Subgroups — #34287341

Order 20224.

898 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	C₂×#19098694	C₂	1	2	yes	maximal
2	#23763275	C₂	1	4	yes	maximal
3	C₄⋊C₄×C₇₉⋊C₈	C₂	1	2	yes	maximal
4	#23763253	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal
6	#23763149	C₂	1	2	yes	maximal
7	#23763146	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#19098694	C₄	1	16	yes	complemented
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
12	C₂×#14579072	C₂²	1	2	yes	—
13	C₂×#14579072	C₂²	1	4	yes	—
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
16	#19098318	C₂²	1	1	yes	characteristic
17	#19098318	C₂²	1	2	yes	—
18	#19098318	C₂²	1	4	yes	—
19	#19098318	C₂²	1	2	yes	—
20	#19098318	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₇₉	C₄	1	2	yes	—
22	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₇₉	C₄	1	4	yes	—
24	C₄×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
27	C₄³×C₇₉	C₂²	1	2	yes	—
28	C₄³×C₇₉	C₂²	1	1	yes	characteristic
29	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
30	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
32	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	16	yes	—
33	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
34	#14579072	—	4	2	no	—
35	#14579072	C₂×C₄	1	16	yes	—
36	C₄×C₇₉⋊C₈	—	4	4	no	—
37	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
38	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₇₉	—	2	8	no	—
40	C₄×Q₈×C₇₉	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
50	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
54	#95×C₇₉	C₂×C₄	1	4	yes	—
55	#95×C₇₉	C₂×C₄	1	2	yes	—
56	C₂×C₇₉⋊C₈	—	2	2	no	—
57	C₂×C₇₉⋊C₈	—	2	4	no	—
58	C₂×C₇₉⋊C₈	—	2	2	no	—
59	C₂×C₇₉⋊C₈	—	2	2	no	—
60	C₂×C₇₉⋊C₈	—	2	2	no	—
61	C₂×C₇₉⋊C₈	—	2	4	no	—
62	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
63	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
64	C₂×C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
65	C₂×Q₈×C₇₉	—	2	4	no	—
66	C₂×Q₈×C₇₉	C₄²	1	4	yes	—
67	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₇₉	—	2	8	no	—
76	C₄⋊C₄×C₇₉	C₄²	1	8	yes	—
77	C₄⋊C₄×C₇₉	C₄²	1	4	yes	—
78	C₄⋊C₄×C₇₉	M₁₆	1	8	yes	—
79	C₄²×C₇₉	—	2	8	no	—
80	C₄²×C₇₉	—	2	4	no	—
81	C₄²×C₇₉	—	2	8	no	—
82	C₄²×C₇₉	—	2	4	no	—
83	C₄²×C₇₉	—	2	2	no	—
84	C₄²×C₇₉	—	2	1	no	—
85	C₄²×C₇₉	—	2	2	no	—
86	C₄²×C₇₉	—	2	2	no	—
87	C₄²×C₇₉	—	2	1	no	—
88	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
89	C₄²×C₇₉	C₂²×C₄	1	8	yes	—
90	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
91	Q₈×C₇₉	—	2	8	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	4	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	—	2	4	no	—
96	C₂×C₄×C₇₉	—	2	4	no	—
97	C₂×C₄×C₇₉	—	2	2	no	—
98	C₂×C₄×C₇₉	—	2	4	no	—
99	C₂×C₄×C₇₉	—	2	2	no	—
100	C₂×C₄×C₇₉	—	2	2	no	—
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₇₉	C₂×M₁₆	1	8	yes	—
107	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₇₉	#131	1	4	yes	—
109	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
112	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
113	C₇₉⋊C₈	—	2	4	no	—
114	C₇₉⋊C₈	—	2	4	no	—
115	C₇₉⋊C₈	—	2	8	no	—
116	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
117	C₂²×C₇₉	#519	1	1	yes	characteristic
118	C₂²×C₇₉	#520	1	1	yes	characteristic
119	C₂²×C₇₉	C₂²×M₁₆	1	2	yes	—
120	C₂²×C₇₉	C₂×#131	1	2	yes	—
121	C₄×C₇₉	—	2	4	no	—
122	C₄×C₇₉	—	2	2	no	—
123	C₄×C₇₉	—	2	4	no	—
124	C₄×C₇₉	—	2	4	no	—
125	C₄×C₇₉	—	2	8	no	—
126	C₄×C₇₉	—	4	2	no	—
127	C₄×C₇₉	C₄×M₁₆	1	4	yes	—
128	C₄×C₇₉	#519	1	2	yes	—
129	C₄×C₇₉	#520	1	2	yes	—
130	#20573	—	79	1	no	maximal, Sylow
131	C₂×C₇₉	#2276	1	1	yes	characteristic
132	C₂×C₇₉	C₂×C₄×M₁₆	1	1	yes	characteristic
133	C₂×C₇₉	C₄×#131	1	1	yes	characteristic
134	C₂×C₇₉	#2984	1	2	yes	—
135	C₂×C₇₉	#2990	1	2	yes	—
136	C₂×#446	—	79	2	no	—
137	C₄²×Q₈	D₇₉	1	1	yes	characteristic
138	#1771	—	79	4	no	—
139	C₈×C₄⋊C₄	—	79	2	no	—
140	#1767	—	79	1	no	—
141	C₄×C₄⋊C₈	—	79	2	no	—
142	#1745	—	79	2	no	—
143	#1744	—	79	1	no	—
144	C₇₉	#20573	1	1	yes	characteristic, complemented, Sylow
145	C₂×C₄×Q₈	Dic₇₉	1	2	yes	—
146	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
147	#446	—	79	16	no	—
148	C₂×C₄⋊C₈	—	79	4	no	—
149	C₂×C₄⋊C₈	—	79	2	no	—
150	C₂×#97	—	79	2	no	—
151	C₂×#97	—	79	4	no	—
152	C₂×C₄×C₈	—	79	4	no	—
153	C₂×C₄×C₈	—	79	2	no	—
154	C₄×C₄⋊C₄	Dic₇₉	1	4	yes	—
155	C₄×C₄⋊C₄	Dic₇₉	1	2	yes	—
156	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
157	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
158	C₄³	C₂×D₇₉	1	1	yes	characteristic
159	C₄³	C₂×D₇₉	1	2	yes	—
160	#336	—	79	1	no	—
161	#336	—	79	1	no	—
162	#336	—	79	2	no	—
163	#336	—	79	2	no	—
164	#336	—	79	4	no	—
165	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
166	C₂²×C₈	—	79	2	no	—
167	C₂²×C₈	—	79	2	no	—
168	C₂²×C₈	—	79	4	no	—
169	C₄×Q₈	—	2	8	no	—
170	C₄×Q₈	C₄×D₇₉	1	8	yes	—
171	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
172	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
175	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
176	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
177	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
178	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
179	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
181	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
184	C₄⋊C₈	—	79	8	no	—
185	C₄⋊C₈	—	79	16	no	—
186	#97	—	316	2	no	—
187	#97	—	79	16	no	—
188	C₄×C₈	—	316	4	no	—
189	C₄×C₈	—	79	8	no	—
190	#95	C₂×Dic₇₉	1	4	yes	—
191	#95	C₂×Dic₇₉	1	2	yes	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	C₄×Dic₇₉	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
195	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
201	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
202	C₂×C₈	—	79	4	no	—
203	C₂×C₈	—	79	4	no	—
204	C₂×C₈	—	79	8	no	—
205	C₂×C₈	—	158	2	no	—
206	C₂×C₈	—	158	4	no	—
207	C₂×C₈	—	158	2	no	—
208	C₂×C₈	—	158	2	no	—
209	C₂×C₈	—	158	2	no	—
210	C₂×C₈	—	158	4	no	—
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	C₄×Dic₇₉	1	8	yes	—
213	C₄⋊C₄	C₄×Dic₇₉	1	4	yes	—
214	C₄⋊C₄	#11929608	1	8	yes	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	8	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	8	no	—
223	C₄²	—	2	4	no	—
224	C₄²	C₂×C₄×D₇₉	1	4	yes	—
225	C₄²	C₂×C₄×D₇₉	1	8	yes	—
226	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
227	Q₈	—	2	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
239	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
240	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
241	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
242	C₂×C₄	C₂×#11929608	1	8	yes	—
243	C₂×C₄	C₂×#11929608	1	4	yes	—
244	C₂×C₄	#14579168	1	4	yes	—
245	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
246	C₂×C₄	#14579207	1	1	yes	characteristic
247	C₂×C₄	#14579208	1	1	yes	characteristic
248	C₂×C₄	#14579209	1	2	yes	—
249	C₈	—	158	4	no	—
250	C₈	—	158	4	no	—
251	C₈	—	158	8	no	—
252	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
253	C₂²	#19099041	1	1	yes	characteristic
254	C₂²	#19099046	1	1	yes	characteristic
255	C₂²	C₂²×#11929608	1	2	yes	—
256	C₂²	C₂×#14579168	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	8	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	C₄×#11929608	1	4	yes	—
264	C₄	#19099039	1	2	yes	—
265	C₄	#19099045	1	2	yes	—
266	C₂	#23765765	1	1	yes	characteristic
267	C₂	C₂×C₄×#11929608	1	1	yes	characteristic
268	C₂	C₄×#14579168	1	1	yes	characteristic
269	C₂	#23772489	1	2	yes	—
270	C₂	#23772525	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Fitting	C₄²×Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂³×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₄²×Q₈	D₇₉
hypocenter	C₇₉	#20573
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
9. #19098694	C₄	semidirect
144. C₇₉	#20573	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098694	2
#23763275	2
C₄⋊C₄×C₇₉⋊C₈	2
#23763253	2
C₄×C₄⋊(C₇₉⋊C₈)	2
#23763149	2
#23763146	2
C₄²×Q₈×C₇₉	2
#20573	79