Subgroups — #25001781

Order 11008.

1142 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	C₂×C₄×C₄⋊Dic₄₃	C₂	1	2	yes	maximal
2	C₂×#14582662	C₂	1	4	yes	maximal
3	C₂²×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₄₃	C₂²	1	4	yes	—
5	C₂²×C₄×Dic₄₃	—	2	2	no	—
6	C₄×C₄⋊Dic₄₃	—	2	4	no	—
7	C₄×C₄⋊Dic₄₃	C₄	1	8	yes	complemented
8	C₂×C₄₃⋊#95	—	2	2	no	—
9	#14582662	C₄	1	16	yes	complemented
10	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×#97×C₄₃	—	2	4	no	—
13	C₂×#97×C₄₃	C₄	1	4	yes	complemented
14	C₂³×Dic₄₃	—	2	4	no	—
15	C₂×C₄⋊Dic₄₃	—	2	4	no	—
16	C₂×C₄⋊Dic₄₃	—	2	8	no	—
17	C₂×C₄⋊Dic₄₃	C₂×C₄	1	16	yes	—
18	C₂×C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
19	C₂×C₄×Dic₄₃	—	2	8	no	—
20	C₂×C₄×Dic₄₃	—	4	8	no	—
21	C₄₃⋊#95	—	2	8	no	—
22	C₂³×C₄×C₄₃	D₄	1	2	yes	—
23	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₄₃	—	2	4	no	—
25	C₂²×C₈×C₄₃	—	2	4	no	—
26	C₂²×C₈×C₄₃	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₄₃	D₄	1	4	yes	—
31	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
32	#97×C₄₃	—	2	8	no	—
33	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	—	2	4	no	—
35	C₂²×C₄×C₄₃	—	2	4	no	—
36	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	#41	1	4	yes	—
45	C₂²×Dic₄₃	—	2	8	no	—
46	C₂²×Dic₄₃	—	2	16	no	—
47	C₂²×Dic₄₃	—	4	8	no	—
48	C₂²×Dic₄₃	—	4	4	no	—
49	C₂²×Dic₄₃	—	4	4	no	—
50	C₂×C₈×C₄₃	—	2	4	no	—
51	C₂×C₈×C₄₃	—	2	4	no	—
52	C₂×C₈×C₄₃	—	2	16	no	—
53	C₂×C₈×C₄₃	C₄²	1	8	yes	complemented
54	C₄²×C₄₃	C₄²	1	4	yes	—
55	C₄²×C₄₃	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₄₃	C₄⋊C₄	1	8	yes	—
57	C₄⋊Dic₄₃	—	2	8	no	—
58	C₄⋊Dic₄₃	—	2	16	no	—
59	C₄⋊Dic₄₃	C₄²	1	16	yes	complemented
60	C₄×Dic₄₃	—	4	16	no	—
61	C₂³×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
62	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂³×C₄₃	C₄×D₄	1	4	yes	—
64	C₂³×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
65	C₂³×C₄₃	#137	1	4	yes	—
66	C₂×C₄×C₄₃	—	2	8	no	—
67	C₂×C₄×C₄₃	—	2	4	no	—
68	C₂×C₄×C₄₃	—	2	4	no	—
69	C₂×C₄×C₄₃	—	2	8	no	—
70	C₂×C₄×C₄₃	#95	1	8	yes	—
71	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
72	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
74	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₄₃	#117	1	4	yes	—
77	C₂×C₄×C₄₃	#117	1	2	yes	—
78	C₂×C₄×C₄₃	C₄×D₄	1	8	yes	—
79	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
80	C₂×Dic₄₃	—	2	16	no	—
81	C₂×Dic₄₃	—	4	8	no	—
82	C₂×Dic₄₃	—	4	8	no	—
83	C₂×Dic₄₃	—	4	16	no	—
84	C₂×Dic₄₃	—	4	16	no	—
85	C₈×C₄₃	—	2	8	no	—
86	#10703	—	43	1	no	maximal, Sylow
87	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
88	C₂²×C₄₃	C₄×C₂²⋊C₄	1	4	yes	—
89	C₂²×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
90	C₂²×C₄₃	#417	1	4	yes	—
91	C₂²×C₄₃	#418	1	2	yes	—
92	C₂²×C₄₃	#419	1	2	yes	—
93	C₂²×C₄₃	#428	1	4	yes	—
94	C₂²×C₄₃	#440	1	8	yes	—
95	C₂²×C₄₃	#441	1	4	yes	—
96	C₂²×C₄₃	#442	1	4	yes	—
97	C₄×C₄₃	—	2	8	no	—
98	C₄×C₄₃	C₄³	1	1	yes	characteristic
99	C₄×C₄₃	#376	1	1	yes	characteristic
100	C₄×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
101	C₄×C₄₃	C₄×C₄⋊C₄	1	4	yes	—
102	Dic₄₃	—	4	16	no	—
103	C₂²×#97	D₄₃	1	1	yes	characteristic
104	C₂×C₄×C₄⋊C₄	—	43	2	no	—
105	C₂×#340	—	43	4	no	—
106	C₂×C₄₃	C₄×#95	1	1	yes	characteristic
107	C₂×C₄₃	#1736	1	2	yes	—
108	C₂×C₄₃	#1762	1	4	yes	—
109	C₂×C₄₃	#1763	1	4	yes	—
110	C₂×C₄₃	#1776	1	4	yes	—
111	C₂³×C₈	C₂×D₄₃	1	2	yes	—
112	C₂²×C₄⋊C₄	—	43	4	no	—
113	C₂²×C₄²	—	86	2	no	—
114	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
115	C₂×#97	—	2	4	no	—
116	C₂×#97	Dic₄₃	1	4	yes	complemented
117	C₄×C₄⋊C₄	—	43	8	no	—
118	C₄×C₄⋊C₄	—	86	4	no	—
119	C₂×#95	—	86	2	no	—
120	#340	—	43	16	no	—
121	C₄₃	#10703	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	—	86	4	no	—
123	C₂³×C₄	D₁₇₂	1	2	yes	—
124	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₄×D₄₃	1	8	yes	—
128	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
129	C₂×C₄⋊C₄	—	43	16	no	—
130	C₂×C₄⋊C₄	—	43	8	no	—
131	C₂×C₄⋊C₄	—	86	8	no	—
132	C₂×C₄⋊C₄	—	86	4	no	—
133	C₂×C₄²	—	86	8	no	—
134	C₂×C₄²	—	172	8	no	—
135	C₂×C₄²	Dic₈₆	1	2	yes	—
136	C₂×C₄²	C₄×D₄₃	1	4	yes	—
137	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
138	C₂×C₄²	C₂²⋊D₄₃	1	4	yes	—
139	#97	—	2	8	no	—
140	#95	—	86	8	no	—
141	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	86	8	no	—
145	C₂²×C₄	—	86	16	no	—
146	C₂²×C₄	—	172	8	no	—
147	C₂²×C₄	—	172	4	no	—
148	C₂²×C₄	—	172	4	no	—
149	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
150	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
151	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
152	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
155	C₂²×C₄	#10621898	1	4	yes	—
156	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	C₄×Dic₄₃	1	8	yes	complemented
162	C₄⋊C₄	—	43	16	no	—
163	C₄⋊C₄	—	86	8	no	—
164	C₄⋊C₄	—	86	16	no	—
165	C₄²	—	172	16	no	—
166	C₄²	C₄×Dic₄₃	1	4	yes	—
167	C₄²	Dic₄₃⋊C₄	1	8	yes	—
168	C₄²	C₂²⋊Dic₄₃	1	4	yes	—
169	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	2	yes	—
170	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
171	C₂³	C₄×D₁₇₂	1	4	yes	—
172	C₂³	#13063353	1	4	yes	—
173	C₂³	#13063358	1	4	yes	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	86	16	no	—
179	C₂×C₄	—	172	8	no	—
180	C₂×C₄	—	172	8	no	—
181	C₂×C₄	—	172	16	no	—
182	C₂×C₄	—	172	16	no	—
183	C₂×C₄	C₄₃⋊#95	1	8	yes	—
184	C₂×C₄	C₄²×D₄₃	1	4	yes	—
185	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
187	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
189	C₂×C₄	#13063282	1	4	yes	—
190	C₂×C₄	#13063284	1	2	yes	—
191	C₂×C₄	C₄×C₂²⋊D₄₃	1	8	yes	—
192	C₂×C₄	C₄×Dic₈₆	1	4	yes	—
193	C₈	—	2	8	no	—
194	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
195	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	4	yes	—
196	C₂²	C₄×C₄⋊Dic₄₃	1	2	yes	—
197	C₂²	#14582901	1	4	yes	—
198	C₂²	#14582905	1	2	yes	—
199	C₂²	#14582909	1	2	yes	—
200	C₂²	#14582941	1	4	yes	—
201	C₂²	#14583001	1	8	yes	—
202	C₂²	#14583007	1	4	yes	—
203	C₂²	#14583012	1	4	yes	—
204	C₄	—	2	8	no	—
205	C₄	—	172	16	no	—
206	C₄	C₄²×Dic₄₃	1	1	yes	characteristic
207	C₄	#14582745	1	1	yes	characteristic
208	C₄	C₄×C₂²⋊Dic₄₃	1	2	yes	—
209	C₄	C₄×Dic₄₃⋊C₄	1	4	yes	—
210	C₂	C₄×C₄₃⋊#95	1	1	yes	characteristic
211	C₂	#19134315	1	2	yes	—
212	C₂	#19134421	1	4	yes	—
213	C₂	#19134426	1	4	yes	—
214	C₂	#19134492	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₇₂
commutator	C₄×C₄₃	C₄³
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂²×#97×C₄₃	C₂
socle, prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂²×#97	D₄₃
hypocenter	C₄₃	#10703
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
7. C₄×C₄⋊Dic₄₃	C₄	semidirect
9. #14582662	C₄	semidirect
13. C₂×#97×C₄₃	C₄	semidirect
53. C₂×C₈×C₄₃	C₄²	semidirect
59. C₄⋊Dic₄₃	C₄²	semidirect
116. C₂×#97	Dic₄₃	semidirect
121. C₄₃	#10703	semidirect
161. C₂×C₈	C₄×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊Dic₄₃	2
C₂×#14582662	2
C₂²×#97×C₄₃	2
#10703	43