Subgroups — #25102036

Order 11008.

668 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135210	C₂	1	4	yes	maximal
2	#19135161	C₂	1	4	yes	maximal, complemented
3	#19134452	C₂	1	2	yes	maximal, complemented
4	#19134361	C₂	1	2	yes	maximal
5	#19134344	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13063236	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#2298	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
10	C₂×#13063236	C₂²	1	4	yes	—
11	C₂×#13063236	C₂²	1	2	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
13	#14582654	C₂²	1	4	yes	—
14	#14582654	C₂²	1	4	yes	—
15	#14582654	C₂²	1	4	yes	—
16	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#388	C₄	1	4	yes	—
18	C₄₃×#387	C₄	1	4	yes	—
19	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
20	C₄×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₄₃×#376	C₂²	1	1	yes	characteristic
22	C₄³×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
24	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
25	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
26	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
27	#13063236	—	2	4	no	—
28	#13063236	—	2	8	no	—
29	C₄×C₄₃⋊C₈	—	2	4	no	—
30	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	#117×C₄₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
41	#95×C₄₃	C₂×C₄	1	4	yes	—
42	#95×C₄₃	C₂×C₄	1	4	yes	—
43	#95×C₄₃	C₂×C₄	1	4	yes	—
44	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₃	—	2	2	no	—
46	C₂²×C₄×C₄₃	—	2	4	no	—
47	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
53	C₄⋊C₄×C₄₃	—	2	4	no	—
54	C₄⋊C₄×C₄₃	—	4	4	no	—
55	C₂²⋊C₄×C₄₃	—	2	4	no	—
56	C₂²⋊C₄×C₄₃	—	4	4	no	—
57	C₄²×C₄₃	—	2	2	no	—
58	C₄²×C₄₃	—	2	4	no	—
59	C₄²×C₄₃	—	2	4	no	—
60	C₄²×C₄₃	—	4	4	no	—
61	C₄²×C₄₃	#41	1	4	yes	—
62	C₂×C₄₃⋊C₈	—	2	4	no	—
63	C₂×C₄₃⋊C₈	—	2	4	no	—
64	C₂×C₄₃⋊C₈	—	2	4	no	—
65	C₂×C₄₃⋊C₈	—	2	4	no	—
66	C₂×C₄₃⋊C₈	—	2	4	no	—
67	C₂×C₄₃⋊C₈	#41	1	8	yes	—
68	C₂³×C₄₃	—	2	1	no	—
69	C₂³×C₄₃	—	2	2	no	—
70	C₂³×C₄₃	—	2	4	no	—
71	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₃	—	2	4	no	—
73	C₂×C₄×C₄₃	—	2	2	no	—
74	C₂×C₄×C₄₃	—	2	1	no	—
75	C₂×C₄×C₄₃	—	2	4	no	—
76	C₂×C₄×C₄₃	—	2	4	no	—
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	—	2	4	no	—
79	C₂×C₄×C₄₃	—	2	4	no	—
80	C₂×C₄×C₄₃	—	2	4	no	—
81	C₂×C₄×C₄₃	—	2	4	no	—
82	C₂×C₄×C₄₃	—	2	4	no	—
83	C₂×C₄×C₄₃	—	4	4	no	—
84	C₂×C₄×C₄₃	#117	1	4	yes	—
85	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
86	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
87	C₄₃⋊C₈	—	2	8	no	—
88	C₄₃⋊C₈	—	4	4	no	—
89	#21433	—	43	1	no	maximal, Sylow
90	C₂²×C₄₃	—	4	2	no	—
91	C₂²×C₄₃	—	4	4	no	—
92	C₂²×C₄₃	—	4	1	no	—
93	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
94	C₂²×C₄₃	C₄×#41	1	2	yes	—
95	C₂²×C₄₃	#568	1	4	yes	—
96	C₄×C₄₃	—	2	4	no	—
97	C₄×C₄₃	—	4	2	no	—
98	C₄×C₄₃	—	4	4	no	—
99	C₄×C₄₃	—	4	4	no	—
100	C₄×C₄₃	—	4	1	no	—
101	C₄×C₄₃	#529	1	4	yes	—
102	#2298	D₄₃	1	1	yes	characteristic
103	#1924	—	43	4	no	—
104	#1917	—	43	4	no	—
105	#1768	—	43	2	no	—
106	#1750	—	43	2	no	—
107	#1745	—	43	1	no	—
108	C₄×#97	—	43	1	no	—
109	C₂×C₄₃	—	8	1	no	—
110	C₂×C₄₃	#2298	1	1	yes	characteristic
111	C₂×C₄₃	#2923	1	2	yes	—
112	C₂×C₄₃	#2967	1	4	yes	—
113	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
114	C₂×C₄⋊C₈	—	43	4	no	—
115	C₂×C₂²⋊C₈	—	43	4	no	—
116	C₂×#97	—	43	4	no	—
117	C₂×#97	—	43	2	no	—
118	C₂×C₄×C₈	—	43	2	no	—
119	#388	Dic₄₃	1	4	yes	—
120	#387	Dic₄₃	1	4	yes	—
121	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
122	C₄×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
123	#376	C₂×D₄₃	1	1	yes	characteristic
124	C₄³	C₂×D₄₃	1	1	yes	characteristic
125	#336	—	43	4	no	—
126	#336	—	43	4	no	—
127	#336	—	43	4	no	—
128	C₄₃	#21433	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
130	C₂²×C₈	—	43	4	no	—
131	C₂²×C₈	—	43	4	no	—
132	#117	—	2	4	no	—
133	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
134	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
135	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
137	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
138	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
139	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
140	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
141	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
142	C₄⋊C₈	—	86	8	no	—
143	C₂²⋊C₈	—	86	8	no	—
144	#97	—	86	8	no	—
145	#97	—	86	4	no	—
146	C₄×C₈	—	86	4	no	—
147	#95	C₂×Dic₄₃	1	4	yes	—
148	#95	C₂×Dic₄₃	1	4	yes	—
149	#95	C₂×Dic₄₃	1	4	yes	—
150	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
154	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
155	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
156	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
158	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
159	C₂×C₈	—	43	8	no	—
160	C₂×C₈	—	86	4	no	—
161	C₂×C₈	—	86	4	no	—
162	C₂×C₈	—	86	4	no	—
163	C₂×C₈	—	86	4	no	—
164	C₂×C₈	—	86	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	4	4	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	2	2	no	—
172	C₄²	—	4	4	no	—
173	C₄²	#10621898	1	4	yes	—
174	C₂³	—	2	1	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	#13063284	1	4	yes	—
191	C₂×C₄	#41×D₄₃	1	4	yes	—
192	C₂×C₄	C₂×#10621898	1	2	yes	—
193	C₈	—	86	8	no	—
194	C₈	—	172	4	no	—
195	C₂²	—	4	1	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	C₂×#13063284	1	1	yes	characteristic
199	C₂²	#41×Dic₄₃	1	2	yes	—
200	C₂²	#14583666	1	4	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	4	1	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	4	no	—
205	C₄	—	4	4	no	—
206	C₄	#14583427	1	4	yes	—
207	C₂	—	8	1	no	—
208	C₂	#19137093	1	1	yes	characteristic
209	C₂	#19143246	1	2	yes	—
210	C₂	#19143582	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂³×C₄
Fitting	C₄₃×#2298	C₂
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2298	D₄₃
hypocenter	C₄₃	#21433
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2298

Complements

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

N	H ≅ G/N	type
2. #19135161	C₂	semidirect
3. #19134452	C₂	semidirect
5. #19134344	C₂	semidirect
6. C₄×#13063236	C₂	semidirect
128. C₄₃	#21433	semidirect

Maximal subgroups

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

subgroup	index
#19135210	2
#19135161	2
#19134452	2
#19134361	2
#19134344	2
C₄×#13063236	2
C₄₃×#2298	2
#21433	43