Subgroups — #25253473

Order 11008.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19146943	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136823	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136793	C₂	1	2	yes	maximal, complemented
5	#19136738	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136691	C₂	1	2	yes	maximal, complemented
7	#19136641	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136514	C₂	1	2	yes	maximal, complemented
9	C₄×#13063270	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14582704	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
13	D₄×Dic₈₆	—	2	1	no	—
14	#14583553	—	2	1	no	—
15	C₂×#13063270	C₂²	1	1	yes	characteristic
16	C₂×#13063270	C₂²	1	2	yes	—
17	#14583304	C₂²	1	1	yes	characteristic
18	#14583279	C₂²	1	1	yes	characteristic
19	#14583213	—	2	2	no	—
20	#14583180	—	2	1	no	—
21	#14583140	C₂²	1	2	yes	—
22	#14583135	C₂²	1	2	yes	—
23	#14583110	—	2	1	no	—
24	#14583057	—	2	2	no	—
25	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
26	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
27	C₂×#13063259	C₂²	1	2	yes	—
28	#14582727	C₂²	1	1	yes	characteristic
29	#14582727	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	#14582704	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×#10621900	—	4	1	no	—
42	C₂²×Dic₈₆	—	2	2	no	—
43	C₂×Dic₁₇₂	—	2	4	no	—
44	C₂×Dic₁₇₂	—	2	2	no	—
45	C₂×Dic₁₇₂	—	2	1	no	—
46	C₂×Dic₁₇₂	—	4	2	no	—
47	C₂×Dic₁₇₂	C₂³	1	2	yes	—
48	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
49	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
50	#13063325	—	2	1	no	—
51	#13063325	—	2	1	no	—
52	#13063324	—	4	1	no	—
53	#13063309	—	4	2	no	—
54	#13063305	—	4	1	no	—
55	#13063304	—	2	2	no	—
56	#13063302	—	4	2	no	—
57	#13063299	—	2	2	no	—
58	C₄×Dic₈₆	—	2	1	no	—
59	C₄×D₁₇₂	—	2	1	no	—
60	D₄×Dic₄₃	—	4	1	no	—
61	C₂×C₄⋊Dic₄₃	—	2	2	no	—
62	#13063270	—	2	2	no	—
63	#13063270	—	4	2	no	—
64	#13063259	—	2	1	no	—
65	#13063259	—	2	2	no	—
66	#13063259	C₂³	1	2	yes	—
67	#13063259	C₂³	1	1	yes	characteristic
68	#13063259	C₂³	1	1	yes	characteristic
69	#13063250	—	2	1	no	—
70	#13063250	—	2	2	no	—
71	#13063247	—	2	2	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	C₂²×Dic₄₃	—	4	2	no	—
86	C₂²×Dic₄₃	—	4	2	no	—
87	#10621900	—	8	4	no	—
88	C₂×D₁₇₂	—	2	1	no	—
89	C₂×C₄×D₄₃	—	4	1	no	—
90	C₂×Dic₈₆	—	2	1	no	—
91	C₂×Dic₈₆	—	2	2	no	—
92	C₂×Dic₈₆	—	4	4	no	—
93	C₂×Dic₈₆	—	4	2	no	—
94	C₂×Dic₈₆	—	4	1	no	—
95	C₂×Dic₈₆	—	8	2	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
103	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂²⋊Dic₄₃	—	4	2	no	—
105	(C₂×D₄₃)⋊C₄	—	4	1	no	—
106	(C₂×D₄₃)⋊C₄	—	4	2	no	—
107	C₄⋊Dic₄₃	—	2	1	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	2	1	no	—
110	C₄⋊Dic₄₃	—	4	2	no	—
111	C₄⋊Dic₄₃	—	4	1	no	—
112	C₄⋊Dic₄₃	—	4	1	no	—
113	C₄⋊Dic₄₃	—	4	1	no	—
114	Dic₄₃⋊C₄	—	4	1	no	—
115	Dic₄₃⋊C₄	—	4	2	no	—
116	C₄×Dic₄₃	—	4	1	no	—
117	Dic₁₇₂	—	2	2	no	—
118	Dic₁₇₂	—	2	1	no	—
119	Dic₁₇₂	—	2	2	no	—
120	Dic₁₇₂	—	4	4	no	—
121	Dic₁₇₂	—	8	1	no	—
122	Dic₁₇₂	#41	1	2	yes	complemented
123	D₃₄₄	—	2	1	no	—
124	D₃₄₄	—	2	1	no	—
125	C₁₆×C₄₃	—	2	1	no	—
126	C₁₆×C₄₃	—	2	1	no	—
127	C₁₆×C₄₃	—	2	2	no	—
128	C₂³×C₄₃	C₂×D₈	1	2	yes	—
129	C₂²×D₄₃	—	4	1	no	—
130	D₄×C₄₃	SD₃₂	1	4	yes	complemented
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₄₃	—	8	4	no	—
138	C₂×Dic₄₃	—	4	1	no	—
139	C₂×Dic₄₃	—	4	1	no	—
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	—	4	1	no	—
143	C₂×Dic₄₃	—	8	4	no	—
144	C₂×Dic₄₃	—	8	4	no	—
145	D₁₇₂	—	4	1	no	—
146	D₁₇₂	—	4	1	no	—
147	C₄×D₄₃	—	8	2	no	—
148	Dic₈₆	—	2	2	no	—
149	Dic₈₆	—	4	1	no	—
150	Dic₈₆	—	4	2	no	—
151	Dic₈₆	—	4	2	no	—
152	Dic₈₆	—	8	4	no	—
153	Dic₈₆	—	8	2	no	—
154	C₈×C₄₃	—	2	2	no	—
155	C₈×C₄₃	—	2	1	no	—
156	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
157	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
158	#33865	—	43	1	no	maximal, Sylow
159	C₂²×C₄₃	—	2	2	no	—
160	C₂²×C₄₃	—	2	2	no	—
161	C₂²×C₄₃	C₂×SD₃₂	1	4	yes	—
162	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₄₃	—	8	1	no	—
164	C₂×D₄₃	—	8	1	no	—
165	C₂×D₄₃	—	8	1	no	—
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	2	2	no	—
168	C₄×C₄₃	C₂×SD₃₂	1	2	yes	—
169	C₄×C₄₃	#546	1	1	yes	characteristic
170	C₄×C₄₃	#577	1	1	yes	characteristic
171	Dic₄₃	—	4	2	no	—
172	Dic₄₃	—	8	2	no	—
173	Dic₄₃	—	8	1	no	—
174	Dic₄₃	—	8	1	no	—
175	Dic₄₃	—	8	2	no	—
176	#3334	—	43	1	no	—
177	D₄×Q₁₆	—	43	1	no	—
178	#2241	—	43	1	no	—
179	#2235	—	43	2	no	—
180	#2225	—	43	1	no	—
181	#2217	—	43	2	no	—
182	#2208	—	43	1	no	—
183	#2185	—	43	2	no	—
184	C₄×SD₃₂	—	43	1	no	—
185	D₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
186	C₂×#358	—	43	2	no	—
187	C₂×C₄₃	—	2	4	no	—
188	C₂×C₄₃	#3294	1	1	yes	characteristic
189	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
190	C₂×C₄₃	#3417	1	1	yes	characteristic
191	D₄₃	—	16	1	no	—
192	C₂²×Q₁₆	—	43	2	no	—
193	D₄×Q₈	—	86	1	no	—
194	#547	—	86	1	no	—
195	C₂×SD₃₂	—	43	1	no	—
196	C₂×SD₃₂	—	43	2	no	—
197	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
198	#500	—	43	1	no	—
199	#494	—	43	1	no	—
200	#480	—	86	2	no	—
201	#474	—	86	1	no	—
202	#467	—	43	2	no	—
203	#466	—	43	2	no	—
204	#462	—	86	1	no	—
205	#451	—	86	2	no	—
206	C₄×Q₁₆	—	43	1	no	—
207	C₄×D₈	—	43	1	no	—
208	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
209	C₂×#107	—	43	2	no	—
210	#367	—	43	1	no	—
211	#367	—	43	2	no	—
212	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
213	#358	—	43	1	no	—
214	#358	—	43	2	no	—
215	#358	—	43	2	no	—
216	#358	—	43	2	no	—
217	#358	—	43	2	no	—
218	#357	—	43	1	no	—
219	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	—
220	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
221	C₄₃	#33865	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	172	1	no	—
223	C₂²×Q₈	—	86	2	no	—
224	C₂×Q₁₆	—	43	1	no	—
225	C₂×Q₁₆	—	43	2	no	—
226	C₂×Q₁₆	—	86	4	no	—
227	C₂×Q₁₆	—	86	1	no	—
228	C₂×Q₁₆	—	86	2	no	—
229	C₂×Q₁₆	—	172	2	no	—
230	C₂×D₈	—	43	1	no	—
231	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
232	C₄⋊Q₈	—	86	1	no	—
233	C₄⋊Q₈	—	86	1	no	—
234	C₄⋊Q₈	—	172	1	no	—
235	#123	—	172	2	no	—
236	C₂²⋊Q₈	—	86	2	no	—
237	C₂²⋊Q₈	—	172	1	no	—
238	C₂²⋊Q₈	—	172	2	no	—
239	#121	—	86	2	no	—
240	C₄×Q₈	—	86	1	no	—
241	C₄×D₄	—	86	1	no	—
242	C₄×D₄	—	172	1	no	—
243	C₄×D₄	D₁₇₂	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	86	2	no	—
245	SD₃₂	—	86	2	no	—
246	SD₃₂	—	172	2	no	—
247	C₂×C₁₆	—	4	2	no	—
248	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
249	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
251	#107	—	43	1	no	—
252	#107	—	43	1	no	—
253	#107	—	43	2	no	—
254	#107	—	86	2	no	—
255	#107	—	86	1	no	—
256	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
257	#103	—	86	1	no	—
258	#103	—	86	2	no	—
259	#102	—	86	1	no	—
260	#102	—	86	2	no	—
261	C₂²⋊C₈	D₁₇₂	1	2	yes	—
262	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
263	#41	—	344	4	no	—
264	C₂×Q₈	—	86	1	no	—
265	C₂×Q₈	—	86	2	no	—
266	C₂×Q₈	—	344	2	no	—
267	C₂×Q₈	—	172	4	no	—
268	C₂×Q₈	—	172	1	no	—
269	C₂×Q₈	—	172	2	no	—
270	C₂×D₄	—	86	1	no	—
271	C₂×D₄	—	172	2	no	—
272	C₂×D₄	D₃₄₄	1	1	yes	characteristic
273	C₂²×C₄	—	172	2	no	—
274	C₂²×C₄	—	172	2	no	—
275	C₂²×C₄	—	172	1	no	—
276	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
277	Q₁₆	—	43	2	no	—
278	Q₁₆	—	86	2	no	—
279	Q₁₆	—	86	1	no	—
280	Q₁₆	—	86	2	no	—
281	Q₁₆	—	344	1	no	—
282	Q₁₆	—	172	4	no	—
283	D₈	—	86	1	no	—
284	D₈	—	86	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
288	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
289	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
290	C₄⋊C₄	—	86	1	no	—
291	C₄⋊C₄	—	86	2	no	—
292	C₄⋊C₄	—	86	1	no	—
293	C₄⋊C₄	—	172	2	no	—
294	C₄⋊C₄	—	172	1	no	—
295	C₄⋊C₄	—	172	1	no	—
296	C₄⋊C₄	—	172	2	no	—
297	C₄⋊C₄	—	172	1	no	—
298	C₄⋊C₄	—	172	1	no	—
299	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
300	C₂²⋊C₄	—	172	2	no	—
301	C₂²⋊C₄	—	172	1	no	—
302	C₂²⋊C₄	—	172	2	no	—
303	C₂²⋊C₄	D₃₄₄	1	2	yes	—
304	C₄²	—	172	1	no	—
305	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₂³	—	172	1	no	—
310	C₂³	C₂×D₃₄₄	1	2	yes	—
311	Q₈	—	86	2	no	—
312	Q₈	—	344	4	no	—
313	Q₈	—	344	2	no	—
314	Q₈	—	172	2	no	—
315	Q₈	—	172	2	no	—
316	Q₈	—	172	1	no	—
317	D₄	—	344	4	no	—
318	D₄	—	172	1	no	—
319	D₄	—	172	1	no	—
320	D₄	#13063270	1	4	yes	complemented
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	344	4	no	—
324	C₂×C₄	—	344	4	no	—
325	C₂×C₄	—	344	2	no	—
326	C₂×C₄	—	172	1	no	—
327	C₂×C₄	—	172	1	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	—	172	2	no	—
330	C₂×C₄	—	172	1	no	—
331	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
332	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
333	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
334	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	2	no	—
337	C₈	#41×D₄₃	1	1	yes	characteristic
338	C₈	#13063370	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	344	1	no	—
342	C₂²	—	344	1	no	—
343	C₂²	—	344	1	no	—
344	C₂²	C₂×#13063270	1	4	yes	—
345	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	344	2	no	—
349	C₄	—	344	1	no	—
350	C₄	—	344	2	no	—
351	C₄	—	344	1	no	—
352	C₄	—	172	2	no	—
353	C₄	C₂×#13063270	1	2	yes	—
354	C₄	#14583544	1	1	yes	characteristic
355	C₄	#14583711	1	1	yes	characteristic
356	C₂	—	2	4	no	—
357	C₂	—	688	1	no	—
358	C₂	#19146409	1	1	yes	characteristic
359	C₂	C₂²×#13063270	1	1	yes	characteristic
360	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	#19146943	C₂
hypercenter	D₄×C₁₆	D₄₃
hypocenter	C₄₃	#33865
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
2. D₄×Dic₁₇₂	C₂	semidirect
3. #19136823	C₂	semidirect
4. #19136793	C₂	semidirect
5. #19136738	C₂	semidirect
6. #19136691	C₂	semidirect
7. #19136641	C₂	semidirect
8. #19136514	C₂	semidirect
9. C₄×#13063270	C₂	semidirect
10. C₂×#14582704	C₂	semidirect
11. D₄×C₁₆×C₄₃	C₂	semidirect
122. Dic₁₇₂	#41	semidirect
130. D₄×C₄₃	SD₃₂	semidirect
185. D₄×C₁₆	D₄₃	semidirect
221. C₄₃	#33865	semidirect
320. D₄	#13063270	semidirect

Maximal subgroups

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

subgroup	index
#19146943	2
D₄×Dic₁₇₂	2
#19136823	2
#19136793	2
#19136738	2
#19136691	2
#19136641	2
#19136514	2
C₄×#13063270	2
C₂×#14582704	2
D₄×C₁₆×C₄₃	2
#33865	43