Subgroups — #25253217

Order 11008.

664 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583339	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136847	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136701	C₂	1	2	yes	maximal, complemented
5	#19136642	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136640	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136535	C₂	1	2	yes	maximal, complemented
8	#19136528	C₂	1	2	yes	maximal, complemented
9	C₄×#13063269	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136427	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3283	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
13	D₄×Dic₈₆	—	2	1	no	—
14	#14583339	—	2	8	no	—
15	C₂×#13063271	C₂²	1	2	yes	—
16	C₂×#13063271	C₂²	1	2	yes	—
17	C₂×#13063271	C₂²	1	2	yes	—
18	C₂×#13063269	C₂²	1	2	yes	—
19	C₂×#13063269	C₂²	1	1	yes	characteristic
20	C₂×#13063265	C₂²	1	2	yes	—
21	#14583279	C₂²	1	1	yes	characteristic
22	#14583140	C₂²	1	2	yes	—
23	#14583110	—	2	1	no	—
24	#14583057	—	2	2	no	—
25	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
26	#14582726	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
28	#14582703	C₂²	1	2	yes	—
29	#14582702	C₂²	1	1	yes	characteristic
30	#14582702	C₂²	1	2	yes	—
31	#14582700	C₂²	1	1	yes	characteristic
32	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
33	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₂×#135×C₄₃	C₂²	1	2	yes	—
35	C₄₃×#547	—	2	1	no	—
36	C₄₃×#494	C₂²	1	1	yes	characteristic
37	C₄₃×#467	C₂²	1	2	yes	—
38	C₄₃×#461	—	2	1	no	—
39	C₄₃×#452	—	2	2	no	—
40	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
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	#13063325	—	2	1	no	—
50	#13063324	—	4	1	no	—
51	#13063304	—	2	2	no	—
52	#13063302	—	4	2	no	—
53	C₄×Dic₈₆	—	2	1	no	—
54	D₄×Dic₄₃	—	4	1	no	—
55	#13063271	—	2	4	no	—
56	#13063271	—	2	4	no	—
57	#13063271	—	4	2	no	—
58	#13063269	—	2	4	no	—
59	#13063269	D₄	1	4	yes	complemented
60	#13063265	—	2	4	no	—
61	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
62	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
64	#13063259	C₂³	1	1	yes	characteristic
65	#13063250	—	2	1	no	—
66	#13063250	—	2	2	no	—
67	C₂×#41×C₄₃	—	2	2	no	—
68	#135×C₄₃	—	2	4	no	—
69	#135×C₄₃	—	4	2	no	—
70	C₂×Q₁₆×C₄₃	—	2	1	no	—
71	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₄₃	—	2	2	no	—
73	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₄₃	—	2	1	no	—
76	#123×C₄₃	—	4	2	no	—
77	C₂²⋊Q₈×C₄₃	—	2	2	no	—
78	#121×C₄₃	—	4	1	no	—
79	C₄×D₄×C₄₃	—	2	1	no	—
80	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₄₃	—	4	1	no	—
82	#107×C₄₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
84	#103×C₄₃	—	2	2	no	—
85	#102×C₄₃	—	2	1	no	—
86	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
87	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	#41×C₄₃	—	4	4	no	—
89	#41×C₄₃	—	8	2	no	—
90	C₂×Q₈×C₄₃	—	2	2	no	—
91	C₂×D₄×C₄₃	—	2	1	no	—
92	C₂×D₄×C₄₃	—	4	2	no	—
93	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	—	4	1	no	—
95	C₂²×C₄×C₄₃	—	4	2	no	—
96	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
97	C₂²×Dic₄₃	—	4	2	no	—
98	C₂×Dic₈₆	—	2	1	no	—
99	C₂×Dic₈₆	—	2	2	no	—
100	C₂×Dic₈₆	—	4	4	no	—
101	C₂×Dic₈₆	—	4	2	no	—
102	C₂×Dic₈₆	—	8	2	no	—
103	Q₁₆×C₄₃	—	2	2	no	—
104	Q₁₆×C₄₃	—	2	2	no	—
105	Q₁₆×C₄₃	—	8	1	no	—
106	SD₁₆×C₄₃	—	4	4	no	—
107	D₈×C₄₃	—	2	1	no	—
108	D₈×C₄₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₄₃	—	2	2	no	—
110	C₂×C₈×C₄₃	—	2	2	no	—
111	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₄₃	—	2	1	no	—
115	C₄⋊C₄×C₄₃	—	4	1	no	—
116	C₄⋊C₄×C₄₃	—	4	2	no	—
117	C₄⋊C₄×C₄₃	—	4	1	no	—
118	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
119	C₂²⋊C₄×C₄₃	—	4	1	no	—
120	C₂²⋊C₄×C₄₃	—	4	2	no	—
121	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
122	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂²⋊Dic₄₃	—	4	2	no	—
124	C₄⋊Dic₄₃	—	2	1	no	—
125	C₄⋊Dic₄₃	—	4	1	no	—
126	C₄⋊Dic₄₃	—	4	1	no	—
127	Dic₄₃⋊C₄	—	4	1	no	—
128	Dic₄₃⋊C₄	—	4	2	no	—
129	C₄×Dic₄₃	—	4	1	no	—
130	Dic₁₇₂	—	2	2	no	—
131	Dic₁₇₂	—	2	1	no	—
132	Dic₁₇₂	—	2	2	no	—
133	Dic₁₇₂	—	4	4	no	—
134	Dic₁₇₂	—	8	1	no	—
135	Dic₁₇₂	C₂×D₄	1	2	yes	—
136	C₄₃⋊C₁₆	—	2	2	no	—
137	C₄₃⋊C₁₆	—	2	1	no	—
138	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
139	C₂³×C₄₃	—	4	1	no	—
140	C₂³×C₄₃	C₂×D₈	1	2	yes	—
141	Q₈×C₄₃	—	4	2	no	—
142	Q₈×C₄₃	—	4	2	no	—
143	D₄×C₄₃	—	2	2	no	—
144	D₄×C₄₃	—	2	2	no	—
145	D₄×C₄₃	—	4	1	no	—
146	D₄×C₄₃	—	8	4	no	—
147	C₂×C₄×C₄₃	—	2	2	no	—
148	C₂×C₄×C₄₃	—	2	2	no	—
149	C₂×C₄×C₄₃	—	4	1	no	—
150	C₂×C₄×C₄₃	—	4	2	no	—
151	C₂×C₄×C₄₃	—	8	4	no	—
152	C₂×C₄×C₄₃	—	8	2	no	—
153	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×Dic₄₃	—	4	1	no	—
158	C₂×Dic₄₃	—	4	1	no	—
159	C₂×Dic₄₃	—	4	2	no	—
160	C₂×Dic₄₃	—	8	4	no	—
161	Dic₈₆	—	2	2	no	—
162	Dic₈₆	—	4	1	no	—
163	Dic₈₆	—	4	2	no	—
164	Dic₈₆	—	4	2	no	—
165	Dic₈₆	—	8	4	no	—
166	C₈×C₄₃	—	2	2	no	—
167	C₈×C₄₃	—	2	1	no	—
168	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
169	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
170	#33842	—	43	1	no	maximal, Sylow
171	C₂²×C₄₃	—	2	2	no	—
172	C₂²×C₄₃	—	2	2	no	—
173	C₂²×C₄₃	—	2	2	no	—
174	C₂²×C₄₃	—	4	2	no	—
175	C₂²×C₄₃	—	8	1	no	—
176	C₂²×C₄₃	—	8	1	no	—
177	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
178	C₄×C₄₃	—	2	1	no	—
179	C₄×C₄₃	—	2	2	no	—
180	C₄×C₄₃	—	8	1	no	—
181	C₄×C₄₃	—	8	2	no	—
182	C₄×C₄₃	#510	1	2	yes	complemented
183	C₄×C₄₃	D₄²	1	1	yes	characteristic
184	C₄×C₄₃	#576	1	1	yes	characteristic
185	Dic₄₃	—	4	2	no	—
186	Dic₄₃	—	8	1	no	—
187	Dic₄₃	—	8	2	no	—
188	C₂×#510	—	43	2	no	—
189	D₄×Q₁₆	—	43	1	no	—
190	#3283	D₄₃	1	1	yes	characteristic
191	#2247	—	43	1	no	—
192	#2219	—	43	2	no	—
193	#2209	—	43	1	no	—
194	#2208	—	43	1	no	—
195	#2189	—	43	2	no	—
196	#2188	—	43	2	no	—
197	C₄×SD₃₂	—	43	1	no	—
198	#2169	—	43	1	no	—
199	C₂×C₄₃	—	4	2	no	—
200	C₂×C₄₃	—	8	2	no	—
201	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
202	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
203	C₂×C₄₃	#3417	1	1	yes	characteristic
204	C₂×#135	C₂×D₄₃	1	2	yes	—
205	C₂²×Q₁₆	—	43	2	no	—
206	D₄×Q₈	—	86	1	no	—
207	#547	—	2	1	no	—
208	#510	—	86	8	no	—
209	C₂×Q₃₂	—	43	2	no	—
210	C₂×Q₃₂	—	43	2	no	—
211	C₂×Q₃₂	—	43	2	no	—
212	C₂×SD₃₂	—	43	1	no	—
213	C₂×SD₃₂	—	43	2	no	—
214	C₂×M₃₂	—	43	2	no	—
215	#494	—	43	1	no	—
216	#494	C₂×D₄₃	1	1	yes	characteristic
217	#467	—	43	2	no	—
218	#467	C₂×D₄₃	1	2	yes	—
219	#462	—	86	1	no	—
220	#461	—	2	1	no	—
221	#452	—	2	2	no	—
222	#451	—	86	2	no	—
223	C₄×Q₁₆	—	43	1	no	—
224	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
225	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
226	#367	—	43	1	no	—
227	C₄⋊C₁₆	—	43	1	no	—
228	#358	—	43	2	no	—
229	#358	—	43	1	no	—
230	#358	—	43	2	no	—
231	#357	—	43	1	no	—
232	C₂²⋊C₁₆	—	43	2	no	—
233	C₄×C₁₆	—	43	1	no	—
234	C₄₃	#33842	1	1	yes	characteristic, complemented, Sylow
235	C₂×#41	—	2	2	no	—
236	C₂²×Q₈	—	86	2	no	—
237	#135	—	2	4	no	—
238	#135	—	4	2	no	—
239	C₂×Q₁₆	—	2	1	no	—
240	C₂×Q₁₆	—	43	1	no	—
241	C₂×Q₁₆	—	43	2	no	—
242	C₂×Q₁₆	—	86	4	no	—
243	C₂×Q₁₆	—	86	1	no	—
244	C₂×Q₁₆	—	86	2	no	—
245	C₂×Q₁₆	—	172	2	no	—
246	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
247	C₂×SD₁₆	—	2	2	no	—
248	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
249	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	86	1	no	—
252	C₄⋊Q₈	—	172	1	no	—
253	#123	—	4	2	no	—
254	C₂²⋊Q₈	—	2	2	no	—
255	C₂²⋊Q₈	—	86	2	no	—
256	C₂²⋊Q₈	—	172	2	no	—
257	#121	—	4	1	no	—
258	C₄×Q₈	—	86	1	no	—
259	C₄×D₄	—	2	1	no	—
260	C₄×D₄	—	172	1	no	—
261	C₄×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
262	C₂×C₄⋊C₄	—	4	1	no	—
263	Q₃₂	—	86	4	no	—
264	Q₃₂	—	86	4	no	—
265	Q₃₂	—	172	2	no	—
266	SD₃₂	—	43	4	no	—
267	SD₃₂	—	86	4	no	—
268	M₃₂	—	86	4	no	—
269	C₂×C₁₆	—	43	1	no	—
270	C₂×C₁₆	—	43	1	no	—
271	C₂×C₁₆	—	43	2	no	—
272	#107	—	43	1	no	—
273	#107	C₂²×D₄₃	1	1	yes	characteristic
274	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
275	#103	—	2	2	no	—
276	#103	—	86	1	no	—
277	#103	—	86	2	no	—
278	#102	—	2	1	no	—
279	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
280	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
281	#41	—	4	4	no	—
282	#41	—	8	2	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂×Q₈	—	86	1	no	—
285	C₂×Q₈	—	86	2	no	—
286	C₂×Q₈	—	344	2	no	—
287	C₂×Q₈	—	172	4	no	—
288	C₂×Q₈	—	172	2	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	4	2	no	—
291	C₂×D₄	#10621882	1	1	yes	characteristic
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	172	2	no	—
295	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
296	Q₁₆	—	2	2	no	—
297	Q₁₆	—	2	2	no	—
298	Q₁₆	—	8	1	no	—
299	Q₁₆	—	43	2	no	—
300	Q₁₆	—	86	2	no	—
301	Q₁₆	—	86	1	no	—
302	Q₁₆	—	86	2	no	—
303	Q₁₆	—	344	1	no	—
304	Q₁₆	—	172	4	no	—
305	SD₁₆	—	4	4	no	—
306	D₈	—	2	1	no	—
307	D₈	D₄×D₄₃	1	2	yes	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
311	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
312	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	86	1	no	—
318	C₄⋊C₄	—	172	1	no	—
319	C₄⋊C₄	—	172	1	no	—
320	C₄⋊C₄	—	172	2	no	—
321	C₄⋊C₄	—	172	1	no	—
322	C₄⋊C₄	#10621882	1	1	yes	characteristic
323	C₂²⋊C₄	—	4	2	no	—
324	C₂²⋊C₄	—	4	1	no	—
325	C₂²⋊C₄	—	172	2	no	—
326	C₂²⋊C₄	#10621882	1	2	yes	—
327	C₄²	—	172	1	no	—
328	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
329	C₁₆	—	43	2	no	—
330	C₁₆	—	86	2	no	—
331	C₁₆	—	86	1	no	—
332	C₂³	—	4	1	no	—
333	C₂³	C₂×#10621882	1	2	yes	—
334	Q₈	—	4	2	no	—
335	Q₈	—	4	2	no	—
336	Q₈	—	86	2	no	—
337	Q₈	—	344	4	no	—
338	Q₈	—	172	2	no	—
339	Q₈	—	172	2	no	—
340	Q₈	—	172	1	no	—
341	D₄	—	2	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	1	no	—
344	D₄	—	8	4	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	4	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	—	344	4	no	—
352	C₂×C₄	—	172	1	no	—
353	C₂×C₄	—	172	2	no	—
354	C₂×C₄	—	172	1	no	—
355	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
356	C₂×C₄	C₂×#10621882	1	2	yes	—
357	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
358	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	2	no	—
361	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
362	C₈	#13063370	1	1	yes	characteristic
363	C₂²	—	2	2	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	8	1	no	—
368	C₂²	—	8	1	no	—
369	C₂²	C₂²×#10621882	1	1	yes	characteristic
370	C₄	—	2	2	no	—
371	C₄	—	2	1	no	—
372	C₄	—	8	2	no	—
373	C₄	—	8	1	no	—
374	C₄	—	344	2	no	—
375	C₄	—	344	1	no	—
376	C₄	—	172	2	no	—
377	C₄	#14583339	1	2	yes	complemented
378	C₄	D₄×C₂²⋊D₄₃	1	1	yes	characteristic
379	C₄	#14583702	1	1	yes	characteristic
380	C₂	—	4	2	no	—
381	C₂	—	8	2	no	—
382	C₂	D₄×#10621882	1	1	yes	characteristic
383	C₂	C₂×#14583339	1	1	yes	characteristic
384	C₂	#19147746	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting, prime radical	C₄₃×#3283	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3283	D₄₃
hypocenter	C₄₃	#33842
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3283

Complements

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

N	H ≅ G/N	type
1. C₂×#14583339	C₂	semidirect
2. D₄×Dic₁₇₂	C₂	semidirect
3. #19136847	C₂	semidirect
4. #19136701	C₂	semidirect
5. #19136642	C₂	semidirect
6. #19136640	C₂	semidirect
7. #19136535	C₂	semidirect
8. #19136528	C₂	semidirect
9. C₄×#13063269	C₂	semidirect
10. #19136427	C₂	semidirect
59. #13063269	D₄	semidirect
182. C₄×C₄₃	#510	semidirect
234. C₄₃	#33842	semidirect
377. C₄	#14583339	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583339	2
D₄×Dic₁₇₂	2
#19136847	2
#19136701	2
#19136642	2
#19136640	2
#19136535	2
#19136528	2
C₄×#13063269	2
#19136427	2
C₄₃×#3283	2
#33842	43