Subgroups — C₂×#23763531 #34288457

Order 20224.

1006 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×#19098429	C₂	1	2	yes	maximal
2	#23763531	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098326	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
6	#19098429	—	2	4	no	—
7	#19098429	C₂²	1	8	yes	—
8	C₂×C₇₉⋊#95	—	2	4	no	—
9	#19098326	C₂²	1	16	yes	—
10	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₉	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₇₉	C₂²	1	4	yes	—
14	C₂³×Dic₇₉	—	2	4	no	—
15	C₂×C₄⋊Dic₇₉	—	2	16	no	—
16	C₂×C₄⋊Dic₇₉	—	2	4	no	—
17	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
18	C₇₉⋊#95	—	2	16	no	—
19	C₂³×C₄×C₇₉	D₄	1	2	yes	—
20	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	—	2	8	no	—
22	C₂²×C₈×C₇₉	C₂×C₄	1	8	yes	—
23	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
24	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₇₉	D₄	1	2	yes	—
26	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
27	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
28	C₄×C₈×C₇₉	C₂×C₄	1	16	yes	complemented
29	C₂²×Dic₇₉	—	2	8	no	—
30	C₂²×Dic₇₉	—	4	8	no	—
31	C₂²×Dic₇₉	—	4	16	no	—
32	C₄⋊Dic₇₉	—	2	16	no	—
33	C₄⋊Dic₇₉	—	4	8	no	—
34	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	—	2	8	no	—
36	C₂²×C₄×C₇₉	C₄⋊C₄	1	8	yes	—
37	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
39	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
41	C₂²×C₄×C₇₉	#41	1	8	yes	—
42	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
43	C₂×C₈×C₇₉	—	2	16	no	—
44	C₂×C₈×C₇₉	—	2	8	no	—
45	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
46	C₄²×C₇₉	C₄⋊C₄	1	8	yes	—
47	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
48	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
49	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
50	C₂³×C₇₉	#124	1	4	yes	—
51	C₂³×C₇₉	#135	1	8	yes	—
52	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	—	2	8	no	—
54	C₂×C₄×C₇₉	—	2	16	no	—
55	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
56	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₇₉	#117	1	8	yes	—
58	C₂×C₄×C₇₉	#124	1	4	yes	—
59	C₂×C₄×C₇₉	#125	1	8	yes	—
60	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
63	C₂×Dic₇₉	—	4	8	no	—
64	C₂×Dic₇₉	—	4	16	no	—
65	C₂×Dic₇₉	—	8	16	no	—
66	C₈×C₇₉	—	2	8	no	—
67	C₂²×C₇₉	#382	1	4	yes	—
68	C₂²×C₇₉	#427	1	8	yes	—
69	C₂²×C₇₉	#488	1	16	yes	—
70	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₇₉	C₂×#124	1	2	yes	—
72	C₂²×C₇₉	C₂×#135	1	4	yes	—
73	C₄×C₇₉	—	2	8	no	—
74	C₄×C₇₉	#382	1	4	yes	—
75	C₄×C₇₉	C₂×#117	1	2	yes	—
76	C₄×C₇₉	C₂×#125	1	2	yes	—
77	Dic₇₉	—	8	8	no	—
78	C₂×#1835	—	79	1	no	maximal, Sylow
79	C₂×C₇₉	#1835	1	8	yes	complemented
80	C₂×C₇₉	C₂×#382	1	1	yes	characteristic
81	C₂×C₇₉	C₂×#427	1	2	yes	—
82	C₂×C₇₉	C₂×#488	1	4	yes	—
83	C₂²×C₄×C₈	D₇₉	1	1	yes	characteristic
84	C₂×#382	—	79	2	no	—
85	#1835	—	79	8	no	—
86	C₂×#340	—	79	4	no	—
87	C₇₉	C₂×#1835	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₈	C₂×D₇₉	1	2	yes	—
89	C₂²×C₄⋊C₄	—	79	4	no	—
90	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
91	C₂×C₄×C₈	Dic₇₉	1	8	yes	complemented
92	C₂×C₄×C₈	C₂×D₇₉	1	4	yes	—
93	#382	—	79	8	no	—
94	#382	—	158	4	no	—
95	C₂×#95	—	158	4	no	—
96	#340	—	79	16	no	—
97	C₂³×C₄	—	158	4	no	—
98	C₂³×C₄	D₃₁₆	1	2	yes	—
99	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
100	C₂²×C₈	—	2	8	no	—
101	C₂²×C₈	C₂×Dic₇₉	1	8	yes	—
102	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
103	C₂×C₄⋊C₄	—	79	8	no	—
104	C₂×C₄⋊C₄	—	158	4	no	—
105	C₂×C₄⋊C₄	—	158	16	no	—
106	C₂×C₄²	Dic₁₅₈	1	4	yes	—
107	C₂×C₄²	D₃₁₆	1	2	yes	—
108	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
109	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
110	C₄×C₈	C₂×Dic₇₉	1	16	yes	complemented
111	#95	—	158	16	no	—
112	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	316	8	no	—
115	C₂²×C₄	—	316	16	no	—
116	C₂²×C₄	—	158	8	no	—
117	C₂²×C₄	C₄⋊Dic₇₉	1	8	yes	—
118	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
119	C₂²×C₄	C₂×D₃₁₆	1	4	yes	—
120	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
121	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
122	C₂²×C₄	#11929624	1	8	yes	—
123	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
124	C₂×C₈	—	2	8	no	—
125	C₂×C₈	—	2	16	no	—
126	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
127	C₄⋊C₄	—	316	8	no	—
128	C₄⋊C₄	—	158	16	no	—
129	C₄²	C₄⋊Dic₇₉	1	8	yes	—
130	C₄²	C₂²×Dic₇₉	1	4	yes	—
131	C₄²	C₂×Dic₁₅₈	1	4	yes	—
132	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
133	C₂³	#14579149	1	4	yes	—
134	C₂³	#14579183	1	8	yes	—
135	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	316	8	no	—
139	C₂×C₄	—	316	16	no	—
140	C₂×C₄	—	632	16	no	—
141	C₂×C₄	C₂×C₄⋊Dic₇₉	1	8	yes	—
142	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
143	C₂×C₄	#14579120	1	8	yes	—
144	C₂×C₄	#14579149	1	4	yes	—
145	C₂×C₄	#14579153	1	8	yes	—
146	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
147	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
148	C₂×C₄	C₂×#11929624	1	4	yes	—
149	C₈	—	2	8	no	—
150	C₂²	#19098429	1	4	yes	—
151	C₂²	#19098601	1	8	yes	—
152	C₂²	#19098918	1	16	yes	—
153	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
154	C₂²	C₂×#14579149	1	2	yes	—
155	C₂²	C₂×#14579183	1	4	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	632	8	no	—
158	C₄	#19098429	1	4	yes	—
159	C₄	C₂×#14579120	1	2	yes	—
160	C₄	C₂×#14579153	1	2	yes	—
161	C₂	#23763531	1	8	yes	direct product factor
162	C₂	C₂×#19098429	1	1	yes	characteristic
163	C₂	C₂×#19098601	1	2	yes	—
164	C₂	C₂×#19098918	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂²×C₄×C₈×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₉
hypocenter	C₇₉	C₂×#1835
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #23763531	C₂	direct (N × H)
12. C₂×C₄×C₈×C₇₉	C₄	semidirect
28. C₄×C₈×C₇₉	C₂×C₄	semidirect
79. C₂×C₇₉	#1835	semidirect
87. C₇₉	C₂×#1835	semidirect
91. C₂×C₄×C₈	Dic₇₉	semidirect
110. C₄×C₈	C₂×Dic₇₉	semidirect
161. C₂	#23763531	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×#19098429	2
#23763531	2
C₂×#19098326	2
C₂²×C₄×C₈×C₇₉	2
C₂×#1835	79

Subgroups — C2×#23763531 #34288457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23763531 #34288457