Subgroups — #34187566

Order 20224.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764353	C₂	1	2	yes	maximal
2	#23763267	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098309	C₂	1	2	yes	maximal
5	C₂×#494×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#14579071	—	2	1	no	—
9	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
10	#19098325	—	2	4	no	—
11	#19098317	—	2	1	no	—
12	#19098314	—	2	2	no	—
13	#19098309	—	2	4	no	—
14	C₂²×Q₁₆×C₇₉	—	2	1	no	—
15	C₂²×Q₁₆×C₇₉	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
17	#494×C₇₉	—	2	2	no	—
18	#494×C₇₉	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	—	2	1	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₂×C₄⋊Dic₇₉	D₄	1	2	yes	—
23	C₂×C₄×Dic₇₉	—	2	1	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
25	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
28	#14579071	—	4	2	no	—
29	C₇₉⋊#95	—	2	1	no	—
30	C₂²×Q₈×C₇₉	—	2	2	no	—
31	C₂×Q₁₆×C₇₉	—	2	4	no	—
32	C₂×Q₁₆×C₇₉	—	2	4	no	—
33	C₂×Q₁₆×C₇₉	—	4	2	no	—
34	C₂×Q₁₆×C₇₉	—	4	2	no	—
35	C₂×Q₁₆×C₇₉	—	4	2	no	—
36	C₂×Q₁₆×C₇₉	—	4	2	no	—
37	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₄⋊Q₈×C₇₉	—	2	2	no	—
40	C₄⋊Q₈×C₇₉	—	2	2	no	—
41	C₄⋊Q₈×C₇₉	—	2	2	no	—
42	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
45	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₇₉	D₄	1	2	yes	—
48	C₂²×Dic₇₉	—	2	2	no	—
49	C₂×C₇₉⋊C₈	—	2	2	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₂×C₇₉⋊C₈	—	4	1	no	—
52	C₂×C₇₉⋊C₈	—	4	1	no	—
53	C₂×C₇₉⋊C₈	—	4	4	no	—
54	C₄⋊Dic₇₉	—	2	4	no	—
55	C₄×Dic₇₉	—	4	1	no	—
56	C₄×Dic₇₉	—	4	1	no	—
57	C₂×Q₈×C₇₉	—	2	4	no	—
58	C₂×Q₈×C₇₉	—	4	2	no	—
59	C₂×Q₈×C₇₉	—	4	2	no	—
60	C₂×Q₈×C₇₉	—	4	2	no	—
61	C₂×Q₈×C₇₉	—	4	2	no	—
62	C₂×Q₈×C₇₉	—	4	2	no	—
63	C₂²×C₄×C₇₉	—	4	2	no	—
64	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
67	Q₁₆×C₇₉	—	4	8	no	—
68	Q₁₆×C₇₉	—	4	8	no	—
69	C₂×C₈×C₇₉	—	2	1	no	—
70	C₂×C₈×C₇₉	—	2	1	no	—
71	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₉	D₈	1	2	yes	—
74	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
75	C₄⋊C₄×C₇₉	—	2	4	no	—
76	C₄⋊C₄×C₇₉	—	2	4	no	—
77	C₄⋊C₄×C₇₉	—	4	2	no	—
78	C₄⋊C₄×C₇₉	—	4	2	no	—
79	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇₉	#41	1	1	yes	characteristic
83	C₂³×C₇₉	#121	1	1	yes	characteristic
84	Q₈×C₇₉	—	4	4	no	—
85	Q₈×C₇₉	—	4	4	no	—
86	Q₈×C₇₉	—	8	2	no	—
87	Q₈×C₇₉	—	8	2	no	—
88	C₂×C₄×C₇₉	—	2	2	no	—
89	C₂×C₄×C₇₉	—	4	4	no	—
90	C₂×C₄×C₇₉	—	8	2	no	—
91	C₂×C₄×C₇₉	—	8	2	no	—
92	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
96	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
97	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₇₉	#137	1	4	yes	—
103	C₂×Dic₇₉	—	4	2	no	—
104	C₂×Dic₇₉	—	4	2	no	—
105	C₂×Dic₇₉	—	4	2	no	—
106	C₈×C₇₉	—	2	2	no	—
107	C₈×C₇₉	#102	1	4	yes	—
108	C₇₉⋊C₈	—	4	2	no	—
109	C₇₉⋊C₈	—	8	2	no	—
110	C₂²×C₇₉	#390	1	1	yes	characteristic
111	C₂²×C₇₉	#459	1	1	yes	characteristic
112	C₂²×C₇₉	#460	1	1	yes	characteristic
113	C₂²×C₇₉	#470	1	2	yes	—
114	C₂²×C₇₉	#473	1	2	yes	—
115	C₄×C₇₉	—	2	1	no	—
116	C₄×C₇₉	—	2	1	no	—
117	C₄×C₇₉	—	8	4	no	—
118	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
119	C₄×C₇₉	#418	1	1	yes	characteristic
120	C₄×C₇₉	#441	1	2	yes	—
121	C₄×C₇₉	#459	1	1	yes	characteristic
122	C₄×C₇₉	#460	1	1	yes	characteristic
123	C₄×C₇₉	#497	1	2	yes	—
124	Dic₇₉	—	8	2	no	—
125	#9742	—	79	1	no	maximal, Sylow
126	C₂×C₇₉	#1689	1	2	yes	—
127	C₂×C₇₉	#1690	1	2	yes	—
128	C₂×C₇₉	#1971	1	1	yes	characteristic
129	C₂×C₇₉	#1979	1	1	yes	characteristic
130	C₂×C₇₉	#1982	1	1	yes	characteristic
131	C₂×#494	D₇₉	1	1	yes	characteristic
132	#1988	—	79	2	no	—
133	#1770	—	79	1	no	—
134	C₂×#334	—	79	1	no	—
135	C₂×#332	—	79	2	no	—
136	C₇₉	#9742	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₁₆	—	2	1	no	—
138	C₂²×Q₁₆	—	2	1	no	—
139	C₂×C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
140	#494	—	2	2	no	—
141	#494	Dic₇₉	1	4	yes	complemented
142	C₂×C₄⋊C₈	—	79	1	no	—
143	C₂×C₄⋊C₈	—	79	2	no	—
144	C₂×#97	—	158	1	no	—
145	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
146	C₄×C₄⋊C₄	—	79	1	no	—
147	#340	—	158	4	no	—
148	#336	—	158	1	no	—
149	#334	—	158	2	no	—
150	#332	—	158	4	no	—
151	C₂²×Q₈	—	2	2	no	—
152	C₂×Q₁₆	—	2	4	no	—
153	C₂×Q₁₆	—	2	4	no	—
154	C₂×Q₁₆	—	4	2	no	—
155	C₂×Q₁₆	—	4	2	no	—
156	C₂×Q₁₆	—	4	2	no	—
157	C₂×Q₁₆	—	4	2	no	—
158	C₂²×C₈	—	158	2	no	—
159	C₂²×C₈	—	158	1	no	—
160	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	C₂×Dic₇₉	1	4	yes	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	79	2	no	—
169	C₂×C₄²	—	158	1	no	—
170	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
171	C₄⋊C₈	—	79	2	no	—
172	C₄⋊C₈	—	158	2	no	—
173	C₄⋊C₈	—	158	2	no	—
174	C₄⋊C₈	—	158	1	no	—
175	#97	—	316	2	no	—
176	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
177	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
178	#95	—	158	1	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	158	2	no	—
187	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
189	C₂²×C₄	#11929623	1	1	yes	characteristic
190	Q₁₆	—	4	8	no	—
191	Q₁₆	—	4	8	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	316	1	no	—
195	C₂×C₈	—	316	1	no	—
196	C₂×C₈	—	316	2	no	—
197	C₂×C₈	—	316	4	no	—
198	C₂×C₈	—	158	2	no	—
199	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
200	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
201	C₂×C₈	#11929609	1	2	yes	—
202	C₂×C₈	#11929612	1	2	yes	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	158	4	no	—
208	C₄²	—	316	1	no	—
209	C₄²	—	316	1	no	—
210	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
211	C₄²	D₄×D₇₉	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
213	C₄²	#11929623	1	1	yes	characteristic
214	C₂³	#14579132	1	1	yes	characteristic
215	Q₈	—	4	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	8	2	no	—
218	Q₈	—	8	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	316	2	no	—
224	C₂×C₄	—	316	2	no	—
225	C₂×C₄	—	316	2	no	—
226	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
227	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
228	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
229	C₂×C₄	#14579132	1	1	yes	characteristic
230	C₂×C₄	#14579148	1	1	yes	characteristic
231	C₂×C₄	#14579157	1	1	yes	characteristic
232	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
233	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
234	C₂×C₄	#14579187	1	1	yes	characteristic
235	C₂×C₄	#14579187	1	1	yes	characteristic
236	C₂×C₄	#14579193	1	4	yes	—
237	C₈	—	2	2	no	—
238	C₈	—	316	2	no	—
239	C₈	—	632	2	no	—
240	C₈	#14579082	1	4	yes	—
241	C₂²	#19098466	1	1	yes	characteristic
242	C₂²	#19098755	1	1	yes	characteristic
243	C₂²	#19098760	1	1	yes	characteristic
244	C₂²	#19098818	1	2	yes	—
245	C₂²	#19098835	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	4	no	—
249	C₄	—	632	2	no	—
250	C₄	C₂×#14579082	1	1	yes	characteristic
251	C₄	#19098568	1	1	yes	characteristic
252	C₄	#19098670	1	2	yes	—
253	C₄	#19098755	1	1	yes	characteristic
254	C₄	#19098760	1	1	yes	characteristic
255	C₄	#19098956	1	2	yes	—
256	C₂	#23762886	1	2	yes	—
257	C₂	#23762891	1	2	yes	—
258	C₂	#23764272	1	1	yes	characteristic
259	C₂	#23764316	1	1	yes	characteristic
260	C₂	#23764328	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579132
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#494×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#121
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#494	D₇₉
hypocenter	C₇₉	#9742
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
18. #494×C₇₉	C₄	semidirect
136. C₇₉	#9742	semidirect
141. #494	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23764353	2
#23763267	2
C₂×#19098314	2
C₂×#19098309	2
C₂×#494×C₇₉	2
#9742	79