Subgroups — #25080495

Order 11008.

702 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144404	C₂	1	2	yes	maximal
2	C₈×#10621898	C₂	1	1	yes	maximal, characteristic
3	#19133962	C₂	1	4	yes	maximal, complemented
4	#19133957	C₂	1	4	yes	maximal, complemented
5	#19133354	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133280	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10621898	C₂²	1	1	yes	characteristic
9	#14583191	—	2	2	no	—
10	#14583186	—	2	2	no	—
11	#14583181	C₂²	1	4	yes	—
12	#14583176	C₂²	1	4	yes	—
13	C₈×Dic₈₆	C₂²	1	2	yes	—
14	C₈×D₁₇₂	C₂²	1	2	yes	—
15	C₈×C₂²⋊D₄₃	—	2	2	no	—
16	#14582952	C₂²	1	1	yes	characteristic
17	#14582950	—	2	1	no	—
18	#14582939	—	2	2	no	—
19	#14582923	C₂²	1	2	yes	—
20	#14582898	—	2	2	no	—
21	C₄×C₈×D₄₃	—	2	1	no	—
22	#14582652	C₂²	1	4	yes	—
23	#14582645	C₂²	1	8	yes	—
24	C₂²×Q₁₆×C₄₃	—	2	2	no	—
25	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
26	C₄₃×#494	—	2	1	no	—
27	C₄₃×#494	—	2	1	no	—
28	C₄₃×#494	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#10621898	D₄	1	1	yes	characteristic
31	C₂×#10621877	—	2	2	no	—
32	C₂×C₈×D₄₃	—	2	2	no	—
33	C₄×Dic₈₆	D₄	1	1	yes	characteristic
34	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
35	C₄×D₁₇₂	D₄	1	1	yes	characteristic
36	C₄×D₁₇₂	C₂³	1	1	yes	characteristic
37	C₄×C₂²⋊D₄₃	—	2	1	no	—
38	C₄×C₂²⋊D₄₃	—	2	1	no	—
39	#13063284	D₄	1	1	yes	characteristic
40	#13063282	—	2	1	no	—
41	C₄²×D₄₃	—	2	1	no	—
42	#13063258	—	2	4	no	—
43	#13063256	—	2	4	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
46	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
47	Dic₄₃⋊C₈	—	2	2	no	—
48	#13063248	—	2	4	no	—
49	#13063245	—	2	4	no	—
50	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
51	(C₂×D₄₃)⋊C₈	—	2	2	no	—
52	C₄×C₄₃⋊C₈	—	2	1	no	—
53	C₈×Dic₄₃	—	2	2	no	—
54	C₂²×Q₈×C₄₃	—	2	2	no	—
55	C₂×Q₁₆×C₄₃	—	2	4	no	—
56	C₂×Q₁₆×C₄₃	—	2	4	no	—
57	C₂×Q₁₆×C₄₃	—	4	8	no	—
58	C₂²×C₈×C₄₃	D₄	1	2	yes	—
59	C₄⋊Q₈×C₄₃	—	2	2	no	—
60	C₄⋊Q₈×C₄₃	—	2	2	no	—
61	C₄⋊Q₈×C₄₃	—	2	2	no	—
62	C₄⋊Q₈×C₄₃	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
64	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₄₃	C₂³	1	2	yes	—
69	C₂×Q₈×C₄₃	—	2	4	no	—
70	C₂×Q₈×C₄₃	—	4	8	no	—
71	C₂×Q₈×C₄₃	—	4	2	no	—
72	C₂²×C₄×C₄₃	—	4	2	no	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
76	C₂×C₂²⋊D₄₃	—	2	1	no	—
77	#10621898	—	2	2	no	—
78	#10621898	Q₁₆	1	4	yes	complemented
79	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×C₄×D₄₃	—	2	1	no	—
82	C₂×C₄×D₄₃	—	2	1	no	—
83	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₄₃	—	4	8	no	—
85	Q₁₆×C₄₃	—	4	8	no	—
86	C₂×C₈×C₄₃	—	2	4	no	—
87	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₄₃	—	2	4	no	—
90	C₄⋊C₄×C₄₃	—	2	4	no	—
91	C₄⋊C₄×C₄₃	—	4	2	no	—
92	C₄⋊C₄×C₄₃	—	4	2	no	—
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₃	#41	1	1	yes	characteristic
96	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₄₃	—	2	1	no	—
98	(C₂×D₄₃)⋊C₄	—	2	1	no	—
99	(C₂×D₄₃)⋊C₄	—	2	1	no	—
100	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
101	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
102	Dic₄₃⋊C₄	—	2	1	no	—
103	Dic₄₃⋊C₄	—	2	1	no	—
104	C₄×Dic₄₃	—	2	1	no	—
105	C₄×Dic₄₃	—	2	1	no	—
106	C₄×Dic₄₃	—	2	1	no	—
107	#10621877	—	4	4	no	—
108	C₂×C₄₃⋊C₈	—	2	2	no	—
109	C₂×C₄₃⋊C₈	—	2	4	no	—
110	C₈×D₄₃	—	4	4	no	—
111	C₂³×C₄₃	#121	1	1	yes	characteristic
112	C₂²×D₄₃	—	2	1	no	—
113	Q₈×C₄₃	—	4	8	no	—
114	Q₈×C₄₃	—	8	4	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	—	8	4	no	—
119	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
123	C₂×C₄×C₄₃	#135	1	2	yes	—
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
127	C₂²⋊D₄₃	—	2	4	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	2	1	no	—
130	C₂×Dic₄₃	—	2	1	no	—
131	D₁₇₂	—	2	1	no	—
132	D₁₇₂	C₂×Q₁₆	1	2	yes	complemented
133	C₄×D₄₃	—	2	4	no	—
134	C₄×D₄₃	—	4	2	no	—
135	C₄×D₄₃	—	4	2	no	—
136	Dic₈₆	—	2	1	no	—
137	Dic₈₆	C₂×Q₁₆	1	2	yes	complemented
138	C₈×C₄₃	—	2	2	no	—
139	C₈×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
140	C₄₃⋊C₈	—	4	4	no	—
141	C₄₃⋊C₈	—	4	2	no	—
142	#19268	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	#467	1	2	yes	—
146	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
147	C₂×D₄₃	—	2	2	no	—
148	C₂×D₄₃	—	4	1	no	—
149	C₂×D₄₃	—	4	1	no	—
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	8	4	no	—
153	C₄×C₄₃	#467	1	2	yes	—
154	C₄×C₄₃	#473	1	2	yes	—
155	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
156	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
157	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
158	Dic₄₃	—	2	2	no	—
159	Dic₄₃	—	4	1	no	—
160	Dic₄₃	—	4	1	no	—
161	C₂×#494	D₄₃	1	1	yes	characteristic, complemented
162	#3079	—	43	2	no	—
163	C₈×#41	—	43	1	no	—
164	#1670	—	43	4	no	—
165	#1669	—	43	4	no	—
166	#1567	—	43	1	no	—
167	#1549	—	43	2	no	—
168	C₂×C₄₃	—	2	2	no	—
169	C₂×C₄₃	#3038	1	1	yes	characteristic
170	C₂×C₄₃	C₂×#467	1	1	yes	characteristic
171	C₂×C₄₃	C₂×#473	1	1	yes	characteristic
172	D₄₃	—	4	2	no	—
173	C₂²×Q₁₆	—	2	2	no	—
174	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
175	C₄×#41	—	43	1	no	—
176	#494	—	2	1	no	—
177	#494	—	2	1	no	—
178	#494	C₂×D₄₃	1	4	yes	—
179	#477	—	86	2	no	—
180	#476	—	86	2	no	—
181	#475	—	43	4	no	—
182	#474	—	43	4	no	—
183	C₈×Q₈	—	43	2	no	—
184	C₈×D₄	—	43	2	no	—
185	C₈×D₄	—	86	2	no	—
186	#431	—	43	1	no	—
187	#431	—	86	1	no	—
188	#428	—	86	2	no	—
189	#423	—	43	2	no	—
190	#417	—	86	2	no	—
191	C₂×C₄×C₈	—	86	1	no	—
192	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
193	#335	—	43	4	no	—
194	#332	—	43	8	no	—
195	C₄₃	#19268	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	—	43	1	no	—
197	C₂²×Q₈	—	2	2	no	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×Q₁₆	—	2	4	no	—
200	C₂×Q₁₆	—	4	8	no	—
201	C₂×M₁₆	—	86	2	no	—
202	C₂²×C₈	—	86	2	no	—
203	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	C₂²×D₄₃	1	4	yes	—
208	C₄×Q₈	—	43	1	no	—
209	C₄×Q₈	—	43	1	no	—
210	C₄×D₄	—	43	1	no	—
211	C₄×D₄	—	43	1	no	—
212	C₄×D₄	—	86	1	no	—
213	C₄×D₄	—	86	1	no	—
214	#117	—	43	1	no	—
215	#117	—	86	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄²	—	86	1	no	—
219	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
220	#107	—	86	4	no	—
221	#106	—	86	4	no	—
222	C₄⋊C₈	—	43	2	no	—
223	C₄⋊C₈	—	43	4	no	—
224	C₄⋊C₈	—	86	2	no	—
225	C₄⋊C₈	—	86	2	no	—
226	#103	—	86	4	no	—
227	#102	—	86	4	no	—
228	C₂²⋊C₈	—	86	1	no	—
229	C₂²⋊C₈	—	86	2	no	—
230	C₄×C₈	—	86	2	no	—
231	C₄×C₈	—	86	1	no	—
232	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
233	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₄₃	1	2	yes	—
235	#41	—	43	4	no	—
236	#41	—	86	2	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	4	8	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	43	1	no	—
241	C₂×D₄	—	43	1	no	—
242	C₂×D₄	—	86	1	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	86	1	no	—
245	C₂²×C₄	—	86	1	no	—
246	C₂²×C₄	—	86	1	no	—
247	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
248	C₂²×C₄	#10621900	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
250	Q₁₆	—	4	8	no	—
251	Q₁₆	—	4	8	no	—
252	M₁₆	—	172	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	86	4	no	—
255	C₂×C₈	—	86	2	no	—
256	C₂×C₈	—	172	4	no	—
257	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
258	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	43	1	no	—
264	C₄⋊C₄	—	43	1	no	—
265	C₄⋊C₄	—	86	1	no	—
266	C₄⋊C₄	—	86	1	no	—
267	C₂²⋊C₄	—	86	1	no	—
268	C₂²⋊C₄	—	86	1	no	—
269	C₂²⋊C₄	—	86	1	no	—
270	C₄²	—	86	1	no	—
271	C₄²	—	86	1	no	—
272	C₄²	—	86	1	no	—
273	C₄²	D₄×D₄₃	1	1	yes	characteristic
274	C₄²	#10621900	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₄²	C₂³×D₄₃	1	1	yes	characteristic
277	C₂³	—	86	1	no	—
278	C₂³	#13063296	1	1	yes	characteristic
279	Q₈	—	4	8	no	—
280	Q₈	—	8	4	no	—
281	Q₈	—	43	2	no	—
282	Q₈	—	86	1	no	—
283	D₄	—	43	2	no	—
284	D₄	—	86	4	no	—
285	D₄	—	86	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	—	86	1	no	—
291	C₂×C₄	—	86	4	no	—
292	C₂×C₄	—	86	1	no	—
293	C₂×C₄	—	86	1	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	#13063296	1	1	yes	characteristic
297	C₂×C₄	#13063296	1	1	yes	characteristic
298	C₂×C₄	#13063296	1	1	yes	characteristic
299	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
300	C₂×C₄	#13063348	1	2	yes	—
301	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
303	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	172	4	no	—
306	C₈	—	172	2	no	—
307	C₈	#13063351	1	4	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	86	2	no	—
311	C₂²	—	172	1	no	—
312	C₂²	—	172	1	no	—
313	C₂²	#14583137	1	2	yes	—
314	C₂²	C₂×#13063296	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	4	no	—
318	C₄	—	86	2	no	—
319	C₄	—	172	1	no	—
320	C₄	—	172	1	no	—
321	C₄	#14583137	1	2	yes	—
322	C₄	#14583171	1	2	yes	—
323	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
324	C₄	C₂×#13063348	1	1	yes	characteristic
325	C₄	C₂×#13063351	1	2	yes	—
326	C₂	—	2	2	no	—
327	C₂	—	172	2	no	—
328	C₂	#19144071	1	1	yes	characteristic
329	C₂	C₂×#14583137	1	1	yes	characteristic
330	C₂	C₂×#14583171	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063296
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#494	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂×#10621898	D₄
hypercenter	C₂×#494	D₄₃
hypocenter	C₄₃	#19268
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #19133962	C₂	semidirect
4. #19133957	C₂	semidirect
5. #19133354	C₂	semidirect
6. #19133280	C₂	semidirect
7. C₂×C₄₃×#494	C₂	semidirect
78. #10621898	Q₁₆	semidirect
132. D₁₇₂	C₂×Q₁₆	semidirect
137. Dic₈₆	C₂×Q₁₆	semidirect
161. C₂×#494	D₄₃	semidirect
195. C₄₃	#19268	semidirect

Maximal subgroups

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

subgroup	index
#19144404	2
C₈×#10621898	2
#19133962	2
#19133957	2
#19133354	2
#19133280	2
C₂×C₄₃×#494	2
#19268	43