Subgroups — #25253293

Order 11008.

676 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583329	C₂	1	2	yes	maximal, complemented
2	#19146317	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136847	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136684	C₂	1	2	yes	maximal, complemented
5	#19136640	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136637	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136535	C₂	1	2	yes	maximal, complemented
8	#19136533	C₂	1	2	yes	maximal, complemented
9	C₄×#13063266	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	2	yes	—
13	#14583554	—	2	1	no	—
14	#14583329	—	2	8	no	—
15	C₂×#13063271	C₂²	1	2	yes	—
16	C₂×#13063271	C₂²	1	2	yes	—
17	C₂×#13063269	C₂²	1	2	yes	—
18	C₂×#13063268	C₂²	1	2	yes	—
19	C₂×#13063266	C₂²	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
21	#14583279	C₂²	1	1	yes	characteristic
22	#14583140	C₂²	1	2	yes	—
23	#14583105	—	2	1	no	—
24	#14583063	—	2	2	no	—
25	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
26	#14582724	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
28	#14582703	C₂²	1	2	yes	—
29	#14582702	C₂²	1	2	yes	—
30	#14582700	C₂²	1	1	yes	characteristic
31	#14582699	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	C₂×#10621898	—	2	2	no	—
43	#13063347	—	2	4	no	—
44	#13063347	—	4	2	no	—
45	C₂×Dic₁₇₂	—	2	1	no	—
46	C₂×Dic₁₇₂	C₂³	1	2	yes	—
47	C₂×#10621873	—	2	2	no	—
48	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
49	#13063325	—	2	1	no	—
50	#13063307	—	4	2	no	—
51	#13063304	—	2	2	no	—
52	#13063296	—	4	1	no	—
53	C₄×D₁₇₂	—	2	1	no	—
54	C₄⋊C₄×D₄₃	—	4	1	no	—
55	#13063271	—	2	4	no	—
56	#13063271	—	4	2	no	—
57	#13063269	—	2	4	no	—
58	#13063268	—	2	4	no	—
59	#13063266	D₄	1	4	yes	complemented
60	C₂×C₄₃⋊C₁₆	—	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	2	no	—
66	#13063247	—	2	1	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	C₂×C₂²⋊D₄₃	—	4	2	no	—
98	#10621898	—	4	4	no	—
99	#10621898	—	8	2	no	—
100	C₂×D₁₇₂	—	2	1	no	—
101	C₂×C₄×D₄₃	—	4	1	no	—
102	C₂×C₄×D₄₃	—	4	2	no	—
103	C₂×Dic₈₆	—	2	2	no	—
104	Q₁₆×C₄₃	—	2	2	no	—
105	Q₁₆×C₄₃	—	2	2	no	—
106	Q₁₆×C₄₃	—	8	1	no	—
107	SD₁₆×C₄₃	—	4	4	no	—
108	D₈×C₄₃	—	2	1	no	—
109	D₈×C₄₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₄₃	—	2	2	no	—
111	C₂×C₈×C₄₃	—	2	2	no	—
112	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₄₃	—	2	1	no	—
116	C₄⋊C₄×C₄₃	—	4	1	no	—
117	C₄⋊C₄×C₄₃	—	4	2	no	—
118	C₄⋊C₄×C₄₃	—	4	1	no	—
119	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
120	C₂²⋊C₄×C₄₃	—	4	1	no	—
121	C₂²⋊C₄×C₄₃	—	4	2	no	—
122	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
123	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
124	C₂²⋊Dic₄₃	—	4	2	no	—
125	(C₂×D₄₃)⋊C₄	—	4	1	no	—
126	C₄⋊Dic₄₃	—	2	1	no	—
127	C₄⋊Dic₄₃	—	4	1	no	—
128	C₄⋊Dic₄₃	—	4	1	no	—
129	Dic₄₃⋊C₄	—	4	2	no	—
130	Dic₁₇₂	—	2	2	no	—
131	Dic₁₇₂	—	2	2	no	—
132	Dic₁₇₂	—	8	1	no	—
133	D₃₄₄	—	2	1	no	—
134	D₃₄₄	C₂×D₄	1	2	yes	—
135	#10621873	—	4	4	no	—
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	C₂²×D₄₃	—	4	1	no	—
142	Q₈×C₄₃	—	4	2	no	—
143	Q₈×C₄₃	—	4	2	no	—
144	D₄×C₄₃	—	2	2	no	—
145	D₄×C₄₃	—	2	2	no	—
146	D₄×C₄₃	—	4	1	no	—
147	D₄×C₄₃	—	8	4	no	—
148	C₂×C₄×C₄₃	—	2	2	no	—
149	C₂×C₄×C₄₃	—	2	2	no	—
150	C₂×C₄×C₄₃	—	4	1	no	—
151	C₂×C₄×C₄₃	—	4	2	no	—
152	C₂×C₄×C₄₃	—	8	4	no	—
153	C₂×C₄×C₄₃	—	8	2	no	—
154	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
158	C₂²⋊D₄₃	—	8	4	no	—
159	C₂×Dic₄₃	—	4	1	no	—
160	C₂×Dic₄₃	—	4	2	no	—
161	D₁₇₂	—	2	2	no	—
162	D₁₇₂	—	4	1	no	—
163	C₄×D₄₃	—	8	4	no	—
164	C₄×D₄₃	—	8	2	no	—
165	Dic₈₆	—	4	2	no	—
166	Dic₈₆	—	4	2	no	—
167	C₈×C₄₃	—	2	2	no	—
168	C₈×C₄₃	—	2	1	no	—
169	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
170	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
171	#33849	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	2	2	no	—
173	C₂²×C₄₃	—	2	2	no	—
174	C₂²×C₄₃	—	2	2	no	—
175	C₂²×C₄₃	—	4	2	no	—
176	C₂²×C₄₃	—	8	1	no	—
177	C₂²×C₄₃	—	8	1	no	—
178	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
179	C₂×D₄₃	—	4	2	no	—
180	C₂×D₄₃	—	8	1	no	—
181	C₂×D₄₃	—	8	1	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	2	2	no	—
184	C₄×C₄₃	—	8	1	no	—
185	C₄×C₄₃	—	8	2	no	—
186	C₄×C₄₃	#508	1	2	yes	complemented
187	C₄×C₄₃	D₄²	1	1	yes	characteristic
188	C₄×C₄₃	#576	1	1	yes	characteristic
189	Dic₄₃	—	8	1	no	—
190	Dic₄₃	—	8	2	no	—
191	C₂×#508	—	43	2	no	—
192	#3283	—	43	1	no	—
193	#3283	D₄₃	1	1	yes	characteristic, complemented
194	#2247	—	43	1	no	—
195	#2216	—	43	2	no	—
196	#2208	—	43	1	no	—
197	#2208	—	43	1	no	—
198	#2189	—	43	2	no	—
199	#2189	—	43	2	no	—
200	C₄×D₁₆	—	43	1	no	—
201	D₄×C₁₆	—	43	1	no	—
202	C₂×C₄₃	—	4	2	no	—
203	C₂×C₄₃	—	8	2	no	—
204	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
205	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
206	C₂×C₄₃	#3417	1	1	yes	characteristic
207	D₄₃	—	8	2	no	—
208	C₂×#135	—	43	2	no	—
209	C₂×#135	C₂×D₄₃	1	2	yes	—
210	#547	—	2	1	no	—
211	#547	—	86	1	no	—
212	#508	—	86	8	no	—
213	C₂×Q₃₂	—	43	2	no	—
214	C₂×Q₃₂	—	43	2	no	—
215	C₂×SD₃₂	—	43	2	no	—
216	C₂×SD₃₂	—	43	2	no	—
217	C₂×D₁₆	—	43	1	no	—
218	C₂²×C₁₆	—	43	2	no	—
219	#494	—	43	1	no	—
220	#494	C₂×D₄₃	1	1	yes	characteristic
221	#467	—	43	2	no	—
222	#467	C₂×D₄₃	1	2	yes	—
223	#461	—	2	1	no	—
224	#461	—	86	1	no	—
225	#452	—	2	2	no	—
226	#452	—	86	2	no	—
227	C₄×D₈	—	43	1	no	—
228	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
229	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
230	#366	—	43	1	no	—
231	C₄⋊C₁₆	—	43	1	no	—
232	#358	—	43	2	no	—
233	#358	—	43	2	no	—
234	#357	—	43	1	no	—
235	#357	—	43	1	no	—
236	C₂²⋊C₁₆	—	43	2	no	—
237	C₄×C₁₆	—	43	1	no	—
238	C₄₃	#33849	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	2	2	no	—
240	C₂×#41	—	86	2	no	—
241	#135	—	2	4	no	—
242	#135	—	4	2	no	—
243	#135	—	86	4	no	—
244	#135	—	172	2	no	—
245	C₂×Q₁₆	—	2	1	no	—
246	C₂×Q₁₆	—	43	2	no	—
247	C₂×Q₁₆	—	86	1	no	—
248	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
249	C₂×SD₁₆	—	2	2	no	—
250	C₂×SD₁₆	—	86	2	no	—
251	C₂×D₈	—	43	1	no	—
252	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
253	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	86	1	no	—
256	#123	—	4	2	no	—
257	#123	—	172	2	no	—
258	C₂²⋊Q₈	—	2	2	no	—
259	C₂²⋊Q₈	—	86	2	no	—
260	#121	—	4	1	no	—
261	#121	—	172	1	no	—
262	C₄×D₄	—	2	1	no	—
263	C₄×D₄	—	86	1	no	—
264	C₄×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
265	C₂×C₄⋊C₄	—	4	1	no	—
266	C₂×C₄⋊C₄	—	172	1	no	—
267	Q₃₂	—	86	4	no	—
268	Q₃₂	—	172	2	no	—
269	SD₃₂	—	86	4	no	—
270	SD₃₂	—	86	4	no	—
271	D₁₆	—	43	4	no	—
272	C₂×C₁₆	—	43	1	no	—
273	C₂×C₁₆	—	43	1	no	—
274	C₂×C₁₆	—	43	2	no	—
275	C₂×C₁₆	—	86	4	no	—
276	#107	—	43	1	no	—
277	#107	C₂²×D₄₃	1	1	yes	characteristic
278	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
279	#103	—	2	2	no	—
280	#103	—	86	2	no	—
281	#102	—	2	1	no	—
282	#102	—	86	1	no	—
283	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
284	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
285	#41	—	4	4	no	—
286	#41	—	8	2	no	—
287	#41	—	344	2	no	—
288	#41	—	172	4	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	86	2	no	—
291	C₂×D₄	—	2	1	no	—
292	C₂×D₄	—	4	2	no	—
293	C₂×D₄	—	86	1	no	—
294	C₂×D₄	—	172	2	no	—
295	C₂×D₄	#10621882	1	1	yes	characteristic
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	172	1	no	—
299	C₂²×C₄	—	172	2	no	—
300	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
301	Q₁₆	—	2	2	no	—
302	Q₁₆	—	2	2	no	—
303	Q₁₆	—	8	1	no	—
304	Q₁₆	—	86	2	no	—
305	Q₁₆	—	86	2	no	—
306	Q₁₆	—	344	1	no	—
307	SD₁₆	—	4	4	no	—
308	SD₁₆	—	172	4	no	—
309	D₈	—	2	1	no	—
310	D₈	—	43	2	no	—
311	D₈	—	86	1	no	—
312	D₈	D₄×D₄₃	1	2	yes	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
316	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
317	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	86	1	no	—
323	C₄⋊C₄	—	172	1	no	—
324	C₄⋊C₄	—	172	2	no	—
325	C₄⋊C₄	—	172	1	no	—
326	C₄⋊C₄	#10621882	1	1	yes	characteristic
327	C₂²⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	4	1	no	—
329	C₂²⋊C₄	—	172	2	no	—
330	C₂²⋊C₄	—	172	1	no	—
331	C₂²⋊C₄	#10621882	1	2	yes	—
332	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
333	C₁₆	—	43	2	no	—
334	C₁₆	—	86	2	no	—
335	C₁₆	—	86	1	no	—
336	C₂³	—	4	1	no	—
337	C₂³	—	172	1	no	—
338	C₂³	C₂×#10621882	1	2	yes	—
339	Q₈	—	4	2	no	—
340	Q₈	—	4	2	no	—
341	Q₈	—	172	2	no	—
342	Q₈	—	172	2	no	—
343	D₄	—	2	2	no	—
344	D₄	—	2	2	no	—
345	D₄	—	4	1	no	—
346	D₄	—	8	4	no	—
347	D₄	—	86	2	no	—
348	D₄	—	344	4	no	—
349	D₄	—	172	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	8	4	no	—
355	C₂×C₄	—	8	2	no	—
356	C₂×C₄	—	344	4	no	—
357	C₂×C₄	—	344	2	no	—
358	C₂×C₄	—	172	1	no	—
359	C₂×C₄	—	172	2	no	—
360	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
361	C₂×C₄	C₂×#10621882	1	2	yes	—
362	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
363	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
364	C₈	—	2	1	no	—
365	C₈	—	2	2	no	—
366	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
367	C₈	#13063370	1	1	yes	characteristic
368	C₂²	—	2	2	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	8	1	no	—
373	C₂²	—	8	1	no	—
374	C₂²	—	344	1	no	—
375	C₂²	—	344	1	no	—
376	C₂²	—	172	2	no	—
377	C₂²	C₂²×#10621882	1	1	yes	characteristic
378	C₄	—	2	2	no	—
379	C₄	—	2	1	no	—
380	C₄	—	8	2	no	—
381	C₄	—	8	1	no	—
382	C₄	—	344	2	no	—
383	C₄	—	344	1	no	—
384	C₄	#14583329	1	2	yes	complemented
385	C₄	D₄×C₂²⋊D₄₃	1	1	yes	characteristic
386	C₄	#14583702	1	1	yes	characteristic
387	C₂	—	4	2	no	—
388	C₂	—	8	2	no	—
389	C₂	—	344	2	no	—
390	C₂	D₄×#10621882	1	1	yes	characteristic
391	C₂	C₂×#14583329	1	1	yes	characteristic
392	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	C₄₃×#3283	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3283	D₄₃
hypocenter	C₄₃	#33849
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×#14583329	C₂	semidirect
2. #19146317	C₂	semidirect
3. #19136847	C₂	semidirect
4. #19136684	C₂	semidirect
5. #19136640	C₂	semidirect
6. #19136637	C₂	semidirect
7. #19136535	C₂	semidirect
8. #19136533	C₂	semidirect
9. C₄×#13063266	C₂	semidirect
10. D₄×C₄₃⋊C₁₆	C₂	semidirect
11. C₄₃×#3283	C₂	semidirect
59. #13063266	D₄	semidirect
186. C₄×C₄₃	#508	semidirect
193. #3283	D₄₃	semidirect
238. C₄₃	#33849	semidirect
384. C₄	#14583329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583329	2
#19146317	2
#19136847	2
#19136684	2
#19136640	2
#19136637	2
#19136535	2
#19136533	2
C₄×#13063266	2
D₄×C₄₃⋊C₁₆	2
C₄₃×#3283	2
#33849	43