Subgroups — #25253525

Order 11008.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147026	C₂	1	1	yes	maximal, characteristic, complemented
2	#19146317	C₂	1	1	yes	maximal, characteristic
3	#19136823	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136804	C₂	1	2	yes	maximal, complemented
5	#19136733	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136686	C₂	1	2	yes	maximal, complemented
7	#19136646	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136537	C₂	1	2	yes	maximal, complemented
9	C₄×#13063270	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136303	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	2	yes	—
13	#14583619	—	2	1	no	—
14	#14583554	—	2	1	no	—
15	C₂×Dic₃₄₄	C₂²	1	2	yes	—
16	C₂×#13063270	C₂²	1	1	yes	characteristic
17	#14583304	C₂²	1	1	yes	characteristic
18	#14583279	C₂²	1	1	yes	characteristic
19	#14583241	—	2	2	no	—
20	#14583185	—	2	1	no	—
21	#14583140	C₂²	1	2	yes	—
22	#14583140	C₂²	1	2	yes	—
23	#14583105	—	2	1	no	—
24	#14583063	—	2	2	no	—
25	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
26	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
27	#14582937	C₂²	1	2	yes	—
28	#14582727	C₂²	1	1	yes	characteristic
29	#14582725	C₂²	1	2	yes	—
30	#14582704	C₂²	1	2	yes	—
31	#14582704	C₂²	1	2	yes	—
32	#14582704	C₂²	1	2	yes	—
33	#14582704	C₂²	1	1	yes	characteristic
34	#14582701	C₂²	1	2	yes	—
35	#14582701	C₂²	1	1	yes	characteristic
36	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
37	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₄₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×#10621898	—	2	2	no	—
42	#13063347	—	2	4	no	—
43	#13063347	—	4	2	no	—
44	C₂×Dic₁₇₂	—	2	1	no	—
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₇₂	C₂³	1	2	yes	—
47	C₂×#10621873	—	2	2	no	—
48	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
49	#13063325	—	2	1	no	—
50	#13063325	—	2	1	no	—
51	#13063318	—	4	2	no	—
52	#13063312	—	4	1	no	—
53	#13063307	—	4	2	no	—
54	#13063304	—	2	2	no	—
55	#13063304	—	2	2	no	—
56	#13063296	—	4	1	no	—
57	C₄×Dic₈₆	—	2	1	no	—
58	#13063290	—	4	1	no	—
59	C₄×D₁₇₂	—	2	1	no	—
60	#13063284	—	2	2	no	—
61	C₄⋊C₄×D₄₃	—	4	1	no	—
62	Dic₃₄₄	—	4	2	no	—
63	#13063270	—	2	2	no	—
64	#13063259	—	2	1	no	—
65	#13063259	C₂³	1	2	yes	—
66	#13063259	C₂³	1	1	yes	characteristic
67	#13063259	C₂³	1	1	yes	characteristic
68	#13063257	—	2	2	no	—
69	#13063250	—	2	2	no	—
70	#13063250	—	2	2	no	—
71	#13063250	—	2	1	no	—
72	#13063247	—	2	1	no	—
73	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
74	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
75	C₂×C₁₆×C₄₃	—	4	2	no	—
76	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
78	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
80	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
84	C₂×C₂²⋊D₄₃	—	4	2	no	—
85	#10621898	—	4	4	no	—
86	#10621898	—	8	2	no	—
87	C₂×D₁₇₂	—	2	1	no	—
88	C₂×C₄×D₄₃	—	4	1	no	—
89	C₂×C₄×D₄₃	—	4	2	no	—
90	C₂×Dic₈₆	—	2	1	no	—
91	C₂×Dic₈₆	—	2	2	no	—
92	C₂×Dic₈₆	—	4	1	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
99	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
100	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂²⋊Dic₄₃	—	4	2	no	—
102	C₂²⋊Dic₄₃	—	4	2	no	—
103	C₂²⋊Dic₄₃	—	4	2	no	—
104	(C₂×D₄₃)⋊C₄	—	4	1	no	—
105	C₄⋊Dic₄₃	—	2	1	no	—
106	C₄⋊Dic₄₃	—	2	1	no	—
107	C₄⋊Dic₄₃	—	2	2	no	—
108	C₄⋊Dic₄₃	—	4	1	no	—
109	C₄⋊Dic₄₃	—	4	1	no	—
110	C₄⋊Dic₄₃	—	4	1	no	—
111	Dic₄₃⋊C₄	—	4	2	no	—
112	Dic₄₃⋊C₄	—	4	2	no	—
113	Dic₄₃⋊C₄	—	4	1	no	—
114	C₄×Dic₄₃	—	4	2	no	—
115	C₄×Dic₄₃	—	4	1	no	—
116	Dic₁₇₂	—	2	1	no	—
117	Dic₁₇₂	—	2	2	no	—
118	Dic₁₇₂	—	2	1	no	—
119	Dic₁₇₂	—	2	2	no	—
120	Dic₁₇₂	—	8	1	no	—
121	D₃₄₄	—	2	1	no	—
122	D₃₄₄	#41	1	2	yes	—
123	#10621873	—	4	4	no	—
124	C₁₆×C₄₃	—	2	1	no	—
125	C₁₆×C₄₃	—	2	1	no	—
126	C₁₆×C₄₃	—	2	2	no	—
127	C₂³×C₄₃	C₂×D₈	1	2	yes	—
128	C₂²×D₄₃	—	4	1	no	—
129	D₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₄₃	—	8	4	no	—
137	C₂×Dic₄₃	—	4	1	no	—
138	C₂×Dic₄₃	—	4	1	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	—	4	1	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	D₁₇₂	—	2	2	no	—
143	D₁₇₂	—	4	1	no	—
144	C₄×D₄₃	—	8	4	no	—
145	C₄×D₄₃	—	8	2	no	—
146	Dic₈₆	—	4	2	no	—
147	Dic₈₆	—	4	1	no	—
148	Dic₈₆	—	4	2	no	—
149	Dic₈₆	—	4	1	no	—
150	Dic₈₆	—	8	2	no	—
151	C₈×C₄₃	—	2	2	no	—
152	C₈×C₄₃	—	2	1	no	—
153	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
154	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
155	#33869	—	43	1	no	maximal, Sylow
156	C₂²×C₄₃	—	2	2	no	—
157	C₂²×C₄₃	—	2	2	no	—
158	C₂²×C₄₃	—	2	2	no	—
159	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
160	C₂×D₄₃	—	4	2	no	—
161	C₂×D₄₃	—	8	1	no	—
162	C₂×D₄₃	—	8	1	no	—
163	C₄×C₄₃	—	2	1	no	—
164	C₄×C₄₃	—	2	2	no	—
165	C₄×C₄₃	#508	1	2	yes	—
166	C₄×C₄₃	#546	1	1	yes	characteristic
167	C₄×C₄₃	#577	1	1	yes	characteristic
168	Dic₄₃	—	8	2	no	—
169	Dic₄₃	—	8	1	no	—
170	Dic₄₃	—	8	1	no	—
171	Dic₄₃	—	8	2	no	—
172	Dic₄₃	—	8	1	no	—
173	#3342	—	43	1	no	—
174	#3283	—	43	1	no	—
175	#2241	—	43	1	no	—
176	#2237	—	43	2	no	—
177	#2224	—	43	1	no	—
178	#2216	—	43	2	no	—
179	#2209	—	43	1	no	—
180	#2189	—	43	2	no	—
181	C₄×SD₃₂	—	43	1	no	—
182	D₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
183	#2138	—	43	2	no	—
184	C₂×C₄₃	—	4	2	no	—
185	C₂×C₄₃	#3294	1	1	yes	characteristic
186	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
187	C₂×C₄₃	#3417	1	1	yes	characteristic
188	D₄₃	—	8	2	no	—
189	C₂×#135	—	43	2	no	—
190	#556	—	86	1	no	—
191	#547	—	86	1	no	—
192	C₂×Q₃₂	—	43	2	no	—
193	C₂×SD₃₂	—	43	1	no	—
194	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
195	#500	—	43	1	no	—
196	#494	—	43	1	no	—
197	#485	—	86	2	no	—
198	#475	—	86	1	no	—
199	#467	—	43	2	no	—
200	#467	—	43	2	no	—
201	#461	—	86	1	no	—
202	#452	—	86	2	no	—
203	C₄×Q₁₆	—	43	1	no	—
204	C₄×D₈	—	43	1	no	—
205	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
206	#427	—	43	2	no	—
207	#367	—	43	1	no	—
208	#366	—	43	2	no	—
209	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
210	#358	—	43	2	no	—
211	#358	—	43	2	no	—
212	#358	—	43	1	no	—
213	#358	—	43	2	no	—
214	#357	—	43	1	no	—
215	#357	—	43	2	no	—
216	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	—
217	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
218	C₄₃	#33869	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	86	2	no	—
220	#135	—	86	4	no	—
221	#135	—	172	2	no	—
222	C₂×Q₁₆	—	43	2	no	—
223	C₂×Q₁₆	—	43	1	no	—
224	C₂×Q₁₆	—	86	1	no	—
225	C₂×SD₁₆	—	86	2	no	—
226	C₂×D₈	—	43	1	no	—
227	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
228	C₄⋊Q₈	—	86	1	no	—
229	C₄⋊Q₈	—	86	1	no	—
230	#126	—	172	2	no	—
231	#124	—	172	1	no	—
232	#123	—	172	2	no	—
233	C₂²⋊Q₈	—	86	2	no	—
234	C₂²⋊Q₈	—	86	2	no	—
235	#121	—	172	1	no	—
236	C₄×Q₈	—	86	1	no	—
237	C₄×Q₈	—	172	1	no	—
238	C₄×D₄	—	86	1	no	—
239	C₄×D₄	D₁₇₂	1	1	yes	characteristic
240	#117	—	86	2	no	—
241	C₂×C₄⋊C₄	—	172	1	no	—
242	Q₃₂	—	172	2	no	—
243	SD₃₂	—	86	2	no	—
244	C₂×C₁₆	—	4	2	no	—
245	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
246	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
247	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
248	#107	—	43	2	no	—
249	#107	—	43	1	no	—
250	#107	—	43	1	no	—
251	#107	—	86	1	no	—
252	#106	—	86	2	no	—
253	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
254	#103	—	86	1	no	—
255	#103	—	86	2	no	—
256	#103	—	86	2	no	—
257	#102	—	86	1	no	—
258	C₂²⋊C₈	D₁₇₂	1	2	yes	—
259	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
260	#41	—	344	2	no	—
261	#41	—	172	4	no	—
262	C₂×Q₈	—	86	1	no	—
263	C₂×Q₈	—	86	2	no	—
264	C₂×Q₈	—	172	1	no	—
265	C₂×D₄	—	86	1	no	—
266	C₂×D₄	—	172	2	no	—
267	C₂×D₄	D₃₄₄	1	1	yes	characteristic
268	C₂²×C₄	—	172	1	no	—
269	C₂²×C₄	—	172	2	no	—
270	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
271	Q₁₆	—	86	1	no	—
272	Q₁₆	—	86	2	no	—
273	Q₁₆	—	86	1	no	—
274	Q₁₆	—	86	2	no	—
275	Q₁₆	—	344	1	no	—
276	SD₁₆	—	172	4	no	—
277	D₈	—	43	2	no	—
278	D₈	—	86	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
282	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
283	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
284	C₄⋊C₄	—	86	1	no	—
285	C₄⋊C₄	—	86	1	no	—
286	C₄⋊C₄	—	86	2	no	—
287	C₄⋊C₄	—	172	1	no	—
288	C₄⋊C₄	—	172	2	no	—
289	C₄⋊C₄	—	172	1	no	—
290	C₄⋊C₄	—	172	2	no	—
291	C₄⋊C₄	—	172	1	no	—
292	C₄⋊C₄	—	172	1	no	—
293	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
294	C₂²⋊C₄	—	172	2	no	—
295	C₂²⋊C₄	—	172	2	no	—
296	C₂²⋊C₄	—	172	2	no	—
297	C₂²⋊C₄	—	172	1	no	—
298	C₂²⋊C₄	D₃₄₄	1	2	yes	—
299	C₄²	—	172	2	no	—
300	C₄²	—	172	1	no	—
301	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
302	C₁₆	—	2	1	no	—
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	2	no	—
305	C₂³	—	172	1	no	—
306	C₂³	C₂×D₃₄₄	1	2	yes	—
307	Q₈	—	344	2	no	—
308	Q₈	—	172	2	no	—
309	Q₈	—	172	1	no	—
310	Q₈	—	172	2	no	—
311	Q₈	—	172	1	no	—
312	D₄	—	2	2	no	—
313	D₄	—	86	2	no	—
314	D₄	—	344	4	no	—
315	D₄	—	172	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	344	4	no	—
319	C₂×C₄	—	344	2	no	—
320	C₂×C₄	—	172	1	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	—	172	1	no	—
323	C₂×C₄	—	172	1	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
326	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
327	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	2	2	no	—
331	C₈	#41×D₄₃	1	1	yes	characteristic
332	C₈	#13063370	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	344	1	no	—
337	C₂²	—	344	1	no	—
338	C₂²	—	172	2	no	—
339	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	344	2	no	—
343	C₄	—	344	1	no	—
344	C₄	—	344	1	no	—
345	C₄	—	344	2	no	—
346	C₄	—	344	1	no	—
347	C₄	#14583331	1	2	yes	—
348	C₄	#14583544	1	1	yes	characteristic
349	C₄	#14583711	1	1	yes	characteristic
350	C₂	—	4	2	no	—
351	C₂	—	344	2	no	—
352	C₂	#19146409	1	1	yes	characteristic
353	C₂	C₂×#14583331	1	1	yes	characteristic
354	C₂	#19147748	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	D₄×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19146317	C₂
hypercenter	D₄×C₁₆	D₄₃
hypocenter	C₄₃	#33869
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #19147026	C₂	semidirect
3. #19136823	C₂	semidirect
4. #19136804	C₂	semidirect
5. #19136733	C₂	semidirect
6. #19136686	C₂	semidirect
7. #19136646	C₂	semidirect
8. #19136537	C₂	semidirect
9. C₄×#13063270	C₂	semidirect
10. #19136303	C₂	semidirect
11. D₄×C₁₆×C₄₃	C₂	semidirect
182. D₄×C₁₆	D₄₃	semidirect
218. C₄₃	#33869	semidirect

Maximal subgroups

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

subgroup	index
#19147026	2
#19146317	2
#19136823	2
#19136804	2
#19136733	2
#19136686	2
#19136646	2
#19136537	2
C₄×#13063270	2
#19136303	2
D₄×C₁₆×C₄₃	2
#33869	43