Subgroups — C₂×#19136958 #25254819

Order 11008.

3654 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×C₄×Dic₈₆	C₂	1	3	yes	maximal
2	#19136958	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊Dic₄₃	C₂	1	3	yes	maximal
4	C₂×#14582750	C₂	1	6	yes	maximal
5	C₂×#14582744	C₂	1	2	yes	maximal
6	C₂×C₄₃×#376	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₈₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₈₆	C₄	1	48	yes	complemented
9	C₂×C₄×Dic₈₆	C₂²	1	24	yes	—
10	C₂²×C₄⋊Dic₄₃	C₂²	1	3	yes	—
11	C₂²×Dic₄₃⋊C₄	C₂²	1	6	yes	—
12	C₂²×C₄×Dic₄₃	C₂²	1	6	yes	—
13	C₂²×C₄×Dic₄₃	C₂²	1	6	yes	—
14	C₄×C₄⋊Dic₄₃	C₂²	1	24	yes	—
15	#14582750	C₂²	1	48	yes	—
16	#14582744	C₂²	1	16	yes	—
17	C₂×C₄₃⋊#95	C₂²	1	6	yes	—
18	C₂²×C₄²×C₄₃	C₂²	1	3	yes	—
19	C₄₃×#376	C₂²	1	8	yes	—
20	C₂×#95×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#95×C₄₃	C₂²	1	3	yes	—
22	C₂²×Dic₈₆	C₂×C₄	1	24	yes	—
23	C₂²×Dic₈₆	C₂³	1	4	yes	—
24	C₂³×Dic₄₃	C₂³	1	6	yes	—
25	C₂³×Dic₄₃	C₂³	1	2	yes	—
26	C₄×Dic₈₆	C₂×C₄	1	192	yes	complemented
27	C₂×C₄⋊Dic₄₃	C₂×C₄	1	24	yes	—
28	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
29	C₂×Dic₄₃⋊C₄	C₂×C₄	1	48	yes	—
30	C₂×Dic₄₃⋊C₄	C₂³	1	24	yes	—
31	C₂×C₄×Dic₄₃	—	4	12	no	—
32	C₂×C₄×Dic₄₃	C₂×C₄	1	48	yes	—
33	C₂×C₄×Dic₄₃	C₂³	1	24	yes	—
34	C₂×C₄×Dic₄₃	C₂³	1	24	yes	—
35	C₄₃⋊#95	C₂³	1	24	yes	—
36	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
37	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
38	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	C₂×C₄	1	24	yes	—
40	C₂×C₄²×C₄₃	C₂³	1	12	yes	—
41	#95×C₄₃	C₂³	1	4	yes	—
42	#95×C₄₃	C₂³	1	12	yes	—
43	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	—	2	6	no	—
45	C₂²×C₄×C₄₃	—	2	12	no	—
46	C₂²×C₄×C₄₃	—	2	6	no	—
47	C₂²×C₄×C₄₃	C₂²×C₄	1	12	yes	—
48	C₂²×C₄×C₄₃	C₂²×C₄	1	12	yes	—
49	C₂²×C₄×C₄₃	C₂²×C₄	1	12	yes	—
50	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
51	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
52	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
53	C₂²×Dic₄₃	—	2	12	no	—
54	C₂²×Dic₄₃	—	2	12	no	—
55	C₂²×Dic₄₃	C₂²×C₄	1	24	yes	—
56	C₂²×Dic₄₃	C₂²×C₄	1	24	yes	—
57	C₂²×Dic₄₃	C₂⁴	1	12	yes	—
58	C₂²×Dic₄₃	C₂⁴	1	4	yes	—
59	C₂×Dic₈₆	C₄²	1	64	yes	complemented
60	C₂×Dic₈₆	C₂²×C₄	1	48	yes	—
61	C₄²×C₄₃	C₂²×C₄	1	48	yes	—
62	C₄⋊Dic₄₃	C₂²×C₄	1	48	yes	—
63	Dic₄₃⋊C₄	C₂²×C₄	1	96	yes	—
64	C₄×Dic₄₃	—	4	24	no	—
65	C₄×Dic₄₃	C₂²×C₄	1	96	yes	—
66	C₂³×C₄₃	C₂³×C₄	1	6	yes	—
67	C₂³×C₄₃	H(5,𝔽₂)	1	2	yes	—
68	C₂³×C₄₃	2₋¹⁺⁴	1	6	yes	—
69	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
70	C₂×C₄×C₄₃	—	2	24	no	—
71	C₂×C₄×C₄₃	—	2	24	no	—
72	C₂×C₄×C₄₃	—	2	6	no	—
73	C₂×C₄×C₄₃	—	2	12	no	—
74	C₂×C₄×C₄₃	—	2	6	no	—
75	C₂×C₄×C₄₃	C₂×C₄²	1	16	yes	—
76	C₂×C₄×C₄₃	C₂³×C₄	1	12	yes	—
77	C₂×C₄×C₄₃	C₂³×C₄	1	12	yes	—
78	C₂×C₄×C₄₃	C₂³×C₄	1	12	yes	—
79	C₂×Dic₄₃	—	2	48	no	—
80	C₂×Dic₄₃	—	2	12	no	—
81	C₂×Dic₄₃	—	2	12	no	—
82	C₂×Dic₄₃	C₂×C₄²	1	32	yes	—
83	C₂×Dic₄₃	C₂³×C₄	1	24	yes	—
84	C₂×Dic₄₃	C₂³×C₄	1	24	yes	—
85	Dic₈₆	C₂×C₄²	1	64	yes	complemented
86	C₂×#2276	—	43	1	no	maximal, Sylow
87	C₂²×C₄₃	C₂²×C₄²	1	4	yes	—
88	C₂²×C₄₃	#519	1	12	yes	—
89	C₂²×C₄₃	#520	1	12	yes	—
90	C₂²×C₄₃	C₂⁴×C₄	1	3	yes	—
91	C₂²×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
92	C₂²×C₄₃	C₂×2₋¹⁺⁴	1	3	yes	—
93	C₄×C₄₃	—	2	12	no	—
94	C₄×C₄₃	—	2	12	no	—
95	C₄×C₄₃	C₂²×C₄²	1	8	yes	—
96	Dic₄₃	—	2	24	no	—
97	Dic₄₃	C₂²×C₄²	1	16	yes	—
98	C₂²×C₄×Q₈	—	43	3	no	—
99	#2276	—	43	16	no	—
100	C₂×C₄×C₄⋊C₄	—	43	3	no	—
101	C₂×C₄×C₄⋊C₄	—	43	6	no	—
102	C₂×#376	—	43	2	no	—
103	C₂×#376	D₄₃	1	1	yes	characteristic
104	C₂×C₄₃	#2276	1	8	yes	complemented
105	C₂×C₄₃	C₂³×C₄²	1	1	yes	characteristic
106	C₂×C₄₃	C₂×#519	1	3	yes	—
107	C₂×C₄₃	C₂×#520	1	3	yes	—
108	C₂³×Q₈	—	43	1	no	—
109	C₂×C₄×Q₈	—	43	24	no	—
110	C₂×C₄×Q₈	—	43	48	no	—
111	C₂²×C₄⋊C₄	—	43	6	no	—
112	C₂²×C₄⋊C₄	—	43	3	no	—
113	C₂²×C₄²	—	43	6	no	—
114	C₂²×C₄²	—	43	6	no	—
115	C₂²×C₄²	C₂×D₄₃	1	3	yes	—
116	C₄×C₄⋊C₄	—	43	24	no	—
117	C₄×C₄⋊C₄	—	43	48	no	—
118	#376	—	43	16	no	—
119	#376	C₂×D₄₃	1	8	yes	—
120	C₂×#95	—	43	6	no	—
121	C₂×#95	C₂×D₄₃	1	1	yes	characteristic
122	C₂×#95	C₂×D₄₃	1	3	yes	—
123	C₄₃	C₂×#2276	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	43	24	no	—
125	C₂²×Q₈	—	43	4	no	—
126	C₂³×C₄	—	43	2	no	—
127	C₂³×C₄	—	43	6	no	—
128	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
129	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
130	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
131	C₄×Q₈	—	43	192	no	—
132	C₂×C₄⋊C₄	—	43	48	no	—
133	C₂×C₄⋊C₄	—	43	24	no	—
134	C₂×C₄⋊C₄	—	43	24	no	—
135	C₂×C₄⋊C₄	—	43	12	no	—
136	C₂×C₄²	—	43	48	no	—
137	C₂×C₄²	—	43	24	no	—
138	C₂×C₄²	—	43	24	no	—
139	C₂×C₄²	—	172	12	no	—
140	C₂×C₄²	C₄×D₄₃	1	24	yes	—
141	C₂×C₄²	C₂²×D₄₃	1	12	yes	—
142	#95	—	43	24	no	—
143	#95	C₂²×D₄₃	1	4	yes	—
144	#95	C₂²×D₄₃	1	12	yes	—
145	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
146	C₂×Q₈	—	43	64	no	—
147	C₂×Q₈	—	43	48	no	—
148	C₂²×C₄	—	2	6	no	—
149	C₂²×C₄	—	2	12	no	—
150	C₂²×C₄	—	2	6	no	—
151	C₂²×C₄	—	43	24	no	—
152	C₂²×C₄	—	43	4	no	—
153	C₂²×C₄	—	43	24	no	—
154	C₂²×C₄	—	43	12	no	—
155	C₂²×C₄	—	86	12	no	—
156	C₂²×C₄	—	86	12	no	—
157	C₂²×C₄	C₂×C₄×D₄₃	1	12	yes	—
158	C₂²×C₄	C₂×C₄×D₄₃	1	12	yes	—
159	C₂²×C₄	C₂×C₄×D₄₃	1	12	yes	—
160	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
161	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
162	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
163	C₄⋊C₄	—	43	96	no	—
164	C₄⋊C₄	—	43	48	no	—
165	C₄²	—	43	96	no	—
166	C₄²	—	172	24	no	—
167	C₄²	C₂×C₄×D₄₃	1	48	yes	—
168	C₂³	C₂²×C₄×D₄₃	1	6	yes	—
169	C₂³	#13063371	1	2	yes	—
170	C₂³	#13063372	1	6	yes	—
171	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
172	Q₈	—	43	64	no	—
173	C₂×C₄	—	2	24	no	—
174	C₂×C₄	—	2	24	no	—
175	C₂×C₄	—	2	6	no	—
176	C₂×C₄	—	2	12	no	—
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	43	32	no	—
179	C₂×C₄	—	43	24	no	—
180	C₂×C₄	—	43	24	no	—
181	C₂×C₄	—	86	48	no	—
182	C₂×C₄	—	86	12	no	—
183	C₂×C₄	—	86	12	no	—
184	C₂×C₄	C₄²×D₄₃	1	16	yes	—
185	C₂×C₄	C₂²×C₄×D₄₃	1	12	yes	—
186	C₂×C₄	C₂²×C₄×D₄₃	1	12	yes	—
187	C₂×C₄	C₂²×C₄×D₄₃	1	12	yes	—
188	C₂²	C₂×C₄²×D₄₃	1	4	yes	—
189	C₂²	#14583376	1	12	yes	—
190	C₂²	#14583379	1	12	yes	—
191	C₂²	C₂³×C₄×D₄₃	1	3	yes	—
192	C₂²	C₂×#13063371	1	1	yes	characteristic
193	C₂²	C₂×#13063372	1	3	yes	—
194	C₄	—	2	12	no	—
195	C₄	—	2	12	no	—
196	C₄	—	43	16	no	—
197	C₄	—	86	24	no	—
198	C₄	C₂×C₄²×D₄₃	1	8	yes	—
199	C₂	#19136958	1	8	yes	direct product factor
200	C₂	C₂²×C₄²×D₄₃	1	1	yes	characteristic
201	C₂	C₂×#14583376	1	3	yes	—
202	C₂	C₂×#14583379	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₃
commutator	C₂×C₄₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#376	C₂
socle, prime radical	C₂⁴×C₄₃	C₂⁴
cosocle	C₂³×C₄₃	C₂⁵
hypercenter	C₂×#376	D₄₃
hypocenter	C₄₃	C₂×#2276
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #19136958	C₂	direct (N × H)
8. C₂×C₄×Dic₈₆	C₄	semidirect
26. C₄×Dic₈₆	C₂×C₄	semidirect
59. C₂×Dic₈₆	C₄²	semidirect
85. Dic₈₆	C₂×C₄²	semidirect
104. C₂×C₄₃	#2276	semidirect
123. C₄₃	C₂×#2276	semidirect
199. C₂	#19136958	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×C₄×Dic₈₆	2
#19136958	2
C₂×C₄×C₄⋊Dic₄₃	2
C₂×#14582750	2
C₂×#14582744	2
C₂×C₄₃×#376	2
C₂×#2276	43

Subgroups — C2×#19136958 #25254819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136958 #25254819