Subgroups — C₂×#24909743 #35559888

Order 21248.

558 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128106	C₂	1	1	yes	maximal, characteristic
2	#24909743	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19127528	C₂	1	4	yes	maximal
4	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	—	2	1	no	—
7	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
8	#19128106	—	2	2	no	—
9	#19128106	C₂²	1	4	yes	—
10	C₂×#14581726	—	2	1	no	—
11	C₂×#14581726	C₂²	1	2	yes	—
12	#19127528	C₂²	1	16	yes	—
13	C₄×C₈₃⋊C₁₆	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₈₃	—	2	1	no	—
15	#500×C₈₃	—	2	2	no	—
16	#500×C₈₃	C₂²	1	4	yes	—
17	C₂×#107×C₈₃	—	2	1	no	—
18	C₂×#107×C₈₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₆₆	—	4	1	no	—
21	#14581792	—	2	4	no	—
22	#14581792	—	4	2	no	—
23	C₂×C₄⋊Dic₈₃	—	2	2	no	—
24	C₂×C₄⋊Dic₈₃	—	4	1	no	—
25	C₂×C₈₃⋊C₁₆	Q₈	1	8	yes	complemented
26	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
27	#14581726	—	2	2	no	—
28	#14581726	—	2	4	no	—
29	#14581726	C₂³	1	4	yes	—
30	C₂²×Q₈×C₈₃	—	4	1	no	—
31	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₈₃	—	2	4	no	—
34	C₄⋊Q₈×C₈₃	—	4	2	no	—
35	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
37	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
38	#107×C₈₃	—	2	2	no	—
39	#107×C₈₃	—	2	4	no	—
40	#107×C₈₃	C₂³	1	4	yes	—
41	C₄×C₈×C₈₃	D₄	1	2	yes	—
42	C₄×C₈×C₈₃	C₂³	1	2	yes	—
43	C₂×Dic₁₆₆	—	4	4	no	—
44	C₂×Dic₁₆₆	—	8	4	no	—
45	C₂²×Dic₈₃	—	4	2	no	—
46	C₄⋊Dic₈₃	—	2	4	no	—
47	C₄⋊Dic₈₃	—	4	4	no	—
48	C₄⋊Dic₈₃	—	4	2	no	—
49	C₈₃⋊C₁₆	C₂×Q₈	1	8	yes	complemented
50	C₂×Q₈×C₈₃	—	4	4	no	—
51	C₂×Q₈×C₈₃	—	8	4	no	—
52	C₂²×C₄×C₈₃	—	4	2	no	—
53	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	Q₁₆	1	4	yes	—
57	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₈₃	C₂×Q₈	1	4	yes	—
60	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₈₃	—	2	4	no	—
62	C₄⋊C₄×C₈₃	—	4	2	no	—
63	C₄⋊C₄×C₈₃	—	4	4	no	—
64	C₄²×C₈₃	D₈	1	2	yes	—
65	C₄²×C₈₃	C₂×D₄	1	2	yes	—
66	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₈₃	—	8	8	no	—
68	C₂×C₄×C₈₃	—	4	4	no	—
69	C₂×C₄×C₈₃	—	8	4	no	—
70	C₂×C₄×C₈₃	SD₃₂	1	4	yes	—
71	C₂×C₄×C₈₃	SD₃₂	1	4	yes	—
72	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₈₃	—	4	4	no	—
79	C₂×Dic₈₃	—	8	4	no	—
80	Dic₁₆₆	—	8	8	no	—
81	C₈×C₈₃	C₄⋊Q₈	1	2	yes	—
82	C₈×C₈₃	C₂×Q₁₆	1	4	yes	—
83	C₈×C₈₃	C₂²×Q₈	1	2	yes	—
84	C₂²×C₈₃	#500	1	2	yes	—
85	C₂²×C₈₃	C₂×SD₃₂	1	2	yes	—
86	C₂²×C₈₃	C₂×SD₃₂	1	2	yes	—
87	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₈₃	—	8	4	no	—
89	C₄×C₈₃	#500	1	2	yes	—
90	C₄×C₈₃	C₂×SD₃₂	1	4	yes	—
91	C₄×C₈₃	C₂×SD₃₂	1	4	yes	—
92	C₄×C₈₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
94	Dic₈₃	—	8	4	no	—
95	C₂×#2254	—	83	1	no	maximal, Sylow
96	C₂×C₈₃	#2254	1	4	yes	complemented
97	C₂×C₈₃	C₂×#500	1	1	yes	characteristic
98	C₂×C₈₃	C₂²×SD₃₂	1	1	yes	characteristic
99	C₂×C₈₃	C₂²×SD₃₂	1	1	yes	characteristic
100	C₂×#500	—	83	1	no	—
101	C₂×#500	D₈₃	1	1	yes	characteristic
102	#2254	—	83	8	no	—
103	C₂×#367	—	83	4	no	—
104	C₂×C₄×C₁₆	—	83	1	no	—
105	C₈₃	C₂×#2254	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	C₂×C₄⋊Q₈	—	166	1	no	—
108	C₂²×C₁₆	—	83	2	no	—
109	#500	—	2	2	no	—
110	#500	—	83	4	no	—
111	#500	—	166	2	no	—
112	#500	C₂×D₈₃	1	4	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	—	83	2	no	—
115	C₂×#107	—	166	1	no	—
116	C₂×#107	C₂×D₈₃	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
118	#367	—	83	16	no	—
119	C₄×C₁₆	—	83	4	no	—
120	C₂²×Q₈	—	4	1	no	—
121	C₂²×Q₈	—	332	1	no	—
122	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	2	no	—
126	C₄⋊Q₈	—	332	2	no	—
127	C₄⋊Q₈	—	166	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	332	1	no	—
131	C₂×C₄⋊C₄	—	166	2	no	—
132	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
133	C₂×C₁₆	—	83	8	no	—
134	C₂×C₁₆	—	83	4	no	—
135	#107	—	2	4	no	—
136	#107	—	2	2	no	—
137	#107	—	83	4	no	—
138	#107	—	166	2	no	—
139	#107	—	166	4	no	—
140	#107	C₂²×D₈₃	1	4	yes	—
141	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
142	C₄×C₈	C₂²×D₈₃	1	2	yes	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	8	4	no	—
145	C₂×Q₈	—	332	4	no	—
146	C₂×Q₈	—	664	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	332	2	no	—
149	C₂²×C₄	#13050575	1	1	yes	characteristic
150	C₂²×C₄	#13050575	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
152	C₂×C₈	#13050580	1	4	yes	—
153	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
154	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
155	C₂×C₈	Q₈×D₈₃	1	4	yes	—
156	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	332	4	no	—
161	C₄⋊C₄	—	332	2	no	—
162	C₄⋊C₄	—	166	4	no	—
163	C₄²	#13050575	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
165	C₁₆	—	83	8	no	—
166	C₂³	C₂×#13050575	1	1	yes	characteristic
167	Q₈	—	8	8	no	—
168	Q₈	—	664	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	332	4	no	—
172	C₂×C₄	—	664	4	no	—
173	C₂×C₄	#14581735	1	4	yes	—
174	C₂×C₄	#14581736	1	4	yes	—
175	C₂×C₄	#14581790	1	2	yes	—
176	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
177	C₂×C₄	C₂×#13050575	1	2	yes	—
178	C₂×C₄	C₂×#13050575	1	2	yes	—
179	C₂×C₄	C₂×#13050580	1	2	yes	—
180	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
181	C₈	#14581790	1	2	yes	—
182	C₈	C₂×#13050580	1	4	yes	—
183	C₈	C₂×Q₈×D₈₃	1	2	yes	—
184	C₂²	#19128104	1	2	yes	—
185	C₂²	C₂×#14581735	1	2	yes	—
186	C₂²	C₂×#14581736	1	2	yes	—
187	C₂²	C₂²×#13050575	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	664	4	no	—
190	C₄	#19128104	1	2	yes	—
191	C₄	C₂×#14581735	1	4	yes	—
192	C₄	C₂×#14581736	1	4	yes	—
193	C₄	C₂×#14581790	1	1	yes	characteristic
194	C₄	C₂²×#13050580	1	1	yes	characteristic
195	C₂	#24909743	1	4	yes	direct product factor
196	C₂	C₂×#19128104	1	1	yes	characteristic
197	C₂	C₂²×#14581735	1	1	yes	characteristic
198	C₂	C₂²×#14581736	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#500×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	C₂×#500	D₈₃
hypocenter	C₈₃	C₂×#2254
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #24909743	C₂	direct (N × H)
25. C₂×C₈₃⋊C₁₆	Q₈	semidirect
49. C₈₃⋊C₁₆	C₂×Q₈	semidirect
96. C₂×C₈₃	#2254	semidirect
105. C₈₃	C₂×#2254	semidirect
195. C₂	#24909743	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×#19128106	2
#24909743	2
C₂×#19127528	2
C₂×C₄×C₈₃⋊C₁₆	2
C₂×#500×C₈₃	2
C₂×#2254	83

Subgroups — C2×#24909743 #35559888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909743 #35559888