Subgroups — #25254401

Order 11008.

520 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147608	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136832	C₂	1	2	yes	maximal, complemented
4	#19136819	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136743	C₂	1	2	yes	maximal, complemented
6	#19136738	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136651	C₂	1	2	yes	maximal, complemented
8	#19136631	C₂	1	1	yes	maximal, characteristic
9	#19136472	C₂	1	2	yes	maximal, complemented
10	C₄×#13063270	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583634	—	2	1	no	—
13	Q₈×D₁₇₂	—	2	1	no	—
14	C₂×#13063270	C₂²	1	1	yes	characteristic
15	C₂×D₆₈₈	C₂²	1	2	yes	—
16	#14583304	C₂²	1	1	yes	characteristic
17	#14583304	C₂²	1	2	yes	—
18	#14583283	C₂²	1	2	yes	—
19	#14583276	C₂²	1	1	yes	characteristic
20	#14583180	—	2	1	no	—
21	#14583180	—	2	2	no	—
22	#14583115	—	2	2	no	—
23	#14583100	—	2	1	no	—
24	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
25	C₄×D₃₄₄	C₂²	1	2	yes	—
26	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
27	C₄×D₃₄₄	C₂²	1	2	yes	—
28	#14582727	C₂²	1	1	yes	characteristic
29	#14582727	C₂²	1	2	yes	—
30	#14582704	C₂²	1	1	yes	characteristic
31	#14582704	C₂²	1	2	yes	—
32	#14582701	C₂²	1	2	yes	—
33	#14582701	C₂²	1	2	yes	—
34	#14582701	C₂²	1	2	yes	—
35	#14582701	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₄₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₄₃	—	4	1	no	—
42	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
43	C₂×#10621873	—	2	2	no	—
44	C₂×D₃₄₄	—	2	1	no	—
45	C₂×D₃₄₄	C₂³	1	2	yes	—
46	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
47	#13063325	—	2	2	no	—
48	#13063325	—	2	1	no	—
49	#13063322	—	2	1	no	—
50	#13063314	—	4	1	no	—
51	#13063313	—	2	2	no	—
52	#13063310	—	4	2	no	—
53	#13063305	—	4	1	no	—
54	#13063305	—	4	2	no	—
55	C₄×Dic₈₆	—	2	1	no	—
56	C₄×D₁₇₂	—	2	2	no	—
57	C₄×D₁₇₂	—	2	1	no	—
58	C₄×D₁₇₂	—	2	2	no	—
59	#13063287	—	4	1	no	—
60	#13063270	—	2	2	no	—
61	D₆₈₈	—	2	4	no	—
62	#13063259	—	2	2	no	—
63	#13063259	—	2	1	no	—
64	#13063259	C₂³	1	2	yes	—
65	#13063259	C₂³	1	1	yes	characteristic
66	#13063259	C₂³	1	1	yes	characteristic
67	#13063250	—	2	1	no	—
68	#13063247	—	2	2	no	—
69	#13063247	—	2	2	no	—
70	#13063247	—	2	1	no	—
71	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
73	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
77	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
80	Q₈×D₄₃	—	8	4	no	—
81	C₂×D₁₇₂	—	2	1	no	—
82	C₂×D₁₇₂	—	2	2	no	—
83	C₂×D₁₇₂	—	4	1	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×C₄×D₄₃	—	4	2	no	—
86	C₂×C₄×D₄₃	—	4	1	no	—
87	C₂×C₄×D₄₃	—	4	2	no	—
88	C₂×Dic₈₆	—	2	1	no	—
89	C₂×Dic₈₆	—	4	2	no	—
90	C₂×Dic₈₆	—	4	1	no	—
91	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
95	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
96	C₄²×C₄₃	C₂×D₄	1	2	yes	—
97	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
98	(C₂×D₄₃)⋊C₄	—	4	2	no	—
99	(C₂×D₄₃)⋊C₄	—	4	1	no	—
100	(C₂×D₄₃)⋊C₄	—	4	2	no	—
101	(C₂×D₄₃)⋊C₄	—	4	2	no	—
102	(C₂×D₄₃)⋊C₄	—	4	2	no	—
103	C₄⋊Dic₄₃	—	2	1	no	—
104	C₄⋊Dic₄₃	—	2	2	no	—
105	C₄⋊Dic₄₃	—	2	1	no	—
106	C₄⋊Dic₄₃	—	4	1	no	—
107	C₄⋊Dic₄₃	—	4	2	no	—
108	Dic₄₃⋊C₄	—	4	1	no	—
109	C₄×Dic₄₃	—	4	1	no	—
110	Dic₁₇₂	—	2	1	no	—
111	Dic₁₇₂	—	2	1	no	—
112	D₃₄₄	—	2	2	no	—
113	D₃₄₄	—	2	1	no	—
114	D₃₄₄	—	2	2	no	—
115	D₃₄₄	—	8	1	no	—
116	D₃₄₄	#41	1	2	yes	—
117	#10621873	—	8	2	no	—
118	C₁₆×C₄₃	—	2	2	no	—
119	C₁₆×C₄₃	—	2	1	no	—
120	C₁₆×C₄₃	—	2	1	no	—
121	C₂²×D₄₃	—	4	1	no	—
122	C₂²×D₄₃	—	4	2	no	—
123	Q₈×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₄₃	—	4	1	no	—
130	C₂×Dic₄₃	—	4	1	no	—
131	C₂×Dic₄₃	—	4	1	no	—
132	C₂×Dic₄₃	—	4	2	no	—
133	D₁₇₂	—	2	2	no	—
134	D₁₇₂	—	4	2	no	—
135	D₁₇₂	—	4	1	no	—
136	D₁₇₂	—	4	2	no	—
137	D₁₇₂	—	8	2	no	—
138	D₁₇₂	—	16	1	no	—
139	C₄×D₄₃	—	8	4	no	—
140	C₄×D₄₃	—	8	2	no	—
141	C₄×D₄₃	—	16	2	no	—
142	Dic₈₆	—	4	1	no	—
143	Dic₈₆	—	4	1	no	—
144	Dic₈₆	—	8	4	no	—
145	Dic₈₆	—	8	2	no	—
146	C₈×C₄₃	—	2	2	no	—
147	C₈×C₄₃	—	2	1	no	—
148	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
150	#33940	—	43	1	no	maximal, Sylow
151	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₄₃	—	4	2	no	—
153	C₂×D₄₃	—	8	1	no	—
154	C₂×D₄₃	—	8	1	no	—
155	C₂×D₄₃	—	8	2	no	—
156	C₂×D₄₃	—	8	2	no	—
157	C₂×D₄₃	—	8	2	no	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	#509	1	2	yes	—
162	C₄×C₄₃	#548	1	1	yes	characteristic
163	C₄×C₄₃	#577	1	1	yes	characteristic
164	Dic₄₃	—	8	1	no	—
165	Dic₄₃	—	8	1	no	—
166	Dic₄₃	—	8	2	no	—
167	Dic₄₃	—	8	1	no	—
168	#3396	—	43	1	no	—
169	Q₈×D₈	—	43	1	no	—
170	#2243	—	43	2	no	—
171	#2240	—	43	1	no	—
172	#2226	—	43	2	no	—
173	#2225	—	43	1	no	—
174	#2210	—	43	2	no	—
175	#2206	—	43	1	no	—
176	#2183	D₄₃	1	1	yes	characteristic, complemented
177	#2177	—	43	2	no	—
178	C₄×SD₃₂	—	43	1	no	—
179	C₂×C₄₃	#3358	1	1	yes	characteristic
180	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
181	C₂×C₄₃	#3416	1	1	yes	characteristic
182	D₄₃	—	8	2	no	—
183	D₄₃	—	16	2	no	—
184	#559	—	86	1	no	—
185	D₄×Q₈	—	86	1	no	—
186	C₂×SD₃₂	—	43	1	no	—
187	C₂×D₁₆	—	43	2	no	—
188	#500	—	43	1	no	—
189	#500	—	43	2	no	—
190	#495	—	43	2	no	—
191	#493	—	43	1	no	—
192	#474	—	86	1	no	—
193	#474	—	86	2	no	—
194	#463	—	86	2	no	—
195	#460	—	86	1	no	—
196	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
197	C₄×Q₁₆	—	43	1	no	—
198	C₄×D₈	—	43	1	no	—
199	C₄×D₈	—	43	2	no	—
200	C₄×D₈	—	43	2	no	—
201	#367	—	43	1	no	—
202	#367	—	43	2	no	—
203	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
204	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
205	#358	—	43	1	no	—
206	#358	—	43	2	no	—
207	#357	—	43	1	no	—
208	#357	—	43	2	no	—
209	#357	—	43	2	no	—
210	#357	—	43	2	no	—
211	#346	C₂×D₄₃	1	2	yes	—
212	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
213	C₄₃	#33940	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	—	172	1	no	—
215	C₂×Q₁₆	—	43	1	no	—
216	C₂×SD₁₆	—	86	2	no	—
217	C₂×D₈	—	43	1	no	—
218	C₂×D₈	—	43	2	no	—
219	C₂×D₈	—	86	1	no	—
220	C₄⋊Q₈	—	86	2	no	—
221	C₄⋊Q₈	—	86	1	no	—
222	#127	—	86	1	no	—
223	#124	—	86	2	no	—
224	#124	—	172	1	no	—
225	#123	—	172	2	no	—
226	C₂²⋊Q₈	—	172	1	no	—
227	C₂²⋊Q₈	—	172	2	no	—
228	C₄×Q₈	—	86	1	no	—
229	C₄×Q₈	D₁₇₂	1	1	yes	characteristic
230	C₄×D₄	—	86	2	no	—
231	C₄×D₄	—	86	1	no	—
232	C₄×D₄	—	86	2	no	—
233	C₄×D₄	—	172	1	no	—
234	SD₃₂	—	86	2	no	—
235	D₁₆	—	86	4	no	—
236	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
239	#107	—	43	2	no	—
240	#107	—	43	1	no	—
241	#107	—	43	1	no	—
242	#107	—	86	2	no	—
243	#107	—	86	1	no	—
244	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
245	C₄⋊C₈	D₁₇₂	1	2	yes	—
246	#103	—	86	1	no	—
247	#102	—	86	2	no	—
248	#102	—	86	2	no	—
249	#102	—	86	1	no	—
250	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₄₃	1	2	yes	—
252	C₂×Q₈	—	86	1	no	—
253	C₂×Q₈	—	344	4	no	—
254	C₂×Q₈	—	172	2	no	—
255	C₂×Q₈	—	172	1	no	—
256	C₂×Q₈	D₃₄₄	1	1	yes	characteristic
257	C₂×D₄	—	86	1	no	—
258	C₂×D₄	—	86	2	no	—
259	C₂×D₄	—	172	1	no	—
260	C₂×D₄	—	172	1	no	—
261	C₂²×C₄	—	172	2	no	—
262	C₂²×C₄	—	172	1	no	—
263	C₂²×C₄	—	172	2	no	—
264	Q₁₆	—	86	1	no	—
265	Q₁₆	—	86	1	no	—
266	SD₁₆	—	344	2	no	—
267	D₈	—	43	2	no	—
268	D₈	—	86	2	no	—
269	D₈	—	86	1	no	—
270	D₈	—	86	2	no	—
271	D₈	—	344	1	no	—
272	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
273	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
275	C₄⋊C₄	—	86	1	no	—
276	C₄⋊C₄	—	86	2	no	—
277	C₄⋊C₄	—	86	1	no	—
278	C₄⋊C₄	—	172	1	no	—
279	C₄⋊C₄	—	172	2	no	—
280	C₄⋊C₄	—	172	1	no	—
281	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
282	C₄⋊C₄	D₃₄₄	1	2	yes	—
283	C₂²⋊C₄	—	172	2	no	—
284	C₂²⋊C₄	—	172	1	no	—
285	C₂²⋊C₄	—	172	2	no	—
286	C₂²⋊C₄	—	172	2	no	—
287	C₂²⋊C₄	—	172	2	no	—
288	C₄²	—	172	1	no	—
289	C₄²	C₂×D₁₇₂	1	2	yes	—
290	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
291	C₁₆	—	2	2	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₂³	—	172	2	no	—
295	C₂³	—	172	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	344	4	no	—
298	Q₈	—	344	2	no	—
299	Q₈	—	172	1	no	—
300	Q₈	—	172	1	no	—
301	D₄	—	86	2	no	—
302	D₄	—	344	2	no	—
303	D₄	—	172	2	no	—
304	D₄	—	172	2	no	—
305	D₄	—	172	1	no	—
306	D₄	—	688	1	no	—
307	C₂×C₄	—	344	4	no	—
308	C₂×C₄	—	344	2	no	—
309	C₂×C₄	—	172	1	no	—
310	C₂×C₄	—	172	1	no	—
311	C₂×C₄	—	172	2	no	—
312	C₂×C₄	—	172	1	no	—
313	C₂×C₄	—	688	2	no	—
314	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
315	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
316	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
317	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	2	no	—
321	C₈	#41×D₄₃	1	1	yes	characteristic
322	C₈	#13063373	1	1	yes	characteristic
323	C₂²	—	344	1	no	—
324	C₂²	—	344	1	no	—
325	C₂²	—	344	2	no	—
326	C₂²	—	344	2	no	—
327	C₂²	—	344	2	no	—
328	C₂²	—	172	2	no	—
329	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	344	2	no	—
334	C₄	—	344	1	no	—
335	C₄	—	344	1	no	—
336	C₄	—	344	1	no	—
337	C₄	#14583337	1	2	yes	—
338	C₄	#14583561	1	1	yes	characteristic
339	C₄	#14583711	1	1	yes	characteristic
340	C₂	—	344	2	no	—
341	C₂	—	688	2	no	—
342	C₂	#19147231	1	1	yes	characteristic
343	C₂	C₂×#14583337	1	1	yes	characteristic
344	C₂	#19147743	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2183	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19136631	C₂
hypercenter	#2183	D₄₃
hypocenter	C₄₃	#33940
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
1. #19147608	C₂	semidirect
2. Q₈×D₃₄₄	C₂	semidirect
3. #19136832	C₂	semidirect
4. #19136819	C₂	semidirect
5. #19136743	C₂	semidirect
6. #19136738	C₂	semidirect
7. #19136651	C₂	semidirect
9. #19136472	C₂	semidirect
10. C₄×#13063270	C₂	semidirect
11. C₄₃×#2183	C₂	semidirect
176. #2183	D₄₃	semidirect
213. C₄₃	#33940	semidirect

Maximal subgroups

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

subgroup	index
#19147608	2
Q₈×D₃₄₄	2
#19136832	2
#19136819	2
#19136743	2
#19136738	2
#19136651	2
#19136631	2
#19136472	2
C₄×#13063270	2
C₄₃×#2183	2
#33940	43