Subgroups — #25259415

Order 11008.

3390 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137772	C₂	1	4	yes	maximal, complemented
2	#19137660	C₂	1	4	yes	maximal, complemented
3	D₄×C₂²⋊Dic₄₃	C₂	1	8	yes	maximal, complemented
4	#19137321	C₂	1	8	yes	maximal, complemented
5	#19137173	C₂	1	4	yes	maximal, complemented
6	#19136993	C₂	1	2	yes	maximal, complemented
7	C₂²×#127×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×D₄×Dic₄₃	C₂²	1	16	yes	complemented
9	C₂²×C₂²⋊Dic₄₃	C₂²	1	8	yes	—
10	#14582802	C₂²	1	16	yes	complemented
11	#14582779	C₂²	1	16	yes	complemented
12	#14582779	C₂²	1	32	yes	complemented
13	#14582779	C₂²	1	16	yes	complemented
14	#14582767	C₂²	1	4	yes	complemented
15	#14582765	C₂²	1	4	yes	complemented
16	#14582762	C₂²	1	4	yes	—
17	#14582759	C₂²	1	4	yes	—
18	#14582755	C₂²	1	8	yes	—
19	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	complemented
20	C₄×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
21	#14582745	C₂²	1	4	yes	complemented
22	C₂³×D₄×C₄₃	C₂²	1	4	yes	—
23	C₂³×D₄×C₄₃	C₂²	1	2	yes	—
24	C₂×#127×C₄₃	C₄	1	16	yes	complemented
25	C₂×#127×C₄₃	C₂²	1	4	yes	—
26	C₂×#127×C₄₃	C₂²	1	2	yes	—
27	C₂×#127×C₄₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
29	C₂³×Dic₄₃	C₂³	1	16	yes	—
30	D₄×Dic₄₃	—	4	32	no	—
31	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
32	C₂×C₂²⋊Dic₄₃	—	4	16	no	—
33	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
34	C₂×C₂²⋊Dic₄₃	C₂³	1	16	yes	—
35	C₂×C₂²⋊Dic₄₃	C₂³	1	32	yes	—
36	C₂×C₄×Dic₄₃	C₂³	1	8	yes	—
37	C₄₃⋊#95	C₂³	1	8	yes	—
38	C₄₃⋊#95	C₂³	1	8	yes	—
39	C₄₃⋊#95	C₂³	1	8	yes	—
40	C₄₃⋊#95	C₂³	1	8	yes	—
41	C₂⁵×C₄₃	C₂³	1	4	yes	—
42	C₂²×D₄×C₄₃	—	4	16	no	—
43	C₂²×D₄×C₄₃	—	4	8	no	—
44	C₂²×D₄×C₄₃	C₂×C₄	1	16	yes	complemented
45	C₂²×D₄×C₄₃	C₂×C₄	1	32	yes	complemented
46	C₂²×D₄×C₄₃	C₂³	1	8	yes	—
47	C₂²×D₄×C₄₃	C₂³	1	4	yes	—
48	C₂²×D₄×C₄₃	C₂³	1	8	yes	—
49	C₂²×D₄×C₄₃	C₂³	1	4	yes	—
50	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
51	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	#127×C₄₃	C₂×C₄	1	16	yes	complemented
53	#127×C₄₃	C₂×C₄	1	32	yes	complemented
54	#127×C₄₃	C₂×C₄	1	16	yes	complemented
55	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	complemented
56	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₂⁴×C₄₃	—	2	16	no	—
60	C₂⁴×C₄₃	—	4	16	no	—
61	C₂⁴×C₄₃	C₂²×C₄	1	16	yes	—
62	C₂⁴×C₄₃	C₂⁴	1	8	yes	—
63	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
64	C₂×D₄×C₄₃	—	2	32	no	—
65	C₂×D₄×C₄₃	—	4	32	no	—
66	C₂×D₄×C₄₃	—	4	4	no	—
67	C₂×D₄×C₄₃	—	4	8	no	—
68	C₂×D₄×C₄₃	—	4	16	no	—
69	C₂×D₄×C₄₃	—	4	4	no	—
70	C₂×D₄×C₄₃	—	4	64	no	—
71	C₂×D₄×C₄₃	C₂²×C₄	1	32	yes	—
72	C₂×D₄×C₄₃	C₂²×C₄	1	16	yes	—
73	C₂×D₄×C₄₃	C₂²×C₄	1	16	yes	—
74	C₂×D₄×C₄₃	C₂²×C₄	1	32	yes	—
75	C₂²×C₄×C₄₃	—	2	8	no	—
76	C₂²×C₄×C₄₃	—	2	4	no	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
79	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
83	C₂²×Dic₄₃	—	4	16	no	—
84	C₂²×Dic₄₃	—	4	32	no	—
85	C₂²×Dic₄₃	C₂⁴	1	16	yes	—
86	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
87	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
88	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
89	C₂²⋊Dic₄₃	—	2	16	no	—
90	C₂²⋊Dic₄₃	—	4	16	no	—
91	C₂²⋊Dic₄₃	—	4	32	no	—
92	C₄⋊Dic₄₃	—	2	16	no	—
93	C₄×Dic₄₃	—	2	16	no	—
94	C₂³×C₄₃	—	2	4	no	—
95	C₂³×C₄₃	—	2	8	no	—
96	C₂³×C₄₃	—	2	32	no	—
97	C₂³×C₄₃	—	2	8	no	—
98	C₂³×C₄₃	—	2	32	no	—
99	C₂³×C₄₃	—	4	32	no	—
100	C₂³×C₄₃	—	8	32	no	—
101	C₂³×C₄₃	C₂³×C₄	1	16	yes	—
102	C₂³×C₄₃	C₂³×C₄	1	2	yes	—
103	C₂³×C₄₃	H(5,𝔽₂)	1	4	yes	—
104	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
105	D₄×C₄₃	—	2	64	no	—
106	D₄×C₄₃	—	4	16	no	—
107	D₄×C₄₃	—	4	16	no	—
108	D₄×C₄₃	—	4	32	no	—
109	C₂×C₄×C₄₃	—	2	8	no	—
110	C₂×C₄×C₄₃	—	2	4	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	16	no	—
114	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
117	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
118	C₂×C₄×C₄₃	H(5,𝔽₂)	1	8	yes	—
119	C₂×Dic₄₃	—	2	16	no	—
120	C₂×Dic₄₃	—	2	16	no	—
121	C₂×Dic₄₃	—	2	16	no	—
122	C₂×Dic₄₃	—	8	32	no	—
123	#34364	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	4	no	—
125	C₂²×C₄₃	—	2	8	no	—
126	C₂²×C₄₃	—	2	16	no	—
127	C₂²×C₄₃	—	2	32	no	—
128	C₂²×C₄₃	—	4	16	no	—
129	C₂²×C₄₃	—	4	16	no	—
130	C₂²×C₄₃	—	8	32	no	—
131	C₂²×C₄₃	#518	1	4	yes	—
132	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
133	C₂²×C₄₃	C₂×H(5,𝔽₂)	1	4	yes	—
134	C₂²×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	—	2	4	no	—
138	C₄×C₄₃	#518	1	8	yes	—
139	Dic₄₃	—	4	16	no	—
140	C₂²×#127	D₄₃	1	1	yes	characteristic
141	#2380	—	43	4	no	—
142	#2366	—	43	4	no	—
143	D₄×C₂²⋊C₄	—	43	8	no	—
144	#2328	—	43	8	no	—
145	#2308	—	43	4	no	—
146	#2283	—	43	2	no	—
147	C₂×C₄₃	—	2	4	no	—
148	C₂×C₄₃	—	4	16	no	—
149	C₂×C₄₃	C₂×#518	1	1	yes	characteristic
150	C₂×C₄₃	C₂×#518	1	2	yes	—
151	C₂×C₄₃	#3543	1	4	yes	—
152	C₂³×D₄	C₂×D₄₃	1	2	yes	—
153	C₂³×D₄	C₂×D₄₃	1	4	yes	—
154	C₂×#127	Dic₄₃	1	16	yes	complemented
155	C₂×#127	C₂×D₄₃	1	4	yes	—
156	C₂×#127	C₂×D₄₃	1	2	yes	—
157	C₂×#127	C₂×D₄₃	1	2	yes	—
158	C₂×C₄×D₄	—	43	16	no	—
159	C₂²×C₂²⋊C₄	—	43	8	no	—
160	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
161	#390	—	43	16	no	—
162	#386	—	43	16	no	—
163	#386	—	43	16	no	—
164	#386	—	43	32	no	—
165	#383	—	43	4	no	—
166	#382	—	43	4	no	—
167	#381	—	43	4	no	—
168	#380	—	43	4	no	—
169	#379	—	43	8	no	—
170	C₄×C₄⋊C₄	—	43	4	no	—
171	C₄×C₂²⋊C₄	—	43	4	no	—
172	#376	—	43	4	no	—
173	C₄₃	#34364	1	1	yes	characteristic, complemented, Sylow
174	C₂⁵	C₂²×D₄₃	1	4	yes	—
175	C₂²×D₄	—	4	16	no	—
176	C₂²×D₄	—	4	8	no	—
177	C₂²×D₄	C₂×Dic₄₃	1	32	yes	complemented
178	C₂²×D₄	C₂×Dic₄₃	1	16	yes	complemented
179	C₂²×D₄	C₂²×D₄₃	1	8	yes	—
180	C₂²×D₄	C₂²×D₄₃	1	4	yes	—
181	C₂²×D₄	C₂²×D₄₃	1	8	yes	—
182	C₂²×D₄	C₂²×D₄₃	1	4	yes	—
183	C₂³×C₄	—	43	16	no	—
184	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
185	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
186	#127	C₂×Dic₄₃	1	16	yes	complemented
187	#127	C₂×Dic₄₃	1	32	yes	complemented
188	#127	C₂×Dic₄₃	1	16	yes	complemented
189	C₄×D₄	—	172	32	no	—
190	C₂×C₄⋊C₄	—	43	8	no	—
191	C₂×C₂²⋊C₄	—	43	8	no	—
192	C₂×C₂²⋊C₄	—	43	16	no	—
193	C₂×C₂²⋊C₄	—	43	32	no	—
194	C₂×C₂²⋊C₄	—	172	16	no	—
195	C₂×C₄²	—	43	8	no	—
196	C₂×C₄²	C₂×Dic₄₃	1	8	yes	complemented
197	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
198	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
200	#95	—	43	8	no	—
201	#95	—	43	8	no	—
202	#95	—	43	8	no	—
203	#95	—	43	8	no	—
204	C₂⁴	—	2	16	no	—
205	C₂⁴	—	4	16	no	—
206	C₂⁴	C₂²×Dic₄₃	1	16	yes	—
207	C₂⁴	C₂³×D₄₃	1	8	yes	—
208	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
209	C₂×D₄	—	2	32	no	—
210	C₂×D₄	—	4	64	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	—	4	8	no	—
213	C₂×D₄	—	4	16	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	4	32	no	—
216	C₂×D₄	C₂²×Dic₄₃	1	32	yes	—
217	C₂×D₄	C₂²×Dic₄₃	1	32	yes	—
218	C₂×D₄	C₂²×Dic₄₃	1	16	yes	—
219	C₂×D₄	C₂²×Dic₄₃	1	16	yes	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	43	16	no	—
223	C₂²×C₄	—	172	16	no	—
224	C₂²×C₄	—	172	32	no	—
225	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
227	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
228	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
229	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
231	C₄⋊C₄	—	86	16	no	—
232	C₂²⋊C₄	—	86	16	no	—
233	C₂²⋊C₄	—	172	16	no	—
234	C₂²⋊C₄	—	172	32	no	—
235	C₄²	—	86	16	no	—
236	C₄²	C₂²×Dic₄₃	1	4	yes	—
237	C₄²	C₂²×Dic₄₃	1	8	yes	—
238	C₄²	C₂²×Dic₄₃	1	4	yes	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	32	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	32	no	—
244	C₂³	—	4	32	no	—
245	C₂³	—	8	32	no	—
246	C₂³	C₂³×Dic₄₃	1	16	yes	—
247	C₂³	C₂³×Dic₄₃	1	2	yes	—
248	C₂³	#13063370	1	4	yes	—
249	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
250	D₄	—	2	64	no	—
251	D₄	—	4	32	no	—
252	D₄	—	4	16	no	—
253	D₄	—	4	16	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	86	16	no	—
260	C₂×C₄	—	86	16	no	—
261	C₂×C₄	—	86	16	no	—
262	C₂×C₄	—	344	32	no	—
263	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
264	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
265	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
266	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
267	C₂×C₄	#13063370	1	8	yes	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	32	no	—
271	C₂²	—	2	16	no	—
272	C₂²	—	4	16	no	—
273	C₂²	—	4	16	no	—
274	C₂²	—	8	32	no	—
275	C₂²	#14583374	1	4	yes	—
276	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
277	C₂²	C₂×#13063370	1	2	yes	—
278	C₂²	C₂×#13063370	1	4	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	172	16	no	—
283	C₄	#14583374	1	8	yes	—
284	C₂	—	2	4	no	—
285	C₂	—	4	16	no	—
286	C₂	C₂×#14583374	1	1	yes	characteristic
287	C₂	C₂×#14583374	1	2	yes	—
288	C₂	#19148928	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Fitting, prime radical	C₂²×#127×C₄₃	C₂
socle, cosocle	C₂³×C₄₃	C₂⁵
hypercenter	C₂²×#127	D₄₃
hypocenter	C₄₃	#34364
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. #19137772	C₂	semidirect
2. #19137660	C₂	semidirect
3. D₄×C₂²⋊Dic₄₃	C₂	semidirect
4. #19137321	C₂	semidirect
5. #19137173	C₂	semidirect
6. #19136993	C₂	semidirect
8. C₂×D₄×Dic₄₃	C₂²	semidirect
10. #14582802	C₂²	semidirect
11. #14582779	C₂²	semidirect
12. #14582779	C₂²	semidirect
13. #14582779	C₂²	semidirect
14. #14582767	C₂²	semidirect
15. #14582765	C₂²	semidirect
19. C₄×C₄⋊Dic₄₃	C₂²	semidirect
21. #14582745	C₂²	semidirect
24. C₂×#127×C₄₃	C₄	semidirect
44. C₂²×D₄×C₄₃	C₂×C₄	semidirect
45. C₂²×D₄×C₄₃	C₂×C₄	semidirect
52. #127×C₄₃	C₂×C₄	semidirect
53. #127×C₄₃	C₂×C₄	semidirect
54. #127×C₄₃	C₂×C₄	semidirect
55. C₂×C₄²×C₄₃	C₂×C₄	semidirect
154. C₂×#127	Dic₄₃	semidirect
173. C₄₃	#34364	semidirect
177. C₂²×D₄	C₂×Dic₄₃	semidirect
178. C₂²×D₄	C₂×Dic₄₃	semidirect
186. #127	C₂×Dic₄₃	semidirect
187. #127	C₂×Dic₄₃	semidirect
188. #127	C₂×Dic₄₃	semidirect
196. C₂×C₄²	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19137772	2
#19137660	2
D₄×C₂²⋊Dic₄₃	2
#19137321	2
#19137173	2
#19136993	2
C₂²×#127×C₄₃	2
#34364	43