Subgroups — C₂×#23763560 #34288671

Order 20224.

1358 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579095	C₂	1	4	yes	maximal
2	C₂×#19098433	C₂	1	2	yes	maximal
3	#23763560	C₂	1	8	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₉	—	2	4	no	—
6	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
7	C₂×#14579095	—	2	16	no	—
8	C₂×#14579095	C₂²	1	16	yes	—
9	#19098433	—	2	4	no	—
10	#19098433	C₂²	1	8	yes	—
11	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
12	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₉	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₇₉	C₂²	1	4	yes	—
15	C₂³×Dic₇₉	—	2	4	no	—
16	C₂×C₄⋊Dic₇₉	—	2	16	no	—
17	C₂×C₄⋊Dic₇₉	—	2	4	no	—
18	C₂×C₄⋊Dic₇₉	—	2	16	no	—
19	C₂×C₄⋊Dic₇₉	—	4	16	no	—
20	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
21	#14579095	—	2	32	no	—
22	C₂³×C₄×C₇₉	D₄	1	2	yes	—
23	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₇₉	D₄	1	8	yes	—
26	C₂²×C₈×C₇₉	Q₈	1	8	yes	—
27	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
28	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₇₉	D₄	1	2	yes	—
30	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
31	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
32	C₄×C₈×C₇₉	C₂×C₄	1	16	yes	complemented
33	C₂²×Dic₇₉	—	2	8	no	—
34	C₂²×Dic₇₉	—	4	8	no	—
35	C₂²×Dic₇₉	—	4	16	no	—
36	C₄⋊Dic₇₉	—	2	16	no	—
37	C₄⋊Dic₇₉	—	4	8	no	—
38	C₄⋊Dic₇₉	—	4	32	no	—
39	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	C₄⋊C₄	1	8	yes	—
41	C₂²×C₄×C₇₉	D₈	1	8	yes	—
42	C₂²×C₄×C₇₉	Q₁₆	1	8	yes	—
43	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
48	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₄⋊C₄	1	32	yes	complemented
51	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
52	C₂×C₈×C₇₉	C₂×D₄	1	8	yes	—
53	C₂×C₈×C₇₉	C₂×Q₈	1	8	yes	—
54	C₄²×C₇₉	C₄⋊C₄	1	8	yes	—
55	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
56	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
57	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
58	C₂³×C₇₉	#127	1	2	yes	—
59	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
60	C₂³×C₇₉	C₂×D₈	1	4	yes	—
61	C₂³×C₇₉	C₂×Q₁₆	1	4	yes	—
62	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#107	1	32	yes	complemented
64	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
65	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
66	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
67	C₂×C₄×C₇₉	#127	1	2	yes	—
68	C₂×C₄×C₇₉	C₄⋊Q₈	1	8	yes	—
69	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
70	C₂×C₄×C₇₉	C₂×D₈	1	8	yes	—
71	C₂×C₄×C₇₉	C₂×Q₁₆	1	8	yes	—
72	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
74	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
75	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
76	C₂×Dic₇₉	—	4	8	no	—
77	C₂×Dic₇₉	—	4	16	no	—
78	C₂×Dic₇₉	—	8	16	no	—
79	C₈×C₇₉	C₂×C₄⋊C₄	1	16	yes	complemented
80	C₂²×C₇₉	#384	1	4	yes	—
81	C₂²×C₇₉	C₂×#107	1	8	yes	—
82	C₂²×C₇₉	#493	1	4	yes	—
83	C₂²×C₇₉	#494	1	4	yes	—
84	C₂²×C₇₉	#500	1	8	yes	—
85	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
87	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₇₉	C₂²×D₈	1	2	yes	—
89	C₂²×C₇₉	C₂²×Q₁₆	1	2	yes	—
90	C₄×C₇₉	#384	1	4	yes	—
91	C₄×C₇₉	C₂×#107	1	16	yes	complemented
92	C₄×C₇₉	C₂²×C₄⋊C₄	1	2	yes	—
93	C₄×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
94	Dic₇₉	—	8	8	no	—
95	C₂×#1845	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	#1845	1	8	yes	complemented
97	C₂×C₇₉	C₂×#384	1	1	yes	characteristic
98	C₂×C₇₉	C₂²×#107	1	2	yes	—
99	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
100	C₂×C₇₉	C₂×#494	1	1	yes	characteristic
101	C₂×C₇₉	C₂×#500	1	2	yes	—
102	C₂²×#107	—	79	4	no	—
103	C₂²×C₄×C₈	D₇₉	1	1	yes	characteristic
104	C₂×#384	—	79	2	no	—
105	#1845	—	79	8	no	—
106	C₇₉	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₈	C₂×D₇₉	1	2	yes	—
108	C₂²×C₄⋊C₄	—	79	4	no	—
109	C₂²×C₄⋊C₄	—	158	4	no	—
110	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
111	C₂×#107	—	79	16	no	—
112	C₂×#107	—	158	16	no	—
113	C₂×C₄×C₈	Dic₇₉	1	8	yes	complemented
114	C₂×C₄×C₈	C₂×D₇₉	1	4	yes	—
115	#384	—	79	8	no	—
116	#384	—	158	4	no	—
117	C₂³×C₄	—	158	4	no	—
118	C₂³×C₄	D₃₁₆	1	2	yes	—
119	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
120	C₂²×C₈	Dic₁₅₈	1	8	yes	—
121	C₂²×C₈	D₃₁₆	1	8	yes	—
122	C₂²×C₈	C₂×Dic₇₉	1	8	yes	—
123	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
124	C₂×C₄⋊C₄	—	316	16	no	—
125	C₂×C₄⋊C₄	—	79	8	no	—
126	C₂×C₄⋊C₄	—	158	4	no	—
127	C₂×C₄⋊C₄	—	158	16	no	—
128	C₂×C₄⋊C₄	—	158	16	no	—
129	C₂×C₄²	Dic₁₅₈	1	4	yes	—
130	C₂×C₄²	D₃₁₆	1	2	yes	—
131	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
132	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
133	#107	—	158	32	no	—
134	C₄×C₈	C₂×Dic₇₉	1	16	yes	complemented
135	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
136	C₂²×C₄	—	316	8	no	—
137	C₂²×C₄	—	316	16	no	—
138	C₂²×C₄	—	158	8	no	—
139	C₂²×C₄	C₄⋊Dic₇₉	1	8	yes	—
140	C₂²×C₄	D₆₃₂	1	8	yes	—
141	C₂²×C₄	Dic₃₁₆	1	8	yes	—
142	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
143	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₃₁₆	1	4	yes	—
145	C₂²×C₄	C₂×D₃₁₆	1	4	yes	—
146	C₂²×C₄	C₂×Dic₁₅₈	1	4	yes	—
147	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
148	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₇₉	1	32	yes	complemented
150	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
151	C₂×C₈	C₂×D₃₁₆	1	8	yes	—
152	C₂×C₈	C₂×Dic₁₅₈	1	8	yes	—
153	C₄⋊C₄	—	316	8	no	—
154	C₄⋊C₄	—	316	32	no	—
155	C₄⋊C₄	—	158	16	no	—
156	C₄²	C₄⋊Dic₇₉	1	8	yes	—
157	C₄²	C₂²×Dic₇₉	1	4	yes	—
158	C₄²	C₂×Dic₁₅₈	1	4	yes	—
159	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
160	C₂³	#14579158	1	2	yes	—
161	C₂³	#14579161	1	2	yes	—
162	C₂³	C₂×D₆₃₂	1	4	yes	—
163	C₂³	C₂×Dic₃₁₆	1	4	yes	—
164	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
165	C₂×C₄	—	316	16	no	—
166	C₂×C₄	—	316	8	no	—
167	C₂×C₄	—	632	16	no	—
168	C₂×C₄	#14579095	1	32	yes	complemented
169	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₇₉	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₇₉	1	8	yes	—
172	C₂×C₄	#14579158	1	2	yes	—
173	C₂×C₄	#14579161	1	2	yes	—
174	C₂×C₄	#14579161	1	8	yes	—
175	C₂×C₄	C₂×D₆₃₂	1	8	yes	—
176	C₂×C₄	C₂×Dic₃₁₆	1	8	yes	—
177	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
179	C₂×C₄	C₂²×Dic₁₅₈	1	2	yes	—
180	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
181	C₈	C₂×C₄⋊Dic₇₉	1	16	yes	complemented
182	C₂²	#19098433	1	4	yes	—
183	C₂²	C₂×#14579095	1	8	yes	—
184	C₂²	#19098940	1	4	yes	—
185	C₂²	#19098943	1	4	yes	—
186	C₂²	#19098968	1	8	yes	—
187	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
188	C₂²	C₂×#14579158	1	1	yes	characteristic
189	C₂²	C₂×#14579161	1	1	yes	characteristic
190	C₂²	C₂²×D₆₃₂	1	2	yes	—
191	C₂²	C₂²×Dic₃₁₆	1	2	yes	—
192	C₄	—	632	8	no	—
193	C₄	#19098433	1	4	yes	—
194	C₄	C₂×#14579095	1	16	yes	complemented
195	C₄	C₂²×C₄⋊Dic₇₉	1	2	yes	—
196	C₄	C₂×#14579161	1	2	yes	—
197	C₂	#23763560	1	8	yes	direct product factor
198	C₂	C₂×#19098433	1	1	yes	characteristic
199	C₂	C₂²×#14579095	1	2	yes	—
200	C₂	C₂×#19098940	1	1	yes	characteristic
201	C₂	C₂×#19098943	1	1	yes	characteristic
202	C₂	C₂×#19098968	1	2	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×#1845
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
3. #23763560	C₂	direct (N × H)
13. C₂×C₄×C₈×C₇₉	C₄	semidirect
32. C₄×C₈×C₇₉	C₂×C₄	semidirect
50. C₂×C₈×C₇₉	C₄⋊C₄	semidirect
63. C₂×C₄×C₇₉	#107	semidirect
79. C₈×C₇₉	C₂×C₄⋊C₄	semidirect
91. C₄×C₇₉	C₂×#107	semidirect
96. C₂×C₇₉	#1845	semidirect
106. C₇₉	C₂×#1845	semidirect
113. C₂×C₄×C₈	Dic₇₉	semidirect
134. C₄×C₈	C₂×Dic₇₉	semidirect
149. C₂×C₈	C₄⋊Dic₇₉	semidirect
168. C₂×C₄	#14579095	semidirect
181. C₈	C₂×C₄⋊Dic₇₉	semidirect
194. C₄	C₂×#14579095	semidirect
197. C₂	#23763560	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14579095	2
C₂×#19098433	2
#23763560	2
C₂²×C₄×C₈×C₇₉	2
C₂×#1845	79

Subgroups — C2×#23763560 #34288671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23763560 #34288671