Subgroups — #34197288

Order 20224.

290 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765199	C₂	1	1	yes	maximal, characteristic
2	C₈×#11929608	C₂	1	1	yes	maximal, characteristic
3	#23761319	C₂	1	2	yes	maximal, complemented
4	#23761316	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₄	1	1	yes	characteristic
7	C₄×#11929608	C₄	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	—
9	C₂²⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	—
10	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
11	#19098285	C₄	1	2	yes	—
12	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₇₉	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₇₉	C₄	1	2	yes	—
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×#11929608	C₈	1	2	yes	—
19	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
20	C₄⋊(C₇₉⋊C₈)	C₈	1	2	yes	—
21	C₂²⋊(C₇₉⋊C₈)	C₈	1	2	yes	—
22	#14579072	C₈	1	2	yes	—
23	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	—	2	4	no	—
29	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	#11929608	—	2	4	no	—
36	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
37	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
38	C₇₉⋊C₁₆	—	2	4	no	—
39	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₉	—	2	2	no	—
43	C₂×C₈×C₇₉	—	2	2	no	—
44	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇₉	C₄²	1	1	yes	characteristic
49	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₁₆×C₇₉	—	2	2	no	—
53	C₁₆×C₇₉	M₁₆	1	4	yes	complemented
54	C₂³×C₇₉	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	M₃₂	1	4	yes	—
62	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₇₉	—	2	1	no	—
66	C₈×C₇₉	—	2	1	no	—
67	C₈×C₇₉	C₂×M₁₆	1	2	yes	—
68	C₈×C₇₉	#131	1	2	yes	—
69	C₇₉⋊C₈	—	2	4	no	—
70	C₇₉⋊C₈	—	2	4	no	—
71	C₂²×C₇₉	—	2	1	no	—
72	C₂²×C₇₉	—	2	1	no	—
73	C₂²×C₇₉	#346	1	2	yes	—
74	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₇₉	—	2	1	no	—
76	C₄×C₇₉	—	2	1	no	—
77	C₄×C₇₉	#346	1	2	yes	—
78	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
79	C₄×C₇₉	#405	1	1	yes	characteristic
80	C₄×C₇₉	C₂×M₃₂	1	2	yes	—
81	C₄×C₇₉	#504	1	2	yes	—
82	#11408	—	79	1	no	maximal, Sylow
83	C₂×C₇₉	—	2	2	no	—
84	C₂×C₇₉	C₈×M₁₆	1	1	yes	characteristic
85	C₂×C₇₉	C₂×#346	1	1	yes	characteristic
86	C₂×C₇₉	#2108	1	1	yes	characteristic
87	#2162	—	79	1	no	—
88	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
89	C₈×M₁₆	—	79	1	no	—
90	#1312	—	79	2	no	—
91	#1311	—	79	2	no	—
92	C₇₉	#11408	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	Dic₇₉	1	2	yes	—
94	#431	—	79	1	no	—
95	C₄×M₁₆	—	79	1	no	—
96	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
97	C₄⋊C₁₆	—	79	2	no	—
98	C₂²⋊C₁₆	—	79	2	no	—
99	C₄×C₁₆	—	79	2	no	—
100	C₄×C₁₆	Dic₇₉	1	2	yes	—
101	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
102	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
103	#322	—	79	2	no	—
104	C₈²	—	79	2	no	—
105	C₂×M₁₆	—	79	2	no	—
106	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	—	79	4	no	—
111	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
113	C₄⋊C₈	—	79	2	no	—
114	C₂²⋊C₈	—	79	2	no	—
115	#97	—	79	2	no	—
116	C₄×C₈	—	79	2	no	—
117	C₄×C₈	—	79	2	no	—
118	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
122	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
123	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
124	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
125	M₁₆	—	158	4	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	79	4	no	—
129	C₂×C₈	—	79	4	no	—
130	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
133	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
134	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
135	C₄²	C₈×D₇₉	1	1	yes	characteristic
136	C₄²	C₈×D₇₉	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
138	C₁₆	—	2	2	no	—
139	C₁₆	—	158	4	no	—
140	C₁₆	#11929608	1	4	yes	complemented
141	C₂³	C₈×Dic₇₉	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	C₈×Dic₇₉	1	1	yes	characteristic
146	C₂×C₄	C₈×Dic₇₉	1	1	yes	characteristic
147	C₂×C₄	C₈×Dic₇₉	1	1	yes	characteristic
148	C₂×C₄	#14579100	1	4	yes	—
149	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
150	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	158	4	no	—
155	C₈	—	158	4	no	—
156	C₈	C₂×#11929608	1	2	yes	—
157	C₈	#14579167	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#19098338	1	2	yes	—
161	C₂²	C₂×C₈×Dic₇₉	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	#19098338	1	2	yes	—
165	C₄	C₄×#11929608	1	1	yes	characteristic
166	C₄	#19098518	1	1	yes	characteristic
167	C₄	C₂×#14579100	1	2	yes	—
168	C₄	#19098979	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	C₈×#11929608	1	1	yes	characteristic
171	C₂	C₂×#19098338	1	1	yes	characteristic
172	C₂	#23764996	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₇₉
commutator, socle	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
prime radical	C₂³×C₇₉	C₄×C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#11408
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #23761319	C₂	semidirect
4. #23761316	C₂	semidirect
53. C₁₆×C₇₉	M₁₆	semidirect
92. C₇₉	#11408	semidirect
140. C₁₆	#11929608	semidirect

Maximal subgroups

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

subgroup	index
#23765199	2
C₈×#11929608	2
#23761319	2
#23761316	2
C₂×C₄×C₁₆×C₇₉	2
#11408	79