Subgroups — #25251278

Order 11008.

542 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583301	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13063347	C₂	1	1	yes	maximal, characteristic
3	#19136760	C₂	1	4	yes	maximal, complemented
4	#19136753	C₂	1	4	yes	maximal, complemented
5	#19136383	C₂	1	2	yes	maximal, complemented
6	#19136308	C₂	1	2	yes	maximal
7	C₄₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	1	yes	characteristic
9	C₂×#13063317	—	2	1	no	—
10	C₄×#10621898	—	2	1	no	—
11	#14583301	—	2	2	no	—
12	#14583301	C₂²	1	4	yes	—
13	C₄×Dic₁₇₂	C₂²	1	2	yes	—
14	C₄×#10621873	—	2	2	no	—
15	C₄×D₃₄₄	C₂²	1	2	yes	—
16	#14582937	C₂²	1	1	yes	characteristic
17	C₂×#13063259	C₂²	1	2	yes	—
18	C₂×#13063257	—	2	1	no	—
19	#14582896	—	2	1	no	—
20	#14582727	C₂²	1	4	yes	—
21	#14582725	C₂²	1	4	yes	—
22	#14582704	C₂²	1	4	yes	—
23	#14582701	C₂²	1	4	yes	—
24	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×C₄⋊C₁₆	C₂²	1	4	yes	—
27	C₂×#10621898	—	2	1	no	—
28	#13063347	—	2	2	no	—
29	#13063347	Q₈	1	4	yes	—
30	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
31	C₂×#10621873	—	2	1	no	—
32	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
33	#13063317	—	2	4	no	—
34	#13063317	—	4	1	no	—
35	#13063317	—	4	1	no	—
36	C₄×Dic₈₆	—	2	2	no	—
37	C₄×D₁₇₂	—	2	2	no	—
38	C₄×C₂²⋊D₄₃	—	4	2	no	—
39	#13063284	—	2	1	no	—
40	#13063282	—	4	1	no	—
41	C₂×C₄⋊Dic₄₃	—	2	2	no	—
42	C₂×Dic₄₃⋊C₄	—	4	2	no	—
43	C₄²×D₄₃	—	4	1	no	—
44	#13063259	—	2	2	no	—
45	#13063259	C₂³	1	4	yes	—
46	#13063259	C₂³	1	2	yes	—
47	#13063257	—	2	4	no	—
48	#13063257	—	2	1	no	—
49	#13063250	—	2	2	no	—
50	#13063247	—	2	2	no	—
51	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
54	M₃₂×C₄₃	—	2	4	no	—
55	C₂×C₁₆×C₄₃	C₂³	1	4	yes	—
56	C₄×C₈×C₄₃	D₄	1	2	yes	—
57	C₄×C₈×C₄₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₄₃	D₈	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₄₃	—	4	1	no	—
61	C₂²×Dic₄₃	—	4	2	no	—
62	#10621898	—	2	4	no	—
63	#10621898	—	4	2	no	—
64	C₂×D₁₇₂	—	2	1	no	—
65	C₂×C₄×D₄₃	—	4	1	no	—
66	C₂×C₄×D₄₃	—	4	2	no	—
67	C₂×Dic₈₆	—	2	1	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₄₃	#41	1	2	yes	—
73	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₄²×C₄₃	D₈	1	1	yes	characteristic
75	C₄²×C₄₃	D₈	1	1	yes	characteristic
76	C₄²×C₄₃	C₂×D₄	1	2	yes	—
77	C₂²⋊Dic₄₃	—	4	1	no	—
78	(C₂×D₄₃)⋊C₄	—	4	2	no	—
79	C₄⋊Dic₄₃	—	2	2	no	—
80	C₄⋊Dic₄₃	—	2	4	no	—
81	C₄⋊Dic₄₃	—	4	2	no	—
82	Dic₄₃⋊C₄	—	4	4	no	—
83	Dic₄₃⋊C₄	—	4	4	no	—
84	Dic₄₃⋊C₄	—	4	2	no	—
85	C₄×Dic₄₃	—	4	1	no	—
86	C₄×Dic₄₃	—	4	2	no	—
87	Dic₁₇₂	—	2	1	no	—
88	Dic₁₇₂	C₂×Q₈	1	2	yes	—
89	D₃₄₄	—	2	1	no	—
90	D₃₄₄	C₂×Q₈	1	2	yes	—
91	#10621873	—	2	4	no	—
92	C₁₆×C₄₃	—	2	4	no	—
93	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
94	C₂²×D₄₃	—	4	1	no	—
95	C₂×C₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₄₃	#137	1	2	yes	—
101	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
102	C₂²⋊D₄₃	—	4	4	no	—
103	C₂×Dic₄₃	—	4	1	no	—
104	C₂×Dic₄₃	—	4	2	no	—
105	C₂×Dic₄₃	—	4	4	no	—
106	C₂×Dic₄₃	—	8	4	no	—
107	D₁₇₂	—	2	2	no	—
108	D₁₇₂	—	4	1	no	—
109	C₄×D₄₃	—	4	4	no	—
110	C₄×D₄₃	—	8	4	no	—
111	Dic₈₆	—	2	2	no	—
112	Dic₈₆	—	4	1	no	—
113	C₈×C₄₃	—	2	2	no	—
114	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
115	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
116	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
117	#33630	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	—	2	2	no	—
119	C₂²×C₄₃	#474	1	2	yes	—
120	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
121	C₂×D₄₃	—	4	2	no	—
122	C₂×D₄₃	—	8	2	no	—
123	C₄×C₄₃	—	2	2	no	—
124	C₄×C₄₃	—	2	2	no	—
125	C₄×C₄₃	#474	1	2	yes	—
126	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
127	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
128	Dic₄₃	—	4	2	no	—
129	Dic₄₃	—	8	2	no	—
130	Dic₄₃	—	8	4	no	—
131	C₂×#499	—	43	1	no	—
132	C₄×#135	—	43	1	no	—
133	#2229	—	43	4	no	—
134	#2228	—	43	4	no	—
135	#2159	—	43	2	no	—
136	#2150	D₄₃	1	1	yes	characteristic, complemented
137	#2139	—	43	2	no	—
138	C₂×C₄₃	—	2	2	no	—
139	C₂×C₄₃	C₂×#474	1	1	yes	characteristic
140	C₂×C₄₃	#3414	1	2	yes	—
141	D₄₃	—	8	2	no	—
142	C₂×#135	—	43	1	no	—
143	C₂×#125	—	86	1	no	—
144	C₄×#41	—	86	1	no	—
145	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
146	#499	—	43	4	no	—
147	#499	—	86	2	no	—
148	C₄×Q₁₆	—	43	2	no	—
149	C₄×SD₁₆	—	86	2	no	—
150	C₄×D₈	—	43	2	no	—
151	#427	—	43	1	no	—
152	C₂×#107	—	43	2	no	—
153	C₂×#106	—	86	1	no	—
154	#416	—	86	1	no	—
155	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
156	#367	—	43	4	no	—
157	#366	—	43	4	no	—
158	C₄⋊C₁₆	C₂×D₄₃	1	4	yes	—
159	#358	—	43	4	no	—
160	#357	—	43	4	no	—
161	C₄₃	#33630	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	86	1	no	—
163	#135	—	43	4	no	—
164	#135	—	86	2	no	—
165	C₂×Q₁₆	—	43	1	no	—
166	C₂×SD₁₆	—	86	1	no	—
167	C₂×D₈	—	43	1	no	—
168	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
170	#125	—	86	4	no	—
171	#125	—	172	1	no	—
172	#125	—	172	1	no	—
173	C₄×Q₈	—	86	2	no	—
174	C₄×D₄	—	86	2	no	—
175	C₄×D₄	—	172	2	no	—
176	#117	—	86	1	no	—
177	#117	—	172	1	no	—
178	C₂×C₄⋊C₄	—	86	2	no	—
179	C₂×C₄⋊C₄	—	172	2	no	—
180	C₂×C₄²	—	172	1	no	—
181	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
182	M₃₂	—	2	4	no	—
183	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
184	#107	—	43	4	no	—
185	#107	—	43	2	no	—
186	#107	—	86	2	no	—
187	#106	—	86	4	no	—
188	#106	—	86	1	no	—
189	#103	—	86	2	no	—
190	#102	—	86	2	no	—
191	C₄×C₈	D₁₇₂	1	2	yes	—
192	C₄×C₈	C₂²×D₄₃	1	2	yes	—
193	#41	—	86	4	no	—
194	#41	—	172	2	no	—
195	C₂×Q₈	—	86	1	no	—
196	C₂×D₄	—	86	1	no	—
197	C₂×D₄	—	172	1	no	—
198	C₂²×C₄	—	172	2	no	—
199	C₂²×C₄	—	172	2	no	—
200	C₂²×C₄	—	172	1	no	—
201	C₂²×C₄	D₃₄₄	1	2	yes	—
202	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
203	Q₁₆	—	43	2	no	—
204	Q₁₆	—	86	1	no	—
205	SD₁₆	—	86	4	no	—
206	D₈	—	43	2	no	—
207	D₈	—	86	1	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
210	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
211	C₂×C₈	#10621900	1	2	yes	—
212	C₂×C₈	Q₈×D₄₃	1	2	yes	—
213	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
214	C₄⋊C₄	—	86	2	no	—
215	C₄⋊C₄	—	86	4	no	—
216	C₄⋊C₄	—	172	2	no	—
217	C₄⋊C₄	—	172	4	no	—
218	C₄⋊C₄	—	172	4	no	—
219	C₄⋊C₄	—	172	2	no	—
220	C₂²⋊C₄	—	172	1	no	—
221	C₂²⋊C₄	—	172	2	no	—
222	C₄²	—	172	1	no	—
223	C₄²	—	172	2	no	—
224	C₄²	D₃₄₄	1	1	yes	characteristic
225	C₄²	D₃₄₄	1	1	yes	characteristic
226	C₄²	C₂×D₁₇₂	1	2	yes	—
227	C₁₆	—	2	4	no	—
228	C₂³	—	172	1	no	—
229	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
230	Q₈	—	86	2	no	—
231	Q₈	—	172	1	no	—
232	D₄	—	86	2	no	—
233	D₄	—	172	4	no	—
234	D₄	—	172	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	344	4	no	—
237	C₂×C₄	—	344	4	no	—
238	C₂×C₄	—	172	2	no	—
239	C₂×C₄	—	172	4	no	—
240	C₂×C₄	—	172	1	no	—
241	C₂×C₄	—	172	4	no	—
242	C₂×C₄	#13063305	1	2	yes	—
243	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
244	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
245	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
246	C₂×C₄	#13063358	1	2	yes	—
247	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	#13063305	1	2	yes	—
250	C₈	C₂×Q₈×D₄₃	1	1	yes	characteristic
251	C₈	C₂×#10621900	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	344	2	no	—
254	C₂²	—	172	2	no	—
255	C₂²	#14583180	1	2	yes	—
256	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	344	2	no	—
260	C₄	—	344	4	no	—
261	C₄	—	172	2	no	—
262	C₄	#14583180	1	2	yes	—
263	C₄	C₂×#13063305	1	1	yes	characteristic
264	C₄	C₂×#13063358	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	344	2	no	—
267	C₂	C₂×#14583180	1	1	yes	characteristic
268	C₂	#19147731	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2150	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#13063347	C₂²
hypercenter	#2150	D₄₃
hypocenter	C₄₃	#33630
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×#14583301	C₂	semidirect
3. #19136760	C₂	semidirect
4. #19136753	C₂	semidirect
5. #19136383	C₂	semidirect
7. C₄₃×#2150	C₂	semidirect
136. #2150	D₄₃	semidirect
161. C₄₃	#33630	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583301	2
C₄×#13063347	2
#19136760	2
#19136753	2
#19136383	2
#19136308	2
C₄₃×#2150	2
#33630	43