37	C₂×C₄⋊Dic₇₉	—	2	8	no	—
38	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
41	C₂×C₂²⋊Dic₇₉	D₄	1	2	yes	—
42	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
43	#14579095	—	2	4	no	—
44	#14579095	—	2	4	no	—
45	#14579071	—	2	4	no	—
46	C₈×Dic₇₉	—	2	4	no	—
47	C₇₉⋊#95	—	2	8	no	—
48	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₇₉	D₄	1	4	yes	—
51	C₂²×C₈×C₇₉	Q₈	1	4	yes	—
52	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
54	#117×C₇₉	—	2	4	no	—
55	#117×C₇₉	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
57	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
58	C₂×C₄²×C₇₉	—	2	2	no	—
59	#107×C₇₉	C₂×C₄	1	8	yes	complemented
60	#106×C₇₉	C₂×C₄	1	8	yes	complemented
61	C₂²×Dic₇₉	—	2	4	no	—
62	C₂²×Dic₇₉	—	4	4	no	—
63	C₂²×Dic₇₉	—	8	4	no	—
64	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
65	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
66	#11929608	—	2	8	no	—
67	C₂×C₇₉⋊C₈	—	2	2	no	—
68	C₂×C₇₉⋊C₈	—	2	2	no	—
69	C₂×C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
70	C₄⋊Dic₇₉	—	2	2	no	—
71	C₄⋊Dic₇₉	—	2	2	no	—
72	C₄⋊Dic₇₉	—	2	4	no	—
73	C₄⋊Dic₇₉	—	2	4	no	—
74	C₄⋊Dic₇₉	—	4	4	no	—
75	C₄⋊Dic₇₉	—	8	8	no	—
76	C₂²⋊Dic₇₉	—	2	4	no	—
77	C₄×Dic₇₉	—	2	2	no	—
78	C₄×Dic₇₉	—	2	2	no	—
79	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	—	2	4	no	—
81	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₇₉	#41	1	2	yes	—
87	C₂²×C₄×C₇₉	#41	1	2	yes	—
88	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₇₉	C₄⋊C₄	1	16	yes	complemented
90	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
96	C₄⋊C₄×C₇₉	—	2	4	no	—
97	C₄⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
98	C₂²⋊C₄×C₇₉	—	2	8	no	—
99	C₄²×C₇₉	—	2	8	no	—
100	C₂³×C₇₉	—	2	2	no	—
101	C₂³×C₇₉	—	2	2	no	—
102	C₂³×C₇₉	C₄×D₄	1	2	yes	—
103	C₂³×C₇₉	#121	1	2	yes	—
104	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
105	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₉	—	2	8	no	—
107	C₂×C₄×C₇₉	—	4	4	no	—
108	C₂×C₄×C₇₉	—	4	4	no	—
109	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇₉	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₇₉	#121	1	2	yes	—
113	C₂×C₄×C₇₉	C₂²⋊Q₈	1	4	yes	—
114	C₂×C₄×C₇₉	#125	1	4	yes	—
115	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
122	C₂×Dic₇₉	—	2	2	no	—
123	C₂×Dic₇₉	—	2	2	no	—
124	C₂×Dic₇₉	—	2	4	no	—
125	C₂×Dic₇₉	—	2	4	no	—
126	C₂×Dic₇₉	—	4	4	no	—
127	C₂×Dic₇₉	—	4	8	no	—
128	C₂×Dic₇₉	—	8	8	no	—
129	C₈×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
130	C₈×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
131	C₇₉⋊C₈	—	2	4	no	—
132	C₂²×C₇₉	—	2	4	no	—
133	C₂²×C₇₉	—	2	4	no	—
134	C₂²×C₇₉	—	2	2	no	—
135	C₂²×C₇₉	—	2	2	no	—
136	C₂²×C₇₉	#389	1	4	yes	—
137	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
139	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₂²×C₇₉	#576	1	2	yes	—
141	C₂²×C₇₉	#578	1	2	yes	—
142	C₄×C₇₉	—	4	8	no	—
143	C₄×C₇₉	#389	1	4	yes	—
144	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₄×C₇₉	C₂×#125	1	1	yes	characteristic
148	Dic₇₉	—	4	2	no	—
149	Dic₇₉	—	4	2	no	—
150	Dic₇₉	—	8	4	no	—
151	#21156	—	79	1	no	maximal, Sylow
152	C₂×C₇₉	—	2	4	no	—
153	C₂×C₇₉	C₂×#389	1	1	yes	characteristic
154	C₂×C₇₉	#2951	1	2	yes	—
155	C₂×C₇₉	#3058	1	1	yes	characteristic
156	C₂×C₇₉	#3059	1	1	yes	characteristic
157	C₂×C₇₉	#3164	1	2	yes	—
158	C₂×#427	D₇₉	1	1	yes	characteristic
159	C₂²×#107	—	79	1	no	—
160	C₂×#405	—	79	1	no	—
161	#2317	—	79	2	no	—
162	#1944	—	79	4	no	—
163	#1942	—	79	4	no	—
164	#1736	—	79	2	no	—
165	C₇₉	#21156	1	1	yes	characteristic, complemented, Sylow
166	C₂²×M₁₆	—	79	1	no	—
167	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
168	C₂×#117	—	79	1	no	—
169	C₂×#117	C₂×D₇₉	1	2	yes	—
170	C₂²×C₄⋊C₄	—	79	2	no	—
171	#427	Dic₇₉	1	8	yes	complemented
172	C₂×#107	—	79	2	no	—
173	C₂×#107	—	79	2	no	—
174	C₂×#107	—	158	4	no	—
175	C₂×#107	C₂×D₇₉	1	2	yes	—
176	C₂×#106	C₂×D₇₉	1	2	yes	—
177	#405	—	158	4	no	—
178	C₂×#97	—	79	2	no	—
179	C₂×C₄×C₈	—	79	2	no	—
180	#389	—	79	8	no	—
181	#385	—	158	4	no	—
182	#384	—	158	2	no	—
183	#382	—	158	2	no	—
184	#340	—	79	8	no	—
185	C₂³×C₄	—	158	2	no	—
186	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
187	C₂×M₁₆	—	79	4	no	—
188	C₂×M₁₆	—	158	2	no	—
189	C₂×M₁₆	—	158	2	no	—
190	C₂²×C₈	—	79	2	no	—
191	C₂²×C₈	Dic₁₅₈	1	4	yes	—
192	C₂²×C₈	D₃₁₆	1	4	yes	—
193	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
194	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
196	#117	—	2	4	no	—
197	#117	—	158	2	no	—
198	#117	—	158	2	no	—
199	#117	C₂×Dic₇₉	1	8	yes	—
200	C₂×C₄⋊C₄	—	79	1	no	—
201	C₂×C₄⋊C₄	—	79	1	no	—
202	C₂×C₄⋊C₄	—	79	4	no	—
203	C₂×C₄⋊C₄	—	158	2	no	—
204	C₂×C₄⋊C₄	—	158	4	no	—
205	C₂×C₄⋊C₄	—	158	4	no	—
206	C₂×C₄⋊C₄	—	158	8	no	—
207	C₂×C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	79	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	79	2	no	—
212	#107	—	158	4	no	—
213	#107	—	158	4	no	—
214	#107	C₂×Dic₇₉	1	8	yes	complemented
215	#106	C₂×Dic₇₉	1	8	yes	complemented
216	#97	—	158	4	no	—
217	C₄×C₈	—	158	4	no	—
218	#95	—	158	8	no	—
219	C₂⁴	D₄×D₇₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	316	4	no	—
222	C₂²×C₄	—	79	2	no	—
223	C₂²×C₄	—	79	2	no	—
224	C₂²×C₄	—	632	4	no	—
225	C₂²×C₄	—	158	4	no	—
226	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
227	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
229	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
230	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
231	C₂²×C₄	#11929623	1	2	yes	—
232	C₂²×C₄	#11929625	1	2	yes	—
233	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
234	M₁₆	—	158	8	no	—
235	C₂×C₈	—	79	4	no	—
236	C₂×C₈	—	158	2	no	—
237	C₂×C₈	—	158	2	no	—
238	C₂×C₈	C₄⋊Dic₇₉	1	16	yes	complemented
239	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
240	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
241	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
242	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
243	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
244	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	316	4	no	—
247	C₄⋊C₄	—	632	8	no	—
248	C₄⋊C₄	—	158	4	no	—
249	C₄⋊C₄	—	158	4	no	—
250	C₄⋊C₄	—	158	2	no	—
251	C₄⋊C₄	—	158	2	no	—
252	C₄⋊C₄	C₂²×Dic₇₉	1	8	yes	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₂²⋊C₄	—	158	4	no	—
255	C₄²	—	2	8	no	—
256	C₄²	—	158	2	no	—
257	C₄²	—	158	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	D₄×Dic₇₉	1	2	yes	—
261	C₂³	#14579136	1	2	yes	—
262	C₂³	#14579160	1	2	yes	—
263	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	316	4	no	—
268	C₂×C₄	—	316	8	no	—
269	C₂×C₄	—	632	8	no	—
270	C₂×C₄	—	158	2	no	—
271	C₂×C₄	—	158	2	no	—
272	C₂×C₄	—	158	4	no	—
273	C₂×C₄	—	158	4	no	—
274	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
275	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
276	C₂×C₄	Q₈×Dic₇₉	1	4	yes	—
277	C₂×C₄	#14579136	1	2	yes	—
278	C₂×C₄	#14579141	1	4	yes	—
279	C₂×C₄	#14579152	1	4	yes	—
280	C₂×C₄	#14579160	1	2	yes	—
281	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
283	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
284	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
285	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
286	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
287	C₈	—	158	4	no	—
288	C₈	C₂×C₄⋊Dic₇₉	1	4	yes	—
289	C₈	C₂×C₄⋊Dic₇₉	1	4	yes	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	#19098462	1	4	yes	—
295	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
296	C₂²	C₂×#14579136	1	1	yes	characteristic
297	C₂²	C₂×#14579160	1	1	yes	characteristic
298	C₂²	#19099365	1	2	yes	—
299	C₂²	#19099377	1	2	yes	—
300	C₄	—	4	8	no	—
301	C₄	—	316	2	no	—
302	C₄	—	316	2	no	—
303	C₄	—	632	4	no	—
304	C₄	#19098462	1	4	yes	—
305	C₄	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
306	C₄	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
307	C₄	C₂×#14579141	1	1	yes	characteristic
308	C₄	C₂×#14579152	1	1	yes	characteristic
309	C₂	—	2	4	no	—
310	C₂	C₂×#19098462	1	1	yes	characteristic
311	C₂	#23772276	1	2	yes	—
312	C₂	#23773049	1	1	yes	characteristic
313	C₂	#23773056	1	1	yes	characteristic
314	C₂	#23773940	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#427×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#427	D₇₉
hypocenter	C₇₉	#21156
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #23764144	C₂	semidirect
5. #23764134	C₂	semidirect
24. #427×C₇₉	C₄	semidirect
59. #107×C₇₉	C₂×C₄	semidirect
60. #106×C₇₉	C₂×C₄	semidirect
89. C₂×C₈×C₇₉	C₄⋊C₄	semidirect
165. C₇₉	#21156	semidirect
171. #427	Dic₇₉	semidirect
214. #107	C₂×Dic₇₉	semidirect
215. #106	C₂×Dic₇₉	semidirect
238. C₂×C₈	C₄⋊Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579095	2
C₂×#19098518	2
#23766023	2
#23764144	2
#23764134	2
#23763121	2
C₂×#427×C₇₉	2
#21156	79