Subgroups — #25013545

Order 11008.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583140	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582702	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#19132686	C₂	1	2	yes	maximal, complemented
5	C₂²×#107×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
7	C₂×#13063304	—	2	1	no	—
8	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	#14583140	—	2	2	no	—
10	#14583140	C₄	1	4	yes	complemented
11	C₂×#13063259	C₂²	1	1	yes	characteristic
12	C₂×#13063250	—	2	1	no	—
13	#14582702	—	2	4	no	—
14	C₂²⋊(C₄₃⋊C₁₆)	—	2	2	no	—
15	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
17	C₂×#107×C₄₃	—	2	1	no	—
18	C₂×#107×C₄₃	—	2	4	no	—
19	C₂×#107×C₄₃	C₂²	1	2	yes	—
20	C₂²×Dic₈₆	—	2	1	no	—
21	C₂×Dic₁₇₂	—	2	2	no	—
22	C₂×Dic₁₇₂	—	2	2	no	—
23	C₂×Dic₁₇₂	—	2	1	no	—
24	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	complemented
25	#13063304	—	2	4	no	—
26	#13063304	—	4	2	no	—
27	C₂×C₄⋊Dic₄₃	—	2	1	no	—
28	C₂×Dic₄₃⋊C₄	—	4	1	no	—
29	C₂×C₂²⋊Dic₄₃	—	4	1	no	—
30	C₂×C₄₃⋊C₁₆	—	2	2	no	—
31	C₂×C₄₃⋊C₁₆	—	2	2	no	—
32	C₂×C₄₃⋊C₁₆	—	2	2	no	—
33	#13063259	—	2	1	no	—
34	#13063259	C₂×C₄	1	2	yes	complemented
35	#13063250	—	2	4	no	—
36	C₂³×C₄×C₄₃	—	4	1	no	—
37	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	—	2	2	no	—
39	C₂²×C₈×C₄₃	—	2	2	no	—
40	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₄₃	D₄	1	2	yes	—
42	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
47	C₂×C₄⋊C₄×C₄₃	—	4	1	no	—
48	#107×C₄₃	—	2	4	no	—
49	#107×C₄₃	—	2	2	no	—
50	#107×C₄₃	D₄	1	4	yes	—
51	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	—	2	1	no	—
53	C₂²×C₄×C₄₃	—	2	1	no	—
54	C₂²×C₄×C₄₃	—	2	1	no	—
55	C₂²×C₄×C₄₃	—	2	1	no	—
56	C₂²×C₄×C₄₃	—	4	4	no	—
57	C₂²×C₄×C₄₃	—	4	2	no	—
58	C₂²×C₄×C₄₃	—	8	2	no	—
59	C₂²×C₄×C₄₃	—	8	2	no	—
60	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₄₃	—	4	1	no	—
65	C₂²×Dic₄₃	—	4	1	no	—
66	C₂×Dic₈₆	—	2	2	no	—
67	C₂×Dic₈₆	—	4	1	no	—
68	C₂×Dic₈₆	—	4	1	no	—
69	C₂×Dic₈₆	—	4	1	no	—
70	C₂×Dic₈₆	—	4	1	no	—
71	C₂×Dic₈₆	—	4	1	no	—
72	C₂×C₈×C₄₃	—	2	1	no	—
73	C₂×C₈×C₄₃	—	2	1	no	—
74	C₂×C₈×C₄₃	—	2	4	no	—
75	C₂×C₈×C₄₃	—	2	4	no	—
76	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₄₃	#41	1	2	yes	—
81	C₄⋊C₄×C₄₃	—	2	4	no	—
82	C₄⋊C₄×C₄₃	—	4	4	no	—
83	C₄⋊C₄×C₄₃	—	4	2	no	—
84	C₂²⋊Dic₄₃	—	4	4	no	—
85	C₄⋊Dic₄₃	—	2	2	no	—
86	C₄⋊Dic₄₃	—	4	1	no	—
87	Dic₄₃⋊C₄	—	4	4	no	—
88	Dic₁₇₂	—	2	4	no	—
89	Dic₁₇₂	—	4	2	no	—
90	C₄₃⋊C₁₆	—	4	2	no	—
91	C₂³×C₄₃	—	2	1	no	—
92	C₂³×C₄₃	—	2	1	no	—
93	C₂³×C₄₃	—	2	1	no	—
94	C₂³×C₄₃	—	2	1	no	—
95	C₂³×C₄₃	#102	1	2	yes	—
96	C₂³×C₄₃	SD₃₂	1	2	yes	—
97	C₂³×C₄₃	Q₃₂	1	2	yes	—
98	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₄₃	—	2	1	no	—
100	C₂×C₄×C₄₃	—	2	1	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	4	4	no	—
106	C₂×C₄×C₄₃	—	8	8	no	—
107	C₂×C₄×C₄₃	—	8	2	no	—
108	C₂×C₄×C₄₃	—	8	2	no	—
109	C₂×C₄×C₄₃	#102	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
116	C₂×Dic₄₃	—	4	2	no	—
117	C₂×Dic₄₃	—	4	2	no	—
118	C₂×Dic₄₃	—	8	1	no	—
119	C₂×Dic₄₃	—	8	1	no	—
120	C₂×Dic₄₃	—	8	1	no	—
121	C₂×Dic₄₃	—	8	1	no	—
122	Dic₈₆	—	4	2	no	—
123	Dic₈₆	—	4	2	no	—
124	Dic₈₆	—	8	1	no	—
125	Dic₈₆	—	8	1	no	—
126	C₈×C₄₃	—	2	2	no	—
127	C₈×C₄₃	C₄×D₄	1	2	yes	—
128	C₈×C₄₃	#121	1	2	yes	—
129	#12993	—	43	1	no	maximal, Sylow
130	C₂²×C₄₃	—	2	2	no	—
131	C₂²×C₄₃	—	2	2	no	—
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	—	2	2	no	—
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	#358	1	4	yes	complemented
137	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₄₃	#447	1	1	yes	characteristic
139	C₂²×C₄₃	#480	1	1	yes	characteristic
140	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
141	C₂²×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
142	C₂²×C₄₃	#509	1	1	yes	characteristic
143	C₂²×C₄₃	#510	1	1	yes	characteristic
144	C₄×C₄₃	—	2	2	no	—
145	C₄×C₄₃	—	8	4	no	—
146	C₄×C₄₃	#386	1	1	yes	characteristic
147	C₄×C₄₃	#417	1	1	yes	characteristic
148	C₄×C₄₃	#448	1	1	yes	characteristic
149	C₄×C₄₃	#481	1	1	yes	characteristic
150	Dic₄₃	—	8	2	no	—
151	Dic₄₃	—	8	2	no	—
152	C₂×#467	—	43	1	no	—
153	C₂²×#107	D₄₃	1	1	yes	characteristic
154	C₂×#358	—	43	2	no	—
155	C₂×C₂²⋊C₁₆	—	43	1	no	—
156	#1380	—	43	2	no	—
157	C₂×C₄₃	—	2	4	no	—
158	C₂×C₄₃	#1874	1	1	yes	characteristic
159	C₂×C₄₃	C₂×#358	1	1	yes	characteristic
160	C₂×C₄₃	#2139	1	1	yes	characteristic
161	C₂×C₄₃	#2187	1	1	yes	characteristic
162	C₂×C₄₃	#2188	1	1	yes	characteristic
163	C₂×C₄₃	#2235	1	1	yes	characteristic
164	C₂×C₄₃	#2236	1	1	yes	characteristic
165	C₂²×Q₁₆	—	43	1	no	—
166	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
167	C₂×C₂²⋊Q₈	—	86	1	no	—
168	C₂²×C₄⋊C₄	—	2	1	no	—
169	C₂²×C₁₆	—	43	2	no	—
170	#467	—	43	4	no	—
171	#467	—	86	2	no	—
172	C₂×#107	—	2	1	no	—
173	C₂×#107	—	2	4	no	—
174	C₂×#107	—	43	1	no	—
175	C₂×#107	C₂×D₄₃	1	2	yes	—
176	C₂×#103	—	86	1	no	—
177	#358	—	86	4	no	—
178	C₂²⋊C₁₆	—	86	2	no	—
179	C₄₃	#12993	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	86	1	no	—
181	C₂³×C₄	—	4	1	no	—
182	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
183	C₂×Q₁₆	—	43	2	no	—
184	C₂×Q₁₆	—	86	1	no	—
185	C₂×Q₁₆	—	86	2	no	—
186	C₂×Q₁₆	—	86	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₄×D₄₃	1	2	yes	—
190	C₂²×C₈	D₁₇₂	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
193	C₂²⋊Q₈	—	86	4	no	—
194	C₂²⋊Q₈	—	172	2	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	4	2	no	—
198	C₂×C₄⋊C₄	—	4	1	no	—
199	C₂×C₄⋊C₄	—	86	1	no	—
200	C₂×C₄⋊C₄	—	172	1	no	—
201	C₂×C₂²⋊C₄	—	172	1	no	—
202	C₂×C₁₆	—	86	2	no	—
203	C₂×C₁₆	—	86	2	no	—
204	C₂×C₁₆	—	86	2	no	—
205	#107	—	2	4	no	—
206	#107	—	2	2	no	—
207	#107	—	43	2	no	—
208	#107	—	86	1	no	—
209	#107	C₂²⋊D₄₃	1	4	yes	—
210	#103	—	86	4	no	—
211	C₂⁴	#10621882	1	1	yes	characteristic
212	C₂×Q₈	—	86	2	no	—
213	C₂×Q₈	—	172	1	no	—
214	C₂×Q₈	—	172	1	no	—
215	C₂×Q₈	—	172	1	no	—
216	C₂×Q₈	—	172	1	no	—
217	C₂×Q₈	—	172	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	8	2	no	—
225	C₂²×C₄	—	8	2	no	—
226	C₂²×C₄	—	172	1	no	—
227	C₂²×C₄	—	172	1	no	—
228	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
229	C₂²×C₄	#10621882	1	1	yes	characteristic
230	C₂²×C₄	#10621884	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
232	Q₁₆	—	86	4	no	—
233	Q₁₆	—	172	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
240	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
241	C₂×C₈	#10621898	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₄⋊C₄	—	172	4	no	—
248	C₄⋊C₄	—	172	1	no	—
249	C₂²⋊C₄	—	172	4	no	—
250	C₁₆	—	172	2	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#13063245	1	2	yes	—
256	C₂³	#13063269	1	2	yes	—
257	C₂³	#13063271	1	2	yes	—
258	C₂³	C₂×#10621882	1	1	yes	characteristic
259	Q₈	—	344	1	no	—
260	Q₈	—	344	1	no	—
261	Q₈	—	172	2	no	—
262	Q₈	—	172	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	8	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	344	1	no	—
274	C₂×C₄	—	344	1	no	—
275	C₂×C₄	—	344	1	no	—
276	C₂×C₄	—	344	1	no	—
277	C₂×C₄	—	172	2	no	—
278	C₂×C₄	—	172	2	no	—
279	C₂×C₄	#13063245	1	2	yes	—
280	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	#13063294	1	1	yes	characteristic
282	C₂×C₄	#13063308	1	1	yes	characteristic
283	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
284	C₂×C₄	#13063351	1	1	yes	characteristic
285	C₂×C₄	#13063356	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	C₄×C₂²⋊D₄₃	1	2	yes	—
288	C₈	#13063299	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#14582702	1	4	yes	complemented
296	C₂²	C₂×#13063245	1	1	yes	characteristic
297	C₂²	#14583033	1	1	yes	characteristic
298	C₂²	#14583212	1	1	yes	characteristic
299	C₂²	C₂×#13063269	1	1	yes	characteristic
300	C₂²	C₂×#13063271	1	1	yes	characteristic
301	C₂²	#14583335	1	1	yes	characteristic
302	C₂²	#14583340	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	8	4	no	—
305	C₄	—	344	2	no	—
306	C₄	—	344	2	no	—
307	C₄	#14582778	1	1	yes	characteristic
308	C₄	#14582898	1	1	yes	characteristic
309	C₄	#14583038	1	1	yes	characteristic
310	C₄	#14583217	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#19134912	1	1	yes	characteristic
313	C₂	C₂×#14582702	1	1	yes	characteristic
314	C₂	#19136305	1	1	yes	characteristic
315	C₂	#19136524	1	1	yes	characteristic
316	C₂	#19136529	1	1	yes	characteristic
317	C₂	#19136792	1	1	yes	characteristic
318	C₂	#19136797	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂²×#107×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂⁴×C₄₃	D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂²×#107	D₄₃
hypocenter	C₄₃	#12993
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. C₂×#14582702	C₂	semidirect
4. #19132686	C₂	semidirect
10. #14583140	C₄	semidirect
24. C₂×Dic₁₇₂	C₂×C₄	semidirect
34. #13063259	C₂×C₄	semidirect
136. C₂²×C₄₃	#358	semidirect
179. C₄₃	#12993	semidirect
295. C₂²	#14582702	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583140	2
C₂×#14582702	2
C₂×C₂²⋊(C₄₃⋊C₁₆)	2
#19132686	2
C₂²×#107×C₄₃	2
#12993	43