Subgroups — C₂×#23761468 #34199831

Order 20224.

222 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098384	C₂	1	2	yes	maximal
2	#23761468	C₂	1	4	yes	maximal, direct product factor
3	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579101	—	2	2	no	—
5	#19098384	—	2	2	no	—
6	#19098384	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	#346×C₇₉	C₄	1	2	yes	—
11	#346×C₇₉	C₂²	1	2	yes	—
12	#14579101	—	2	4	no	—
13	#14579101	—	4	2	no	—
14	C₂×C₇₉⋊C₁₆	—	4	2	no	—
15	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
18	C₂×C₁₆×C₇₉	—	2	1	no	—
19	C₂×C₁₆×C₇₉	—	2	1	no	—
20	C₂×C₁₆×C₇₉	C₈	1	2	yes	—
21	C₂×C₁₆×C₇₉	C₈	1	2	yes	—
22	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
26	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
28	C₇₉⋊C₁₆	—	4	4	no	—
29	C₂²×C₄×C₇₉	—	2	1	no	—
30	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
38	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
39	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₇₉	—	2	1	no	—
42	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
44	C₁₆×C₇₉	—	2	2	no	—
45	C₁₆×C₇₉	C₂×C₈	1	4	yes	—
46	C₂³×C₇₉	#108	1	1	yes	characteristic
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	—	2	1	no	—
49	C₂×C₄×C₇₉	—	4	2	no	—
50	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#108	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₈×C₇₉	C₄⋊C₈	1	1	yes	characteristic
57	C₈×C₇₉	C₄⋊C₈	1	1	yes	characteristic
58	C₈×C₇₉	#106	1	2	yes	—
59	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	#334	1	1	yes	characteristic
64	C₂²×C₇₉	#334	1	1	yes	characteristic
65	C₂²×C₇₉	C₂×#108	1	1	yes	characteristic
66	C₄×C₇₉	—	4	1	no	—
67	C₄×C₇₉	—	4	1	no	—
68	C₄×C₇₉	#334	1	1	yes	characteristic
69	C₄×C₇₉	#334	1	1	yes	characteristic
70	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
72	C₂×#1369	—	79	1	no	maximal, Sylow
73	C₂×C₇₉	—	2	1	no	—
74	C₂×C₇₉	—	2	1	no	—
75	C₂×C₇₉	#1369	1	2	yes	complemented
76	C₂×C₇₉	C₂×#334	1	1	yes	characteristic
77	C₂×#364	—	79	2	no	—
78	C₂×#346	D₇₉	1	1	yes	characteristic
79	#1369	—	79	4	no	—
80	C₇₉	C₂×#1369	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	158	2	no	—
82	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
83	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
85	#364	—	79	4	no	—
86	#364	—	158	2	no	—
87	#346	Dic₇₉	1	2	yes	—
88	#346	C₂×D₇₉	1	2	yes	—
89	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
91	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
92	M₃₂	—	316	2	no	—
93	M₃₂	—	158	4	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	316	2	no	—
97	C₂×C₁₆	C₇₉⋊C₈	1	2	yes	—
98	C₂×C₁₆	C₇₉⋊C₈	1	2	yes	—
99	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
100	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
101	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
102	C₄×C₈	D₃₁₆	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
104	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
107	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
108	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
109	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
110	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
111	C₂×C₈	#11929608	1	1	yes	characteristic
112	C₂×C₈	#11929608	1	1	yes	characteristic
113	C₂×C₈	#11929613	1	2	yes	—
114	C₂×C₈	#11929613	1	2	yes	—
115	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
116	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
119	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	316	4	no	—
122	C₁₆	C₂×C₇₉⋊C₈	1	4	yes	—
123	C₂³	#14579097	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
128	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
129	C₂×C₄	#14579093	1	1	yes	characteristic
130	C₂×C₄	#14579093	1	1	yes	characteristic
131	C₂×C₄	#14579097	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
133	C₈	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
134	C₈	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
135	C₈	#14579093	1	2	yes	—
136	C₈	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
137	C₈	C₂×#11929608	1	1	yes	characteristic
138	C₈	C₂×#11929613	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	#19098314	1	1	yes	characteristic
141	C₂²	#19098314	1	1	yes	characteristic
142	C₂²	C₂×#14579097	1	1	yes	characteristic
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	#19098314	1	1	yes	characteristic
146	C₄	#19098314	1	1	yes	characteristic
147	C₄	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
148	C₄	C₂×#14579093	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#23761468	1	2	yes	direct product factor
152	C₂	C₂×#19098314	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579093
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle	C₂²×C₇₉	#334
prime radical	C₂³×C₇₉	#108
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	C₂×#1369
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #23761468	C₂	direct (N × H)
75. C₂×C₇₉	#1369	semidirect
80. C₇₉	C₂×#1369	semidirect
151. C₂	#23761468	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098384	2
#23761468	2
C₂×#346×C₇₉	2
C₂×#1369	79

Subgroups — C2×#23761468 #34199831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761468 #34199831