Subgroups — #34175142

Order 20224.

234 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₉⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
6	#19098339	—	2	1	no	—
7	#19098339	—	2	1	no	—
8	#19098314	—	2	1	no	—
9	#19098314	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#334×C₇₉	—	2	1	no	—
13	#334×C₇₉	C₄	1	2	yes	—
14	C₂²×C₇₉⋊C₈	—	4	1	no	—
15	C₂×C₇₉⋊C₁₆	—	2	2	no	—
16	C₂×C₇₉⋊C₁₆	—	2	2	no	—
17	C₂×C₇₉⋊C₁₆	C₈	1	4	yes	complemented
18	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
19	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
20	C₂²×C₈×C₇₉	—	4	1	no	—
21	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
24	C₄⋊C₈×C₇₉	—	2	2	no	—
25	C₄⋊C₈×C₇₉	—	4	1	no	—
26	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₂×C₇₉⋊C₈	—	4	2	no	—
31	C₂×C₇₉⋊C₈	—	8	2	no	—
32	C₇₉⋊C₁₆	—	2	4	no	—
33	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₇₉	—	2	2	no	—
37	C₂×C₈×C₇₉	—	4	2	no	—
38	C₂×C₈×C₇₉	—	8	2	no	—
39	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
45	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₉	D₈	1	1	yes	characteristic
48	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
49	C₂³×C₇₉	#106	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#108	1	2	yes	—
60	C₈×C₇₉	—	2	1	no	—
61	C₈×C₇₉	—	2	1	no	—
62	C₈×C₇₉	—	8	2	no	—
63	C₈×C₇₉	C₄×C₈	1	1	yes	characteristic
64	C₈×C₇₉	#97	1	1	yes	characteristic
65	C₈×C₇₉	C₂²⋊C₈	1	2	yes	—
66	C₇₉⋊C₈	—	8	2	no	—
67	C₂²×C₇₉	#334	1	2	yes	—
68	C₂²×C₇₉	#340	1	1	yes	characteristic
69	C₂²×C₇₉	#365	1	2	yes	—
70	C₂²×C₇₉	#365	1	2	yes	—
71	C₄×C₇₉	—	2	1	no	—
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	—	2	1	no	—
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	#335	1	2	yes	—
76	C₄×C₇₉	#336	1	1	yes	characteristic
77	C₄×C₇₉	#341	1	1	yes	characteristic
78	#7156	—	79	1	no	maximal, Sylow
79	C₂×C₇₉	#1277	1	1	yes	characteristic
80	C₂×C₇₉	#1368	1	4	yes	—
81	C₂×C₇₉	#1383	1	1	yes	characteristic
82	C₂×C₇₉	#1383	1	1	yes	characteristic
83	C₂×#346	—	79	1	no	—
84	C₂×#334	—	79	1	no	—
85	C₂×#334	D₇₉	1	1	yes	characteristic
86	C₇₉	#7156	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₁₆	—	79	2	no	—
88	C₂×C₄⋊C₈	—	2	1	no	—
89	C₂×C₄⋊C₈	—	158	1	no	—
90	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
91	#346	—	158	1	no	—
92	#346	—	158	1	no	—
93	#334	—	2	1	no	—
94	#334	—	79	2	no	—
95	#334	—	158	1	no	—
96	#334	Dic₇₉	1	2	yes	—
97	C₂²×C₈	—	4	1	no	—
98	C₂²×C₈	—	316	1	no	—
99	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
100	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
102	C₂×C₁₆	—	79	4	no	—
103	C₂×C₁₆	—	158	2	no	—
104	C₂×C₁₆	—	158	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄⋊C₈	—	316	1	no	—
108	C₄⋊C₈	—	158	2	no	—
109	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
110	C₄×C₈	D₃₁₆	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
113	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
114	C₂²×C₄	#11929611	1	1	yes	characteristic
115	C₂²×C₄	#11929612	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	8	2	no	—
119	C₂×C₈	—	316	2	no	—
120	C₂×C₈	—	632	2	no	—
121	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
122	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
123	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
125	C₂×C₈	C₈×D₇₉	1	2	yes	—
126	C₂×C₈	#11929607	1	2	yes	—
127	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
128	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
129	C₄²	#11929609	1	1	yes	characteristic
130	C₄²	#11929614	1	1	yes	characteristic
131	C₁₆	—	158	4	no	—
132	C₂³	#14579092	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
136	C₂×C₄	#14579081	1	1	yes	characteristic
137	C₂×C₄	#14579082	1	1	yes	characteristic
138	C₂×C₄	#14579084	1	1	yes	characteristic
139	C₂×C₄	#14579085	1	1	yes	characteristic
140	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
141	C₂×C₄	#14579094	1	1	yes	characteristic
142	C₂×C₄	#14579096	1	2	yes	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	8	2	no	—
146	C₈	—	632	2	no	—
147	C₈	C₈×Dic₇₉	1	1	yes	characteristic
148	C₈	#14579071	1	1	yes	characteristic
149	C₈	(C₂×D₇₉)⋊C₈	1	2	yes	—
150	C₂²	#19098313	1	2	yes	—
151	C₂²	#19098325	1	1	yes	characteristic
152	C₂²	#19098385	1	2	yes	—
153	C₂²	#19098387	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#19098315	1	2	yes	—
159	C₄	#19098317	1	1	yes	characteristic
160	C₄	#19098327	1	1	yes	characteristic
161	C₂	#23761234	1	1	yes	characteristic
162	C₂	#23761465	1	4	yes	—
163	C₂	#23761498	1	1	yes	characteristic
164	C₂	#23761500	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579092
commutator	C₈×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×#334×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#106
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×#334	D₇₉
hypocenter	C₇₉	#7156
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
17. C₂×C₇₉⋊C₁₆	C₈	semidirect
86. C₇₉	#7156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098339	2
C₂×#19098314	2
C₂×#334×C₇₉	2
#7156	79