Subgroups — C₂×#19136882 #25252095

Order 11008.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583304	C₂	1	1	yes	maximal, characteristic
2	#19136882	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14582726	C₂	1	2	yes	maximal
4	C₂×#14582724	C₂	1	2	yes	maximal
5	C₂×#14582675	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₃×#499	C₂	1	1	yes	maximal, characteristic
7	C₂×#13063325	—	2	1	no	—
8	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	#14583304	—	2	2	no	—
10	#14583304	C₂²	1	4	yes	—
11	C₂×#13063259	—	2	1	no	—
12	C₂×#13063259	C₂²	1	2	yes	—
13	#14582726	C₂²	1	8	yes	—
14	#14582724	C₂²	1	8	yes	—
15	#14582675	C₂²	1	4	yes	—
16	C₂×#125×C₄₃	—	2	1	no	—
17	C₄₃×#499	—	2	2	no	—
18	C₄₃×#499	C₂²	1	4	yes	—
19	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#106×C₄₃	—	2	1	no	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₈₆	—	4	1	no	—
24	#13063325	—	2	4	no	—
25	#13063325	—	4	2	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	—	4	1	no	—
28	C₂×C₄₃⋊C₁₆	Q₈	1	8	yes	complemented
29	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
30	#13063259	—	2	2	no	—
31	#13063259	—	2	4	no	—
32	#13063259	C₂³	1	4	yes	—
33	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
35	#125×C₄₃	—	2	4	no	—
36	#125×C₄₃	—	4	2	no	—
37	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₃	—	4	1	no	—
40	C₂×C₄⋊C₄×C₄₃	—	4	1	no	—
41	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
42	#107×C₄₃	—	2	1	no	—
43	#107×C₄₃	—	2	1	no	—
44	#107×C₄₃	C₂³	1	2	yes	—
45	#107×C₄₃	C₂³	1	2	yes	—
46	#106×C₄₃	—	2	4	no	—
47	C₄×C₈×C₄₃	D₄	1	2	yes	—
48	C₄×C₈×C₄₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₄₃	—	4	1	no	—
50	C₂²×C₄×C₄₃	—	4	1	no	—
51	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₄₃	—	4	2	no	—
55	C₂×Dic₈₆	—	4	4	no	—
56	C₂×Dic₈₆	—	8	4	no	—
57	C₂×C₈×C₄₃	Q₁₆	1	4	yes	—
58	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
61	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₄₃	—	2	2	no	—
63	C₄⋊C₄×C₄₃	—	2	2	no	—
64	C₄⋊C₄×C₄₃	—	4	1	no	—
65	C₄⋊C₄×C₄₃	—	4	1	no	—
66	C₄⋊C₄×C₄₃	—	4	4	no	—
67	C₄⋊C₄×C₄₃	—	4	4	no	—
68	C₄²×C₄₃	D₈	1	2	yes	—
69	C₄²×C₄₃	C₂×D₄	1	2	yes	—
70	C₄⋊Dic₄₃	—	2	4	no	—
71	C₄⋊Dic₄₃	—	4	2	no	—
72	C₄⋊Dic₄₃	—	4	4	no	—
73	C₄₃⋊C₁₆	C₂×Q₈	1	8	yes	complemented
74	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₄₃	—	2	2	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	4	2	no	—
78	C₂×C₄×C₄₃	—	4	2	no	—
79	C₂×C₄×C₄₃	—	8	2	no	—
80	C₂×C₄×C₄₃	—	8	2	no	—
81	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×Dic₄₃	—	4	4	no	—
88	C₂×Dic₄₃	—	8	4	no	—
89	Dic₈₆	—	8	8	no	—
90	C₈×C₄₃	C₄⋊Q₈	1	2	yes	—
91	C₈×C₄₃	C₂×Q₁₆	1	4	yes	—
92	C₈×C₄₃	C₂²×Q₈	1	2	yes	—
93	C₂×#2255	—	43	1	no	maximal, Sylow
94	C₂²×C₄₃	#500	1	2	yes	—
95	C₂²×C₄₃	#509	1	2	yes	—
96	C₂²×C₄₃	#510	1	2	yes	—
97	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₄₃	—	2	2	no	—
99	C₄×C₄₃	—	2	2	no	—
100	C₄×C₄₃	—	8	2	no	—
101	C₄×C₄₃	—	8	2	no	—
102	C₄×C₄₃	#500	1	2	yes	—
103	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
105	Dic₄₃	—	8	4	no	—
106	C₂×#500	—	43	1	no	—
107	C₂×#499	D₄₃	1	1	yes	characteristic
108	#2255	—	43	8	no	—
109	C₂×#367	—	43	2	no	—
110	C₂×#366	—	43	2	no	—
111	C₂×#346	—	43	1	no	—
112	C₂×C₄₃	#2255	1	4	yes	complemented
113	C₂×C₄₃	C₂×#500	1	1	yes	characteristic
114	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
115	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
116	C₂×C₄⋊Q₈	—	86	1	no	—
117	C₂×#125	—	2	1	no	—
118	C₂²×C₁₆	—	43	2	no	—
119	#500	—	43	4	no	—
120	#500	—	86	2	no	—
121	#499	—	2	2	no	—
122	#499	C₂×D₄₃	1	4	yes	—
123	C₂×#107	—	43	2	no	—
124	C₂×#107	—	86	1	no	—
125	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
126	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
127	C₂×#106	—	2	1	no	—
128	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
129	#367	—	43	8	no	—
130	#366	—	43	8	no	—
131	#346	—	43	4	no	—
132	C₄₃	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	172	1	no	—
134	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	86	4	no	—
137	C₄⋊Q₈	—	172	2	no	—
138	#125	—	2	4	no	—
139	#125	—	4	2	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	86	2	no	—
145	C₂×C₄⋊C₄	—	172	1	no	—
146	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
147	C₂×C₁₆	—	43	8	no	—
148	C₂×C₁₆	—	43	4	no	—
149	#107	—	2	1	no	—
150	#107	—	2	1	no	—
151	#107	—	43	4	no	—
152	#107	—	86	2	no	—
153	#107	—	86	4	no	—
154	#107	C₂²×D₄₃	1	2	yes	—
155	#107	C₂²×D₄₃	1	2	yes	—
156	#106	—	2	4	no	—
157	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
158	C₄×C₈	C₂²×D₄₃	1	2	yes	—
159	C₂×Q₈	—	344	4	no	—
160	C₂×Q₈	—	172	4	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	172	2	no	—
164	C₂²×C₄	#10621882	1	1	yes	characteristic
165	C₂²×C₄	#10621882	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
167	C₂×C₈	#10621885	1	4	yes	—
168	C₂×C₈	Q₈×D₄₃	1	4	yes	—
169	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
171	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	86	4	no	—
179	C₄⋊C₄	—	172	2	no	—
180	C₄⋊C₄	—	172	4	no	—
181	C₄²	#10621882	1	2	yes	—
182	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
183	C₁₆	—	43	8	no	—
184	C₂³	C₂×#10621882	1	1	yes	characteristic
185	Q₈	—	344	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	344	4	no	—
193	C₂×C₄	—	172	4	no	—
194	C₂×C₄	#13063323	1	2	yes	—
195	C₂×C₄	C₂×#10621882	1	2	yes	—
196	C₂×C₄	C₂×#10621882	1	2	yes	—
197	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
198	C₂×C₄	C₂×#10621885	1	2	yes	—
199	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
200	C₈	#13063323	1	2	yes	—
201	C₈	C₂×#10621885	1	4	yes	—
202	C₈	C₂×Q₈×D₄₃	1	2	yes	—
203	C₂²	#14583302	1	2	yes	—
204	C₂²	#14583334	1	2	yes	—
205	C₂²	#14583339	1	2	yes	—
206	C₂²	C₂²×#10621882	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	2	no	—
210	C₄	—	8	2	no	—
211	C₄	—	344	4	no	—
212	C₄	#14583302	1	2	yes	—
213	C₄	C₂×#13063323	1	1	yes	characteristic
214	C₄	C₂²×#10621885	1	1	yes	characteristic
215	C₂	#19136882	1	4	yes	direct product factor
216	C₂	C₂×#14583302	1	1	yes	characteristic
217	C₂	C₂×#14583334	1	1	yes	characteristic
218	C₂	C₂×#14583339	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#499	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
hypercenter	C₂×#499	D₄₃
hypocenter	C₄₃	C₂×#2255
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #19136882	C₂	direct (N × H)
28. C₂×C₄₃⋊C₁₆	Q₈	semidirect
73. C₄₃⋊C₁₆	C₂×Q₈	semidirect
112. C₂×C₄₃	#2255	semidirect
132. C₄₃	C₂×#2255	semidirect
215. C₂	#19136882	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×#14583304	2
#19136882	2
C₂×#14582726	2
C₂×#14582724	2
C₂×#14582675	2
C₂×C₄₃×#499	2
C₂×#2255	43

Subgroups — C2×#19136882 #25252095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136882 #25252095