Subgroups — #34203365

Order 20224.

410 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765021	C₂	1	1	yes	maximal, characteristic
2	#23764996	C₂	1	2	yes	maximal, complemented
3	#23763200	C₂	1	1	yes	maximal, characteristic
4	#23761485	C₂	1	2	yes	maximal, complemented
5	C₂×#504×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	#19098547	C₄	1	1	yes	characteristic
8	#19098537	C₄	1	1	yes	characteristic
9	#19098518	C₂²	1	2	yes	—
10	#19098345	C₄	1	4	yes	complemented
11	#19098338	—	2	2	no	—
12	C₁₆×Dic₇₉	—	2	2	no	—
13	#19098334	C₄	1	2	yes	—
14	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
15	#504×C₇₉	—	2	2	no	—
16	#504×C₇₉	—	2	2	no	—
17	C₂×M₃₂×C₇₉	C₄	1	2	yes	complemented
18	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₇₉	C₄	1	2	yes	complemented
20	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#11929608	C₂×C₄	1	2	yes	—
22	#14579120	C₂×C₄	1	2	yes	—
23	#14579101	—	2	2	no	—
24	#14579101	C₂×C₄	1	4	yes	—
25	C₂×C₇₉⋊C₁₆	—	2	2	no	—
26	#14579080	C₈	1	2	yes	—
27	#14579078	C₈	1	2	yes	—
28	#14579076	C₈	1	4	yes	—
29	#14579071	—	2	2	no	—
30	C₈×Dic₇₉	—	2	2	no	—
31	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
32	#131×C₇₉	—	2	2	no	—
33	#131×C₇₉	—	2	2	no	—
34	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
38	M₃₂×C₇₉	—	2	2	no	—
39	M₃₂×C₇₉	—	2	4	no	—
40	C₂×C₁₆×C₇₉	—	2	2	no	—
41	C₂×C₁₆×C₇₉	—	2	2	no	—
42	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
46	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
47	#11929608	—	2	2	no	—
48	#11929608	C₂×C₈	1	4	yes	—
49	C₂×C₇₉⋊C₈	—	2	2	no	—
50	C₄⋊Dic₇₉	—	2	2	no	—
51	C₂²⋊Dic₇₉	C₂×C₈	1	4	yes	—
52	C₄×Dic₇₉	—	2	2	no	—
53	C₇₉⋊C₁₆	—	2	4	no	—
54	#41×C₇₉	—	2	2	no	—
55	#41×C₇₉	—	2	2	no	—
56	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₇₉	C₂×C₈	1	2	yes	—
58	C₂×D₄×C₇₉	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₇₉	—	2	4	no	—
62	M₁₆×C₇₉	—	2	2	no	—
63	C₂×C₈×C₇₉	—	2	2	no	—
64	C₂×C₈×C₇₉	—	2	2	no	—
65	C₂×C₈×C₇₉	C₄²	1	2	yes	—
66	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₇₉	#41	1	2	yes	—
70	C₁₆×C₇₉	—	2	2	no	—
71	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₁₆×C₇₉	C₂²⋊C₄	1	2	yes	—
73	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂³×C₇₉	C₄×C₈	1	2	yes	—
75	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
76	Q₈×C₇₉	—	2	2	no	—
77	D₄×C₇₉	—	2	4	no	—
78	D₄×C₇₉	—	2	2	no	—
79	C₂×C₄×C₇₉	—	2	2	no	—
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₇₉	#131	1	2	yes	—
86	C₂×Dic₇₉	—	2	4	no	—
87	C₈×C₇₉	—	2	2	no	—
88	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₇₉	#117	1	1	yes	characteristic
90	C₈×C₇₉	C₄×D₄	1	2	yes	—
91	C₇₉⋊C₈	—	2	4	no	—
92	C₂²×C₇₉	—	2	2	no	—
93	C₂²×C₇₉	—	4	1	no	—
94	C₂²×C₇₉	—	4	2	no	—
95	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
96	C₂²×C₇₉	C₈×D₄	1	2	yes	—
97	C₄×C₇₉	—	2	2	no	—
98	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₇₉	#405	1	1	yes	characteristic
100	C₄×C₇₉	C₈×D₄	1	2	yes	—
101	Dic₇₉	—	4	4	no	—
102	#12613	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	—	2	2	no	—
104	C₂×C₇₉	—	2	1	no	—
105	C₂×C₇₉	—	4	2	no	—
106	C₂×C₇₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#504	D₇₉	1	1	yes	characteristic
108	#2115	—	79	1	no	—
109	#2108	—	79	2	no	—
110	#1757	—	79	1	no	—
111	#1378	—	79	2	no	—
112	C₇₉	#12613	1	1	yes	characteristic, complemented, Sylow
113	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
114	#504	—	2	2	no	—
115	#504	—	2	2	no	—
116	C₂×M₃₂	—	79	2	no	—
117	C₂×M₃₂	Dic₇₉	1	2	yes	complemented
118	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
119	C₂²×C₁₆	Dic₇₉	1	2	yes	complemented
120	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
121	#413	—	79	1	no	—
122	#410	—	79	1	no	—
123	#405	—	79	2	no	—
124	#349	—	79	4	no	—
125	#346	—	158	2	no	—
126	C₄×C₁₆	—	158	2	no	—
127	#344	—	79	2	no	—
128	C₂×#41	C₄×D₇₉	1	1	yes	characteristic
129	#131	—	2	2	no	—
130	#131	—	2	2	no	—
131	C₂×M₁₆	—	79	2	no	—
132	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
133	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
134	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
135	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
136	#117	—	79	2	no	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	4	no	—
139	M₃₂	—	79	4	no	—
140	M₃₂	—	158	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	158	2	no	—
144	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
145	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
146	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
147	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
148	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
149	#101	—	79	2	no	—
150	#100	—	79	2	no	—
151	#99	—	79	4	no	—
152	#97	—	158	2	no	—
153	C₄×C₈	—	158	2	no	—
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	C₂×Q₈	C₈×D₇₉	1	1	yes	characteristic
157	C₂×D₄	C₈×D₇₉	1	2	yes	—
158	C₂×D₄	C₈×D₇₉	1	1	yes	characteristic
159	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
160	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	79	4	no	—
164	M₁₆	—	158	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	158	2	no	—
168	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
169	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
170	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
172	C₂×C₈	#11929624	1	2	yes	—
173	C₄⋊C₄	—	158	2	no	—
174	C₂²⋊C₄	—	79	4	no	—
175	C₄²	—	158	2	no	—
176	C₁₆	—	2	2	no	—
177	C₁₆	—	158	4	no	—
178	C₁₆	C₂²⋊Dic₇₉	1	1	yes	characteristic
179	C₁₆	C₂²⋊Dic₇₉	1	2	yes	—
180	C₁₆	C₂²⋊Dic₇₉	1	1	yes	characteristic
181	C₂³	C₈×Dic₇₉	1	2	yes	—
182	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
183	Q₈	—	2	2	no	—
184	D₄	—	2	4	no	—
185	D₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	158	4	no	—
189	C₂×C₄	C₈×Dic₇₉	1	2	yes	—
190	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
191	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
192	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
193	C₂×C₄	#14579167	1	2	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	158	4	no	—
196	C₈	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
197	C₈	#14579120	1	1	yes	characteristic
198	C₈	C₄×C₂²⋊D₇₉	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂×C₈×Dic₇₉	1	1	yes	characteristic
203	C₂²	C₈×C₂²⋊D₇₉	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	316	4	no	—
206	C₄	C₄×C₂²⋊Dic₇₉	1	1	yes	characteristic
207	C₄	#19098518	1	1	yes	characteristic
208	C₄	C₈×C₂²⋊D₇₉	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	C₈×C₂²⋊Dic₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊Dic₇₉
commutator	C₂²×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#504×C₇₉	C₂
socle	C₂×C₇₉	C₈×C₂²⋊C₄
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#504	D₇₉
hypocenter	C₇₉	#12613
LCS	G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #23764996	C₂	semidirect
4. #23761485	C₂	semidirect
10. #19098345	C₄	semidirect
17. C₂×M₃₂×C₇₉	C₄	semidirect
19. C₂²×C₁₆×C₇₉	C₄	semidirect
112. C₇₉	#12613	semidirect
117. C₂×M₃₂	Dic₇₉	semidirect
119. C₂²×C₁₆	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23765021	2
#23764996	2
#23763200	2
#23761485	2
C₂×#504×C₇₉	2
#12613	79