Subgroups — #25251193

Order 11008.

636 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583342	C₂	1	1	yes	maximal, characteristic, complemented
2	#19147697	C₂	1	1	yes	maximal, characteristic, complemented
3	#19144316	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583175	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136726	C₂	1	4	yes	maximal, complemented
6	#19136721	C₂	1	4	yes	maximal, complemented
7	#19136342	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14582714	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583699	—	2	1	no	—
11	C₂×#13063358	—	2	1	no	—
12	C₂×#13063347	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
14	C₂×#13063331	C₂²	1	1	yes	characteristic
15	#14583342	—	2	2	no	—
16	#14583342	—	2	2	no	—
17	C₂×Dic₃₄₄	C₂²	1	2	yes	complemented
18	C₂×#13063270	C₂²	1	2	yes	complemented
19	#14583317	—	2	2	no	—
20	#14583315	C₂²	1	4	yes	—
21	C₂×#13063265	C₂²	1	1	yes	characteristic
22	M₃₂×D₄₃	—	2	2	no	—
23	#14583175	—	2	2	no	—
24	#14583175	C₂²	1	4	yes	—
25	#14583175	C₂²	1	2	yes	—
26	#14583170	C₂²	1	2	yes	—
27	#14583163	—	2	2	no	—
28	#14582881	—	2	1	no	—
29	C₂×#13063243	—	2	1	no	—
30	#14582714	C₂²	1	4	yes	complemented
31	#14582714	C₂²	1	2	yes	complemented
32	#14582711	C₂²	1	2	yes	complemented
33	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
36	C₄₃×#368	C₂²	1	4	yes	complemented
37	C₂×#10621898	—	2	1	no	—
38	C₂×#10621898	D₄	1	1	yes	characteristic
39	C₂²×Dic₈₆	—	2	1	no	—
40	#13063358	—	2	2	no	—
41	#13063358	—	2	4	no	—
42	#13063358	—	4	2	no	—
43	#13063358	—	4	1	no	—
44	#13063353	—	2	2	no	—
45	#13063353	—	4	1	no	—
46	#13063347	—	2	1	no	—
47	#13063347	—	2	1	no	—
48	#13063347	—	2	1	no	—
49	#13063347	—	2	1	no	—
50	#13063347	—	4	2	no	—
51	C₂×Dic₁₇₂	—	2	1	no	—
52	C₂×Dic₁₇₂	—	2	1	no	—
53	C₂×Dic₁₇₂	—	2	1	no	—
54	C₂×Dic₁₇₂	—	2	1	no	—
55	C₂×Dic₁₇₂	—	2	1	no	—
56	C₂×Dic₁₇₂	—	4	1	no	—
57	C₂×Dic₁₇₂	C₂³	1	2	yes	—
58	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
59	C₂×#10621873	—	2	1	no	—
60	C₂×#10621873	—	4	1	no	—
61	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
62	#13063331	—	2	1	no	—
63	#13063331	—	2	1	no	—
64	#13063331	D₄	1	2	yes	—
65	#13063331	C₂³	1	2	yes	—
66	C₂×#10621877	D₄	1	1	yes	characteristic
67	C₂×#10621872	—	2	1	no	—
68	C₂×C₈×D₄₃	—	2	1	no	—
69	Dic₃₄₄	—	2	4	no	—
70	#13063270	—	2	4	no	—
71	#13063265	—	2	1	no	—
72	#13063265	C₂³	1	2	yes	—
73	#13063264	—	2	4	no	—
74	C₂×C₄₃⋊C₁₆	—	2	1	no	—
75	C₁₆×D₄₃	—	2	4	no	—
76	#13063243	—	2	4	no	—
77	#13063243	—	2	2	no	—
78	#13063241	—	2	2	no	—
79	C₂×M₁₆×C₄₃	—	2	1	no	—
80	C₂×M₁₆×C₄₃	—	2	1	no	—
81	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	M₃₂×C₄₃	D₄	1	2	yes	—
83	M₃₂×C₄₃	D₄	1	2	yes	—
84	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
85	#108×C₄₃	—	2	1	no	—
86	#108×C₄₃	—	2	1	no	—
87	#108×C₄₃	C₂³	1	2	yes	—
88	#108×C₄₃	C₂³	1	2	yes	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₃	—	2	1	no	—
91	C₂×C₂²⋊D₄₃	—	4	1	no	—
92	C₂²×Dic₄₃	—	4	1	no	—
93	#10621898	—	2	1	no	—
94	#10621898	—	2	1	no	—
95	#10621898	—	4	1	no	—
96	#10621898	—	4	1	no	—
97	#10621898	—	4	1	no	—
98	#10621898	—	4	1	no	—
99	#10621898	D₈	1	2	yes	—
100	#10621898	C₂×D₄	1	2	yes	—
101	C₂×D₁₇₂	—	2	1	no	—
102	C₂×D₁₇₂	D₈	1	1	yes	characteristic
103	C₂×C₄×D₄₃	—	2	1	no	—
104	C₂×C₄×D₄₃	—	4	1	no	—
105	C₂×Dic₈₆	—	2	1	no	—
106	C₂×Dic₈₆	—	2	2	no	—
107	C₂×Dic₈₆	—	4	1	no	—
108	C₂×Dic₈₆	—	4	1	no	—
109	C₂×Dic₈₆	—	4	1	no	—
110	C₂×Dic₈₆	—	4	1	no	—
111	C₂×Dic₈₆	—	4	1	no	—
112	C₂×Dic₈₆	D₈	1	1	yes	characteristic
113	M₁₆×C₄₃	—	2	2	no	—
114	M₁₆×C₄₃	—	2	2	no	—
115	M₁₆×C₄₃	—	4	1	no	—
116	M₁₆×C₄₃	—	4	1	no	—
117	C₂×C₈×C₄₃	—	4	1	no	—
118	C₂×C₈×C₄₃	—	4	1	no	—
119	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
123	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
124	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
125	#10621877	—	2	1	no	—
126	#10621877	C₂×D₄	1	2	yes	—
127	C₂×C₄₃⋊C₈	—	2	1	no	—
128	Dic₁₇₂	—	2	2	no	—
129	Dic₁₇₂	—	2	1	no	—
130	Dic₁₇₂	—	2	2	no	—
131	Dic₁₇₂	—	2	1	no	—
132	Dic₁₇₂	—	4	4	no	—
133	Dic₁₇₂	—	4	2	no	—
134	Dic₁₇₂	—	4	1	no	—
135	Dic₁₇₂	—	4	1	no	—
136	D₃₄₄	—	2	1	no	—
137	D₃₄₄	—	2	1	no	—
138	D₃₄₄	—	4	2	no	—
139	#10621873	—	4	4	no	—
140	#10621873	—	4	2	no	—
141	#10621873	—	4	2	no	—
142	#10621873	—	4	1	no	—
143	#10621873	—	4	1	no	—
144	#10621872	—	2	2	no	—
145	#10621872	—	2	2	no	—
146	C₈×D₄₃	—	2	2	no	—
147	C₈×D₄₃	—	2	2	no	—
148	C₁₆×C₄₃	C₂×D₄	1	4	yes	—
149	C₄₃⋊C₁₆	—	2	2	no	—
150	C₂³×C₄₃	#135	1	1	yes	characteristic
151	C₂²×D₄₃	—	2	1	no	—
152	C₂²×D₄₃	—	4	1	no	—
153	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
155	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
158	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
159	C₂²⋊D₄₃	—	2	2	no	—
160	C₂²⋊D₄₃	—	2	2	no	—
161	C₂²⋊D₄₃	—	8	1	no	—
162	C₂²⋊D₄₃	—	8	1	no	—
163	C₂×Dic₄₃	—	2	1	no	—
164	C₂×Dic₄₃	—	4	2	no	—
165	C₂×Dic₄₃	—	4	1	no	—
166	C₂×Dic₄₃	—	8	1	no	—
167	C₂×Dic₄₃	—	8	1	no	—
168	D₁₇₂	—	2	1	no	—
169	D₁₇₂	—	4	1	no	—
170	D₁₇₂	—	4	1	no	—
171	D₁₇₂	C₂×D₈	1	2	yes	—
172	C₄×D₄₃	—	2	2	no	—
173	C₄×D₄₃	—	2	2	no	—
174	C₄×D₄₃	—	8	1	no	—
175	C₄×D₄₃	—	8	1	no	—
176	Dic₈₆	—	2	1	no	—
177	Dic₈₆	—	4	1	no	—
178	Dic₈₆	—	4	2	no	—
179	Dic₈₆	—	4	2	no	—
180	Dic₈₆	—	4	1	no	—
181	Dic₈₆	—	8	1	no	—
182	Dic₈₆	—	8	1	no	—
183	Dic₈₆	C₂×D₈	1	2	yes	—
184	C₈×C₄₃	—	4	2	no	—
185	C₈×C₄₃	—	4	2	no	—
186	C₈×C₄₃	#121	1	1	yes	characteristic
187	C₈×C₄₃	#121	1	1	yes	characteristic
188	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
189	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
190	C₄₃⋊C₈	—	2	2	no	—
191	#33619	—	43	1	no	maximal, Sylow
192	C₂²×C₄₃	—	4	1	no	—
193	C₂²×C₄₃	#466	1	1	yes	characteristic
194	C₂²×C₄₃	#466	1	1	yes	characteristic
195	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
196	C₂×D₄₃	—	2	2	no	—
197	C₂×D₄₃	—	8	1	no	—
198	C₂×D₄₃	—	8	1	no	—
199	C₂×D₄₃	—	16	1	no	—
200	C₄×C₄₃	#466	1	1	yes	characteristic
201	C₄×C₄₃	#466	1	1	yes	characteristic
202	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
203	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
204	Dic₄₃	—	2	2	no	—
205	Dic₄₃	—	8	2	no	—
206	Dic₄₃	—	8	1	no	—
207	C₂×#510	—	43	1	no	—
208	#3407	—	43	1	no	—
209	#3067	—	43	1	no	—
210	C₂×#473	—	43	1	no	—
211	#2223	—	43	4	no	—
212	#2222	—	43	4	no	—
213	#2161	D₄₃	1	1	yes	characteristic, complemented
214	#2148	—	43	1	no	—
215	C₂×#362	—	43	1	no	—
216	C₂×C₄₃	—	2	1	no	—
217	C₂×C₄₃	—	2	1	no	—
218	C₂×C₄₃	—	2	1	no	—
219	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
220	D₄₃	—	4	2	no	—
221	D₄₃	—	16	1	no	—
222	#575	—	86	1	no	—
223	C₂×#137	—	86	1	no	—
224	C₂×#135	—	43	1	no	—
225	C₂²×Q₁₆	—	43	1	no	—
226	C₂×#131	—	43	1	no	—
227	#510	—	86	2	no	—
228	#510	—	86	2	no	—
229	C₂×Q₃₂	—	43	2	no	—
230	C₂×SD₃₂	—	43	2	no	—
231	#504	—	43	4	no	—
232	#504	—	86	2	no	—
233	C₂×M₃₂	—	43	1	no	—
234	C₂×M₃₂	—	86	2	no	—
235	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
236	#473	—	43	2	no	—
237	#473	—	43	4	no	—
238	#473	—	86	2	no	—
239	#472	—	43	2	no	—
240	#471	—	86	2	no	—
241	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
242	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
243	#413	—	86	1	no	—
244	C₂×#101	—	86	1	no	—
245	#368	C₂×D₄₃	1	4	yes	complemented
246	#362	—	43	4	no	—
247	#362	—	43	2	no	—
248	#361	—	43	2	no	—
249	C₄₃	#33619	1	1	yes	characteristic, complemented, Sylow
250	C₂×#41	—	43	1	no	—
251	C₂×#41	—	86	1	no	—
252	C₂²×Q₈	—	86	1	no	—
253	#137	—	86	4	no	—
254	#137	—	86	2	no	—
255	#137	—	172	1	no	—
256	#137	—	172	2	no	—
257	AGL(1,ℤ/8)	—	86	2	no	—
258	AGL(1,ℤ/8)	—	172	1	no	—
259	#135	—	86	1	no	—
260	#135	—	86	1	no	—
261	#135	—	86	1	no	—
262	#135	—	86	1	no	—
263	#135	—	172	2	no	—
264	C₂×Q₁₆	—	43	2	no	—
265	C₂×Q₁₆	—	43	1	no	—
266	C₂×Q₁₆	—	86	1	no	—
267	C₂×Q₁₆	—	86	1	no	—
268	C₂×Q₁₆	—	86	1	no	—
269	C₂×Q₁₆	—	86	1	no	—
270	C₂×Q₁₆	—	86	1	no	—
271	C₂×Q₁₆	—	172	1	no	—
272	C₂×SD₁₆	—	86	1	no	—
273	C₂×SD₁₆	—	172	1	no	—
274	C₂×D₈	—	43	1	no	—
275	#131	—	43	2	no	—
276	#131	—	43	2	no	—
277	#131	—	86	1	no	—
278	#131	—	86	1	no	—
279	C₂×M₁₆	—	2	1	no	—
280	C₂×M₁₆	—	2	1	no	—
281	C₂×M₁₆	—	43	1	no	—
282	C₂×M₁₆	—	86	1	no	—
283	C₂²×C₈	—	86	1	no	—
284	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
285	Q₃₂	—	86	4	no	—
286	SD₃₂	—	86	4	no	—
287	M₃₂	—	43	2	no	—
288	M₃₂	—	86	1	no	—
289	M₃₂	—	86	4	no	—
290	M₃₂	D₁₇₂	1	2	yes	—
291	M₃₂	D₁₇₂	1	2	yes	—
292	C₂×C₁₆	—	86	1	no	—
293	C₂×C₁₆	—	86	4	no	—
294	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
295	#108	—	2	1	no	—
296	#108	—	2	1	no	—
297	#108	C₂²×D₄₃	1	2	yes	—
298	#108	C₂²×D₄₃	1	2	yes	—
299	#101	—	86	4	no	—
300	#101	—	86	2	no	—
301	#100	—	86	2	no	—
302	#41	—	43	2	no	—
303	#41	—	43	2	no	—
304	#41	—	86	1	no	—
305	#41	—	86	1	no	—
306	#41	—	172	1	no	—
307	#41	—	172	1	no	—
308	#41	—	172	1	no	—
309	#41	—	172	1	no	—
310	C₂×Q₈	—	43	1	no	—
311	C₂×Q₈	—	86	1	no	—
312	C₂×Q₈	—	86	2	no	—
313	C₂×Q₈	—	172	1	no	—
314	C₂×Q₈	—	172	1	no	—
315	C₂×Q₈	—	172	1	no	—
316	C₂×Q₈	—	172	1	no	—
317	C₂×Q₈	—	172	1	no	—
318	C₂×D₄	—	43	1	no	—
319	C₂×D₄	—	86	1	no	—
320	C₂×D₄	—	86	1	no	—
321	C₂×D₄	—	172	1	no	—
322	C₂²×C₄	—	86	1	no	—
323	C₂²×C₄	—	172	1	no	—
324	C₂²×C₄	—	172	1	no	—
325	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
326	Q₁₆	—	86	2	no	—
327	Q₁₆	—	86	1	no	—
328	Q₁₆	—	86	2	no	—
329	Q₁₆	—	86	1	no	—
330	Q₁₆	—	172	4	no	—
331	Q₁₆	—	172	2	no	—
332	Q₁₆	—	172	1	no	—
333	Q₁₆	—	172	1	no	—
334	SD₁₆	—	172	4	no	—
335	SD₁₆	—	172	2	no	—
336	SD₁₆	—	172	2	no	—
337	SD₁₆	—	172	1	no	—
338	SD₁₆	—	172	1	no	—
339	D₈	—	86	1	no	—
340	D₈	—	86	1	no	—
341	D₈	—	172	2	no	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	4	1	no	—
345	M₁₆	—	4	1	no	—
346	M₁₆	—	43	2	no	—
347	M₁₆	—	86	1	no	—
348	M₁₆	—	86	2	no	—
349	M₁₆	—	86	2	no	—
350	C₂×C₈	—	4	1	no	—
351	C₂×C₈	—	4	1	no	—
352	C₂×C₈	—	86	1	no	—
353	C₂×C₈	—	86	2	no	—
354	C₂×C₈	—	86	2	no	—
355	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
356	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
357	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
358	C₂×C₈	#10621902	1	1	yes	characteristic
359	C₂×C₈	#10621902	1	1	yes	characteristic
360	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
361	C₁₆	—	86	2	no	—
362	C₁₆	C₂×D₁₇₂	1	4	yes	—
363	C₂³	—	86	1	no	—
364	C₂³	—	172	1	no	—
365	C₂³	#13063348	1	1	yes	characteristic
366	Q₈	—	43	2	no	—
367	Q₈	—	86	1	no	—
368	Q₈	—	344	1	no	—
369	Q₈	—	344	1	no	—
370	Q₈	—	172	1	no	—
371	Q₈	—	172	2	no	—
372	Q₈	—	172	2	no	—
373	Q₈	—	172	1	no	—
374	D₄	—	43	2	no	—
375	D₄	—	86	2	no	—
376	D₄	—	86	2	no	—
377	D₄	—	86	1	no	—
378	D₄	—	344	1	no	—
379	D₄	—	344	1	no	—
380	D₄	—	172	1	no	—
381	D₄	—	172	1	no	—
382	C₂×C₄	—	86	1	no	—
383	C₂×C₄	—	86	2	no	—
384	C₂×C₄	—	86	2	no	—
385	C₂×C₄	—	344	1	no	—
386	C₂×C₄	—	344	1	no	—
387	C₂×C₄	—	344	1	no	—
388	C₂×C₄	—	344	1	no	—
389	C₂×C₄	—	172	1	no	—
390	C₂×C₄	—	172	2	no	—
391	C₂×C₄	#13063300	1	1	yes	characteristic
392	C₂×C₄	#13063300	1	1	yes	characteristic
393	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
394	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
395	C₂×C₄	#13063348	1	1	yes	characteristic
396	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
397	C₈	—	4	2	no	—
398	C₈	—	4	2	no	—
399	C₈	—	86	2	no	—
400	C₈	#13063300	1	1	yes	characteristic
401	C₈	#13063300	1	1	yes	characteristic
402	C₈	C₂²×D₁₇₂	1	1	yes	characteristic
403	C₈	C₂×#10621902	1	1	yes	characteristic
404	C₂²	—	4	1	no	—
405	C₂²	—	86	2	no	—
406	C₂²	—	344	1	no	—
407	C₂²	—	344	1	no	—
408	C₂²	—	688	1	no	—
409	C₂²	#14583136	1	1	yes	characteristic
410	C₂²	#14583136	1	1	yes	characteristic
411	C₂²	C₂×#13063348	1	1	yes	characteristic
412	C₄	—	86	2	no	—
413	C₄	—	344	1	no	—
414	C₄	—	344	2	no	—
415	C₄	#14583136	1	1	yes	characteristic
416	C₄	#14583136	1	1	yes	characteristic
417	C₄	C₂×#13063300	1	1	yes	characteristic
418	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
419	C₂	—	2	1	no	—
420	C₂	—	2	1	no	—
421	C₂	—	2	1	no	—
422	C₂	—	172	2	no	—
423	C₂	—	688	1	no	—
424	C₂	C₂×#14583136	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14583136
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2161	C₂
socle	C₂×C₄₃	C₂×#466
prime radical	#19144316	C₂
hypercenter	#2161	D₄₃
hypocenter	C₄₃	#33619
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

Complements

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

N	H ≅ G/N	type
1. C₂×#14583342	C₂	semidirect
2. #19147697	C₂	semidirect
4. C₂×#14583175	C₂	semidirect
5. #19136726	C₂	semidirect
6. #19136721	C₂	semidirect
7. #19136342	C₂	semidirect
8. C₂×#14582714	C₂	semidirect
9. C₄₃×#2161	C₂	semidirect
17. C₂×Dic₃₄₄	C₂²	semidirect
18. C₂×#13063270	C₂²	semidirect
30. #14582714	C₂²	semidirect
31. #14582714	C₂²	semidirect
32. #14582711	C₂²	semidirect
36. C₄₃×#368	C₂²	semidirect
213. #2161	D₄₃	semidirect
245. #368	C₂×D₄₃	semidirect
249. C₄₃	#33619	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583342	2
#19147697	2
#19144316	2
C₂×#14583175	2
#19136726	2
#19136721	2
#19136342	2
C₂×#14582714	2
C₄₃×#2161	2
#33619	43