Subgroups — C₂×#19134666 #25096302

Order 11008.

1526 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
2	#19134666	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14582654	C₂	1	4	yes	maximal, complemented
4	C₂²×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₃⋊C₈	C₂²	1	4	yes	—
6	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
7	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	8	yes	—
8	#14582654	C₂²	1	16	yes	complemented
9	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
10	C₂×#117×C₄₃	—	2	8	no	—
11	C₂×#117×C₄₃	C₄	1	4	yes	—
12	C₂×#117×C₄₃	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
14	C₂²×C₂²⋊C₄×C₄₃	C₄	1	2	yes	—
15	C₂²×C₄²×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₄₃⋊C₈	—	2	8	no	—
17	C₂²×C₄₃⋊C₈	—	2	16	no	—
18	C₂²×C₄₃⋊C₈	C₂³	1	8	yes	—
19	C₂²⋊(C₄₃⋊C₈)	—	2	16	no	—
20	C₂⁵×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₃	—	2	8	no	—
22	C₂³×C₄×C₄₃	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₃	D₄	1	8	yes	—
26	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
27	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	#117×C₄₃	—	2	16	no	—
29	#117×C₄₃	—	2	16	no	—
30	C₂×C₄⋊C₄×C₄₃	—	2	8	no	—
31	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
32	C₂×C₂²⋊C₄×C₄₃	—	2	8	no	—
33	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₄₃	—	2	8	no	—
35	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
37	C₂⁴×C₄₃	—	2	4	no	—
38	C₂⁴×C₄₃	—	2	8	no	—
39	C₂⁴×C₄₃	C₂²⋊C₄	1	4	yes	—
40	C₂⁴×C₄₃	C₂²×C₄	1	2	yes	—
41	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	—	2	8	no	—
43	C₂²×C₄×C₄₃	—	2	32	no	—
44	C₂²×C₄×C₄₃	—	2	8	no	—
45	C₂²×C₄×C₄₃	—	2	8	no	—
46	C₂²×C₄×C₄₃	—	2	4	no	—
47	C₂²×C₄×C₄₃	—	2	16	no	—
48	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
49	C₂²×C₄×C₄₃	C₂²⋊C₄	1	8	yes	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	8	yes	—
54	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₄₃	#41	1	8	yes	—
56	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₄₃	—	2	16	no	—
58	C₂²⋊C₄×C₄₃	—	2	16	no	—
59	C₄²×C₄₃	—	2	8	no	—
60	C₄²×C₄₃	—	2	8	no	—
61	C₂×C₄₃⋊C₈	—	2	8	no	—
62	C₂×C₄₃⋊C₈	—	2	16	no	—
63	C₂×C₄₃⋊C₈	—	4	16	no	—
64	C₂³×C₄₃	—	2	4	no	—
65	C₂³×C₄₃	—	2	8	no	—
66	C₂³×C₄₃	—	2	8	no	—
67	C₂³×C₄₃	—	2	16	no	—
68	C₂³×C₄₃	—	4	16	no	—
69	C₂³×C₄₃	C₂×C₂²⋊C₄	1	4	yes	—
70	C₂³×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
71	C₂³×C₄₃	#117	1	4	yes	—
72	C₂³×C₄₃	#131	1	8	yes	—
73	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	—	2	4	no	—
75	C₂×C₄×C₄₃	—	2	8	no	—
76	C₂×C₄×C₄₃	—	2	16	no	—
77	C₂×C₄×C₄₃	—	2	16	no	—
78	C₂×C₄×C₄₃	—	2	8	no	—
79	C₂×C₄×C₄₃	—	2	16	no	—
80	C₂×C₄×C₄₃	—	4	16	no	—
81	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	4	yes	—
82	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₄₃	#117	1	4	yes	—
84	C₂×C₄×C₄₃	#123	1	16	yes	—
85	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
87	C₄₃⋊C₈	—	4	8	no	—
88	C₂×#1818	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	—	2	8	no	—
90	C₂²×C₄₃	—	2	4	no	—
91	C₂²×C₄₃	—	4	16	no	—
92	C₂²×C₄₃	—	4	8	no	—
93	C₂²×C₄₃	#381	1	4	yes	—
94	C₂²×C₄₃	#408	1	8	yes	—
95	C₂²×C₄₃	#433	1	16	yes	—
96	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₄₃	C₂×#117	1	2	yes	—
98	C₂²×C₄₃	C₂×#131	1	4	yes	—
99	C₄×C₄₃	—	2	4	no	—
100	C₄×C₄₃	—	4	8	no	—
101	C₄×C₄₃	#381	1	4	yes	—
102	C₄×C₄₃	C₂×#123	1	4	yes	—
103	C₂²×#117	D₄₃	1	1	yes	characteristic
104	C₂²×C₂²⋊C₈	—	43	2	no	—
105	#1818	—	43	8	no	—
106	C₂×#336	—	43	4	no	—
107	C₂×C₄₃	—	4	4	no	—
108	C₂×C₄₃	#1818	1	8	yes	complemented
109	C₂×C₄₃	C₂×#381	1	1	yes	characteristic
110	C₂×C₄₃	C₂×#408	1	2	yes	—
111	C₂×C₄₃	C₂×#433	1	4	yes	—
112	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
113	C₂³×C₈	—	43	4	no	—
114	C₂×#117	—	2	8	no	—
115	C₂×#117	Dic₄₃	1	4	yes	—
116	C₂×#117	C₂×D₄₃	1	4	yes	—
117	C₂²×C₄⋊C₄	Dic₄₃	1	2	yes	—
118	C₂²×C₂²⋊C₄	Dic₄₃	1	2	yes	—
119	C₂²×C₄²	C₂×D₄₃	1	2	yes	—
120	C₂×C₂²⋊C₈	—	43	8	no	—
121	C₂×C₂²⋊C₈	—	86	8	no	—
122	#336	—	43	16	no	—
123	C₄₃	C₂×#1818	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	C₂×Dic₄₃	1	1	yes	characteristic
125	C₂³×C₄	—	2	8	no	—
126	C₂³×C₄	C₂×Dic₄₃	1	4	yes	—
127	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
128	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
129	C₂³×C₄	C₂²⋊D₄₃	1	8	yes	—
130	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
131	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
132	C₂²×C₈	—	43	8	no	—
133	C₂²×C₈	—	86	16	no	—
134	C₂²×C₈	—	86	8	no	—
135	#117	—	2	16	no	—
136	#117	—	2	16	no	—
137	C₂×C₄⋊C₄	—	2	8	no	—
138	C₂×C₄⋊C₄	C₂×Dic₄₃	1	8	yes	—
139	C₂×C₂²⋊C₄	—	2	8	no	—
140	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	8	yes	—
141	C₂×C₄²	—	2	8	no	—
142	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
143	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
144	C₂²⋊C₈	—	86	16	no	—
145	C₂⁴	—	2	4	no	—
146	C₂⁴	—	2	8	no	—
147	C₂⁴	C₂²⋊Dic₄₃	1	4	yes	—
148	C₂⁴	C₂²×Dic₄₃	1	2	yes	—
149	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	32	no	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	2	16	no	—
156	C₂²×C₄	C₂²⋊Dic₄₃	1	4	yes	—
157	C₂²×C₄	C₂²⋊Dic₄₃	1	8	yes	—
158	C₂²×C₄	#10621898	1	8	yes	—
159	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
160	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
161	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₂²⋊D₄₃	1	8	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
164	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
165	C₂×C₈	—	86	8	no	—
166	C₂×C₈	—	86	16	no	—
167	C₂×C₈	—	172	16	no	—
168	C₄⋊C₄	—	2	16	no	—
169	C₂²⋊C₄	—	2	16	no	—
170	C₄²	—	2	8	no	—
171	C₄²	—	2	8	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	16	no	—
174	C₂³	—	2	8	no	—
175	C₂³	—	2	8	no	—
176	C₂³	—	4	16	no	—
177	C₂³	C₂×C₂²⋊Dic₄₃	1	4	yes	—
178	C₂³	C₂×C₂²⋊Dic₄₃	1	2	yes	—
179	C₂³	#13063284	1	4	yes	—
180	C₂³	#13063332	1	8	yes	—
181	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	16	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	16	no	—
188	C₂×C₄	—	4	16	no	—
189	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	4	yes	—
190	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	2	yes	—
191	C₂×C₄	#13063284	1	4	yes	—
192	C₂×C₄	#13063308	1	16	yes	—
193	C₂×C₄	C₂²×C₂²⋊D₄₃	1	2	yes	—
194	C₂×C₄	C₂×#10621898	1	4	yes	—
195	C₈	—	172	8	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	4	8	no	—
199	C₂²	—	4	16	no	—
200	C₂²	#14582762	1	4	yes	—
201	C₂²	#14582866	1	8	yes	—
202	C₂²	#14582960	1	16	yes	—
203	C₂²	C₂²×C₂²⋊Dic₄₃	1	1	yes	characteristic
204	C₂²	C₂×#13063284	1	2	yes	—
205	C₂²	C₂×#13063332	1	4	yes	—
206	C₄	—	2	4	no	—
207	C₄	—	4	8	no	—
208	C₄	#14582762	1	4	yes	—
209	C₄	C₂×#13063308	1	4	yes	—
210	C₂	—	4	4	no	—
211	C₂	#19134666	1	8	yes	direct product factor
212	C₂	C₂×#14582762	1	1	yes	characteristic
213	C₂	C₂×#14582866	1	2	yes	—
214	C₂	C₂×#14582960	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₃
commutator	C₂³×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂²×#117×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂²×C₄
prime radical	C₂⁵×C₄₃	C₂×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×#117	D₄₃
hypocenter	C₄₃	C₂×#1818
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. #19134666	C₂	direct (N × H)
3. C₂×#14582654	C₂	semidirect
8. #14582654	C₂²	semidirect
108. C₂×C₄₃	#1818	semidirect
123. C₄₃	C₂×#1818	semidirect
211. C₂	#19134666	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×C₂²⋊(C₄₃⋊C₈)	2
#19134666	2
C₂×#14582654	2
C₂²×#117×C₄₃	2
C₂×#1818	43

Subgroups — C2×#19134666 #25096302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19134666 #25096302