Subgroups — #34291630

Order 20224.

982 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579093	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098518	C₂	1	1	yes	maximal, characteristic
3	#23766023	C₂	1	2	yes	maximal
4	#23764139	C₂	1	4	yes	maximal, complemented
5	#23764129	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×#14579093	—	2	4	no	—
12	C₂×#14579093	C₂²	1	2	yes	—
13	C₂×#14579093	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	2	yes	—
39	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
40	C₂×C₂²⋊Dic₇₉	D₄	1	2	yes	—
41	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
42	#14579093	—	2	4	no	—
43	#14579093	—	2	4	no	—
44	#14579071	—	2	4	no	—
45	C₈×Dic₇₉	—	2	4	no	—
46	C₇₉⋊#95	—	2	8	no	—
47	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₇₉	D₄	1	4	yes	—
50	C₂²×C₈×C₇₉	Q₈	1	4	yes	—
51	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	#117×C₇₉	—	2	4	no	—
54	#117×C₇₉	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
57	C₂×C₄²×C₇₉	—	2	2	no	—
58	#107×C₇₉	C₂×C₄	1	8	yes	complemented
59	#106×C₇₉	C₂×C₄	1	8	yes	complemented
60	C₂²×Dic₇₉	—	2	4	no	—
61	C₂²×Dic₇₉	—	4	4	no	—
62	C₂²×Dic₇₉	—	8	4	no	—
63	C₂²×Dic₇₉	C₂×D₄	1	4	yes	—
64	#11929608	—	2	8	no	—
65	C₂×C₇₉⋊C₈	—	2	2	no	—
66	C₂×C₇₉⋊C₈	—	2	2	no	—
67	C₂×C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
68	C₄⋊Dic₇₉	—	2	4	no	—
69	C₄⋊Dic₇₉	—	2	4	no	—
70	C₄⋊Dic₇₉	—	2	4	no	—
71	C₄⋊Dic₇₉	—	4	4	no	—
72	C₄⋊Dic₇₉	—	8	8	no	—
73	C₂²⋊Dic₇₉	—	2	4	no	—
74	C₄×Dic₇₉	—	2	2	no	—
75	C₄×Dic₇₉	—	2	2	no	—
76	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	—	2	4	no	—
78	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₇₉	#41	1	2	yes	—
84	C₂²×C₄×C₇₉	#41	1	2	yes	—
85	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₇₉	C₄⋊C₄	1	16	yes	complemented
87	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
93	C₄⋊C₄×C₇₉	—	2	4	no	—
94	C₄⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
95	C₂²⋊C₄×C₇₉	—	2	8	no	—
96	C₄²×C₇₉	—	2	8	no	—
97	C₂³×C₇₉	—	2	4	no	—
98	C₂³×C₇₉	C₄×D₄	1	2	yes	—
99	C₂³×C₇₉	#121	1	2	yes	—
100	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
101	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇₉	—	2	8	no	—
103	C₂×C₄×C₇₉	—	4	4	no	—
104	C₂×C₄×C₇₉	—	4	4	no	—
105	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₇₉	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₇₉	#121	1	2	yes	—
109	C₂×C₄×C₇₉	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₇₉	#125	1	4	yes	—
111	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₇₉	—	2	4	no	—
119	C₂×Dic₇₉	—	2	8	no	—
120	C₂×Dic₇₉	—	4	4	no	—
121	C₂×Dic₇₉	—	4	8	no	—
122	C₂×Dic₇₉	—	8	8	no	—
123	C₈×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
124	C₈×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
125	C₇₉⋊C₈	—	2	4	no	—
126	C₂²×C₇₉	—	2	4	no	—
127	C₂²×C₇₉	—	2	4	no	—
128	C₂²×C₇₉	—	2	4	no	—
129	C₂²×C₇₉	#389	1	4	yes	—
130	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
132	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₂²×C₇₉	#577	1	4	yes	—
134	C₄×C₇₉	—	4	8	no	—
135	C₄×C₇₉	#389	1	4	yes	—
136	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₄×C₇₉	C₂×#125	1	1	yes	characteristic
140	Dic₇₉	—	4	4	no	—
141	Dic₇₉	—	8	4	no	—
142	#21155	—	79	1	no	maximal, Sylow
143	C₂×C₇₉	—	2	4	no	—
144	C₂×C₇₉	C₂×#389	1	1	yes	characteristic
145	C₂×C₇₉	#2952	1	2	yes	—
146	C₂×C₇₉	#3057	1	2	yes	—
147	C₂×C₇₉	#3163	1	2	yes	—
148	C₂×#427	D₇₉	1	1	yes	characteristic
149	C₂²×#106	—	79	1	no	—
150	C₂×#405	—	79	1	no	—
151	#2317	—	79	2	no	—
152	#1943	—	79	4	no	—
153	#1941	—	79	4	no	—
154	#1736	—	79	2	no	—
155	C₇₉	#21155	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	—	79	1	no	—
157	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
158	C₂×#117	—	79	1	no	—
159	C₂×#117	C₂×D₇₉	1	2	yes	—
160	C₂²×C₄⋊C₄	—	79	2	no	—
161	#427	Dic₇₉	1	8	yes	complemented
162	C₂×#107	C₂×D₇₉	1	2	yes	—
163	C₂×#106	—	79	2	no	—
164	C₂×#106	—	79	2	no	—
165	C₂×#106	—	158	4	no	—
166	C₂×#106	C₂×D₇₉	1	2	yes	—
167	#405	—	158	4	no	—
168	C₂×#97	—	79	2	no	—
169	C₂×C₄×C₈	—	79	2	no	—
170	#389	—	79	8	no	—
171	#385	—	158	4	no	—
172	#384	—	158	2	no	—
173	#382	—	158	2	no	—
174	#340	—	79	8	no	—
175	C₂³×C₄	—	158	2	no	—
176	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
177	C₂×M₁₆	—	79	4	no	—
178	C₂×M₁₆	—	158	2	no	—
179	C₂×M₁₆	—	158	2	no	—
180	C₂²×C₈	—	79	2	no	—
181	C₂²×C₈	Dic₁₅₈	1	4	yes	—
182	C₂²×C₈	D₃₁₆	1	4	yes	—
183	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
184	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
186	#117	—	2	4	no	—
187	#117	—	158	2	no	—
188	#117	—	158	2	no	—
189	#117	C₂×Dic₇₉	1	8	yes	—
190	C₂×C₄⋊C₄	—	79	2	no	—
191	C₂×C₄⋊C₄	—	79	4	no	—
192	C₂×C₄⋊C₄	—	158	2	no	—
193	C₂×C₄⋊C₄	—	158	4	no	—
194	C₂×C₄⋊C₄	—	158	4	no	—
195	C₂×C₄⋊C₄	—	158	8	no	—
196	C₂×C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	79	2	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	79	2	no	—
201	#107	C₂×Dic₇₉	1	8	yes	complemented
202	#106	—	158	4	no	—
203	#106	—	158	4	no	—
204	#106	C₂×Dic₇₉	1	8	yes	complemented
205	#97	—	158	4	no	—
206	C₄×C₈	—	158	4	no	—
207	#95	—	158	8	no	—
208	C₂⁴	D₄×D₇₉	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	316	4	no	—
211	C₂²×C₄	—	79	4	no	—
212	C₂²×C₄	—	632	4	no	—
213	C₂²×C₄	—	158	4	no	—
214	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
215	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
217	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
218	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
219	C₂²×C₄	#11929623	1	2	yes	—
220	C₂²×C₄	#11929625	1	2	yes	—
221	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
222	M₁₆	—	158	8	no	—
223	C₂×C₈	—	79	4	no	—
224	C₂×C₈	—	158	2	no	—
225	C₂×C₈	—	158	2	no	—
226	C₂×C₈	C₄⋊Dic₇₉	1	16	yes	complemented
227	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
228	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
229	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
230	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
231	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
232	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	316	4	no	—
235	C₄⋊C₄	—	632	8	no	—
236	C₄⋊C₄	—	158	4	no	—
237	C₄⋊C₄	—	158	4	no	—
238	C₄⋊C₄	—	158	4	no	—
239	C₄⋊C₄	C₂²×Dic₇₉	1	8	yes	—
240	C₂²⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	158	4	no	—
242	C₄²	—	2	8	no	—
243	C₄²	—	158	2	no	—
244	C₄²	—	158	2	no	—
245	C₂³	—	2	4	no	—
246	C₂³	D₄×Dic₇₉	1	2	yes	—
247	C₂³	#14579136	1	2	yes	—
248	C₂³	#14579160	1	2	yes	—
249	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	316	4	no	—
254	C₂×C₄	—	316	8	no	—
255	C₂×C₄	—	632	8	no	—
256	C₂×C₄	—	158	4	no	—
257	C₂×C₄	—	158	8	no	—
258	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
259	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
260	C₂×C₄	Q₈×Dic₇₉	1	4	yes	—
261	C₂×C₄	#14579136	1	2	yes	—
262	C₂×C₄	#14579141	1	4	yes	—
263	C₂×C₄	#14579152	1	4	yes	—
264	C₂×C₄	#14579160	1	2	yes	—
265	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
267	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
268	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
270	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
271	C₈	—	158	4	no	—
272	C₈	C₂×C₄⋊Dic₇₉	1	4	yes	—
273	C₈	C₂×C₄⋊Dic₇₉	1	4	yes	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	#19098462	1	4	yes	—
278	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
279	C₂²	C₂×#14579136	1	1	yes	characteristic
280	C₂²	C₂×#14579160	1	1	yes	characteristic
281	C₂²	#19099370	1	4	yes	—
282	C₄	—	4	8	no	—
283	C₄	—	316	4	no	—
284	C₄	—	632	4	no	—
285	C₄	#19098462	1	4	yes	—
286	C₄	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
287	C₄	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
288	C₄	C₂×#14579141	1	1	yes	characteristic
289	C₄	C₂×#14579152	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	C₂×#19098462	1	1	yes	characteristic
292	C₂	#23772284	1	2	yes	—
293	C₂	#23773040	1	2	yes	—
294	C₂	#23773932	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	#21155
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. #23764139	C₂	semidirect
5. #23764129	C₂	semidirect
24. #427×C₇₉	C₄	semidirect
58. #107×C₇₉	C₂×C₄	semidirect
59. #106×C₇₉	C₂×C₄	semidirect
86. C₂×C₈×C₇₉	C₄⋊C₄	semidirect
155. C₇₉	#21155	semidirect
161. #427	Dic₇₉	semidirect
201. #107	C₂×Dic₇₉	semidirect
204. #106	C₂×Dic₇₉	semidirect
226. 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×#14579093	2
C₂×#19098518	2
#23766023	2
#23764139	2
#23764129	2
#23763121	2
C₂×#427×C₇₉	2
#21155	79