Subgroups — #34196919

Order 20224.

722 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098605	C₂	1	2	yes	maximal
2	C₂²×#14579095	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098331	C₂	1	2	yes	maximal
4	#23763096	C₂	1	1	yes	maximal, characteristic
5	C₂×#341×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	1	yes	characteristic
7	C₂×#14579120	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₇₉	C₂²	1	2	yes	—
9	#19098605	—	2	4	no	—
10	#19098605	—	2	4	no	—
11	C₂×#14579095	—	2	4	no	—
12	C₂×#14579095	—	2	2	no	—
13	C₂×#14579095	—	2	2	no	—
14	C₂×#14579093	—	2	2	no	—
15	#19098331	—	2	4	no	—
16	#19098317	—	2	2	no	—
17	C₂²×M₁₆×C₇₉	C₂²	1	2	yes	—
18	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#341×C₇₉	C₄	1	4	yes	complemented
20	C₂³×Dic₇₉	—	2	2	no	—
21	C₂×#11929608	—	2	1	no	—
22	C₂×#11929608	—	2	2	no	—
23	C₂×#11929608	—	2	2	no	—
24	C₂×#11929608	Q₈	1	2	yes	—
25	C₂²×C₇₉⋊C₈	—	2	1	no	—
26	#14579120	—	2	2	no	—
27	#14579120	—	2	2	no	—
28	C₂×C₄⋊Dic₇₉	—	2	2	no	—
29	C₂×C₄⋊Dic₇₉	—	2	2	no	—
30	C₂×C₄⋊Dic₇₉	—	2	2	no	—
31	C₂×C₄⋊Dic₇₉	—	2	2	no	—
32	C₂×C₄⋊Dic₇₉	—	2	2	no	—
33	C₂×C₄⋊Dic₇₉	D₄	1	4	yes	—
34	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₇₉	—	2	1	no	—
37	C₂×C₄×Dic₇₉	—	2	1	no	—
38	#14579095	—	2	8	no	—
39	#14579095	—	2	8	no	—
40	#14579093	—	2	8	no	—
41	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₇₉	—	2	2	no	—
43	C₂×M₁₆×C₇₉	—	2	2	no	—
44	C₂×M₁₆×C₇₉	—	2	2	no	—
45	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
46	C₂×M₁₆×C₇₉	D₄	1	4	yes	—
47	C₂×M₁₆×C₇₉	Q₈	1	4	yes	—
48	C₂²×C₈×C₇₉	—	2	2	no	—
49	C₂²×C₈×C₇₉	—	2	2	no	—
50	C₂²×C₈×C₇₉	—	2	2	no	—
51	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₇₉	D₄	1	2	yes	—
53	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
54	C₂²×Dic₇₉	—	2	1	no	—
55	C₂²×Dic₇₉	—	2	1	no	—
56	C₂²×Dic₇₉	—	2	4	no	—
57	C₂²×Dic₇₉	—	4	2	no	—
58	C₂²×Dic₇₉	—	4	2	no	—
59	C₂²×Dic₇₉	—	8	4	no	—
60	#11929608	—	2	4	no	—
61	#11929608	—	4	2	no	—
62	C₂×C₇₉⋊C₈	—	2	2	no	—
63	C₂×C₇₉⋊C₈	—	4	1	no	—
64	C₂×C₇₉⋊C₈	—	4	1	no	—
65	C₄⋊Dic₇₉	—	2	2	no	—
66	C₄⋊Dic₇₉	—	2	4	no	—
67	C₄⋊Dic₇₉	—	2	4	no	—
68	C₄⋊Dic₇₉	—	2	2	no	—
69	C₄⋊Dic₇₉	—	4	2	no	—
70	C₄⋊Dic₇₉	—	4	2	no	—
71	C₂²⋊Dic₇₉	—	4	2	no	—
72	C₄×Dic₇₉	—	4	1	no	—
73	C₄×Dic₇₉	—	4	1	no	—
74	C₂⁴×C₇₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	#41	1	2	yes	—
84	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
85	M₁₆×C₇₉	—	2	4	no	—
86	M₁₆×C₇₉	—	2	4	no	—
87	M₁₆×C₇₉	—	2	4	no	—
88	M₁₆×C₇₉	C₄⋊C₄	1	8	yes	—
89	C₂×C₈×C₇₉	—	2	4	no	—
90	C₂×C₈×C₇₉	—	2	4	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	4	no	—
93	C₂×C₈×C₇₉	—	2	4	no	—
94	C₂×C₈×C₇₉	—	2	4	no	—
95	C₂×C₈×C₇₉	—	2	2	no	—
96	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
97	C₂³×C₇₉	—	2	2	no	—
98	C₂³×C₇₉	—	2	2	no	—
99	C₂³×C₇₉	C₄×Q₈	1	1	yes	characteristic
100	C₂³×C₇₉	#121	1	2	yes	—
101	C₂³×C₇₉	#121	1	1	yes	characteristic
102	C₂³×C₇₉	#125	1	2	yes	—
103	C₂³×C₇₉	#125	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#120	1	2	yes	—
110	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#123	1	2	yes	—
117	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×Dic₇₉	—	4	1	no	—
123	C₂×Dic₇₉	—	4	1	no	—
124	C₂×Dic₇₉	—	4	4	no	—
125	C₂×Dic₇₉	—	4	2	no	—
126	C₂×Dic₇₉	—	4	2	no	—
127	C₂×Dic₇₉	—	8	4	no	—
128	C₂×Dic₇₉	—	8	4	no	—
129	C₂×Dic₇₉	—	8	4	no	—
130	C₈×C₇₉	—	2	4	no	—
131	C₈×C₇₉	—	2	8	no	—
132	C₇₉⋊C₈	—	4	2	no	—
133	C₂²×C₇₉	—	2	4	no	—
134	C₂²×C₇₉	—	2	2	no	—
135	C₂²×C₇₉	—	2	2	no	—
136	C₂²×C₇₉	—	2	2	no	—
137	C₂²×C₇₉	—	2	2	no	—
138	C₂²×C₇₉	#389	1	2	yes	—
139	C₂²×C₇₉	#389	1	1	yes	characteristic
140	C₂²×C₇₉	#392	1	1	yes	characteristic
141	C₂²×C₇₉	#398	1	2	yes	—
142	C₂²×C₇₉	#398	1	1	yes	characteristic
143	C₂²×C₇₉	#472	1	2	yes	—
144	C₂²×C₇₉	#473	1	2	yes	—
145	C₄×C₇₉	#384	1	1	yes	characteristic
146	C₄×C₇₉	#385	1	1	yes	characteristic
147	C₄×C₇₉	#385	1	2	yes	—
148	C₄×C₇₉	#395	1	1	yes	characteristic
149	C₄×C₇₉	#395	1	2	yes	—
150	C₄×C₇₉	#399	1	1	yes	characteristic
151	Dic₇₉	—	8	1	no	—
152	Dic₇₉	—	8	4	no	—
153	Dic₇₉	—	8	1	no	—
154	#11331	—	79	1	no	maximal, Sylow
155	C₂×C₇₉	—	2	2	no	—
156	C₂×C₇₉	—	2	2	no	—
157	C₂×C₇₉	#1444	1	1	yes	characteristic
158	C₂×C₇₉	#1952	1	2	yes	—
159	C₂×C₇₉	#2018	1	1	yes	characteristic
160	C₂×C₇₉	#2020	1	1	yes	characteristic
161	C₂×C₇₉	#2089	1	2	yes	—
162	C₂×#428	—	79	2	no	—
163	C₂²×#107	—	79	1	no	—
164	C₂×#343	—	79	2	no	—
165	C₂×#341	D₇₉	1	1	yes	characteristic
166	#1729	—	79	1	no	—
167	C₇₉	#11331	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	—	79	1	no	—
169	C₂²×M₁₆	C₂×D₇₉	1	2	yes	—
170	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
171	C₂×#117	—	79	1	no	—
172	C₂²×C₄⋊C₄	—	79	2	no	—
173	#428	—	158	4	no	—
174	#428	—	158	4	no	—
175	C₂×#107	—	158	4	no	—
176	C₂×#107	—	158	2	no	—
177	C₂×#107	—	158	2	no	—
178	C₂×#106	—	158	2	no	—
179	#343	—	158	4	no	—
180	#341	Dic₇₉	1	4	yes	complemented
181	#336	—	158	2	no	—
182	C₂³×C₄	—	158	2	no	—
183	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	79	2	no	—
188	C₂×M₁₆	—	158	2	no	—
189	C₂×M₁₆	—	158	2	no	—
190	C₂×M₁₆	—	158	1	no	—
191	C₂×M₁₆	Dic₁₅₈	1	4	yes	—
192	C₂×M₁₆	D₃₁₆	1	4	yes	—
193	C₂×M₁₆	C₂×Dic₇₉	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	158	1	no	—
198	C₂²×C₈	Dic₁₅₈	1	2	yes	—
199	C₂²×C₈	D₃₁₆	1	2	yes	—
200	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
201	#117	—	158	2	no	—
202	#117	—	158	2	no	—
203	C₂×C₄⋊C₄	—	79	4	no	—
204	C₂×C₄⋊C₄	—	79	1	no	—
205	C₂×C₄⋊C₄	—	79	1	no	—
206	C₂×C₄⋊C₄	—	158	2	no	—
207	C₂×C₄⋊C₄	—	158	2	no	—
208	C₂×C₄⋊C₄	—	158	2	no	—
209	C₂×C₄⋊C₄	—	158	2	no	—
210	C₂×C₄⋊C₄	—	158	2	no	—
211	C₂×C₂²⋊C₄	—	158	1	no	—
212	C₂×C₄²	—	158	1	no	—
213	#107	—	158	8	no	—
214	#107	—	158	8	no	—
215	#106	—	158	8	no	—
216	C₂⁴	#11929625	1	1	yes	characteristic
217	C₂²×C₄	—	316	2	no	—
218	C₂²×C₄	—	316	2	no	—
219	C₂²×C₄	—	632	4	no	—
220	C₂²×C₄	—	158	1	no	—
221	C₂²×C₄	—	158	1	no	—
222	C₂²×C₄	—	158	4	no	—
223	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₇₉	1	2	yes	—
225	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
227	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
228	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
230	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
231	C₂²×C₄	#11929623	1	2	yes	—
232	C₂²×C₄	#11929623	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	316	2	no	—
237	M₁₆	—	158	4	no	—
238	M₁₆	C₄⋊Dic₇₉	1	8	yes	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	316	1	no	—
247	C₂×C₈	—	316	1	no	—
248	C₂×C₈	—	158	2	no	—
249	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
250	C₄⋊C₄	—	316	2	no	—
251	C₄⋊C₄	—	316	2	no	—
252	C₄⋊C₄	—	158	2	no	—
253	C₄⋊C₄	—	158	4	no	—
254	C₄⋊C₄	—	158	4	no	—
255	C₄⋊C₄	—	158	2	no	—
256	C₂²⋊C₄	—	316	2	no	—
257	C₄²	—	316	1	no	—
258	C₄²	—	316	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	Q₈×Dic₇₉	1	1	yes	characteristic
262	C₂³	#14579136	1	2	yes	—
263	C₂³	#14579136	1	1	yes	characteristic
264	C₂³	#14579152	1	2	yes	—
265	C₂³	#14579152	1	1	yes	characteristic
266	C₂×C₄	—	316	1	no	—
267	C₂×C₄	—	316	1	no	—
268	C₂×C₄	—	316	4	no	—
269	C₂×C₄	—	316	2	no	—
270	C₂×C₄	—	316	2	no	—
271	C₂×C₄	—	632	4	no	—
272	C₂×C₄	—	632	4	no	—
273	C₂×C₄	—	632	4	no	—
274	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
276	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
277	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
278	C₂×C₄	#14579131	1	1	yes	characteristic
279	C₂×C₄	#14579131	1	2	yes	—
280	C₂×C₄	#14579138	1	2	yes	—
281	C₂×C₄	#14579138	1	2	yes	—
282	C₂×C₄	#14579138	1	2	yes	—
283	C₂×C₄	#14579141	1	1	yes	characteristic
284	C₂×C₄	#14579141	1	2	yes	—
285	C₂×C₄	#14579145	1	1	yes	characteristic
286	C₂×C₄	#14579145	1	2	yes	—
287	C₂×C₄	#14579158	1	1	yes	characteristic
288	C₂×C₄	#14579160	1	2	yes	—
289	C₂×C₄	#14579160	1	1	yes	characteristic
290	C₂×C₄	#14579161	1	2	yes	—
291	C₂×C₄	#14579161	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	2	8	no	—
294	C₈	—	316	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	#19098462	1	2	yes	—
301	C₂²	#19098462	1	1	yes	characteristic
302	C₂²	#19098473	1	1	yes	characteristic
303	C₂²	#19098495	1	2	yes	—
304	C₂²	#19098495	1	1	yes	characteristic
305	C₂²	#19098834	1	2	yes	—
306	C₂²	#19098839	1	2	yes	—
307	C₄	—	632	1	no	—
308	C₄	—	632	4	no	—
309	C₄	—	632	1	no	—
310	C₄	#19098433	1	1	yes	characteristic
311	C₄	#19098438	1	1	yes	characteristic
312	C₄	#19098438	1	2	yes	—
313	C₄	#19098484	1	1	yes	characteristic
314	C₄	#19098484	1	2	yes	—
315	C₄	#19098500	1	1	yes	characteristic
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	#23761665	1	1	yes	characteristic
319	C₂	#23764186	1	2	yes	—
320	C₂	#23764510	1	1	yes	characteristic
321	C₂	#23764522	1	1	yes	characteristic
322	C₂	#23764907	1	2	yes	—

	subgroup	G/H
center	C₂³	#14579136
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×#341×C₇₉	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂⁴×C₇₉	#41
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#341	D₇₉
hypocenter	C₇₉	#11331
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
19. #341×C₇₉	C₄	semidirect
167. C₇₉	#11331	semidirect
180. #341	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098605	2
C₂²×#14579095	2
C₂×#19098331	2
#23763096	2
C₂×#341×C₇₉	2
#11331	79