Subgroups — #34187611

Order 20224.

406 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
2	#23762148	C₂	1	1	yes	maximal, characteristic
3	#23761899	C₂	1	2	yes	maximal
4	#23761803	C₂	1	2	yes	maximal
5	C₇₉×#3160	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₉	—	2	1	no	—
9	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
10	#19098317	—	2	1	no	—
11	#19098316	—	2	1	no	—
12	#19098314	—	2	1	no	—
13	#19098300	—	2	4	no	—
14	Q₈⋊(C₇₉⋊C₈)	—	2	4	no	—
15	#533×C₇₉	C₂²	1	2	yes	—
16	#500×C₇₉	C₄	1	1	yes	characteristic, complemented
17	#500×C₇₉	C₄	1	1	yes	characteristic, complemented
18	#499×C₇₉	—	2	1	no	—
19	#499×C₇₉	—	2	1	no	—
20	#498×C₇₉	C₄	1	2	yes	complemented
21	#427×C₇₉	—	2	1	no	—
22	#427×C₇₉	—	2	1	no	—
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₂²×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₄⋊Dic₇₉	—	2	1	no	—
26	C₂×C₄×Dic₇₉	—	2	1	no	—
27	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
28	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
29	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
30	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
31	C₄×C₇₉⋊C₈	—	2	2	no	—
32	C₈×Dic₇₉	—	4	2	no	—
33	C₇₉⋊#95	D₄	1	2	yes	—
34	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
37	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
38	#125×C₇₉	—	2	2	no	—
39	C₂²⋊Q₈×C₇₉	—	2	2	no	—
40	C₂²⋊Q₈×C₇₉	—	2	2	no	—
41	C₄×Q₈×C₇₉	—	2	2	no	—
42	C₄×Q₈×C₇₉	—	2	2	no	—
43	#117×C₇₉	—	2	2	no	—
44	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
45	#107×C₇₉	—	2	1	no	—
46	#107×C₇₉	—	2	1	no	—
47	#107×C₇₉	—	2	1	no	—
48	#107×C₇₉	—	2	1	no	—
49	#106×C₇₉	—	2	2	no	—
50	#106×C₇₉	—	2	2	no	—
51	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
55	C₂²×Dic₇₉	—	2	2	no	—
56	C₂×C₇₉⋊C₈	—	2	2	no	—
57	C₂×C₇₉⋊C₈	—	4	2	no	—
58	C₂×C₇₉⋊C₈	—	4	1	no	—
59	C₂×C₇₉⋊C₈	—	4	1	no	—
60	C₄⋊Dic₇₉	—	2	4	no	—
61	C₄×Dic₇₉	—	4	1	no	—
62	C₄×Dic₇₉	—	4	1	no	—
63	C₂×Q₈×C₇₉	—	2	2	no	—
64	C₂×Q₈×C₇₉	—	2	2	no	—
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
68	C₂×C₈×C₇₉	—	2	1	no	—
69	C₂×C₈×C₇₉	—	2	1	no	—
70	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
75	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
76	C₄⋊C₄×C₇₉	—	2	2	no	—
77	C₄⋊C₄×C₇₉	—	2	2	no	—
78	C₄⋊C₄×C₇₉	—	2	2	no	—
79	C₄⋊C₄×C₇₉	—	2	2	no	—
80	C₄⋊C₄×C₇₉	—	4	2	no	—
81	C₄⋊C₄×C₇₉	—	4	2	no	—
82	C₂²⋊C₄×C₇₉	—	4	2	no	—
83	C₄²×C₇₉	—	4	2	no	—
84	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇₉	#41	1	1	yes	characteristic
88	C₂³×C₇₉	#121	1	1	yes	characteristic
89	Q₈×C₇₉	—	4	4	no	—
90	Q₈×C₇₉	—	8	2	no	—
91	C₂×C₄×C₇₉	—	2	2	no	—
92	C₂×C₄×C₇₉	—	4	1	no	—
93	C₂×C₄×C₇₉	—	4	2	no	—
94	C₂×C₄×C₇₉	—	4	2	no	—
95	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
104	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
105	C₂×Dic₇₉	—	2	4	no	—
106	C₂×Dic₇₉	—	4	2	no	—
107	C₂×Dic₇₉	—	4	2	no	—
108	C₈×C₇₉	—	2	1	no	—
109	C₈×C₇₉	—	2	1	no	—
110	C₈×C₇₉	#103	1	2	yes	—
111	C₈×C₇₉	#103	1	2	yes	—
112	C₇₉⋊C₈	—	4	2	no	—
113	C₇₉⋊C₈	—	8	2	no	—
114	C₂²×C₇₉	—	2	1	no	—
115	C₂²×C₇₉	—	2	1	no	—
116	C₂²×C₇₉	#390	1	1	yes	characteristic
117	C₂²×C₇₉	#461	1	1	yes	characteristic
118	C₂²×C₇₉	#462	1	1	yes	characteristic
119	C₄×C₇₉	—	2	1	no	—
120	C₄×C₇₉	—	2	1	no	—
121	C₄×C₇₉	—	2	1	no	—
122	C₄×C₇₉	—	8	2	no	—
123	C₄×C₇₉	—	8	2	no	—
124	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
125	C₄×C₇₉	#419	1	1	yes	characteristic
126	C₄×C₇₉	#443	1	2	yes	—
127	C₄×C₇₉	#461	1	1	yes	characteristic
128	C₄×C₇₉	#462	1	1	yes	characteristic
129	Dic₇₉	—	4	4	no	—
130	#9749	—	79	1	no	maximal, Sylow
131	C₂×C₇₉	—	2	1	no	—
132	C₂×C₇₉	—	2	1	no	—
133	C₂×C₇₉	#1974	1	1	yes	characteristic
134	C₂×C₇₉	#1976	1	1	yes	characteristic
135	C₂×C₇₉	#1981	1	1	yes	characteristic
136	#3160	D₇₉	1	1	yes	characteristic
137	C₈×C₄⋊C₄	—	79	1	no	—
138	#1564	—	79	1	no	—
139	#1503	—	79	2	no	—
140	#1481	—	79	2	no	—
141	C₇₉	#9749	1	1	yes	characteristic, complemented, Sylow
142	#533	C₂×D₇₉	1	2	yes	—
143	#500	Dic₇₉	1	1	yes	characteristic, complemented
144	#500	Dic₇₉	1	1	yes	characteristic, complemented
145	#499	—	2	1	no	—
146	#499	—	2	1	no	—
147	#498	Dic₇₉	1	2	yes	complemented
148	#431	—	79	2	no	—
149	#427	—	2	1	no	—
150	#427	—	2	1	no	—
151	C₂×C₄⋊C₈	—	79	1	no	—
152	C₂×C₄×C₈	—	158	1	no	—
153	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
154	C₄×C₄⋊C₄	—	79	1	no	—
155	#336	—	158	1	no	—
156	#335	—	158	1	no	—
157	#334	—	158	1	no	—
158	#328	—	158	4	no	—
159	Q₈⋊C₈	—	158	4	no	—
160	C₂²×C₈	—	158	1	no	—
161	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
163	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
164	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
165	#125	—	2	2	no	—
166	C₂²⋊Q₈	—	2	2	no	—
167	C₂²⋊Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	#117	—	2	2	no	—
171	C₂×C₄⋊C₄	—	158	1	no	—
172	C₂×C₄²	—	158	1	no	—
173	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
174	#107	—	2	1	no	—
175	#107	—	2	1	no	—
176	#107	—	2	1	no	—
177	#107	—	2	1	no	—
178	#106	—	2	2	no	—
179	#106	—	2	2	no	—
180	C₄⋊C₈	—	79	2	no	—
181	C₄⋊C₈	—	158	2	no	—
182	C₄⋊C₈	—	158	1	no	—
183	C₂²⋊C₈	—	158	2	no	—
184	C₄×C₈	—	316	2	no	—
185	C₄×C₈	—	158	2	no	—
186	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
190	#95	—	79	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂²×C₄	—	158	2	no	—
194	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
196	C₂²×C₄	#11929623	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	316	1	no	—
200	C₂×C₈	—	316	1	no	—
201	C₂×C₈	—	316	2	no	—
202	C₂×C₈	—	158	2	no	—
203	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
207	C₂×C₈	#11929611	1	2	yes	—
208	C₂×C₈	#11929614	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	158	4	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₄²	—	4	2	no	—
218	C₄²	—	316	1	no	—
219	C₄²	—	316	1	no	—
220	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
221	C₄²	D₄×D₇₉	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
223	C₄²	#11929623	1	1	yes	characteristic
224	C₂³	#14579132	1	1	yes	characteristic
225	Q₈	—	4	4	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	316	2	no	—
232	C₂×C₄	—	316	2	no	—
233	C₂×C₄	—	158	4	no	—
234	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
235	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
237	C₂×C₄	#14579132	1	1	yes	characteristic
238	C₂×C₄	#14579148	1	1	yes	characteristic
239	C₂×C₄	#14579157	1	1	yes	characteristic
240	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
241	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
242	C₂×C₄	#14579192	1	1	yes	characteristic
243	C₂×C₄	#14579192	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	316	2	no	—
247	C₈	—	632	2	no	—
248	C₈	#14579085	1	2	yes	—
249	C₈	#14579085	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#19098466	1	1	yes	characteristic
253	C₂²	#19098765	1	1	yes	characteristic
254	C₂²	#19098770	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	8	2	no	—
259	C₄	—	8	2	no	—
260	C₄	—	316	4	no	—
261	C₄	C₂×#14579085	1	1	yes	characteristic
262	C₄	#19098572	1	1	yes	characteristic
263	C₄	#19098680	1	2	yes	—
264	C₄	#19098765	1	1	yes	characteristic
265	C₄	#19098770	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#23764289	1	1	yes	characteristic
269	C₂	#23764299	1	1	yes	characteristic
270	C₂	#23764324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579132
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3160	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#121
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3160	D₇₉
hypocenter	C₇₉	#9749
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
16. #500×C₇₉	C₄	semidirect
17. #500×C₇₉	C₄	semidirect
20. #498×C₇₉	C₄	semidirect
141. C₇₉	#9749	semidirect
143. #500	Dic₇₉	semidirect
144. #500	Dic₇₉	semidirect
147. #498	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Dic₇₉	2
#23762148	2
#23761899	2
#23761803	2
C₇₉×#3160	2
#9749	79