Subgroups — #25253432

Order 11008.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19146988	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	#19136819	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136781	C₂	1	2	yes	maximal, complemented
5	#19136743	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136691	C₂	1	2	yes	maximal, complemented
7	#19136636	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136526	C₂	1	2	yes	maximal, complemented
9	C₄×#13063270	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14582701	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₄₄	C₂²	1	2	yes	—
13	#14583606	—	2	1	no	—
14	D₄×D₁₇₂	—	2	1	no	—
15	C₂×#13063270	C₂²	1	1	yes	characteristic
16	C₂×#13063270	C₂²	1	2	yes	complemented
17	#14583304	C₂²	1	1	yes	characteristic, complemented
18	#14583276	C₂²	1	1	yes	characteristic
19	#14583235	—	2	2	no	—
20	#14583195	—	2	1	no	—
21	#14583140	C₂²	1	2	yes	complemented
22	#14583135	C₂²	1	2	yes	—
23	#14583095	—	2	1	no	—
24	#14583035	—	2	2	no	—
25	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic, complemented
26	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
27	C₂×#13063259	C₂²	1	2	yes	complemented
28	#14582727	C₂²	1	1	yes	characteristic, complemented
29	#14582727	C₂²	1	2	yes	complemented
30	#14582704	C₂²	1	1	yes	characteristic, complemented
31	#14582701	C₂²	1	2	yes	—
32	#14582701	C₂²	1	1	yes	characteristic
33	#14582701	C₂²	1	2	yes	complemented
34	#14582701	C₂²	1	2	yes	complemented
35	#14582701	C₂²	1	2	yes	complemented
36	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	complemented
37	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
39	C₄₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₁₇₂	—	2	2	no	—
42	C₂×D₄×D₄₃	—	4	1	no	—
43	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
44	C₂×D₃₄₄	—	2	4	no	—
45	C₂×D₃₄₄	—	2	2	no	—
46	C₂×D₃₄₄	—	2	1	no	—
47	C₂×D₃₄₄	—	4	2	no	—
48	C₂×D₃₄₄	C₂³	1	2	yes	—
49	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
50	#13063325	—	2	1	no	—
51	#13063322	—	2	1	no	—
52	#13063316	—	4	1	no	—
53	#13063304	—	2	2	no	—
54	#13063302	—	4	2	no	—
55	#13063300	—	4	1	no	—
56	#13063299	—	2	2	no	—
57	#13063295	—	4	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	2	no	—
70	#13063250	—	2	1	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	C₂³×D₄₃	—	4	2	no	—
83	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
85	C₂×C₂²⋊D₄₃	—	4	2	no	—
86	C₂²×Dic₄₃	—	4	2	no	—
87	D₄×D₄₃	—	8	4	no	—
88	C₂×D₁₇₂	—	2	1	no	—
89	C₂×D₁₇₂	—	2	2	no	—
90	C₂×D₁₇₂	—	4	2	no	—
91	C₂×D₁₇₂	—	4	1	no	—
92	C₂×D₁₇₂	—	4	1	no	—
93	C₂×D₁₇₂	—	4	4	no	—
94	C₂×D₁₇₂	—	8	2	no	—
95	C₂×C₄×D₄₃	—	4	1	no	—
96	C₂×Dic₈₆	—	2	1	no	—
97	C₂×Dic₈₆	—	4	1	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
104	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₄₃	—	4	2	no	—
107	(C₂×D₄₃)⋊C₄	—	4	1	no	—
108	(C₂×D₄₃)⋊C₄	—	4	2	no	—
109	C₄⋊Dic₄₃	—	2	1	no	—
110	C₄⋊Dic₄₃	—	2	1	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	4	2	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	1	no	—
118	Dic₁₇₂	—	2	1	no	—
119	D₃₄₄	—	2	2	no	—
120	D₃₄₄	—	2	1	no	—
121	D₃₄₄	—	2	2	no	—
122	D₃₄₄	—	4	4	no	—
123	D₃₄₄	—	8	1	no	—
124	D₃₄₄	#41	1	2	yes	—
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	C₂²×D₄₃	—	4	2	no	—
131	C₂²×D₄₃	—	8	4	no	—
132	C₂²×D₄₃	—	8	4	no	—
133	C₂²×D₄₃	—	16	2	no	—
134	D₄×C₄₃	SD₃₂	1	4	yes	complemented
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₄₃	—	8	4	no	—
142	C₂×Dic₄₃	—	4	1	no	—
143	C₂×Dic₄₃	—	4	1	no	—
144	C₂×Dic₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	4	1	no	—
146	C₂×Dic₄₃	—	16	2	no	—
147	D₁₇₂	—	2	2	no	—
148	D₁₇₂	—	4	2	no	—
149	D₁₇₂	—	4	1	no	—
150	D₁₇₂	—	4	2	no	—
151	D₁₇₂	—	8	4	no	—
152	D₁₇₂	—	8	2	no	—
153	D₁₇₂	—	16	1	no	—
154	C₄×D₄₃	—	8	2	no	—
155	Dic₈₆	—	4	1	no	—
156	Dic₈₆	—	4	1	no	—
157	Dic₈₆	—	16	1	no	—
158	C₈×C₄₃	—	2	2	no	—
159	C₈×C₄₃	—	2	1	no	—
160	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
161	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
162	#33862	—	43	1	no	maximal, Sylow
163	C₂²×C₄₃	—	2	2	no	—
164	C₂²×C₄₃	—	2	2	no	—
165	C₂²×C₄₃	C₂×SD₃₂	1	4	yes	—
166	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₄₃	—	4	2	no	—
168	C₂×D₄₃	—	8	1	no	—
169	C₂×D₄₃	—	8	1	no	—
170	C₂×D₄₃	—	8	2	no	—
171	C₂×D₄₃	—	8	2	no	—
172	C₂×D₄₃	—	8	2	no	—
173	C₂×D₄₃	—	16	8	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	2	no	—
176	C₄×C₄₃	C₂×SD₃₂	1	2	yes	—
177	C₄×C₄₃	#546	1	1	yes	characteristic
178	C₄×C₄₃	#577	1	1	yes	characteristic
179	Dic₄₃	—	8	1	no	—
180	Dic₄₃	—	8	2	no	—
181	Dic₄₃	—	8	1	no	—
182	Dic₄₃	—	8	1	no	—
183	#3338	—	43	1	no	—
184	D₄×D₈	—	43	1	no	—
185	#2240	—	43	1	no	—
186	#2233	—	43	2	no	—
187	#2226	—	43	1	no	—
188	#2217	—	43	2	no	—
189	#2207	—	43	1	no	—
190	#2187	—	43	2	no	—
191	C₄×SD₃₂	—	43	1	no	—
192	D₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
193	C₂×#357	—	43	2	no	—
194	C₂×C₄₃	—	2	4	no	—
195	C₂×C₄₃	#3294	1	1	yes	characteristic
196	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
197	C₂×C₄₃	#3415	1	1	yes	characteristic
198	D₄₃	—	8	2	no	—
199	D₄₃	—	16	2	no	—
200	C₂²×D₈	—	43	2	no	—
201	#554	—	86	1	no	—
202	D₄²	—	86	1	no	—
203	C₂×SD₃₂	—	43	1	no	—
204	C₂×SD₃₂	—	43	2	no	—
205	C₂²×C₁₆	C₂×D₄₃	1	2	yes	complemented
206	#500	—	43	1	no	—
207	#493	—	43	1	no	—
208	#484	—	86	2	no	—
209	#477	—	86	1	no	—
210	#467	—	43	2	no	—
211	#466	—	43	2	no	—
212	#459	—	86	1	no	—
213	#447	—	86	2	no	—
214	C₄×Q₁₆	—	43	1	no	—
215	C₄×D₈	—	43	1	no	—
216	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
217	C₂×#107	—	43	2	no	—
218	#367	—	43	1	no	—
219	#367	—	43	2	no	—
220	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
221	#358	—	43	1	no	—
222	#357	—	43	1	no	—
223	#357	—	43	2	no	—
224	#357	—	43	2	no	—
225	#357	—	43	2	no	—
226	#357	—	43	2	no	—
227	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	complemented
228	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
229	C₄₃	#33862	1	1	yes	characteristic, complemented, Sylow
230	C₂²×D₄	—	86	2	no	—
231	C₂²×D₄	—	172	1	no	—
232	C₂×Q₁₆	—	43	1	no	—
233	C₂×D₈	—	43	1	no	—
234	C₂×D₈	—	43	2	no	—
235	C₂×D₈	—	86	4	no	—
236	C₂×D₈	—	86	1	no	—
237	C₂×D₈	—	86	2	no	—
238	C₂×D₈	—	172	2	no	—
239	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
240	C₄⋊Q₈	—	86	1	no	—
241	#127	—	86	1	no	—
242	#125	—	172	1	no	—
243	C₂²⋊Q₈	—	86	2	no	—
244	C₂²⋊Q₈	—	172	2	no	—
245	#121	—	86	2	no	—
246	#121	—	172	1	no	—
247	#120	—	172	2	no	—
248	C₄×Q₈	—	86	1	no	—
249	C₄×D₄	—	86	1	no	—
250	C₄×D₄	—	172	1	no	—
251	C₄×D₄	D₁₇₂	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	86	2	no	—
253	SD₃₂	—	86	2	no	—
254	SD₃₂	—	172	2	no	—
255	C₂×C₁₆	—	4	2	no	—
256	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
257	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
259	#107	—	43	1	no	—
260	#107	—	43	2	no	—
261	#107	—	43	1	no	—
262	#107	—	86	2	no	—
263	#107	—	86	1	no	—
264	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
265	#103	—	86	1	no	—
266	#103	—	86	2	no	—
267	#102	—	86	1	no	—
268	#102	—	86	2	no	—
269	C₂²⋊C₈	D₁₇₂	1	2	yes	—
270	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
271	C₂⁴	—	172	2	no	—
272	C₂×Q₈	—	86	1	no	—
273	C₂×Q₈	—	172	1	no	—
274	C₂×D₄	—	86	1	no	—
275	C₂×D₄	—	86	2	no	—
276	C₂×D₄	—	344	2	no	—
277	C₂×D₄	—	344	4	no	—
278	C₂×D₄	—	172	4	no	—
279	C₂×D₄	—	172	2	no	—
280	C₂×D₄	—	172	2	no	—
281	C₂×D₄	—	172	1	no	—
282	C₂×D₄	—	172	1	no	—
283	C₂×D₄	D₃₄₄	1	1	yes	characteristic
284	C₂²×C₄	—	172	1	no	—
285	C₂²×C₄	—	172	2	no	—
286	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
287	Q₁₆	—	86	1	no	—
288	Q₁₆	—	86	1	no	—
289	D₈	—	43	2	no	—
290	D₈	—	86	2	no	—
291	D₈	—	86	1	no	—
292	D₈	—	86	2	no	—
293	D₈	—	344	1	no	—
294	D₈	—	172	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
298	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
299	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
300	C₄⋊C₄	—	86	1	no	—
301	C₄⋊C₄	—	86	1	no	—
302	C₄⋊C₄	—	86	2	no	—
303	C₄⋊C₄	—	172	2	no	—
304	C₄⋊C₄	—	172	1	no	—
305	C₄⋊C₄	—	172	1	no	—
306	C₄⋊C₄	—	172	2	no	—
307	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
308	C₂²⋊C₄	—	172	2	no	—
309	C₂²⋊C₄	—	172	1	no	—
310	C₂²⋊C₄	—	172	2	no	—
311	C₂²⋊C₄	D₃₄₄	1	2	yes	—
312	C₄²	—	172	1	no	—
313	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	2	2	no	—
317	C₂³	—	344	4	no	—
318	C₂³	—	344	4	no	—
319	C₂³	—	172	2	no	—
320	C₂³	—	172	1	no	—
321	C₂³	—	688	2	no	—
322	C₂³	C₂×D₃₄₄	1	2	yes	—
323	Q₈	—	172	1	no	—
324	Q₈	—	172	1	no	—
325	Q₈	—	688	1	no	—
326	D₄	—	86	2	no	—
327	D₄	—	344	4	no	—
328	D₄	—	344	4	no	—
329	D₄	—	344	2	no	—
330	D₄	—	172	2	no	—
331	D₄	—	172	1	no	—
332	D₄	—	172	2	no	—
333	D₄	—	688	1	no	—
334	D₄	#13063270	1	4	yes	complemented
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	344	2	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	—	172	1	no	—
340	C₂×C₄	—	172	1	no	—
341	C₂×C₄	—	172	2	no	—
342	C₂×C₄	—	688	2	no	—
343	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
344	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
345	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
346	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	2	2	no	—
349	C₈	#41×D₄₃	1	1	yes	characteristic
350	C₈	#13063370	1	1	yes	characteristic
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	344	1	no	—
354	C₂²	—	344	1	no	—
355	C₂²	—	344	2	no	—
356	C₂²	—	344	2	no	—
357	C₂²	—	344	2	no	—
358	C₂²	—	172	2	no	—
359	C₂²	—	688	8	no	—
360	C₂²	C₂×#13063270	1	4	yes	—
361	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	1	no	—
364	C₄	—	344	1	no	—
365	C₄	—	344	1	no	—
366	C₄	—	344	1	no	—
367	C₄	—	344	2	no	—
368	C₄	C₂×#13063270	1	2	yes	—
369	C₄	#14583544	1	1	yes	characteristic
370	C₄	#14583711	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	—	344	2	no	—
373	C₂	—	688	2	no	—
374	C₂	#19146409	1	1	yes	characteristic
375	C₂	C₂²×#13063270	1	1	yes	characteristic
376	C₂	#19147736	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	D₄×D₃₄₄	C₂
hypercenter	D₄×C₁₆	D₄₃
hypocenter	C₄₃	#33862
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. #19146988	C₂	semidirect
3. #19136819	C₂	semidirect
4. #19136781	C₂	semidirect
5. #19136743	C₂	semidirect
6. #19136691	C₂	semidirect
7. #19136636	C₂	semidirect
8. #19136526	C₂	semidirect
9. C₄×#13063270	C₂	semidirect
10. C₂×#14582701	C₂	semidirect
11. D₄×C₁₆×C₄₃	C₂	semidirect
16. C₂×#13063270	C₂²	semidirect
17. #14583304	C₂²	semidirect
21. #14583140	C₂²	semidirect
25. C₄×Dic₁₇₂	C₂²	semidirect
27. C₂×#13063259	C₂²	semidirect
28. #14582727	C₂²	semidirect
29. #14582727	C₂²	semidirect
30. #14582704	C₂²	semidirect
33. #14582701	C₂²	semidirect
34. #14582701	C₂²	semidirect
35. #14582701	C₂²	semidirect
36. C₂²×C₁₆×C₄₃	C₂²	semidirect
38. C₄₃×C₄⋊C₁₆	C₂²	semidirect
39. C₄₃×C₂²⋊C₁₆	C₂²	semidirect
40. C₄×C₁₆×C₄₃	C₂²	semidirect
134. D₄×C₄₃	SD₃₂	semidirect
192. D₄×C₁₆	D₄₃	semidirect
205. C₂²×C₁₆	C₂×D₄₃	semidirect
220. C₄⋊C₁₆	C₂×D₄₃	semidirect
227. C₂²⋊C₁₆	C₂×D₄₃	semidirect
228. C₄×C₁₆	C₂×D₄₃	semidirect
229. C₄₃	#33862	semidirect
334. D₄	#13063270	semidirect

Maximal subgroups

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

subgroup	index
#19146988	2
D₄×D₃₄₄	2
#19136819	2
#19136781	2
#19136743	2
#19136691	2
#19136636	2
#19136526	2
C₄×#13063270	2
C₂×#14582701	2
D₄×C₁₆×C₄₃	2
#33862	43