Subgroups — #24999159

Order 11008.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583246	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583194	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14583099	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582647	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582642	C₂	1	1	yes	maximal, characteristic
6	#19132430	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621884	—	2	1	no	—
9	C₂×#13063325	C₂²	1	1	yes	characteristic
10	C₂×#13063322	C₂²	1	1	yes	characteristic
11	#14583246	—	2	2	no	—
12	#14583246	—	2	2	no	—
13	#14583194	—	2	2	no	—
14	#14583194	—	2	2	no	—
15	#14583099	—	2	2	no	—
16	#14583099	—	2	2	no	—
17	C₂×#13063258	—	2	1	no	—
18	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#13063249	—	2	1	no	—
21	C₂×#13063247	—	2	2	no	—
22	C₂×#13063245	—	2	1	no	—
23	#14582647	—	2	2	no	—
24	#14582642	—	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×#336	—	2	2	no	—
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂²×D₁₇₂	—	2	1	no	—
31	C₂²×D₁₇₂	—	2	1	no	—
32	C₂²×D₁₇₂	—	2	1	no	—
33	C₂×#10621884	—	2	4	no	—
34	C₂×#10621884	—	4	2	no	—
35	C₂×#10621884	—	4	2	no	—
36	C₂²×C₄₃⋊C₈	—	2	1	no	—
37	C₂²×C₄₃⋊C₈	—	2	1	no	—
38	#13063325	—	2	1	no	—
39	#13063325	—	2	1	no	—
40	#13063325	—	2	1	no	—
41	#13063325	D₄	1	2	yes	—
42	#13063322	—	2	1	no	—
43	#13063322	—	2	1	no	—
44	#13063322	—	2	1	no	—
45	#13063322	D₄	1	2	yes	—
46	C₂×C₄⋊Dic₄₃	—	2	1	no	—
47	C₂×C₄⋊Dic₄₃	—	2	1	no	—
48	#13063258	—	2	4	no	—
49	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
50	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
51	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
52	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
53	#13063249	—	2	4	no	—
54	#13063247	—	2	8	no	—
55	#13063245	—	2	4	no	—
56	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	—	2	1	no	—
58	C₂²×C₈×C₄₃	D₄	1	2	yes	—
59	C₄×Q₈×C₄₃	—	2	1	no	—
60	C₄×Q₈×C₄₃	—	2	1	no	—
61	C₄×Q₈×C₄₃	—	2	1	no	—
62	C₄×Q₈×C₄₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	—	2	1	no	—
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₃	—	2	1	no	—
68	C₄⋊C₈×C₄₃	—	2	1	no	—
69	C₄×C₈×C₄₃	—	2	1	no	—
70	C₄×C₈×C₄₃	—	2	1	no	—
71	C₂³×D₄₃	—	4	1	no	—
72	C₂×Q₈×C₄₃	—	2	1	no	—
73	C₂×Q₈×C₄₃	—	2	1	no	—
74	C₂×Q₈×C₄₃	—	2	1	no	—
75	C₂×Q₈×C₄₃	—	2	1	no	—
76	C₂×Q₈×C₄₃	—	2	1	no	—
77	C₂×Q₈×C₄₃	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₄₃	—	2	1	no	—
79	C₂²×C₄×C₄₃	—	2	1	no	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
83	C₂²×Dic₄₃	—	4	1	no	—
84	C₂×D₁₇₂	—	2	2	no	—
85	C₂×D₁₇₂	—	2	2	no	—
86	C₂×D₁₇₂	—	2	2	no	—
87	C₂×D₁₇₂	—	4	1	no	—
88	C₂×D₁₇₂	—	4	1	no	—
89	C₂×D₁₇₂	—	4	1	no	—
90	C₂×D₁₇₂	—	4	1	no	—
91	C₂×D₁₇₂	—	4	2	no	—
92	C₂×D₁₇₂	—	4	1	no	—
93	C₂×D₁₇₂	—	4	1	no	—
94	C₂×D₁₇₂	—	4	2	no	—
95	C₂×D₁₇₂	—	4	1	no	—
96	C₂×D₁₇₂	—	8	2	no	—
97	C₂×Dic₈₆	—	2	2	no	—
98	C₂×Dic₈₆	—	4	1	no	—
99	C₂×Dic₈₆	—	8	2	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	4	1	no	—
104	C₂×C₈×C₄₃	—	4	2	no	—
105	C₄⋊C₄×C₄₃	—	2	1	no	—
106	C₄⋊C₄×C₄₃	—	2	1	no	—
107	C₄⋊C₄×C₄₃	—	4	1	no	—
108	C₄⋊C₄×C₄₃	—	4	1	no	—
109	C₄²×C₄₃	—	4	1	no	—
110	C₄²×C₄₃	—	4	1	no	—
111	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
115	#10621884	—	4	8	no	—
116	C₄⋊Dic₄₃	—	2	2	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	C₄⋊Dic₄₃	—	4	1	no	—
119	C₄⋊Dic₄₃	—	4	1	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	—	4	1	no	—
125	C₂×C₄₃⋊C₈	—	4	1	no	—
126	C₂³×C₄₃	#121	1	1	yes	characteristic
127	C₂²×D₄₃	—	4	2	no	—
128	C₂²×D₄₃	—	8	2	no	—
129	C₂²×D₄₃	—	8	1	no	—
130	C₂²×D₄₃	—	8	1	no	—
131	C₂²×D₄₃	—	8	2	no	—
132	Q₈×C₄₃	—	2	2	no	—
133	Q₈×C₄₃	—	2	2	no	—
134	Q₈×C₄₃	—	4	1	no	—
135	Q₈×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	—	4	1	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₄×C₄₃	#135	1	2	yes	—
150	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
151	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₄₃	#137	1	2	yes	—
153	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	8	2	no	—
156	D₁₇₂	—	4	2	no	—
157	D₁₇₂	—	4	4	no	—
158	D₁₇₂	—	4	2	no	—
159	D₁₇₂	—	8	2	no	—
160	D₁₇₂	—	8	2	no	—
161	D₁₇₂	—	8	1	no	—
162	D₁₇₂	—	8	1	no	—
163	D₁₇₂	—	8	1	no	—
164	D₁₇₂	—	8	1	no	—
165	Dic₈₆	—	8	2	no	—
166	Dic₈₆	—	8	2	no	—
167	C₈×C₄₃	—	4	2	no	—
168	C₈×C₄₃	—	8	1	no	—
169	C₄₃⋊C₈	—	4	2	no	—
170	C₄₃⋊C₈	—	4	2	no	—
171	#10269	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	#392	1	1	yes	characteristic
173	C₂²×C₄₃	#464	1	2	yes	—
174	C₂²×C₄₃	#465	1	1	yes	characteristic
175	C₂²×C₄₃	#468	1	1	yes	characteristic
176	C₂²×C₄₃	#472	1	2	yes	—
177	C₂×D₄₃	—	8	2	no	—
178	C₂×D₄₃	—	8	2	no	—
179	C₂×D₄₃	—	8	2	no	—
180	C₂×D₄₃	—	16	2	no	—
181	C₂×D₄₃	—	16	2	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	2	1	no	—
184	C₄×C₄₃	—	4	2	no	—
185	C₄×C₄₃	—	8	1	no	—
186	C₄×C₄₃	#448	1	1	yes	characteristic
187	C₄×C₄₃	#449	1	1	yes	characteristic
188	C₄×C₄₃	#450	1	1	yes	characteristic
189	C₄×C₄₃	#452	1	2	yes	—
190	C₄×C₄₃	#452	1	1	yes	characteristic
191	C₄×C₄₃	#495	1	1	yes	characteristic
192	C₄×C₄₃	#497	1	1	yes	characteristic
193	Dic₄₃	—	8	2	no	—
194	C₂×#486	—	43	1	no	—
195	C₂×#477	—	43	1	no	—
196	C₂×#460	—	43	1	no	—
197	#1984	D₄₃	1	1	yes	characteristic, complemented
198	C₂×#333	—	43	1	no	—
199	C₂×#331	—	43	1	no	—
200	#1277	—	43	1	no	—
201	C₂×C₄₃	#1678	1	2	yes	—
202	C₂×C₄₃	#1698	1	2	yes	—
203	C₂×C₄₃	#2000	1	1	yes	characteristic
204	C₂×C₄₃	#2015	1	1	yes	characteristic
205	C₂×C₄₃	#2018	1	1	yes	characteristic
206	D₄₃	—	16	2	no	—
207	C₂²×SD₁₆	—	86	1	no	—
208	C₂×C₄⋊Q₈	—	43	1	no	—
209	C₂×#127	—	43	1	no	—
210	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
211	#486	—	86	2	no	—
212	#486	—	86	2	no	—
213	#477	—	86	2	no	—
214	#477	—	86	2	no	—
215	#460	—	86	2	no	—
216	#460	—	86	2	no	—
217	C₂×#107	—	86	1	no	—
218	C₂×C₄⋊C₈	—	43	1	no	—
219	C₂×C₄⋊C₈	—	43	1	no	—
220	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
221	C₂×#103	—	86	1	no	—
222	C₂×#102	—	86	2	no	—
223	C₂×#102	—	86	1	no	—
224	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
225	#336	—	2	2	no	—
226	#333	—	86	2	no	—
227	#331	—	86	2	no	—
228	C₄₃	#10269	1	1	yes	characteristic, complemented, Sylow
229	C₂²×Q₈	—	86	1	no	—
230	C₂²×Q₈	D₁₇₂	1	1	yes	characteristic
231	C₂²×D₄	—	86	1	no	—
232	C₂²×D₄	—	86	1	no	—
233	C₂²×D₄	—	86	1	no	—
234	C₂×SD₁₆	—	86	4	no	—
235	C₂×SD₁₆	—	172	2	no	—
236	C₂×SD₁₆	—	172	2	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	86	1	no	—
239	C₂²×C₈	—	86	1	no	—
240	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
241	C₄⋊Q₈	—	43	2	no	—
242	C₄⋊Q₈	—	86	1	no	—
243	C₄⋊Q₈	—	86	1	no	—
244	C₄⋊Q₈	—	86	1	no	—
245	#127	—	43	2	no	—
246	#127	—	86	1	no	—
247	#127	—	86	1	no	—
248	#127	—	86	1	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	86	1	no	—
255	C₂×C₄⋊C₄	—	86	1	no	—
256	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
259	#107	—	86	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	43	2	no	—
263	C₄⋊C₈	—	43	2	no	—
264	C₄⋊C₈	—	86	1	no	—
265	C₄⋊C₈	—	86	1	no	—
266	#103	—	86	4	no	—
267	#102	—	86	4	no	—
268	#102	—	86	8	no	—
269	C₄×C₈	—	2	1	no	—
270	C₄×C₈	—	2	1	no	—
271	C₂⁴	—	172	1	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	86	2	no	—
278	C₂×Q₈	—	344	2	no	—
279	C₂×Q₈	—	172	1	no	—
280	C₂×Q₈	#10621873	1	2	yes	—
281	C₂×D₄	—	86	2	no	—
282	C₂×D₄	—	86	2	no	—
283	C₂×D₄	—	86	2	no	—
284	C₂×D₄	—	344	2	no	—
285	C₂×D₄	—	172	1	no	—
286	C₂×D₄	—	172	1	no	—
287	C₂×D₄	—	172	1	no	—
288	C₂×D₄	—	172	1	no	—
289	C₂×D₄	—	172	2	no	—
290	C₂×D₄	—	172	1	no	—
291	C₂×D₄	—	172	1	no	—
292	C₂×D₄	—	172	2	no	—
293	C₂×D₄	—	172	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	172	1	no	—
297	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
298	C₂²×C₄	#10621898	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
300	SD₁₆	—	172	8	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	—	86	2	no	—
307	C₂×C₈	—	86	2	no	—
308	C₂×C₈	—	172	1	no	—
309	C₂×C₈	—	172	1	no	—
310	C₂×C₈	—	172	1	no	—
311	C₂×C₈	—	172	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	86	2	no	—
317	C₄⋊C₄	—	86	2	no	—
318	C₄⋊C₄	—	172	1	no	—
319	C₄⋊C₄	—	172	1	no	—
320	C₄²	—	4	1	no	—
321	C₄²	—	4	1	no	—
322	C₄²	D₄×D₄₃	1	1	yes	characteristic
323	C₄²	D₄×D₄₃	1	1	yes	characteristic
324	C₄²	D₄×D₄₃	1	1	yes	characteristic
325	C₄²	D₄×D₄₃	1	1	yes	characteristic
326	C₂³	—	344	2	no	—
327	C₂³	—	344	1	no	—
328	C₂³	—	344	1	no	—
329	C₂³	—	344	2	no	—
330	C₂³	—	172	2	no	—
331	C₂³	#13063299	1	1	yes	characteristic
332	Q₈	—	2	2	no	—
333	Q₈	—	2	2	no	—
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	344	2	no	—
337	Q₈	—	344	2	no	—
338	D₄	—	344	2	no	—
339	D₄	—	344	2	no	—
340	D₄	—	344	1	no	—
341	D₄	—	344	1	no	—
342	D₄	—	344	1	no	—
343	D₄	—	344	1	no	—
344	D₄	—	172	2	no	—
345	D₄	—	172	4	no	—
346	D₄	—	172	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	344	2	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	#13063293	1	1	yes	characteristic
356	C₂×C₄	#13063295	1	1	yes	characteristic
357	C₂×C₄	#13063295	1	1	yes	characteristic
358	C₂×C₄	#13063298	1	1	yes	characteristic
359	C₂×C₄	#13063298	1	1	yes	characteristic
360	C₂×C₄	#13063321	1	1	yes	characteristic
361	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
362	C₂×C₄	#13063345	1	2	yes	—
363	C₂×C₄	#13063347	1	1	yes	characteristic
364	C₂×C₄	#13063353	1	1	yes	characteristic
365	C₂×C₄	#13063356	1	2	yes	—
366	C₂×C₄	#13063358	1	1	yes	characteristic
367	C₈	—	4	2	no	—
368	C₈	—	8	1	no	—
369	C₈	—	172	2	no	—
370	C₈	—	172	2	no	—
371	C₂²	—	344	2	no	—
372	C₂²	—	344	2	no	—
373	C₂²	—	344	2	no	—
374	C₂²	—	688	2	no	—
375	C₂²	—	688	2	no	—
376	C₂²	#14582808	1	1	yes	characteristic
377	C₂²	#14583121	1	2	yes	—
378	C₂²	#14583128	1	1	yes	characteristic
379	C₂²	#14583145	1	1	yes	characteristic
380	C₂²	#14583169	1	2	yes	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	4	2	no	—
384	C₄	—	8	1	no	—
385	C₄	—	344	2	no	—
386	C₄	#14583040	1	1	yes	characteristic
387	C₄	#14583046	1	1	yes	characteristic
388	C₄	#14583052	1	1	yes	characteristic
389	C₄	#14583059	1	2	yes	—
390	C₄	#14583063	1	1	yes	characteristic
391	C₄	#14583281	1	1	yes	characteristic
392	C₄	#14583291	1	1	yes	characteristic
393	C₂	—	688	2	no	—
394	C₂	#19134011	1	2	yes	—
395	C₂	#19134135	1	2	yes	—
396	C₂	#19135600	1	1	yes	characteristic
397	C₂	#19135688	1	1	yes	characteristic
398	C₂	#19135704	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063299
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1984	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂×#13063322	C₂²
hypercenter	#1984	D₄₃
hypocenter	C₄₃	#10269
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

Complements

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

N	H ≅ G/N	type
2. C₂×#14583194	C₂	semidirect
4. C₂×#14582647	C₂	semidirect
6. #19132430	C₂	semidirect
7. C₄₃×#1984	C₂	semidirect
197. #1984	D₄₃	semidirect
228. C₄₃	#10269	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583246	2
C₂×#14583194	2
C₂×#14583099	2
C₂×#14582647	2
C₂×#14582642	2
#19132430	2
C₄₃×#1984	2
#10269	43