Subgroups — #25196854

Order 11008.

706 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142958	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063325	C₂	1	1	yes	maximal, characteristic
3	#19136083	C₂	1	4	yes	maximal
4	#19135556	C₂	1	1	yes	maximal, characteristic
5	#19135556	C₂	1	1	yes	maximal, characteristic
6	#19135167	C₂	1	4	yes	maximal
7	#19134755	C₂	1	1	yes	maximal, characteristic
8	#19134723	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×#13063317	—	2	1	no	—
12	C₂×C₄×Dic₈₆	—	2	1	no	—
13	C₂×#13063259	C₂²	1	4	yes	—
14	#14582805	—	2	1	no	—
15	#14582797	—	2	1	no	—
16	#14582797	—	2	1	no	—
17	#14582797	—	2	1	no	—
18	#14582797	C₂²	1	2	yes	—
19	#14582797	C₂²	1	2	yes	—
20	#14582797	C₂²	1	2	yes	—
21	#14582769	C₂²	1	1	yes	characteristic
22	#14582769	C₂²	1	1	yes	characteristic
23	#14582769	C₂²	1	1	yes	characteristic
24	#14582765	—	2	1	no	—
25	C₄×C₄⋊Dic₄₃	—	2	1	no	—
26	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
27	#14582662	C₂²	1	4	yes	—
28	#14582662	C₂²	1	4	yes	—
29	#14582662	C₂²	1	4	yes	—
30	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₄³×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#336	C₂²	1	2	yes	—
35	C₂²×Dic₈₆	—	2	1	no	—
36	#13063325	—	2	2	no	—
37	#13063325	Q₈	1	4	yes	—
38	#13063317	—	4	4	no	—
39	C₄×Dic₈₆	—	4	4	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	—	2	1	no	—
42	C₂×C₄⋊Dic₄₃	—	2	1	no	—
43	C₂×C₄⋊Dic₄₃	—	2	1	no	—
44	C₂×C₄⋊Dic₄₃	—	2	1	no	—
45	C₂×C₄⋊Dic₄₃	—	2	1	no	—
46	C₂×C₄⋊Dic₄₃	—	2	1	no	—
47	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
51	C₂×Dic₄₃⋊C₄	—	2	2	no	—
52	C₂×Dic₄₃⋊C₄	—	2	2	no	—
53	C₂×Dic₄₃⋊C₄	—	2	2	no	—
54	C₂×C₄×Dic₄₃	—	2	2	no	—
55	#13063259	—	2	8	no	—
56	C₄₃⋊#95	—	2	2	no	—
57	C₄₃⋊#95	—	2	2	no	—
58	C₄₃⋊#95	—	2	2	no	—
59	C₄₃⋊#95	—	2	2	no	—
60	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₃	—	2	4	no	—
70	C₄×C₈×C₄₃	—	2	2	no	—
71	C₄×C₈×C₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₃	—	2	2	no	—
80	C₂²×Dic₄₃	—	2	2	no	—
81	C₂²×Dic₄₃	—	2	2	no	—
82	C₂²×Dic₄₃	—	2	2	no	—
83	C₂×Dic₈₆	—	2	4	no	—
84	C₂×Dic₈₆	—	4	2	no	—
85	C₂×Dic₈₆	—	4	2	no	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	—	2	2	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₂×C₈×C₄₃	#41	1	4	yes	—
92	C₄²×C₄₃	—	2	2	no	—
93	C₄²×C₄₃	—	2	2	no	—
94	C₄²×C₄₃	—	2	2	no	—
95	C₄²×C₄₃	—	2	2	no	—
96	C₄²×C₄₃	—	2	2	no	—
97	C₄²×C₄₃	D₈	1	4	yes	—
98	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
99	C₄²×C₄₃	#41	1	2	yes	—
100	C₄⋊Dic₄₃	—	2	4	no	—
101	C₄⋊Dic₄₃	—	2	2	no	—
102	C₄⋊Dic₄₃	—	2	4	no	—
103	C₄⋊Dic₄₃	—	2	4	no	—
104	C₄⋊Dic₄₃	—	2	4	no	—
105	C₄⋊Dic₄₃	—	4	2	no	—
106	C₄⋊Dic₄₃	—	4	2	no	—
107	C₄⋊Dic₄₃	—	4	2	no	—
108	C₄⋊Dic₄₃	—	4	2	no	—
109	C₄⋊Dic₄₃	—	4	2	no	—
110	C₄⋊Dic₄₃	—	4	2	no	—
111	C₄⋊Dic₄₃	C₂×Q₈	1	4	yes	—
112	Dic₄₃⋊C₄	—	4	4	no	—
113	Dic₄₃⋊C₄	—	4	4	no	—
114	Dic₄₃⋊C₄	—	4	4	no	—
115	C₄×Dic₄₃	—	4	4	no	—
116	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₄₃	#135	1	2	yes	—
133	C₂×C₄×C₄₃	#137	1	2	yes	—
134	C₂×C₄×C₄₃	#137	1	2	yes	—
135	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
138	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
139	C₂×Dic₄₃	—	2	4	no	—
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	—	4	2	no	—
143	C₂×Dic₄₃	—	4	2	no	—
144	C₂×Dic₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	4	2	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	Dic₈₆	—	4	8	no	—
152	C₈×C₄₃	—	2	4	no	—
153	C₈×C₄₃	—	4	2	no	—
154	#29178	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×C₄₃	#532	1	1	yes	characteristic
157	C₂²×C₄₃	#536	1	1	yes	characteristic
158	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
160	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	4	1	no	—
165	C₄×C₄₃	—	4	1	no	—
166	C₄×C₄₃	#474	1	4	yes	—
167	C₄×C₄₃	#479	1	4	yes	—
168	C₄×C₄₃	#552	1	2	yes	—
169	C₄×C₄₃	#557	1	2	yes	—
170	Dic₄₃	—	4	4	no	—
171	Dic₄₃	—	8	2	no	—
172	Dic₄₃	—	8	2	no	—
173	Dic₄₃	—	8	2	no	—
174	#2859	—	43	1	no	—
175	C₄×C₄⋊Q₈	—	43	1	no	—
176	#2086	—	43	4	no	—
177	#1990	—	43	1	no	—
178	#1990	—	43	1	no	—
179	#1918	—	43	4	no	—
180	#1845	—	43	1	no	—
181	#1834	—	43	1	no	—
182	C₄×C₄⋊C₈	D₄₃	1	1	yes	characteristic
183	C₂×C₄₃	#2565	1	1	yes	characteristic
184	C₂×C₄₃	C₂×#474	1	1	yes	characteristic
185	C₂×C₄₃	C₂×#479	1	1	yes	characteristic
186	C₂×C₄₃	#3169	1	1	yes	characteristic
187	C₂×C₄₃	#3185	1	1	yes	characteristic
188	C₂×C₄₃	#3226	1	1	yes	characteristic
189	C₂×C₄₃	#3231	1	1	yes	characteristic
190	C₂×C₄⋊Q₈	—	43	1	no	—
191	C₂×#125	—	86	1	no	—
192	C₂×C₄×Q₈	—	86	1	no	—
193	C₂×#107	—	43	4	no	—
194	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
195	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
197	#391	—	86	1	no	—
198	#389	—	43	2	no	—
199	#389	—	43	2	no	—
200	#389	—	43	2	no	—
201	#389	—	86	1	no	—
202	#389	—	86	1	no	—
203	#389	—	86	1	no	—
204	#384	—	43	1	no	—
205	#384	—	43	1	no	—
206	#384	—	43	1	no	—
207	#382	—	86	1	no	—
208	C₄×C₄⋊C₄	—	43	2	no	—
209	C₄×C₄⋊C₄	—	86	1	no	—
210	C₄³	C₂×D₄₃	1	1	yes	characteristic
211	#340	—	43	4	no	—
212	#340	—	43	4	no	—
213	#340	—	43	4	no	—
214	#336	C₂×D₄₃	1	2	yes	—
215	C₄₃	#29178	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	86	1	no	—
217	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
218	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
219	C₄⋊Q₈	—	43	4	no	—
220	C₄⋊Q₈	—	86	2	no	—
221	#125	—	172	4	no	—
222	C₄×Q₈	—	172	4	no	—
223	C₂×C₄⋊C₄	—	43	2	no	—
224	C₂×C₄⋊C₄	—	43	2	no	—
225	C₂×C₄⋊C₄	—	43	2	no	—
226	C₂×C₄⋊C₄	—	43	2	no	—
227	C₂×C₄⋊C₄	—	86	1	no	—
228	C₂×C₄⋊C₄	—	86	2	no	—
229	C₂×C₄⋊C₄	—	86	1	no	—
230	C₂×C₄⋊C₄	—	86	2	no	—
231	C₂×C₄⋊C₄	—	86	2	no	—
232	C₂×C₄⋊C₄	—	86	1	no	—
233	C₂×C₄⋊C₄	—	86	1	no	—
234	C₂×C₄⋊C₄	—	86	1	no	—
235	C₂×C₄⋊C₄	—	86	1	no	—
236	C₂×C₄⋊C₄	—	86	1	no	—
237	C₂×C₄²	—	86	2	no	—
238	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
239	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
240	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
241	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
245	#107	—	86	8	no	—
246	C₄⋊C₈	—	2	4	no	—
247	C₄×C₈	—	2	2	no	—
248	C₄×C₈	—	2	2	no	—
249	#95	—	86	2	no	—
250	#95	—	86	2	no	—
251	#95	—	86	2	no	—
252	#95	—	86	2	no	—
253	C₂×Q₈	—	86	4	no	—
254	C₂×Q₈	—	172	2	no	—
255	C₂×Q₈	—	172	2	no	—
256	C₂²×C₄	—	86	2	no	—
257	C₂²×C₄	—	86	2	no	—
258	C₂²×C₄	—	86	2	no	—
259	C₂²×C₄	—	86	2	no	—
260	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	#10621898	1	4	yes	—
273	C₄⋊C₄	—	43	4	no	—
274	C₄⋊C₄	—	86	4	no	—
275	C₄⋊C₄	—	86	2	no	—
276	C₄⋊C₄	—	86	4	no	—
277	C₄⋊C₄	—	86	4	no	—
278	C₄⋊C₄	—	86	4	no	—
279	C₄⋊C₄	—	172	4	no	—
280	C₄⋊C₄	—	172	2	no	—
281	C₄⋊C₄	—	172	4	no	—
282	C₄⋊C₄	—	172	4	no	—
283	C₄⋊C₄	—	172	2	no	—
284	C₄⋊C₄	—	172	2	no	—
285	C₄⋊C₄	—	172	2	no	—
286	C₄⋊C₄	—	172	2	no	—
287	C₄⋊C₄	—	172	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	172	4	no	—
294	C₄²	D₃₄₄	1	4	yes	—
295	C₄²	#10621900	1	2	yes	—
296	C₄²	Q₈×D₄₃	1	2	yes	—
297	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
298	Q₈	—	172	8	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	86	4	no	—
310	C₂×C₄	—	172	2	no	—
311	C₂×C₄	—	172	2	no	—
312	C₂×C₄	—	172	2	no	—
313	C₂×C₄	—	172	2	no	—
314	C₂×C₄	—	172	2	no	—
315	C₂×C₄	—	172	2	no	—
316	C₂×C₄	—	172	2	no	—
317	C₂×C₄	—	172	2	no	—
318	C₂×C₄	—	172	2	no	—
319	C₂×C₄	—	172	2	no	—
320	C₂×C₄	—	172	2	no	—
321	C₂×C₄	#13063305	1	2	yes	—
322	C₂×C₄	#13063305	1	2	yes	—
323	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
324	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
325	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
326	C₂×C₄	#13063347	1	2	yes	—
327	C₂×C₄	#13063358	1	2	yes	—
328	C₂×C₄	#13063358	1	2	yes	—
329	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
330	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
331	C₂×C₄	C₂×#10621898	1	2	yes	—
332	C₂×C₄	#13063372	1	2	yes	—
333	C₈	—	2	4	no	—
334	C₈	—	4	2	no	—
335	C₂²	C₂×#13063305	1	1	yes	characteristic
336	C₂²	#14583449	1	1	yes	characteristic
337	C₂²	#14583475	1	1	yes	characteristic
338	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
339	C₂²	C₂×#13063347	1	1	yes	characteristic
340	C₂²	C₂×#13063358	1	1	yes	characteristic
341	C₂²	C₂×#13063358	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	344	2	no	—
347	C₄	—	344	2	no	—
348	C₄	—	344	2	no	—
349	C₄	—	172	4	no	—
350	C₄	#14583180	1	4	yes	—
351	C₄	#14583207	1	4	yes	—
352	C₄	#14583589	1	2	yes	—
353	C₄	#14583627	1	2	yes	—
354	C₂	#19139786	1	1	yes	characteristic
355	C₂	C₂×#14583180	1	1	yes	characteristic
356	C₂	C₂×#14583207	1	1	yes	characteristic
357	C₂	#19145174	1	1	yes	characteristic
358	C₂	#19145340	1	1	yes	characteristic
359	C₂	#19145718	1	1	yes	characteristic
360	C₂	#19145756	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×C₄⋊C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₄×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#29178
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
215. C₄₃	#29178	semidirect

Maximal subgroups

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

subgroup	index
#19142958	2
C₄×#13063325	2
#19136083	2
#19135556	2
#19135167	2
#19134755	2
#19134723	2
C₄×C₄⋊C₈×C₄₃	2
#29178	43