Subgroups — #24999192

Order 11008.

626 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583278	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583180	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583176	C₂	1	1	yes	maximal, characteristic
4	#19135388	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582646	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132429	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621885	—	2	2	no	—
9	C₂×#13063325	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
11	#14583278	—	2	2	no	—
12	#14583278	—	2	2	no	—
13	#14583180	—	2	2	no	—
14	#14583180	—	2	2	no	—
15	#14583176	—	2	2	no	—
16	#14583176	—	2	2	no	—
17	C₂×#13063259	—	2	1	no	—
18	C₂×#13063258	—	2	1	no	—
19	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#13063247	—	2	1	no	—
21	C₂×#13063245	—	2	1	no	—
22	C₂×(C₂×D₄₃)⋊C₈	—	2	2	no	—
23	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
24	#14582646	—	2	2	no	—
25	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#332	—	2	2	no	—
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₃	—	2	1	no	—
32	C₂×#10621885	—	2	8	no	—
33	C₂×#10621885	—	4	4	no	—
34	C₂×#10621885	—	4	4	no	—
35	C₂²×C₄₃⋊C₈	—	2	1	no	—
36	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
37	#13063325	—	2	1	no	—
38	#13063325	—	2	1	no	—
39	#13063325	—	2	1	no	—
40	#13063325	D₄	1	2	yes	—
41	C₄×D₁₇₂	—	2	1	no	—
42	C₄×D₁₇₂	—	2	1	no	—
43	C₄×D₁₇₂	—	2	1	no	—
44	C₄×D₁₇₂	—	2	1	no	—
45	C₂×C₄⋊Dic₄₃	—	2	1	no	—
46	C₂×C₄⋊Dic₄₃	—	2	1	no	—
47	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
48	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
49	#13063259	—	2	4	no	—
50	#13063258	—	2	4	no	—
51	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
52	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
53	#13063247	—	2	4	no	—
54	#13063245	—	2	4	no	—
55	(C₂×D₄₃)⋊C₈	—	4	4	no	—
56	C₄×C₄₃⋊C₈	—	2	1	no	—
57	C₄×C₄₃⋊C₈	—	2	1	no	—
58	C₂²×Q₈×C₄₃	—	2	1	no	—
59	C₂²×C₈×C₄₃	—	2	1	no	—
60	C₂²×C₈×C₄₃	—	2	1	no	—
61	C₄⋊Q₈×C₄₃	—	2	1	no	—
62	C₄⋊Q₈×C₄₃	—	2	1	no	—
63	C₄⋊Q₈×C₄₃	—	2	1	no	—
64	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
67	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₄₃	—	2	1	no	—
69	C₄⋊C₈×C₄₃	—	2	1	no	—
70	C₄⋊C₈×C₄₃	—	2	1	no	—
71	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
72	C₂³×D₄₃	—	2	1	no	—
73	C₂×Q₈×C₄₃	—	2	2	no	—
74	C₂×Q₈×C₄₃	—	4	1	no	—
75	C₂×Q₈×C₄₃	—	4	2	no	—
76	C₂×Q₈×C₄₃	—	4	2	no	—
77	C₂²×C₄×C₄₃	—	4	1	no	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
81	C₂²×Dic₄₃	—	2	1	no	—
82	C₂²×Dic₄₃	—	4	1	no	—
83	C₂×D₁₇₂	—	2	1	no	—
84	C₂×D₁₇₂	—	2	1	no	—
85	C₂×D₁₇₂	—	2	1	no	—
86	C₂×D₁₇₂	—	2	1	no	—
87	C₂×D₁₇₂	—	2	1	no	—
88	C₂×D₁₇₂	Q₁₆	1	2	yes	complemented
89	C₂×C₄×D₄₃	—	4	1	no	—
90	C₂×C₄×D₄₃	—	4	1	no	—
91	C₂×C₄×D₄₃	—	4	2	no	—
92	C₂×C₄×D₄₃	—	4	2	no	—
93	C₂×Dic₈₆	—	2	2	no	—
94	C₂×Dic₈₆	—	4	2	no	—
95	C₂×Dic₈₆	—	4	2	no	—
96	C₂×Dic₈₆	—	4	1	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	4	1	no	—
99	C₂×C₈×C₄₃	—	4	1	no	—
100	C₂×C₈×C₄₃	—	4	1	no	—
101	C₂×C₈×C₄₃	—	4	2	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	4	1	no	—
105	C₄⋊C₄×C₄₃	—	4	1	no	—
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
110	#10621885	—	4	16	no	—
111	(C₂×D₄₃)⋊C₄	—	4	1	no	—
112	(C₂×D₄₃)⋊C₄	—	4	1	no	—
113	C₄⋊Dic₄₃	—	2	2	no	—
114	C₄⋊Dic₄₃	—	2	1	no	—
115	C₄⋊Dic₄₃	—	2	1	no	—
116	C₄⋊Dic₄₃	—	2	2	no	—
117	C₄⋊Dic₄₃	—	4	1	no	—
118	C₄⋊Dic₄₃	—	4	1	no	—
119	C₂×C₄₃⋊C₈	—	2	2	no	—
120	C₂×C₄₃⋊C₈	—	2	2	no	—
121	C₂×C₄₃⋊C₈	—	2	2	no	—
122	C₂×C₄₃⋊C₈	—	4	1	no	—
123	C₂×C₄₃⋊C₈	—	4	1	no	—
124	C₂×C₄₃⋊C₈	D₈	1	4	yes	—
125	C₂³×C₄₃	#121	1	1	yes	characteristic
126	C₂²×D₄₃	—	2	2	no	—
127	C₂²×D₄₃	—	4	1	no	—
128	C₂²×D₄₃	—	4	2	no	—
129	C₂²×D₄₃	—	4	1	no	—
130	C₂²×D₄₃	—	4	2	no	—
131	Q₈×C₄₃	—	4	4	no	—
132	Q₈×C₄₃	—	8	1	no	—
133	Q₈×C₄₃	—	8	1	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	8	2	no	—
137	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
146	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
148	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	4	1	no	—
151	C₂×Dic₄₃	—	4	2	no	—
152	C₂×Dic₄₃	—	8	2	no	—
153	D₁₇₂	—	2	2	no	—
154	D₁₇₂	—	2	2	no	—
155	D₁₇₂	—	4	1	no	—
156	D₁₇₂	—	4	1	no	—
157	C₄×D₄₃	—	8	2	no	—
158	C₄×D₄₃	—	8	1	no	—
159	C₄×D₄₃	—	8	1	no	—
160	Dic₈₆	—	4	4	no	—
161	Dic₈₆	—	8	1	no	—
162	Dic₈₆	—	8	1	no	—
163	C₈×C₄₃	—	4	2	no	—
164	C₈×C₄₃	—	8	1	no	—
165	C₄₃⋊C₈	—	2	4	no	—
166	C₄₃⋊C₈	—	4	2	no	—
167	#10274	—	43	1	no	maximal, Sylow
168	C₂²×C₄₃	#392	1	1	yes	characteristic
169	C₂²×C₄₃	#459	1	2	yes	—
170	C₂²×C₄₃	#467	1	1	yes	characteristic
171	C₂²×C₄₃	#470	1	1	yes	characteristic
172	C₂²×C₄₃	#473	1	2	yes	—
173	C₂×D₄₃	—	4	4	no	—
174	C₂×D₄₃	—	4	2	no	—
175	C₂×D₄₃	—	8	2	no	—
176	C₂×D₄₃	—	8	1	no	—
177	C₂×D₄₃	—	8	1	no	—
178	C₄×C₄₃	—	2	1	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	8	2	no	—
181	C₄×C₄₃	#447	1	2	yes	—
182	C₄×C₄₃	#451	1	1	yes	characteristic
183	C₄×C₄₃	#451	1	1	yes	characteristic
184	C₄×C₄₃	#452	1	1	yes	characteristic
185	C₄×C₄₃	#452	1	1	yes	characteristic
186	C₄×C₄₃	#493	1	1	yes	characteristic
187	C₄×C₄₃	#496	1	1	yes	characteristic
188	Dic₄₃	—	8	2	no	—
189	Dic₄₃	—	8	1	no	—
190	C₂×#494	—	43	1	no	—
191	C₂×#474	—	43	1	no	—
192	C₂×#474	—	43	1	no	—
193	#1954	—	43	1	no	—
194	C₂×#332	—	43	1	no	—
195	C₂×#332	D₄₃	1	1	yes	characteristic, complemented
196	#1277	—	43	1	no	—
197	C₂×C₄₃	#1669	1	2	yes	—
198	C₂×C₄₃	#1689	1	2	yes	—
199	C₂×C₄₃	#2001	1	1	yes	characteristic
200	C₂×C₄₃	#2011	1	1	yes	characteristic
201	C₂×C₄₃	#2020	1	1	yes	characteristic
202	D₄₃	—	8	2	no	—
203	C₂²×Q₁₆	—	86	2	no	—
204	C₂×C₄⋊Q₈	—	43	1	no	—
205	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
206	C₂×C₄×D₄	—	43	1	no	—
207	#494	—	86	2	no	—
208	#494	—	86	2	no	—
209	#474	—	86	2	no	—
210	#474	—	86	2	no	—
211	#474	—	86	2	no	—
212	#474	—	86	2	no	—
213	C₂×#107	—	86	1	no	—
214	C₂×#107	—	86	1	no	—
215	C₂×C₄⋊C₈	—	43	1	no	—
216	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
217	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
218	C₂×#102	—	86	1	no	—
219	C₂×#102	—	86	1	no	—
220	C₂×C₂²⋊C₈	—	86	2	no	—
221	C₂×C₄×C₈	—	43	1	no	—
222	#332	—	2	2	no	—
223	#332	—	86	2	no	—
224	C₄₃	#10274	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	2	1	no	—
226	C₂²×Q₈	—	86	1	no	—
227	C₂²×D₄	—	43	1	no	—
228	C₂³×C₄	—	86	1	no	—
229	C₂×Q₁₆	—	86	8	no	—
230	C₂×Q₁₆	—	172	4	no	—
231	C₂×Q₁₆	—	172	4	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	2	1	no	—
234	C₂²×C₈	—	43	2	no	—
235	C₂²×C₈	—	86	1	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	43	2	no	—
240	C₄⋊Q₈	—	86	1	no	—
241	C₄⋊Q₈	—	86	1	no	—
242	C₄⋊Q₈	—	86	1	no	—
243	C₄⋊Q₈	D₁₇₂	1	2	yes	—
244	C₄×D₄	—	86	1	no	—
245	C₄×D₄	—	86	1	no	—
246	C₄×D₄	—	86	1	no	—
247	C₄×D₄	—	86	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	43	1	no	—
251	C₂×C₄⋊C₄	—	86	1	no	—
252	C₂×C₄⋊C₄	—	86	1	no	—
253	C₂×C₂²⋊C₄	—	86	1	no	—
254	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
255	#107	—	86	4	no	—
256	#107	—	86	4	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	43	2	no	—
261	C₄⋊C₈	—	86	1	no	—
262	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
263	#102	—	86	4	no	—
264	#102	—	86	4	no	—
265	C₂²⋊C₈	—	172	4	no	—
266	C₄×C₈	—	86	1	no	—
267	C₄×C₈	—	86	1	no	—
268	C₂⁴	—	86	1	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	4	1	no	—
271	C₂×Q₈	—	4	2	no	—
272	C₂×Q₈	—	4	2	no	—
273	C₂×Q₈	—	86	2	no	—
274	C₂×Q₈	—	172	2	no	—
275	C₂×Q₈	—	172	2	no	—
276	C₂×Q₈	—	172	1	no	—
277	C₂×D₄	—	43	2	no	—
278	C₂×D₄	—	86	1	no	—
279	C₂×D₄	—	86	1	no	—
280	C₂×D₄	—	86	1	no	—
281	C₂×D₄	—	86	1	no	—
282	C₂×D₄	—	86	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	86	1	no	—
285	C₂²×C₄	—	172	1	no	—
286	C₂²×C₄	—	172	1	no	—
287	C₂²×C₄	—	172	1	no	—
288	C₂²×C₄	—	172	2	no	—
289	C₂²×C₄	—	172	2	no	—
290	C₂²×C₄	#10621898	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
293	Q₁₆	—	172	16	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	4	1	no	—
296	C₂×C₈	—	4	1	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	4	2	no	—
299	C₂×C₈	—	43	4	no	—
300	C₂×C₈	—	86	2	no	—
301	C₂×C₈	—	86	2	no	—
302	C₂×C₈	—	86	2	no	—
303	C₂×C₈	—	172	1	no	—
304	C₂×C₈	—	172	1	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	86	2	no	—
310	C₄⋊C₄	—	86	1	no	—
311	C₄⋊C₄	—	86	1	no	—
312	C₄⋊C₄	—	86	2	no	—
313	C₄⋊C₄	—	172	1	no	—
314	C₄⋊C₄	—	172	1	no	—
315	C₂²⋊C₄	—	172	1	no	—
316	C₂²⋊C₄	—	172	1	no	—
317	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
318	C₄²	D₄×D₄₃	1	1	yes	characteristic
319	C₄²	D₄×D₄₃	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
321	C₂³	—	86	2	no	—
322	C₂³	—	172	1	no	—
323	C₂³	—	172	2	no	—
324	C₂³	—	172	1	no	—
325	C₂³	—	172	2	no	—
326	C₂³	#13063298	1	1	yes	characteristic
327	Q₈	—	4	4	no	—
328	Q₈	—	8	1	no	—
329	Q₈	—	8	1	no	—
330	Q₈	—	344	1	no	—
331	Q₈	—	344	1	no	—
332	Q₈	—	172	4	no	—
333	D₄	—	86	2	no	—
334	D₄	—	86	2	no	—
335	D₄	—	172	1	no	—
336	D₄	—	172	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	—	344	2	no	—
341	C₂×C₄	—	344	1	no	—
342	C₂×C₄	—	344	1	no	—
343	C₂×C₄	—	344	2	no	—
344	C₂×C₄	—	172	1	no	—
345	C₂×C₄	—	172	2	no	—
346	C₂×C₄	—	172	2	no	—
347	C₂×C₄	#13063293	1	1	yes	characteristic
348	C₂×C₄	#13063295	1	1	yes	characteristic
349	C₂×C₄	#13063295	1	1	yes	characteristic
350	C₂×C₄	#13063298	1	1	yes	characteristic
351	C₂×C₄	#13063299	1	1	yes	characteristic
352	C₂×C₄	#13063321	1	1	yes	characteristic
353	C₂×C₄	D₈×D₄₃	1	2	yes	—
354	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
355	C₂×C₄	#13063346	1	1	yes	characteristic
356	C₂×C₄	#13063352	1	2	yes	—
357	C₂×C₄	#13063354	1	1	yes	characteristic
358	C₂×C₄	#13063357	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	8	1	no	—
361	C₈	—	86	4	no	—
362	C₈	—	172	2	no	—
363	C₂²	—	344	2	no	—
364	C₂²	—	344	1	no	—
365	C₂²	—	344	1	no	—
366	C₂²	—	172	4	no	—
367	C₂²	—	172	2	no	—
368	C₂²	#14582808	1	1	yes	characteristic
369	C₂²	#14583093	1	2	yes	—
370	C₂²	#14583139	1	1	yes	characteristic
371	C₂²	#14583156	1	1	yes	characteristic
372	C₂²	#14583173	1	2	yes	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	344	1	no	—
377	C₄	—	344	2	no	—
378	C₄	#14583034	1	2	yes	—
379	C₄	#14583053	1	1	yes	characteristic
380	C₄	#14583056	1	1	yes	characteristic
381	C₄	#14583058	1	1	yes	characteristic
382	C₄	#14583064	1	1	yes	characteristic
383	C₄	#14583274	1	1	yes	characteristic
384	C₄	#14583285	1	1	yes	characteristic
385	C₂	—	344	2	no	—
386	C₂	#19133959	1	2	yes	—
387	C₂	#19134083	1	2	yes	—
388	C₂	#19135604	1	1	yes	characteristic
389	C₂	#19135667	1	1	yes	characteristic
390	C₂	#19135715	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063298
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#332	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	C₂×#332	D₄₃
hypocenter	C₄₃	#10274
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
1. C₂×#14583278	C₂	semidirect
5. C₂×#14582646	C₂	semidirect
6. #19132429	C₂	semidirect
7. C₂×C₄₃×#332	C₂	semidirect
88. C₂×D₁₇₂	Q₁₆	semidirect
195. C₂×#332	D₄₃	semidirect
224. C₄₃	#10274	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583278	2
C₂×#14583180	2
C₂×#14583176	2
#19135388	2
C₂×#14582646	2
#19132429	2
C₂×C₄₃×#332	2
#10274	43