Subgroups — #25015166

Order 11008.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063261	C₂	1	1	yes	maximal, characteristic
3	#19132690	C₂	1	4	yes	maximal
4	C₄₃×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13063261	C₂²	1	2	yes	—
7	C₄×#10621877	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
10	#14582664	—	2	1	no	—
11	C₄₃×#533	—	2	1	no	—
12	C₄₃×#500	C₄	1	1	yes	characteristic
13	C₄₃×#500	C₄	1	1	yes	characteristic
14	C₄₃×#499	—	2	1	no	—
15	C₄₃×#499	C₄	1	2	yes	—
16	C₄₃×#498	—	2	1	no	—
17	C₄₃×#427	—	2	1	no	—
18	C₄₃×#427	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#10621877	—	2	2	no	—
21	C₂×C₄₃⋊C₁₆	—	2	2	no	—
22	C₂×C₄₃⋊C₁₆	—	2	4	no	—
23	#13063261	—	2	2	no	—
24	#13063261	Q₈	1	4	yes	complemented
25	#13063236	—	4	1	no	—
26	C₄×C₄₃⋊C₈	—	4	1	no	—
27	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₄₃	—	2	1	no	—
30	C₄⋊Q₈×C₄₃	—	2	1	no	—
31	#125×C₄₃	—	2	2	no	—
32	C₂²⋊Q₈×C₄₃	—	4	1	no	—
33	C₂²⋊Q₈×C₄₃	—	4	1	no	—
34	C₄×Q₈×C₄₃	—	4	1	no	—
35	C₄×Q₈×C₄₃	—	4	1	no	—
36	#117×C₄₃	—	2	2	no	—
37	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
38	#107×C₄₃	—	2	1	no	—
39	#107×C₄₃	—	2	1	no	—
40	#107×C₄₃	C₂×C₄	1	2	yes	—
41	#107×C₄₃	C₂×C₄	1	2	yes	—
42	#106×C₄₃	—	2	2	no	—
43	#106×C₄₃	—	2	2	no	—
44	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₄₃	—	4	1	no	—
49	C₂×Q₈×C₄₃	—	4	1	no	—
50	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₄₃	—	2	2	no	—
54	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₄₃	#41	1	2	yes	—
60	C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₄⋊C₄×C₄₃	—	2	2	no	—
62	C₄⋊C₄×C₄₃	—	4	2	no	—
63	C₄⋊C₄×C₄₃	—	4	2	no	—
64	C₄⋊C₄×C₄₃	—	4	1	no	—
65	C₄⋊C₄×C₄₃	—	4	1	no	—
66	C₂²⋊C₄×C₄₃	—	4	2	no	—
67	C₄²×C₄₃	—	4	2	no	—
68	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₄₃	D₈	1	1	yes	characteristic
71	C₄²×C₄₃	D₈	1	1	yes	characteristic
72	#10621877	—	2	4	no	—
73	#10621877	—	4	2	no	—
74	C₂×C₄₃⋊C₈	—	4	2	no	—
75	C₄₃⋊C₁₆	—	4	2	no	—
76	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
77	Q₈×C₄₃	—	8	2	no	—
78	Q₈×C₄₃	—	8	2	no	—
79	C₂×C₄×C₄₃	—	2	2	no	—
80	C₂×C₄×C₄₃	—	4	1	no	—
81	C₂×C₄×C₄₃	—	4	2	no	—
82	C₂×C₄×C₄₃	—	4	2	no	—
83	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₄₃	#137	1	2	yes	—
91	C₈×C₄₃	—	2	1	no	—
92	C₈×C₄₃	—	2	1	no	—
93	C₈×C₄₃	C₄×Q₈	1	2	yes	—
94	C₈×C₄₃	#124	1	1	yes	characteristic
95	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₄₃⋊C₈	—	4	4	no	—
97	#13274	—	43	1	no	maximal, Sylow
98	C₂²×C₄₃	—	2	1	no	—
99	C₂²×C₄₃	—	2	1	no	—
100	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
101	C₂²×C₄₃	#475	1	2	yes	—
102	C₄×C₄₃	—	2	1	no	—
103	C₄×C₄₃	—	2	1	no	—
104	C₄×C₄₃	—	2	1	no	—
105	C₄×C₄₃	—	8	2	no	—
106	C₄×C₄₃	—	8	2	no	—
107	C₄×C₄₃	#359	1	2	yes	—
108	C₄×C₄₃	#391	1	1	yes	characteristic
109	C₄×C₄₃	#419	1	1	yes	characteristic
110	C₄×C₄₃	#474	1	2	yes	—
111	#3160	D₄₃	1	1	yes	characteristic
112	C₂×C₄⋊C₁₆	—	43	1	no	—
113	C₄×#108	—	43	1	no	—
114	#1382	—	43	4	no	—
115	C₂×C₄₃	—	2	2	no	—
116	C₂×C₄₃	#1988	1	1	yes	characteristic
117	C₂×C₄₃	C₂×#359	1	1	yes	characteristic
118	C₂×C₄₃	#2143	1	1	yes	characteristic
119	#533	—	2	1	no	—
120	C₂²×C₁₆	—	43	2	no	—
121	#500	Dic₄₃	1	1	yes	characteristic
122	#500	Dic₄₃	1	1	yes	characteristic
123	#499	—	2	1	no	—
124	#499	Dic₄₃	1	2	yes	—
125	#498	—	2	1	no	—
126	C₂×#108	—	43	2	no	—
127	#427	—	2	1	no	—
128	#427	C₂×D₄₃	1	2	yes	—
129	C₄×M₁₆	—	86	1	no	—
130	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	86	1	no	—
132	C₄⋊C₁₆	—	86	1	no	—
133	#341	—	86	1	no	—
134	C₄₃	#13274	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	86	2	no	—
136	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	1	no	—
139	C₄⋊Q₈	—	2	1	no	—
140	#125	—	2	2	no	—
141	C₂²⋊Q₈	—	4	1	no	—
142	C₂²⋊Q₈	—	4	1	no	—
143	C₄×Q₈	—	4	1	no	—
144	C₄×Q₈	—	4	1	no	—
145	#117	—	2	2	no	—
146	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
147	C₂×C₁₆	—	86	4	no	—
148	C₂×C₁₆	—	86	2	no	—
149	#108	—	43	4	no	—
150	#108	—	86	2	no	—
151	#107	—	2	1	no	—
152	#107	—	2	1	no	—
153	#107	C₂×Dic₄₃	1	2	yes	—
154	#107	C₂×Dic₄₃	1	2	yes	—
155	#106	—	2	2	no	—
156	#106	—	2	2	no	—
157	#97	—	172	1	no	—
158	C₄×C₈	—	172	1	no	—
159	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
160	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	#10621883	1	1	yes	characteristic
166	C₂²×C₄	#10621883	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
168	M₁₆	—	86	4	no	—
169	M₁₆	—	172	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	172	2	no	—
172	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
174	C₂×C₈	Q₈×D₄₃	1	2	yes	—
175	C₂×C₈	#10621902	1	2	yes	—
176	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	#10621882	1	1	yes	characteristic
187	C₄²	#10621882	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
190	C₁₆	—	172	2	no	—
191	C₂³	C₂×#10621883	1	1	yes	characteristic
192	Q₈	—	8	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#13063246	1	1	yes	characteristic
199	C₂×C₄	#13063246	1	1	yes	characteristic
200	C₂×C₄	#13063246	1	1	yes	characteristic
201	C₂×C₄	#13063246	1	1	yes	characteristic
202	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
203	C₂×C₄	#13063301	1	2	yes	—
204	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
205	C₂×C₄	#13063356	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	172	4	no	—
209	C₈	Q₈×Dic₄₃	1	2	yes	—
210	C₈	#13063314	1	1	yes	characteristic
211	C₈	#13063323	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#13063246	1	1	yes	characteristic
215	C₂²	#14583181	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	8	2	no	—
220	C₄	—	8	2	no	—
221	C₄	#14582706	1	2	yes	—
222	C₄	#14582806	1	1	yes	characteristic
223	C₄	#14582908	1	1	yes	characteristic
224	C₄	#14583176	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#19135548	1	1	yes	characteristic
227	C₂	C₂×#14582706	1	1	yes	characteristic
228	C₂	#19136323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#3160	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#3160	D₄₃
hypocenter	C₄₃	#13274
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
24. #13063261	Q₈	semidirect
134. C₄₃	#13274	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₃⋊C₁₆)	2
C₄×#13063261	2
#19132690	2
C₄₃×#3160	2
#13274	43