Subgroups — #34288682

Order 20224.

1126 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098601	C₂	1	4	yes	maximal
2	#23765807	C₂	1	2	yes	maximal
3	#23763564	C₂	1	4	yes	maximal, complemented
4	#23763560	C₂	1	2	yes	maximal, complemented
5	#23763557	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579120	C₂²	1	4	yes	—
8	#19098601	—	2	16	no	—
9	C₂×#14579095	C₂²	1	8	yes	—
10	C₂×#14579093	C₂²	1	8	yes	—
11	#19098433	C₂²	1	4	yes	—
12	#19098429	C₂²	1	4	yes	—
13	#19098423	—	2	4	no	—
14	C₄×C₄⋊Dic₇₉	—	2	4	no	—
15	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₉	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
21	#14579120	—	2	16	no	—
22	C₂×C₄⋊Dic₇₉	—	2	8	no	—
23	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
24	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
25	C₂×C₄×Dic₇₉	—	2	4	no	—
26	#14579095	—	2	16	no	—
27	#14579093	—	2	16	no	—
28	C₇₉⋊#95	—	2	8	no	—
29	C₂³×C₄×C₇₉	D₄	1	2	yes	—
30	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₇₉	D₄	1	8	yes	—
33	C₂²×C₈×C₇₉	Q₈	1	8	yes	—
34	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
36	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₇₉	D₄	1	2	yes	—
38	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
39	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₇₉	C₂×C₄	1	8	yes	—
44	C₂²×Dic₇₉	—	2	8	no	—
45	C₄⋊Dic₇₉	—	2	16	no	—
46	C₄⋊Dic₇₉	—	4	16	no	—
47	C₂²⋊Dic₇₉	—	4	8	no	—
48	C₄×Dic₇₉	—	4	8	no	—
49	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	—	2	8	no	—
51	C₂²×C₄×C₇₉	C₄⋊C₄	1	8	yes	—
52	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
56	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₄⋊C₄	1	32	yes	complemented
61	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
66	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
67	C₄²×C₇₉	C₄⋊C₄	1	8	yes	—
68	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
69	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
70	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
71	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
72	C₂³×C₇₉	—	2	4	no	—
73	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₇₉	#127	1	2	yes	—
75	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
76	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₇₉	—	2	4	no	—
78	C₂×C₄×C₇₉	—	2	16	no	—
79	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
81	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₇₉	#127	1	2	yes	—
84	C₂×C₄×C₇₉	C₄⋊Q₈	1	8	yes	—
85	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₇₉	#135	1	8	yes	—
87	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
89	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
90	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
91	C₂×Dic₇₉	—	4	8	no	—
92	C₂×Dic₇₉	—	4	16	no	—
93	C₈×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
94	C₈×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
95	C₂²×C₇₉	—	2	4	no	—
96	C₂²×C₇₉	—	2	4	no	—
97	C₂²×C₇₉	—	2	4	no	—
98	C₂²×C₇₉	#384	1	4	yes	—
99	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
101	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₂²×C₇₉	C₂×#135	1	4	yes	—
103	C₄×C₇₉	—	2	4	no	—
104	C₄×C₇₉	#384	1	4	yes	—
105	C₄×C₇₉	#427	1	8	yes	complemented
106	C₄×C₇₉	C₂²×C₄⋊C₄	1	2	yes	—
107	C₄×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
108	Dic₇₉	—	8	8	no	—
109	#20797	—	79	1	no	maximal, Sylow
110	C₂×C₇₉	—	2	4	no	—
111	C₂×C₇₉	C₂×#384	1	1	yes	characteristic
112	C₂×C₇₉	C₂×#427	1	2	yes	—
113	C₂×C₇₉	#3147	1	2	yes	—
114	C₂×C₇₉	#3160	1	2	yes	—
115	C₂×#427	—	79	4	no	—
116	C₂²×C₄×C₈	D₇₉	1	1	yes	characteristic
117	#2285	—	79	2	no	—
118	#1846	—	79	4	no	—
119	#1845	—	79	2	no	—
120	#1844	—	79	2	no	—
121	C₇₉	#20797	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₈	C₂×D₇₉	1	2	yes	—
123	C₂×#117	—	79	4	no	—
124	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
125	#427	—	158	16	no	—
126	C₂×#107	—	79	8	no	—
127	C₂×#106	—	79	8	no	—
128	C₂×C₄×C₈	Dic₇₉	1	8	yes	complemented
129	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
132	#384	—	79	4	no	—
133	#382	—	79	4	no	—
134	#380	—	158	4	no	—
135	C₄×C₄⋊C₄	—	158	4	no	—
136	C₂³×C₄	D₃₁₆	1	2	yes	—
137	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
138	C₂²×C₈	Dic₁₅₈	1	8	yes	—
139	C₂²×C₈	D₃₁₆	1	8	yes	—
140	C₂²×C₈	C₂×Dic₇₉	1	8	yes	—
141	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
142	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
143	#117	—	158	16	no	—
144	C₂×C₄⋊C₄	—	79	8	no	—
145	C₂×C₄⋊C₄	—	158	8	no	—
146	C₂×C₂²⋊C₄	—	158	4	no	—
147	C₂×C₄²	—	158	4	no	—
148	C₂×C₄²	Dic₁₅₈	1	4	yes	—
149	C₂×C₄²	D₃₁₆	1	2	yes	—
150	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
151	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
153	#107	—	158	16	no	—
154	#106	—	158	16	no	—
155	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
156	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
157	C₄×C₈	C₂×Dic₇₉	1	8	yes	—
158	#95	—	158	8	no	—
159	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	158	8	no	—
162	C₂²×C₄	C₄⋊Dic₇₉	1	8	yes	—
163	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
164	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
165	C₂²×C₄	C₂×D₃₁₆	1	4	yes	—
166	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
167	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
168	C₂²×C₄	C₂×Dic₁₅₈	1	4	yes	—
169	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
170	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
171	C₂×C₈	C₄⋊Dic₇₉	1	32	yes	complemented
172	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
173	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
174	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
175	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
176	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
177	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
178	C₄⋊C₄	—	316	16	no	—
179	C₄⋊C₄	—	158	16	no	—
180	C₂²⋊C₄	—	316	8	no	—
181	C₄²	—	316	8	no	—
182	C₄²	C₄⋊Dic₇₉	1	8	yes	—
183	C₄²	C₂²×Dic₇₉	1	2	yes	—
184	C₄²	C₂²×Dic₇₉	1	2	yes	—
185	C₄²	C₂×Dic₁₅₈	1	2	yes	—
186	C₄²	C₂×Dic₁₅₈	1	2	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
189	C₂³	#14579158	1	2	yes	—
190	C₂³	#14579161	1	2	yes	—
191	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	316	16	no	—
195	C₂×C₄	—	316	8	no	—
196	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
197	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
198	C₂×C₄	C₂×C₄⋊Dic₇₉	1	8	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
200	C₂×C₄	#14579158	1	2	yes	—
201	C₂×C₄	#14579161	1	2	yes	—
202	C₂×C₄	#14579161	1	8	yes	—
203	C₂×C₄	#14579183	1	8	yes	—
204	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
206	C₂×C₄	C₂²×Dic₁₅₈	1	2	yes	—
207	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
208	C₈	C₂×C₄⋊Dic₇₉	1	8	yes	—
209	C₈	C₂×C₄⋊Dic₇₉	1	8	yes	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	#19098433	1	4	yes	—
214	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
215	C₂²	C₂×#14579158	1	1	yes	characteristic
216	C₂²	C₂×#14579161	1	1	yes	characteristic
217	C₂²	C₂×#14579183	1	4	yes	—
218	C₄	—	2	4	no	—
219	C₄	—	632	8	no	—
220	C₄	#19098433	1	4	yes	—
221	C₄	#19098601	1	8	yes	complemented
222	C₄	C₂²×C₄⋊Dic₇₉	1	2	yes	—
223	C₄	C₂×#14579161	1	2	yes	—
224	C₂	—	2	4	no	—
225	C₂	C₂×#19098433	1	1	yes	characteristic
226	C₂	C₂×#19098601	1	2	yes	—
227	C₂	#23773828	1	2	yes	—
228	C₂	#23773914	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂²×C₄×C₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₉
hypocenter	C₇₉	#20797
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #23763564	C₂	semidirect
4. #23763560	C₂	semidirect
5. #23763557	C₂	semidirect
17. C₂×C₄×C₈×C₇₉	C₄	semidirect
60. C₂×C₈×C₇₉	C₄⋊C₄	semidirect
105. C₄×C₇₉	#427	semidirect
121. C₇₉	#20797	semidirect
128. C₂×C₄×C₈	Dic₇₉	semidirect
171. C₂×C₈	C₄⋊Dic₇₉	semidirect
221. C₄	#19098601	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098601	2
#23765807	2
#23763564	2
#23763560	2
#23763557	2
C₂²×C₄×C₈×C₇₉	2
#20797	79