Subgroups — #24996512

Order 11008.

492 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	#19145154	C₂	1	1	yes	maximal, characteristic
2	#19135572	C₂	1	1	yes	maximal, characteristic, complemented
3	#19135423	C₂	1	1	yes	maximal, characteristic
4	#19133354	C₂	1	1	yes	maximal, characteristic, complemented
5	#19133051	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133047	C₂	1	1	yes	maximal, characteristic
7	C₄×#97×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583456	C₂²	1	1	yes	characteristic
9	#14583449	C₂²	1	1	yes	characteristic
10	#14583297	C₄	1	1	yes	characteristic, complemented
11	#14583272	C₄	1	1	yes	characteristic, complemented
12	#14583250	C₄	1	1	yes	characteristic, complemented
13	#14583246	C₄	1	1	yes	characteristic, complemented
14	#14583128	—	2	2	no	—
15	C₄×#10621873	—	2	2	no	—
16	#14582923	C₂²	1	1	yes	characteristic
17	#14582923	C₂²	1	1	yes	characteristic
18	C₂×#13063253	—	2	2	no	—
19	#14582659	—	2	2	no	—
20	#14582652	—	2	2	no	—
21	#14582631	—	2	2	no	—
22	#14582628	—	2	2	no	—
23	C₂×#97×C₄₃	—	2	2	no	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₄³×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621898	—	2	1	no	—
28	C₂×#10621873	—	2	2	no	—
29	C₂×#10621877	—	2	1	no	—
30	C₂×#10621877	—	2	1	no	—
31	#13063325	C₂×C₄	1	1	yes	characteristic
32	#13063325	C₂×C₄	1	1	yes	characteristic
33	#13063325	C₂×C₄	1	1	yes	characteristic
34	#13063322	C₂×C₄	1	1	yes	characteristic
35	#13063317	—	2	1	no	—
36	#13063313	—	2	1	no	—
37	#13063304	—	2	1	no	—
38	#13063304	—	2	1	no	—
39	#13063299	—	2	1	no	—
40	#13063299	—	2	1	no	—
41	C₄×Dic₈₆	—	2	1	no	—
42	C₄×Dic₈₆	—	2	1	no	—
43	C₄×D₁₇₂	—	2	1	no	—
44	C₄×D₁₇₂	—	2	1	no	—
45	#13063284	—	2	1	no	—
46	#13063257	—	2	2	no	—
47	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
50	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
51	#13063253	—	4	4	no	—
52	#13063250	—	2	2	no	—
53	#13063247	—	2	2	no	—
54	C₂²×C₈×C₄₃	—	2	1	no	—
55	C₂²×C₈×C₄₃	D₄	1	2	yes	—
56	C₂×C₄²×C₄₃	—	2	1	no	—
57	C₂×C₄²×C₄₃	—	2	1	no	—
58	C₂×C₄²×C₄₃	D₄	1	2	yes	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	#97×C₄₃	—	2	8	no	—
61	C₄×C₈×C₄₃	—	2	1	no	—
62	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₃	D₄	1	2	yes	—
65	C₄×C₈×C₄₃	D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₃	—	4	1	no	—
71	#10621898	—	4	2	no	—
72	#10621898	—	4	2	no	—
73	C₂×D₁₇₂	—	2	1	no	—
74	C₂×D₁₇₂	—	2	1	no	—
75	C₂×D₁₇₂	—	2	1	no	—
76	C₂×C₄×D₄₃	—	4	1	no	—
77	C₂×Dic₈₆	—	2	1	no	—
78	C₂×Dic₈₆	—	2	1	no	—
79	C₂×Dic₈₆	—	2	1	no	—
80	C₂×C₈×C₄₃	—	2	4	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	4	no	—
83	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₄₃	—	2	4	no	—
86	C₄²×C₄₃	—	2	2	no	—
87	C₄²×C₄₃	—	2	2	no	—
88	C₄²×C₄₃	—	4	2	no	—
89	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₄₃	#41	1	1	yes	characteristic
93	C₂²⋊Dic₄₃	—	4	1	no	—
94	(C₂×D₄₃)⋊C₄	—	4	1	no	—
95	(C₂×D₄₃)⋊C₄	—	4	1	no	—
96	C₄⋊Dic₄₃	—	2	1	no	—
97	C₄⋊Dic₄₃	—	2	1	no	—
98	C₄⋊Dic₄₃	—	2	1	no	—
99	C₄⋊Dic₄₃	—	2	1	no	—
100	C₄⋊Dic₄₃	—	2	1	no	—
101	C₄⋊Dic₄₃	—	2	1	no	—
102	Dic₄₃⋊C₄	—	4	1	no	—
103	Dic₄₃⋊C₄	—	4	1	no	—
104	C₄×Dic₄₃	—	4	1	no	—
105	#10621877	—	4	2	no	—
106	#10621877	—	4	2	no	—
107	C₂×C₄₃⋊C₈	—	4	1	no	—
108	C₂×C₄₃⋊C₈	—	4	1	no	—
109	#10621873	—	4	4	no	—
110	C₂³×C₄₃	#121	1	1	yes	characteristic
111	C₂²×D₄₃	—	4	1	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	4	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₄₃	#137	1	2	yes	—
125	C₂²⋊D₄₃	—	8	2	no	—
126	C₂×Dic₄₃	—	4	1	no	—
127	C₂×Dic₄₃	—	4	1	no	—
128	C₂×Dic₄₃	—	4	1	no	—
129	D₁₇₂	—	4	2	no	—
130	D₁₇₂	—	4	2	no	—
131	D₁₇₂	—	4	2	no	—
132	C₄×D₄₃	—	8	1	no	—
133	C₄×D₄₃	—	8	1	no	—
134	Dic₈₆	—	4	2	no	—
135	Dic₈₆	—	4	2	no	—
136	Dic₈₆	—	4	2	no	—
137	C₈×C₄₃	—	2	4	no	—
138	C₈×C₄₃	—	2	2	no	—
139	C₈×C₄₃	#104	1	4	yes	complemented
140	C₄₃⋊C₈	—	8	1	no	—
141	C₄₃⋊C₈	—	8	1	no	—
142	#9851	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	#390	1	1	yes	characteristic
146	C₂²×C₄₃	#468	1	2	yes	—
147	C₂×D₄₃	—	8	1	no	—
148	C₂×D₄₃	—	8	1	no	—
149	C₂×D₄₃	—	8	1	no	—
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	4	4	no	—
153	C₄×C₄₃	C₂×#104	1	2	yes	—
154	C₄×C₄₃	#440	1	2	yes	—
155	C₄×C₄₃	#444	1	2	yes	—
156	C₄×C₄₃	#496	1	1	yes	characteristic
157	C₄×C₄₃	#497	1	1	yes	characteristic
158	Dic₄₃	—	8	1	no	—
159	Dic₄₃	—	8	1	no	—
160	Dic₄₃	—	8	1	no	—
161	#3167	—	43	1	no	—
162	#1994	—	43	1	no	—
163	#1962	—	43	1	no	—
164	C₄×#97	D₄₃	1	1	yes	characteristic, complemented
165	#1567	—	43	1	no	—
166	#1493	—	43	1	no	—
167	#1492	—	43	1	no	—
168	C₂×C₄₃	—	2	2	no	—
169	C₂×C₄₃	#1962	1	1	yes	characteristic
170	C₂×C₄₃	#1973	1	1	yes	characteristic
171	C₂×C₄₃	#1975	1	1	yes	characteristic
172	D₄₃	—	16	1	no	—
173	#533	—	43	1	no	—
174	#532	—	43	1	no	—
175	#498	—	43	1	no	—
176	#492	—	43	1	no	—
177	#487	—	43	1	no	—
178	#486	—	43	1	no	—
179	#465	—	86	2	no	—
180	C₄×SD₁₆	—	86	2	no	—
181	#423	—	43	1	no	—
182	#423	—	43	1	no	—
183	C₂×#104	—	86	2	no	—
184	C₂×#97	—	2	2	no	—
185	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
187	C₄³	C₂×D₄₃	1	1	yes	characteristic
188	#339	—	86	2	no	—
189	#335	—	86	2	no	—
190	Q₈⋊C₈	—	86	2	no	—
191	D₄⋊C₈	—	86	2	no	—
192	C₄₃	#9851	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	86	1	no	—
194	C₂×SD₁₆	—	86	2	no	—
195	C₂×M₁₆	—	86	1	no	—
196	C₂×M₁₆	—	86	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
199	C₄⋊Q₈	—	43	1	no	—
200	C₄⋊Q₈	—	43	1	no	—
201	C₄⋊Q₈	—	43	1	no	—
202	#127	—	43	1	no	—
203	#125	—	86	1	no	—
204	#124	—	86	1	no	—
205	C₂²⋊Q₈	—	86	1	no	—
206	C₂²⋊Q₈	—	86	1	no	—
207	#121	—	86	1	no	—
208	#121	—	86	1	no	—
209	C₄×Q₈	—	86	1	no	—
210	C₄×Q₈	—	86	1	no	—
211	C₄×D₄	—	86	1	no	—
212	C₄×D₄	—	86	1	no	—
213	#117	—	86	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	D₁₇₂	1	2	yes	—
217	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
218	#106	—	86	2	no	—
219	C₄⋊C₈	—	86	1	no	—
220	C₄⋊C₈	—	86	1	no	—
221	C₄⋊C₈	—	86	1	no	—
222	C₄⋊C₈	—	86	1	no	—
223	#104	—	172	4	no	—
224	#103	—	86	2	no	—
225	#102	—	86	2	no	—
226	#97	—	2	8	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
229	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
230	C₄×C₈	D₁₇₂	1	2	yes	—
231	C₄×C₈	D₁₇₂	1	2	yes	—
232	#41	—	172	2	no	—
233	#41	—	172	2	no	—
234	C₂×Q₈	—	86	1	no	—
235	C₂×Q₈	—	86	1	no	—
236	C₂×Q₈	—	86	1	no	—
237	C₂×D₄	—	86	1	no	—
238	C₂×D₄	—	86	1	no	—
239	C₂×D₄	—	86	1	no	—
240	C₂×D₄	—	172	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	172	1	no	—
243	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
244	C₂²×C₄	#10621898	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
246	SD₁₆	—	172	4	no	—
247	M₁₆	—	172	2	no	—
248	M₁₆	—	172	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	172	1	no	—
253	C₂×C₈	—	172	1	no	—
254	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
255	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
256	C₄⋊C₄	—	86	1	no	—
257	C₄⋊C₄	—	86	1	no	—
258	C₄⋊C₄	—	86	1	no	—
259	C₄⋊C₄	—	86	1	no	—
260	C₄⋊C₄	—	86	1	no	—
261	C₄⋊C₄	—	86	1	no	—
262	C₄⋊C₄	—	172	1	no	—
263	C₄⋊C₄	—	172	1	no	—
264	C₂²⋊C₄	—	172	1	no	—
265	C₂²⋊C₄	—	172	1	no	—
266	C₂²⋊C₄	—	172	1	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	172	1	no	—
272	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
273	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
274	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
275	C₄²	#10621898	1	1	yes	characteristic
276	C₂³	—	172	1	no	—
277	C₂³	#13063299	1	1	yes	characteristic
278	Q₈	—	172	2	no	—
279	Q₈	—	172	2	no	—
280	Q₈	—	172	2	no	—
281	D₄	—	344	2	no	—
282	D₄	—	172	2	no	—
283	D₄	—	172	2	no	—
284	D₄	—	172	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	344	1	no	—
291	C₂×C₄	—	344	1	no	—
292	C₂×C₄	—	172	1	no	—
293	C₂×C₄	—	172	1	no	—
294	C₂×C₄	—	172	1	no	—
295	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
297	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
298	C₂×C₄	#13063299	1	1	yes	characteristic
299	C₂×C₄	#13063313	1	1	yes	characteristic
300	C₂×C₄	#13063322	1	1	yes	characteristic
301	C₂×C₄	#13063353	1	2	yes	—
302	C₂×C₄	#13063358	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	2	2	no	—
305	C₈	—	344	1	no	—
306	C₈	—	344	1	no	—
307	C₈	#13063253	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	344	1	no	—
311	C₂²	—	344	1	no	—
312	C₂²	—	344	1	no	—
313	C₂²	#14582801	1	1	yes	characteristic
314	C₂²	#14583145	1	2	yes	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	4	no	—
318	C₄	—	344	1	no	—
319	C₄	—	344	1	no	—
320	C₄	—	344	1	no	—
321	C₄	C₂×#13063253	1	2	yes	—
322	C₄	#14583001	1	2	yes	—
323	C₄	#14583022	1	2	yes	—
324	C₄	#14583288	1	1	yes	characteristic
325	C₄	#14583293	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	688	1	no	—
328	C₂	#19135423	1	1	yes	characteristic
329	C₂	#19135476	1	1	yes	characteristic
330	C₂	#19135488	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063322
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#97×C₄₃	C₂
socle	C₂²×C₄₃	#390
prime radical	#14583449	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#97	D₄₃
hypocenter	C₄₃	#9851
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. #19135572	C₂	semidirect
4. #19133354	C₂	semidirect
5. #19133051	C₂	semidirect
7. C₄×#97×C₄₃	C₂	semidirect
10. #14583297	C₄	semidirect
11. #14583272	C₄	semidirect
12. #14583250	C₄	semidirect
13. #14583246	C₄	semidirect
139. C₈×C₄₃	#104	semidirect
164. C₄×#97	D₄₃	semidirect
192. C₄₃	#9851	semidirect
307. C₈	#13063253	semidirect

Maximal subgroups

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

subgroup	index
#19145154	2
#19135572	2
#19135423	2
#19133354	2
#19133051	2
#19133047	2
C₄×#97×C₄₃	2
#9851	43