Subgroups — #25253425

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	D₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	#19136820	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136780	C₂	1	2	yes	maximal, complemented
4	#19136739	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136693	C₂	1	2	yes	maximal, complemented
6	#19136635	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136523	C₂	1	2	yes	maximal, complemented
8	C₄×#13063268	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14582699	C₂	1	2	yes	maximal, complemented
11	C₄₃×#3338	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₄₄	C₂²	1	2	yes	—
13	D₄×D₁₇₂	—	2	1	no	—
14	C₂×#13063268	C₂²	1	1	yes	characteristic
15	C₂×#13063268	C₂²	1	2	yes	complemented
16	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
17	#14583276	C₂²	1	1	yes	characteristic
18	#14583135	C₂²	1	2	yes	—
19	#14583095	—	2	1	no	—
20	#14583035	—	2	2	no	—
21	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
22	#14582726	C₂²	1	1	yes	characteristic, complemented
23	#14582726	C₂²	1	2	yes	complemented
24	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
25	#14582703	C₂²	1	1	yes	characteristic, complemented
26	#14582699	C₂²	1	2	yes	—
27	#14582699	C₂²	1	1	yes	characteristic
28	#14582699	C₂²	1	2	yes	complemented
29	#14582699	C₂²	1	2	yes	complemented
30	#14582699	C₂²	1	2	yes	complemented
31	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	complemented
32	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
33	C₄₃×#554	—	2	1	no	—
34	C₄₃×#500	C₂²	1	1	yes	characteristic, complemented
35	C₄₃×#484	—	2	2	no	—
36	C₄₃×#477	—	2	1	no	—
37	C₄₃×#467	C₂²	1	2	yes	complemented
38	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
39	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
40	C₂×#107×C₄₃	C₂²	1	2	yes	complemented
41	C₂²×D₁₇₂	—	2	2	no	—
42	C₂×D₄×D₄₃	—	4	1	no	—
43	C₂×D₃₄₄	—	2	4	no	—
44	C₂×D₃₄₄	—	2	2	no	—
45	C₂×D₃₄₄	—	2	1	no	—
46	C₂×D₃₄₄	—	4	2	no	—
47	C₂×D₃₄₄	C₂³	1	2	yes	—
48	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
49	#13063322	—	2	1	no	—
50	#13063300	—	4	1	no	—
51	#13063299	—	2	2	no	—
52	#13063295	—	4	2	no	—
53	C₄×D₁₇₂	—	2	1	no	—
54	#13063268	—	2	2	no	—
55	#13063268	—	4	2	no	—
56	C₂×C₄₃⋊C₁₆	—	4	2	no	—
57	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
60	#13063259	C₂³	1	1	yes	characteristic
61	#13063247	—	2	2	no	—
62	#13063247	—	2	1	no	—
63	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₃	—	2	1	no	—
66	#125×C₄₃	—	4	1	no	—
67	C₂²⋊Q₈×C₄₃	—	2	2	no	—
68	C₂²⋊Q₈×C₄₃	—	4	2	no	—
69	C₄×Q₈×C₄₃	—	2	1	no	—
70	C₄×D₄×C₄₃	—	4	1	no	—
71	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
73	#107×C₄₃	—	2	1	no	—
74	#107×C₄₃	—	2	2	no	—
75	#107×C₄₃	C₂³	1	2	yes	—
76	#107×C₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
78	#103×C₄₃	—	2	1	no	—
79	#103×C₄₃	—	2	2	no	—
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×Q₈×C₄₃	—	2	1	no	—
84	C₂×Q₈×C₄₃	—	4	1	no	—
85	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	—	4	2	no	—
87	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₂²⋊D₄₃	—	4	2	no	—
89	D₄×D₄₃	—	8	4	no	—
90	C₂×D₁₇₂	—	2	1	no	—
91	C₂×D₁₇₂	—	2	2	no	—
92	C₂×D₁₇₂	—	4	2	no	—
93	C₂×D₁₇₂	—	4	1	no	—
94	C₂×D₁₇₂	—	4	1	no	—
95	C₂×D₁₇₂	—	4	4	no	—
96	C₂×D₁₇₂	—	8	2	no	—
97	C₂×C₄×D₄₃	—	4	1	no	—
98	Q₁₆×C₄₃	—	2	1	no	—
99	Q₁₆×C₄₃	—	2	1	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₄₃	—	2	1	no	—
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	4	2	no	—
108	C₄⋊C₄×C₄₃	—	4	1	no	—
109	C₄⋊C₄×C₄₃	—	4	1	no	—
110	C₄⋊C₄×C₄₃	—	4	2	no	—
111	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
112	C₂²⋊C₄×C₄₃	—	4	2	no	—
113	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
114	C₄²×C₄₃	—	4	1	no	—
115	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
116	(C₂×D₄₃)⋊C₄	—	4	1	no	—
117	(C₂×D₄₃)⋊C₄	—	4	2	no	—
118	C₄⋊Dic₄₃	—	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	C₄₃⋊C₁₆	—	2	1	no	—
126	C₄₃⋊C₁₆	—	2	1	no	—
127	C₄₃⋊C₁₆	—	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	Q₈×C₄₃	—	4	1	no	—
135	Q₈×C₄₃	—	4	1	no	—
136	Q₈×C₄₃	—	16	1	no	—
137	D₄×C₄₃	SD₃₂	1	4	yes	complemented
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	4	1	no	—
141	C₂×C₄×C₄₃	—	4	1	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	16	2	no	—
144	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₄₃	—	8	4	no	—
149	C₂×Dic₄₃	—	4	1	no	—
150	D₁₇₂	—	2	2	no	—
151	D₁₇₂	—	4	2	no	—
152	D₁₇₂	—	4	1	no	—
153	D₁₇₂	—	4	2	no	—
154	D₁₇₂	—	8	4	no	—
155	D₁₇₂	—	8	2	no	—
156	D₁₇₂	—	16	1	no	—
157	C₄×D₄₃	—	8	2	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	—	8	1	no	—
177	C₄×C₄₃	—	8	2	no	—
178	C₄×C₄₃	—	8	1	no	—
179	C₄×C₄₃	C₂×SD₃₂	1	2	yes	—
180	C₄×C₄₃	#546	1	1	yes	characteristic
181	C₄×C₄₃	#577	1	1	yes	characteristic
182	Dic₄₃	—	8	1	no	—
183	#3338	D₄₃	1	1	yes	characteristic, complemented
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	—	43	1	no	—
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	—	2	1	no	—
202	D₄²	—	86	1	no	—
203	C₂×SD₃₂	—	43	1	no	—
204	C₂×SD₃₂	—	43	2	no	—
205	C₂²×C₁₆	—	43	2	no	—
206	#500	C₂×D₄₃	1	1	yes	characteristic, complemented
207	#493	—	43	1	no	—
208	#484	—	2	2	no	—
209	#477	—	2	1	no	—
210	#467	C₂×D₄₃	1	2	yes	complemented
211	#466	—	43	2	no	—
212	#459	—	86	1	no	—
213	#447	—	86	2	no	—
214	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
215	C₄×D₈	—	43	1	no	—
216	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
217	C₂×#107	C₂×D₄₃	1	2	yes	complemented
218	#367	—	43	1	no	—
219	#367	—	43	2	no	—
220	C₄⋊C₁₆	—	43	1	no	—
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	—	43	2	no	—
228	C₄×C₁₆	—	43	1	no	—
229	C₄₃	#33862	1	1	yes	characteristic, complemented, Sylow
230	C₂²×D₄	—	86	2	no	—
231	C₂²×D₄	—	172	1	no	—
232	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
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	—	2	1	no	—
241	#127	—	86	1	no	—
242	#125	—	4	1	no	—
243	C₂²⋊Q₈	—	2	2	no	—
244	C₂²⋊Q₈	—	4	2	no	—
245	#121	—	86	2	no	—
246	#121	—	172	1	no	—
247	#120	—	172	2	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×D₄	—	4	1	no	—
250	C₄×D₄	—	86	1	no	—
251	C₄×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	2	2	no	—
253	SD₃₂	—	86	2	no	—
254	SD₃₂	—	172	2	no	—
255	C₂×C₁₆	—	43	1	no	—
256	C₂×C₁₆	—	43	1	no	—
257	C₂×C₁₆	—	43	2	no	—
258	C₂×C₁₆	—	172	2	no	—
259	#107	—	2	2	no	—
260	#107	—	2	1	no	—
261	#107	—	43	1	no	—
262	#107	C₂²×D₄₃	1	1	yes	characteristic
263	#107	C₂²×D₄₃	1	2	yes	—
264	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
265	#103	—	2	1	no	—
266	#103	—	2	2	no	—
267	#102	—	86	1	no	—
268	#102	—	86	2	no	—
269	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
270	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
271	C₂⁴	—	172	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	4	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	#10621882	1	1	yes	characteristic
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	—	172	1	no	—
286	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
287	Q₁₆	—	2	1	no	—
288	Q₁₆	—	2	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×C₂²⋊D₄₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
299	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	86	1	no	—
307	C₄⋊C₄	#10621882	1	1	yes	characteristic
308	C₂²⋊C₄	—	4	2	no	—
309	C₂²⋊C₄	—	172	1	no	—
310	C₂²⋊C₄	—	172	2	no	—
311	C₂²⋊C₄	#10621882	1	2	yes	—
312	C₄²	—	4	1	no	—
313	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
314	C₁₆	—	86	1	no	—
315	C₁₆	—	86	1	no	—
316	C₁₆	—	86	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×#10621882	1	2	yes	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	16	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	#13063268	1	4	yes	complemented
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	16	2	no	—
341	C₂×C₄	—	344	2	no	—
342	C₂×C₄	—	172	1	no	—
343	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
344	C₂×C₄	C₂×#10621882	1	2	yes	—
345	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
346	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	2	2	no	—
349	C₈	#41×D₄₃	1	1	yes	characteristic
350	C₈	#13063371	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×#13063268	1	4	yes	—
361	C₂²	C₂²×#10621882	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	344	1	no	—
368	C₄	C₂×#13063268	1	2	yes	—
369	C₄	#14583537	1	1	yes	characteristic
370	C₄	#14583707	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	—	344	2	no	—
373	C₂	—	688	2	no	—
374	C₂	#19146402	1	1	yes	characteristic
375	C₂	C₂²×#13063268	1	1	yes	characteristic
376	C₂	#19147734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3338	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	D₄×D₃₄₄	C₂
hypercenter	#3338	D₄₃
hypocenter	C₄₃	#33862
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3338

Complements

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

N	H ≅ G/N	type
2. #19136820	C₂	semidirect
3. #19136780	C₂	semidirect
4. #19136739	C₂	semidirect
5. #19136693	C₂	semidirect
6. #19136635	C₂	semidirect
7. #19136523	C₂	semidirect
8. C₄×#13063268	C₂	semidirect
9. D₄×C₄₃⋊C₁₆	C₂	semidirect
10. C₂×#14582699	C₂	semidirect
11. C₄₃×#3338	C₂	semidirect
15. C₂×#13063268	C₂²	semidirect
16. C₂²×C₄₃⋊C₁₆	C₂²	semidirect
22. #14582726	C₂²	semidirect
23. #14582726	C₂²	semidirect
24. C₄⋊(C₄₃⋊C₁₆)	C₂²	semidirect
25. #14582703	C₂²	semidirect
28. #14582699	C₂²	semidirect
29. #14582699	C₂²	semidirect
30. #14582699	C₂²	semidirect
31. C₂²⋊(C₄₃⋊C₁₆)	C₂²	semidirect
32. C₄×C₄₃⋊C₁₆	C₂²	semidirect
34. C₄₃×#500	C₂²	semidirect
37. C₄₃×#467	C₂²	semidirect
38. C₄×Q₁₆×C₄₃	C₂²	semidirect
40. C₂×#107×C₄₃	C₂²	semidirect
137. D₄×C₄₃	SD₃₂	semidirect
183. #3338	D₄₃	semidirect
206. #500	C₂×D₄₃	semidirect
210. #467	C₂×D₄₃	semidirect
214. C₄×Q₁₆	C₂×D₄₃	semidirect
217. C₂×#107	C₂×D₄₃	semidirect
229. C₄₃	#33862	semidirect
334. D₄	#13063268	semidirect

Maximal subgroups

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

subgroup	index
D₄×D₃₄₄	2
#19136820	2
#19136780	2
#19136739	2
#19136693	2
#19136635	2
#19136523	2
C₄×#13063268	2
D₄×C₄₃⋊C₁₆	2
C₂×#14582699	2
C₄₃×#3338	2
#33862	43