Subgroups — #25015064

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583276	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582699	C₂	1	4	yes	maximal, complemented
4	C₄×#107×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₄₄	—	2	1	no	—
6	C₂²×D₃₄₄	C₂²	1	2	yes	—
7	C₂×#13063322	—	2	1	no	—
8	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	#14583276	—	2	2	no	—
10	#14583276	C₄	1	4	yes	complemented
11	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
12	#14582699	—	2	8	no	—
13	C₂×#107×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
16	C₄₃×#340	—	2	1	no	—
17	C₂²×D₁₇₂	—	2	2	no	—
18	C₂²×D₁₇₂	—	4	1	no	—
19	C₂²×D₁₇₂	—	4	1	no	—
20	C₂×D₃₄₄	—	2	4	no	—
21	C₂×D₃₄₄	—	2	4	no	—
22	C₂×D₃₄₄	—	2	4	no	—
23	C₂×D₃₄₄	—	2	2	no	—
24	C₂×D₃₄₄	—	4	4	no	—
25	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
26	#13063322	—	2	4	no	—
27	#13063322	—	4	2	no	—
28	C₂×C₄₃⋊C₁₆	—	2	4	no	—
29	C₂×C₄₃⋊C₁₆	—	2	2	no	—
30	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
33	C₂×C₄²×C₄₃	—	4	1	no	—
34	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
35	#107×C₄₃	—	2	2	no	—
36	#107×C₄₃	D₄	1	4	yes	complemented
37	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₄₃	D₄	1	2	yes	—
39	#95×C₄₃	—	4	1	no	—
40	C₂³×D₄₃	—	4	2	no	—
41	C₂²×C₄×C₄₃	—	4	2	no	—
42	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂×D₁₇₂	—	2	4	no	—
46	C₂×D₁₇₂	—	4	4	no	—
47	C₂×D₁₇₂	—	4	4	no	—
48	C₂×D₁₇₂	—	4	2	no	—
49	C₂×D₁₇₂	—	4	4	no	—
50	C₂×D₁₇₂	—	4	4	no	—
51	C₂×D₁₇₂	—	8	4	no	—
52	C₂×D₁₇₂	—	8	2	no	—
53	C₂×D₁₇₂	—	8	2	no	—
54	C₂×C₈×C₄₃	—	2	2	no	—
55	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₃	#41	1	2	yes	—
60	C₄⋊C₄×C₄₃	—	2	4	no	—
61	C₄⋊C₄×C₄₃	—	4	2	no	—
62	C₄²×C₄₃	—	8	2	no	—
63	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₄₃	SD₁₆	1	2	yes	—
65	D₃₄₄	—	2	8	no	—
66	D₃₄₄	—	4	8	no	—
67	D₃₄₄	—	4	2	no	—
68	D₃₄₄	—	4	2	no	—
69	C₄₃⋊C₁₆	—	4	2	no	—
70	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₄₃	—	4	4	no	—
72	C₂²×D₄₃	—	8	4	no	—
73	C₂²×D₄₃	—	8	2	no	—
74	C₂²×D₄₃	—	8	2	no	—
75	C₂²×D₄₃	—	8	4	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	4	4	no	—
78	C₂×C₄×C₄₃	—	8	2	no	—
79	C₂×C₄×C₄₃	—	8	2	no	—
80	C₂×C₄×C₄₃	#102	1	2	yes	—
81	C₂×C₄×C₄₃	#102	1	2	yes	—
82	C₂×C₄×C₄₃	D₁₆	1	2	yes	—
83	C₂×C₄×C₄₃	SD₃₂	1	2	yes	—
84	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#121	1	2	yes	—
86	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
88	D₁₇₂	—	4	4	no	—
89	D₁₇₂	—	4	4	no	—
90	D₁₇₂	—	8	8	no	—
91	D₁₇₂	—	8	8	no	—
92	D₁₇₂	—	8	4	no	—
93	C₈×C₄₃	—	2	2	no	—
94	C₈×C₄₃	C₄×D₄	1	2	yes	—
95	C₈×C₄₃	#124	1	1	yes	characteristic
96	C₈×C₄₃	#127	1	1	yes	characteristic
97	#13252	—	43	1	no	maximal, Sylow
98	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
99	C₂²×C₄₃	#459	1	2	yes	—
100	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
101	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
102	C₂²×C₄₃	#509	1	2	yes	—
103	C₂×D₄₃	—	8	4	no	—
104	C₂×D₄₃	—	8	4	no	—
105	C₂×D₄₃	—	8	4	no	—
106	C₂×D₄₃	—	16	4	no	—
107	C₂×D₄₃	—	16	4	no	—
108	C₄×C₄₃	—	2	2	no	—
109	C₄×C₄₃	—	8	4	no	—
110	C₄×C₄₃	#357	1	4	yes	complemented
111	C₄×C₄₃	#390	1	1	yes	characteristic
112	C₄×C₄₃	#418	1	1	yes	characteristic
113	C₄×C₄₃	#460	1	2	yes	—
114	C₂×#493	—	43	1	no	—
115	C₂×C₄⋊C₁₆	—	43	1	no	—
116	C₂×#357	—	43	4	no	—
117	C₄×#107	D₄₃	1	1	yes	characteristic, complemented
118	C₂×C₄₃	#1982	1	1	yes	characteristic
119	C₂×C₄₃	C₂×#357	1	1	yes	characteristic
120	C₂×C₄₃	#2140	1	1	yes	characteristic
121	C₂×C₄₃	#2206	1	2	yes	—
122	C₂×C₄₃	#2207	1	2	yes	—
123	D₄₃	—	16	4	no	—
124	C₂²×D₈	—	43	2	no	—
125	C₂²×D₈	—	86	1	no	—
126	C₂×#127	—	86	1	no	—
127	C₂²×C₁₆	—	43	2	no	—
128	#493	—	43	4	no	—
129	#493	—	86	2	no	—
130	C₂×#107	C₂×D₄₃	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₁₆	—	86	2	no	—
134	#357	—	86	8	no	—
135	#340	—	2	1	no	—
136	C₄₃	#13252	1	1	yes	characteristic, complemented, Sylow
137	C₂²×D₄	—	86	2	no	—
138	C₂²×D₄	—	172	1	no	—
139	C₂²×D₄	—	172	1	no	—
140	C₂×D₈	—	43	4	no	—
141	C₂×D₈	—	86	4	no	—
142	C₂×D₈	—	86	4	no	—
143	C₂×D₈	—	86	2	no	—
144	C₂×D₈	—	86	4	no	—
145	C₂×D₈	—	172	4	no	—
146	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
148	#127	—	86	4	no	—
149	#127	—	172	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄²	—	4	1	no	—
152	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
153	C₂×C₁₆	—	86	4	no	—
154	C₂×C₁₆	—	86	2	no	—
155	#107	—	2	2	no	—
156	#107	D₁₇₂	1	4	yes	complemented
157	C₄×C₈	C₄×D₄₃	1	2	yes	—
158	C₄×C₈	D₁₇₂	1	2	yes	—
159	#95	—	4	1	no	—
160	C₂⁴	—	172	2	no	—
161	C₂×D₄	—	86	4	no	—
162	C₂×D₄	—	344	4	no	—
163	C₂×D₄	—	344	2	no	—
164	C₂×D₄	—	344	2	no	—
165	C₂×D₄	—	172	4	no	—
166	C₂×D₄	—	172	4	no	—
167	C₂×D₄	—	172	4	no	—
168	C₂×D₄	—	172	2	no	—
169	C₂×D₄	—	172	4	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	#10621882	1	1	yes	characteristic
172	C₂²×C₄	#10621882	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
174	D₈	—	86	8	no	—
175	D₈	—	172	8	no	—
176	D₈	—	172	2	no	—
177	D₈	—	172	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
180	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
181	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
182	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
183	C₂×C₈	#10621898	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄²	—	8	2	no	—
187	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
188	C₄²	#10621884	1	2	yes	—
189	C₁₆	—	172	2	no	—
190	C₂³	—	344	2	no	—
191	C₂³	—	344	2	no	—
192	C₂³	—	344	4	no	—
193	C₂³	—	344	4	no	—
194	C₂³	—	172	4	no	—
195	C₂³	C₂×#10621882	1	1	yes	characteristic
196	D₄	—	344	8	no	—
197	D₄	—	344	4	no	—
198	D₄	—	344	8	no	—
199	D₄	—	172	4	no	—
200	D₄	—	172	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	#13063245	1	2	yes	—
206	C₂×C₄	#13063245	1	2	yes	—
207	C₂×C₄	#13063266	1	2	yes	—
208	C₂×C₄	#13063268	1	2	yes	—
209	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
210	C₂×C₄	#13063299	1	2	yes	—
211	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
212	C₂×C₄	#13063350	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₄×D₁₇₂	1	2	yes	—
215	C₈	#13063313	1	1	yes	characteristic
216	C₈	#13063322	1	1	yes	characteristic
217	C₂²	—	344	4	no	—
218	C₂²	—	344	4	no	—
219	C₂²	—	344	4	no	—
220	C₂²	—	688	4	no	—
221	C₂²	—	688	4	no	—
222	C₂²	C₂×#13063245	1	1	yes	characteristic
223	C₂²	#14583094	1	2	yes	—
224	C₂²	C₂×#13063266	1	1	yes	characteristic
225	C₂²	C₂×#13063268	1	1	yes	characteristic
226	C₂²	#14583334	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	8	4	no	—
229	C₄	#14582699	1	4	yes	complemented
230	C₄	#14582801	1	1	yes	characteristic
231	C₄	#14582903	1	1	yes	characteristic
232	C₄	#14583099	1	2	yes	—
233	C₂	—	688	4	no	—
234	C₂	#19135522	1	1	yes	characteristic
235	C₂	C₂×#14582699	1	1	yes	characteristic
236	C₂	#19136310	1	1	yes	characteristic
237	C₂	#19136630	1	2	yes	—
238	C₂	#19136635	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄×#107×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂×#14583276	C₂
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄×#107	D₄₃
hypocenter	C₄₃	#13252
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	semidirect
3. C₂×#14582699	C₂	semidirect
4. C₄×#107×C₄₃	C₂	semidirect
10. #14583276	C₄	semidirect
36. #107×C₄₃	D₄	semidirect
110. C₄×C₄₃	#357	semidirect
117. C₄×#107	D₄₃	semidirect
136. C₄₃	#13252	semidirect
156. #107	D₁₇₂	semidirect
229. C₄	#14582699	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583276	2
C₂×C₄⋊(C₄₃⋊C₁₆)	2
C₂×#14582699	2
C₄×#107×C₄₃	2
#13252	43