Subgroups — #34188082

Order 20224.

694 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764207	C₂	1	2	yes	maximal, complemented
2	#23764207	C₂	1	2	yes	maximal, complemented
3	#23763528	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#493×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579082	—	2	4	no	—
7	C₂×#14579082	—	2	4	no	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	#19098433	C₂²	1	2	yes	—
11	#19098326	—	2	2	no	—
12	#19098285	—	2	2	no	—
13	C₂²×D₈×C₇₉	—	2	2	no	—
14	C₂×#127×C₇₉	C₂²	1	2	yes	—
15	#493×C₇₉	—	2	2	no	—
16	#493×C₇₉	C₄	1	4	yes	complemented
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
19	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	C₂×C₄⋊Dic₇₉	—	2	2	no	—
22	C₂×C₄⋊Dic₇₉	—	2	2	no	—
23	#14579082	—	4	8	no	—
24	#14579082	—	4	8	no	—
25	C₄×C₇₉⋊C₈	—	2	1	no	—
26	C₄×C₇₉⋊C₈	—	2	1	no	—
27	C₄×C₇₉⋊C₈	—	2	1	no	—
28	C₄×C₇₉⋊C₈	—	2	1	no	—
29	C₂²×D₄×C₇₉	—	2	2	no	—
30	C₂²×D₄×C₇₉	—	2	2	no	—
31	C₂²×D₄×C₇₉	—	2	2	no	—
32	C₂×D₈×C₇₉	—	2	8	no	—
33	C₂×D₈×C₇₉	—	4	8	no	—
34	C₂²×C₈×C₇₉	D₄	1	2	yes	—
35	#127×C₇₉	—	2	2	no	—
36	#127×C₇₉	—	2	2	no	—
37	#127×C₇₉	—	2	2	no	—
38	#127×C₇₉	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₇₉	D₄	1	2	yes	—
42	C₂²×Dic₇₉	—	4	2	no	—
43	C₂×C₇₉⋊C₈	—	2	2	no	—
44	C₂×C₇₉⋊C₈	—	2	2	no	—
45	C₂×C₇₉⋊C₈	—	2	2	no	—
46	C₂×C₇₉⋊C₈	—	2	4	no	—
47	C₂×C₇₉⋊C₈	D₈	1	4	yes	complemented
48	C₄⋊Dic₇₉	—	4	4	no	—
49	C₄⋊Dic₇₉	—	4	4	no	—
50	C₄⋊Dic₇₉	—	4	4	no	—
51	C₂⁴×C₇₉	—	4	2	no	—
52	C₂×D₄×C₇₉	—	2	4	no	—
53	C₂×D₄×C₇₉	—	2	4	no	—
54	C₂×D₄×C₇₉	—	2	4	no	—
55	C₂×D₄×C₇₉	—	4	2	no	—
56	C₂×D₄×C₇₉	—	4	2	no	—
57	C₂×D₄×C₇₉	—	4	2	no	—
58	C₂×D₄×C₇₉	—	4	4	no	—
59	C₂×D₄×C₇₉	—	4	4	no	—
60	C₂×D₄×C₇₉	—	4	4	no	—
61	C₂×D₄×C₇₉	—	4	4	no	—
62	C₂×D₄×C₇₉	—	4	4	no	—
63	C₂×D₄×C₇₉	—	4	4	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	C₂×D₄	1	1	yes	characteristic
67	D₈×C₇₉	—	4	16	no	—
68	C₂×C₈×C₇₉	—	2	4	no	—
69	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
70	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₇₉	#41	1	1	yes	characteristic
73	C₄²×C₇₉	#41	1	1	yes	characteristic
74	C₂³×C₇₉	—	4	4	no	—
75	C₂³×C₇₉	—	8	2	no	—
76	C₂³×C₇₉	—	8	2	no	—
77	C₂³×C₇₉	—	8	2	no	—
78	C₂³×C₇₉	—	8	2	no	—
79	C₂³×C₇₉	—	8	2	no	—
80	C₂³×C₇₉	—	8	2	no	—
81	C₂³×C₇₉	#127	1	1	yes	characteristic
82	D₄×C₇₉	—	4	8	no	—
83	D₄×C₇₉	—	4	8	no	—
84	D₄×C₇₉	—	4	8	no	—
85	D₄×C₇₉	—	8	4	no	—
86	D₄×C₇₉	—	8	4	no	—
87	D₄×C₇₉	—	8	4	no	—
88	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
94	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₇₉	#135	1	2	yes	—
96	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
97	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×Dic₇₉	—	8	2	no	—
99	C₂×Dic₇₉	—	8	2	no	—
100	C₂×Dic₇₉	—	8	2	no	—
101	C₈×C₇₉	—	2	4	no	—
102	C₇₉⋊C₈	—	2	4	no	—
103	C₇₉⋊C₈	—	4	2	no	—
104	C₂²×C₇₉	—	8	4	no	—
105	C₂²×C₇₉	—	8	4	no	—
106	C₂²×C₇₉	—	8	4	no	—
107	C₂²×C₇₉	—	16	2	no	—
108	C₂²×C₇₉	—	16	2	no	—
109	C₂²×C₇₉	—	16	2	no	—
110	C₂²×C₇₉	—	16	2	no	—
111	C₂²×C₇₉	#390	1	1	yes	characteristic
112	C₂²×C₇₉	#493	1	1	yes	characteristic
113	C₂²×C₇₉	#495	1	1	yes	characteristic
114	C₂²×C₇₉	#496	1	2	yes	—
115	C₂²×C₇₉	#496	1	2	yes	—
116	C₄×C₇₉	#418	1	2	yes	—
117	C₄×C₇₉	C₄×D₈	1	2	yes	—
118	C₄×C₇₉	#442	1	2	yes	—
119	C₄×C₇₉	#466	1	2	yes	—
120	C₄×C₇₉	#469	1	2	yes	—
121	C₄×C₇₉	#490	1	2	yes	—
122	Dic₇₉	—	16	2	no	—
123	#9832	—	79	1	no	maximal, Sylow
124	C₂×C₇₉	—	16	4	no	—
125	C₂×C₇₉	#1717	1	2	yes	—
126	C₂×C₇₉	#1721	1	2	yes	—
127	C₂×C₇₉	#1961	1	1	yes	characteristic
128	C₂×C₇₉	#1970	1	1	yes	characteristic
129	C₂×C₇₉	#1972	1	1	yes	characteristic
130	C₂×#493	D₇₉	1	1	yes	characteristic
131	#1957	—	79	2	no	—
132	#1957	—	79	2	no	—
133	#1834	—	79	1	no	—
134	C₂×#322	—	79	1	no	—
135	C₇₉	#9832	1	1	yes	characteristic, complemented, Sylow
136	C₂²×D₈	—	2	2	no	—
137	C₂×#127	C₂×D₇₉	1	2	yes	—
138	#493	—	2	2	no	—
139	#493	Dic₇₉	1	4	yes	complemented
140	C₂×#102	—	158	4	no	—
141	C₂×#102	—	158	4	no	—
142	C₂×C₄×C₈	—	79	1	no	—
143	C₂×C₄×C₈	—	79	1	no	—
144	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
145	#384	—	79	2	no	—
146	#340	—	158	2	no	—
147	#322	—	158	2	no	—
148	C₂²×D₄	—	2	2	no	—
149	C₂²×D₄	—	2	2	no	—
150	C₂²×D₄	—	2	2	no	—
151	C₂×D₈	—	2	8	no	—
152	C₂×D₈	—	4	8	no	—
153	C₂²×C₈	—	79	2	no	—
154	C₂²×C₈	—	79	2	no	—
155	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	—	2	2	no	—
159	#127	C₂×Dic₇₉	1	4	yes	—
160	C₂×C₄⋊C₄	—	158	2	no	—
161	C₂×C₄⋊C₄	—	158	2	no	—
162	C₂×C₄⋊C₄	—	158	2	no	—
163	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
164	#102	—	316	8	no	—
165	#102	—	316	8	no	—
166	C₄×C₈	—	158	1	no	—
167	C₄×C₈	—	158	1	no	—
168	C₄×C₈	—	158	1	no	—
169	C₄×C₈	—	158	1	no	—
170	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
171	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
172	C₂⁴	—	4	2	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂×D₄	—	4	4	no	—
184	C₂×D₄	—	4	4	no	—
185	C₂²×C₄	—	316	2	no	—
186	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
187	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
189	D₈	—	4	16	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	79	4	no	—
192	C₂×C₈	—	158	2	no	—
193	C₂×C₈	—	158	4	no	—
194	C₂×C₈	—	158	2	no	—
195	C₂×C₈	—	158	2	no	—
196	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
197	C₄⋊C₄	—	316	4	no	—
198	C₄⋊C₄	—	316	4	no	—
199	C₄⋊C₄	—	316	4	no	—
200	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
202	C₄²	#11929623	1	1	yes	characteristic
203	C₄²	#11929623	1	1	yes	characteristic
204	C₂³	—	4	4	no	—
205	C₂³	—	8	2	no	—
206	C₂³	—	8	2	no	—
207	C₂³	—	8	2	no	—
208	C₂³	—	8	2	no	—
209	C₂³	—	8	2	no	—
210	C₂³	—	8	2	no	—
211	C₂³	#14579157	1	1	yes	characteristic
212	D₄	—	4	8	no	—
213	D₄	—	4	8	no	—
214	D₄	—	4	8	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	4	no	—
217	D₄	—	8	4	no	—
218	C₂×C₄	—	632	2	no	—
219	C₂×C₄	—	632	2	no	—
220	C₂×C₄	—	632	2	no	—
221	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
222	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
223	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
224	C₂×C₄	#14579132	1	1	yes	characteristic
225	C₂×C₄	#14579132	1	1	yes	characteristic
226	C₂×C₄	#14579148	1	1	yes	characteristic
227	C₂×C₄	D₈×D₇₉	1	2	yes	—
228	C₂×C₄	#14579180	1	2	yes	—
229	C₂×C₄	#14579185	1	4	yes	—
230	C₂×C₄	#14579187	1	4	yes	—
231	C₈	—	2	4	no	—
232	C₈	—	316	2	no	—
233	C₈	—	158	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	—	8	4	no	—
237	C₂²	—	16	2	no	—
238	C₂²	—	16	2	no	—
239	C₂²	—	16	2	no	—
240	C₂²	—	16	2	no	—
241	C₂²	#19098466	1	1	yes	characteristic
242	C₂²	#19098938	1	1	yes	characteristic
243	C₂²	#19098946	1	1	yes	characteristic
244	C₂²	#19098950	1	2	yes	—
245	C₂²	#19098951	1	2	yes	—
246	C₄	—	1264	2	no	—
247	C₄	#19098568	1	2	yes	—
248	C₄	D₈×Dic₇₉	1	2	yes	—
249	C₄	#19098675	1	2	yes	—
250	C₄	#19098796	1	2	yes	—
251	C₄	#19098813	1	2	yes	—
252	C₄	#19098927	1	2	yes	—
253	C₂	—	16	4	no	—
254	C₂	#23763038	1	2	yes	—
255	C₂	#23763065	1	2	yes	—
256	C₂	#23764225	1	1	yes	characteristic
257	C₂	#23764267	1	1	yes	characteristic
258	C₂	#23764277	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting, prime radical	C₂×#493×C₇₉	C₂
socle	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#493	D₇₉
hypocenter	C₇₉	#9832
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #23764207	C₂	semidirect
2. #23764207	C₂	semidirect
3. #23763528	C₂	semidirect
4. C₂×#19098285	C₂	semidirect
16. #493×C₇₉	C₄	semidirect
47. C₂×C₇₉⋊C₈	D₈	semidirect
135. C₇₉	#9832	semidirect
139. #493	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23764207	2
#23763528	2
C₂×#19098285	2
C₂×#493×C₇₉	2
#9832	79