Subgroups — #25254662

Order 11008.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 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
2	#19147601	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136866	C₂	1	2	yes	maximal, complemented
4	#19136851	C₂	1	1	yes	maximal, characteristic
5	#19136658	C₂	1	2	yes	maximal, complemented
6	#19136646	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136641	C₂	1	2	yes	maximal, complemented
8	#19136636	C₂	1	1	yes	maximal, characteristic
9	#19136461	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583634	—	2	1	no	—
13	#14583628	—	2	1	no	—
14	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₄₄	C₂²	1	2	yes	—
16	C₂×#13063270	C₂²	1	2	yes	—
17	C₂×#13063270	C₂²	1	2	yes	—
18	C₂×#13063270	C₂²	1	2	yes	—
19	#14583283	C₂²	1	2	yes	—
20	#14583279	C₂²	1	1	yes	characteristic
21	#14583279	C₂²	1	2	yes	—
22	#14583276	C₂²	1	1	yes	characteristic
23	#14583115	—	2	2	no	—
24	#14583110	—	2	1	no	—
25	#14583110	—	2	2	no	—
26	#14583100	—	2	1	no	—
27	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₇₂	C₂²	1	2	yes	—
29	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
30	C₄×D₃₄₄	C₂²	1	2	yes	—
31	#14582725	C₂²	1	1	yes	characteristic
32	#14582704	C₂²	1	1	yes	characteristic
33	#14582704	C₂²	1	2	yes	—
34	#14582704	C₂²	1	1	yes	characteristic
35	#14582701	C₂²	1	2	yes	—
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×Dic₁₇₂	—	2	2	no	—
42	C₂×Dic₁₇₂	—	2	1	no	—
43	C₂×Dic₁₇₂	C₂³	1	2	yes	—
44	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×#10621873	—	2	2	no	—
47	C₂×D₃₄₄	—	2	1	no	—
48	C₂×D₃₄₄	C₂³	1	2	yes	—
49	#13063325	—	2	2	no	—
50	#13063325	—	2	1	no	—
51	#13063322	—	2	1	no	—
52	#13063316	—	4	1	no	—
53	#13063316	—	4	2	no	—
54	#13063314	—	4	1	no	—
55	#13063313	—	2	2	no	—
56	#13063310	—	4	2	no	—
57	Q₈×Dic₄₃	—	4	1	no	—
58	C₄×Dic₈₆	—	2	1	no	—
59	C₄×Dic₈₆	—	2	2	no	—
60	C₄×Dic₈₆	—	2	1	no	—
61	C₄×D₁₇₂	—	2	2	no	—
62	#13063287	—	4	1	no	—
63	Dic₃₄₄	—	2	2	no	—
64	Dic₃₄₄	—	4	2	no	—
65	#13063270	—	2	4	no	—
66	#13063270	—	4	2	no	—
67	#13063270	—	4	2	no	—
68	#13063259	C₂³	1	1	yes	characteristic
69	#13063259	C₂³	1	1	yes	characteristic
70	#13063250	—	2	2	no	—
71	#13063250	—	2	1	no	—
72	#13063250	—	2	1	no	—
73	#13063247	—	2	2	no	—
74	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
75	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
77	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
79	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
80	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₄₃	C₂³	1	2	yes	—
82	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
83	C₂×D₁₇₂	—	2	2	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×D₁₇₂	—	4	1	no	—
86	C₂×C₄×D₄₃	—	4	2	no	—
87	C₂×Dic₈₆	—	2	1	no	—
88	C₂×Dic₈₆	—	2	2	no	—
89	C₂×Dic₈₆	—	2	1	no	—
90	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
94	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
95	C₄²×C₄₃	C₂×D₄	1	2	yes	—
96	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
97	(C₂×D₄₃)⋊C₄	—	4	2	no	—
98	(C₂×D₄₃)⋊C₄	—	4	2	no	—
99	(C₂×D₄₃)⋊C₄	—	4	2	no	—
100	C₄⋊Dic₄₃	—	2	1	no	—
101	C₄⋊Dic₄₃	—	2	1	no	—
102	C₄⋊Dic₄₃	—	4	2	no	—
103	C₄⋊Dic₄₃	—	4	1	no	—
104	C₄⋊Dic₄₃	—	4	1	no	—
105	C₄⋊Dic₄₃	—	4	2	no	—
106	Dic₄₃⋊C₄	—	4	1	no	—
107	Dic₄₃⋊C₄	—	4	1	no	—
108	Dic₄₃⋊C₄	—	4	2	no	—
109	C₄×Dic₄₃	—	4	1	no	—
110	C₄×Dic₄₃	—	4	2	no	—
111	C₄×Dic₄₃	—	4	1	no	—
112	Dic₁₇₂	—	2	2	no	—
113	Dic₁₇₂	—	2	1	no	—
114	Dic₁₇₂	—	2	1	no	—
115	Dic₁₇₂	—	2	1	no	—
116	Dic₁₇₂	—	2	2	no	—
117	Dic₁₇₂	—	2	1	no	—
118	Dic₁₇₂	—	8	2	no	—
119	Dic₁₇₂	—	8	1	no	—
120	D₃₄₄	—	2	2	no	—
121	D₃₄₄	—	2	2	no	—
122	D₃₄₄	—	8	1	no	—
123	#10621873	—	8	2	no	—
124	C₁₆×C₄₃	—	2	1	no	—
125	C₁₆×C₄₃	—	2	2	no	—
126	C₁₆×C₄₃	#41	1	2	yes	—
127	C₂²×D₄₃	—	4	2	no	—
128	Q₈×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₄₃	—	4	1	no	—
135	C₂×Dic₄₃	—	4	1	no	—
136	C₂×Dic₄₃	—	4	2	no	—
137	C₂×Dic₄₃	—	4	1	no	—
138	C₂×Dic₄₃	—	4	1	no	—
139	D₁₇₂	—	4	2	no	—
140	D₁₇₂	—	4	2	no	—
141	D₁₇₂	—	8	2	no	—
142	D₁₇₂	—	16	1	no	—
143	C₄×D₄₃	—	16	2	no	—
144	Dic₈₆	—	4	2	no	—
145	Dic₈₆	—	4	1	no	—
146	Dic₈₆	—	4	1	no	—
147	Dic₈₆	—	4	1	no	—
148	Dic₈₆	—	4	1	no	—
149	Dic₈₆	—	4	2	no	—
150	C₈×C₄₃	—	2	2	no	—
151	C₈×C₄₃	—	2	1	no	—
152	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
154	#33967	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₄₃	—	8	2	no	—
157	C₂×D₄₃	—	8	2	no	—
158	C₂×D₄₃	—	8	2	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	2	2	no	—
161	C₄×C₄₃	—	2	1	no	—
162	C₄×C₄₃	#510	1	2	yes	—
163	C₄×C₄₃	#550	1	1	yes	characteristic
164	C₄×C₄₃	#576	1	1	yes	characteristic
165	Dic₄₃	—	8	1	no	—
166	Dic₄₃	—	8	2	no	—
167	Dic₄₃	—	8	1	no	—
168	Dic₄₃	—	8	1	no	—
169	Dic₄₃	—	8	1	no	—
170	#3396	—	43	1	no	—
171	#3395	—	43	1	no	—
172	#2251	—	43	2	no	—
173	#2248	—	43	1	no	—
174	#2211	—	43	2	no	—
175	#2209	—	43	1	no	—
176	#2208	—	43	2	no	—
177	#2207	—	43	1	no	—
178	#2183	D₄₃	1	1	yes	characteristic, complemented
179	#2175	—	43	2	no	—
180	C₄×Q₃₂	—	43	1	no	—
181	C₂×C₄₃	#3391	1	1	yes	characteristic
182	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
183	C₂×C₄₃	#3416	1	1	yes	characteristic
184	D₄₃	—	16	2	no	—
185	#559	—	86	1	no	—
186	#557	—	86	1	no	—
187	C₂×Q₃₂	—	43	1	no	—
188	C₂×Q₃₂	—	43	2	no	—
189	C₂×SD₃₂	—	43	2	no	—
190	C₂×SD₃₂	—	43	2	no	—
191	C₂×SD₃₂	—	43	2	no	—
192	#495	—	43	2	no	—
193	#494	—	43	2	no	—
194	#494	—	43	1	no	—
195	#493	—	43	1	no	—
196	#463	—	86	2	no	—
197	#462	—	86	1	no	—
198	#462	—	86	2	no	—
199	#460	—	86	1	no	—
200	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
201	C₄×Q₁₆	—	43	1	no	—
202	C₄×Q₁₆	—	43	2	no	—
203	C₄×Q₁₆	—	43	1	no	—
204	C₄×D₈	—	43	2	no	—
205	#366	—	43	1	no	—
206	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
207	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
208	#358	—	43	1	no	—
209	#358	—	43	1	no	—
210	#358	—	43	2	no	—
211	#357	—	43	2	no	—
212	#346	C₂×D₄₃	1	2	yes	—
213	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
214	C₄₃	#33967	1	1	yes	characteristic, complemented, Sylow
215	C₂×Q₁₆	—	43	2	no	—
216	C₂×Q₁₆	—	43	1	no	—
217	C₂×Q₁₆	—	43	1	no	—
218	C₂×Q₁₆	—	86	2	no	—
219	C₂×Q₁₆	—	86	1	no	—
220	C₂×SD₁₆	—	86	2	no	—
221	C₂×D₈	—	43	2	no	—
222	C₂×D₈	—	86	1	no	—
223	C₄⋊Q₈	—	86	2	no	—
224	C₄⋊Q₈	—	86	1	no	—
225	#127	—	86	1	no	—
226	#125	—	172	1	no	—
227	#125	—	172	2	no	—
228	#124	—	86	2	no	—
229	#124	—	172	1	no	—
230	#123	—	172	2	no	—
231	C₄×Q₈	—	86	2	no	—
232	C₄×Q₈	—	86	1	no	—
233	C₄×Q₈	—	86	1	no	—
234	C₄×Q₈	—	172	1	no	—
235	C₄×Q₈	D₁₇₂	1	1	yes	characteristic
236	C₄×D₄	—	86	2	no	—
237	C₄×D₄	—	172	1	no	—
238	Q₃₂	—	86	2	no	—
239	Q₃₂	—	172	2	no	—
240	SD₃₂	—	86	4	no	—
241	SD₃₂	—	172	2	no	—
242	SD₃₂	—	172	2	no	—
243	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
244	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
245	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
246	#107	—	43	1	no	—
247	#107	—	43	1	no	—
248	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
249	C₄⋊C₈	D₁₇₂	1	2	yes	—
250	#103	—	86	2	no	—
251	#103	—	86	1	no	—
252	#103	—	86	1	no	—
253	#102	—	86	2	no	—
254	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₄₃	1	2	yes	—
256	C₂×Q₈	—	86	1	no	—
257	C₂×Q₈	—	86	2	no	—
258	C₂×Q₈	—	86	1	no	—
259	C₂×Q₈	D₃₄₄	1	1	yes	characteristic
260	C₂×D₄	—	86	2	no	—
261	C₂×D₄	—	172	1	no	—
262	C₂×D₄	—	172	1	no	—
263	C₂²×C₄	—	172	2	no	—
264	Q₁₆	—	86	2	no	—
265	Q₁₆	—	86	1	no	—
266	Q₁₆	—	86	1	no	—
267	Q₁₆	—	86	1	no	—
268	Q₁₆	—	86	2	no	—
269	Q₁₆	—	86	1	no	—
270	Q₁₆	—	344	1	no	—
271	Q₁₆	—	344	2	no	—
272	SD₁₆	—	344	2	no	—
273	D₈	—	86	2	no	—
274	D₈	—	86	2	no	—
275	D₈	—	344	1	no	—
276	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
277	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
279	C₄⋊C₄	—	86	1	no	—
280	C₄⋊C₄	—	86	1	no	—
281	C₄⋊C₄	—	172	1	no	—
282	C₄⋊C₄	—	172	2	no	—
283	C₄⋊C₄	—	172	1	no	—
284	C₄⋊C₄	—	172	2	no	—
285	C₄⋊C₄	—	172	1	no	—
286	C₄⋊C₄	—	172	2	no	—
287	C₄⋊C₄	—	172	1	no	—
288	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
289	C₄⋊C₄	D₃₄₄	1	2	yes	—
290	C₂²⋊C₄	—	172	2	no	—
291	C₂²⋊C₄	—	172	2	no	—
292	C₂²⋊C₄	—	172	2	no	—
293	C₄²	—	172	1	no	—
294	C₄²	—	172	2	no	—
295	C₄²	—	172	1	no	—
296	C₄²	C₂×D₁₇₂	1	2	yes	—
297	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	#10621902	1	2	yes	—
301	C₂³	—	172	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	172	2	no	—
304	Q₈	—	172	1	no	—
305	Q₈	—	172	1	no	—
306	Q₈	—	172	1	no	—
307	Q₈	—	172	2	no	—
308	Q₈	—	172	1	no	—
309	D₄	—	344	2	no	—
310	D₄	—	172	2	no	—
311	D₄	—	172	2	no	—
312	D₄	—	688	1	no	—
313	C₂×C₄	—	172	1	no	—
314	C₂×C₄	—	172	1	no	—
315	C₂×C₄	—	172	1	no	—
316	C₂×C₄	—	172	2	no	—
317	C₂×C₄	—	172	1	no	—
318	C₂×C₄	—	688	2	no	—
319	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
320	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
321	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
322	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₈	C₂×#10621902	1	1	yes	characteristic
327	C₈	#13063371	1	1	yes	characteristic
328	C₂²	—	344	2	no	—
329	C₂²	—	344	2	no	—
330	C₂²	—	344	2	no	—
331	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	344	1	no	—
336	C₄	—	344	2	no	—
337	C₄	—	344	1	no	—
338	C₄	—	344	1	no	—
339	C₄	—	344	1	no	—
340	C₄	#14583342	1	2	yes	—
341	C₄	#14583571	1	1	yes	characteristic
342	C₄	#14583704	1	1	yes	characteristic
343	C₂	—	688	2	no	—
344	C₂	#19147570	1	1	yes	characteristic
345	C₂	C₂×#14583342	1	1	yes	characteristic
346	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	#14583276	C₂²
hypercenter	#2183	D₄₃
hypocenter	C₄₃	#33967
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
2. #19147601	C₂	semidirect
3. #19136866	C₂	semidirect
5. #19136658	C₂	semidirect
6. #19136646	C₂	semidirect
7. #19136641	C₂	semidirect
9. #19136461	C₂	semidirect
10. C₄×Dic₃₄₄	C₂	semidirect
11. C₄₃×#2183	C₂	semidirect
178. #2183	D₄₃	semidirect
214. C₄₃	#33967	semidirect

Maximal subgroups

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

subgroup	index
#19147608	2
#19147601	2
#19136866	2
#19136851	2
#19136658	2
#19136646	2
#19136641	2
#19136636	2
#19136461	2
C₄×Dic₃₄₄	2
C₄₃×#2183	2
#33967	43