Subgroups — #25196435

Order 11008.

810 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142993	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063325	C₂	1	1	yes	maximal, characteristic
3	#19135737	C₂	1	4	yes	maximal
4	#19135397	C₂	1	2	yes	maximal
5	#19134723	C₂	1	2	yes	maximal
6	#19134601	C₂	1	4	yes	maximal
7	C₄×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063325	—	2	1	no	—
9	C₂×#13063325	—	2	1	no	—
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×#13063325	C₂²	1	2	yes	—
12	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
13	C₂×#13063250	C₂²	1	8	yes	—
14	#14582805	C₂²	1	2	yes	—
15	#14582797	—	2	1	no	—
16	#14582797	—	2	1	no	—
17	#14582769	—	2	1	no	—
18	#14582769	—	2	1	no	—
19	#14582769	C₂²	1	1	yes	characteristic
20	#14582769	C₂²	1	2	yes	—
21	#14582769	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₄₃	—	2	1	no	—
23	C₄×C₄⋊Dic₄₃	—	2	1	no	—
24	#14582662	C₂²	1	8	yes	—
25	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
27	C₄³×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#336	C₂²	1	2	yes	—
29	C₂²×Dic₈₆	—	2	2	no	—
30	C₂²×Dic₈₆	C₂³	1	2	yes	—
31	#13063325	—	2	2	no	—
32	#13063325	—	4	4	no	—
33	#13063325	—	4	4	no	—
34	#13063325	—	4	4	no	—
35	#13063325	Q₈	1	4	yes	complemented
36	C₄×Dic₈₆	—	2	8	no	—
37	C₂×C₄⋊Dic₄₃	—	2	1	no	—
38	C₂×C₄⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	—	2	2	no	—
42	C₂×C₄⋊Dic₄₃	—	2	2	no	—
43	C₂×C₄⋊Dic₄₃	—	2	2	no	—
44	C₂×C₄⋊Dic₄₃	—	2	2	no	—
45	C₂×C₄⋊Dic₄₃	—	2	2	no	—
46	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
48	C₂×Dic₄₃⋊C₄	—	2	2	no	—
49	C₂×C₄×Dic₄₃	—	2	2	no	—
50	#13063250	—	4	8	no	—
51	C₄₃⋊#95	—	2	2	no	—
52	C₄₃⋊#95	—	2	2	no	—
53	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
54	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₃	—	2	4	no	—
62	C₄×C₈×C₄₃	—	2	4	no	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	2	2	no	—
71	C₂²×Dic₄₃	—	2	2	no	—
72	C₂²×Dic₄₃	—	2	4	no	—
73	C₂×Dic₈₆	—	2	4	no	—
74	C₂×Dic₈₆	—	2	4	no	—
75	C₂×Dic₈₆	—	2	2	no	—
76	C₂×Dic₈₆	—	4	4	no	—
77	C₂×Dic₈₆	—	4	4	no	—
78	C₂×Dic₈₆	—	4	4	no	—
79	C₂×Dic₈₆	C₂×Q₈	1	4	yes	—
80	C₂×C₈×C₄₃	—	2	4	no	—
81	C₂×C₈×C₄₃	—	2	4	no	—
82	C₂×C₈×C₄₃	—	2	4	no	—
83	C₄²×C₄₃	—	2	2	no	—
84	C₄²×C₄₃	—	2	2	no	—
85	C₄²×C₄₃	—	2	2	no	—
86	C₄²×C₄₃	SD₁₆	1	4	yes	—
87	C₄²×C₄₃	Q₁₆	1	4	yes	—
88	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
89	C₄²×C₄₃	#41	1	4	yes	—
90	C₄²×C₄₃	#41	1	2	yes	—
91	C₄⋊Dic₄₃	—	2	4	no	—
92	C₄⋊Dic₄₃	—	2	4	no	—
93	C₄⋊Dic₄₃	—	2	4	no	—
94	C₄⋊Dic₄₃	—	2	8	no	—
95	C₄⋊Dic₄₃	—	4	4	no	—
96	C₄⋊Dic₄₃	—	4	4	no	—
97	C₄⋊Dic₄₃	—	4	4	no	—
98	C₄⋊Dic₄₃	—	4	4	no	—
99	C₄⋊Dic₄₃	—	4	2	no	—
100	C₄⋊Dic₄₃	—	4	4	no	—
101	C₄⋊Dic₄₃	—	4	2	no	—
102	Dic₄₃⋊C₄	—	4	4	no	—
103	C₄×Dic₄₃	—	4	4	no	—
104	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₃	#124	1	2	yes	—
114	C₂×C₄×C₄₃	#124	1	2	yes	—
115	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
121	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₄₃	#137	1	2	yes	—
123	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
127	C₂×Dic₄₃	—	2	4	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	C₂×Dic₄₃	—	4	2	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	—	4	4	no	—
132	C₂×Dic₄₃	—	4	4	no	—
133	C₂×Dic₄₃	—	4	4	no	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	—	4	2	no	—
136	Dic₈₆	—	2	8	no	—
137	Dic₈₆	—	4	4	no	—
138	Dic₈₆	—	8	8	no	—
139	C₈×C₄₃	—	4	4	no	—
140	#29141	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
143	C₂²×C₄₃	#535	1	1	yes	characteristic
144	C₂²×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
145	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
146	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	2	2	no	—
150	C₄×C₄₃	—	2	2	no	—
151	C₄×C₄₃	#475	1	4	yes	—
152	C₄×C₄₃	#477	1	4	yes	—
153	C₄×C₄₃	#487	1	4	yes	—
154	C₄×C₄₃	#554	1	2	yes	—
155	C₄×C₄₃	#555	1	2	yes	—
156	Dic₄₃	—	4	4	no	—
157	Dic₄₃	—	8	2	no	—
158	Dic₄₃	—	8	4	no	—
159	#2868	—	43	1	no	—
160	C₄×C₄⋊Q₈	—	43	1	no	—
161	#2024	—	43	4	no	—
162	#1956	—	43	2	no	—
163	#1834	—	43	2	no	—
164	#1803	—	43	4	no	—
165	C₄×C₄⋊C₈	D₄₃	1	1	yes	characteristic
166	C₂×C₄₃	#2563	1	1	yes	characteristic
167	C₂×C₄₃	C₂×#475	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#477	1	1	yes	characteristic
169	C₂×C₄₃	C₂×#487	1	1	yes	characteristic
170	C₂×C₄₃	#3141	1	1	yes	characteristic
171	C₂×C₄₃	#3208	1	1	yes	characteristic
172	C₂×C₄₃	#3209	1	1	yes	characteristic
173	C₂×C₄⋊Q₈	—	43	2	no	—
174	C₂×C₄⋊Q₈	—	43	1	no	—
175	C₂×C₄⋊Q₈	—	86	1	no	—
176	C₂×C₄⋊Q₈	—	86	1	no	—
177	C₂×C₄×Q₈	—	43	2	no	—
178	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
179	C₂×#103	—	43	8	no	—
180	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
181	#391	—	43	2	no	—
182	#389	—	86	1	no	—
183	#389	—	86	1	no	—
184	#384	—	43	1	no	—
185	#384	—	43	2	no	—
186	#384	—	43	2	no	—
187	#384	—	86	1	no	—
188	#384	—	86	1	no	—
189	C₄×C₄⋊C₄	—	86	1	no	—
190	C₄×C₄⋊C₄	—	86	1	no	—
191	C₄³	C₂×D₄₃	1	1	yes	characteristic
192	#340	—	43	8	no	—
193	#336	C₂×D₄₃	1	2	yes	—
194	C₄₃	#29141	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	43	2	no	—
196	C₂²×Q₈	—	86	2	no	—
197	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
198	C₄⋊Q₈	—	43	4	no	—
199	C₄⋊Q₈	—	86	2	no	—
200	C₄⋊Q₈	—	172	4	no	—
201	C₄⋊Q₈	—	172	4	no	—
202	C₄⋊Q₈	—	172	4	no	—
203	C₄×Q₈	—	86	8	no	—
204	C₂×C₄⋊C₄	—	43	2	no	—
205	C₂×C₄⋊C₄	—	43	4	no	—
206	C₂×C₄⋊C₄	—	86	1	no	—
207	C₂×C₄⋊C₄	—	86	1	no	—
208	C₂×C₄⋊C₄	—	86	2	no	—
209	C₂×C₄⋊C₄	—	86	2	no	—
210	C₂×C₄⋊C₄	—	86	2	no	—
211	C₂×C₄⋊C₄	—	86	2	no	—
212	C₂×C₄⋊C₄	—	86	2	no	—
213	C₂×C₄⋊C₄	—	86	2	no	—
214	C₂×C₄⋊C₄	—	86	1	no	—
215	C₂×C₄⋊C₄	—	86	1	no	—
216	C₂×C₄²	—	86	2	no	—
217	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
218	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
219	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
220	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
224	C₄⋊C₈	—	2	4	no	—
225	#103	—	172	8	no	—
226	C₄×C₈	—	2	4	no	—
227	#95	—	86	2	no	—
228	#95	—	86	2	no	—
229	C₂×Q₈	—	43	4	no	—
230	C₂×Q₈	—	86	2	no	—
231	C₂×Q₈	—	86	4	no	—
232	C₂×Q₈	—	86	4	no	—
233	C₂×Q₈	—	172	4	no	—
234	C₂×Q₈	—	172	4	no	—
235	C₂×Q₈	—	172	4	no	—
236	C₂²×C₄	—	86	2	no	—
237	C₂²×C₄	—	86	2	no	—
238	C₂²×C₄	—	86	4	no	—
239	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
245	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₄⋊C₄	—	86	4	no	—
250	C₄⋊C₄	—	86	4	no	—
251	C₄⋊C₄	—	86	4	no	—
252	C₄⋊C₄	—	86	8	no	—
253	C₄⋊C₄	—	172	4	no	—
254	C₄⋊C₄	—	172	4	no	—
255	C₄⋊C₄	—	172	4	no	—
256	C₄⋊C₄	—	172	4	no	—
257	C₄⋊C₄	—	172	4	no	—
258	C₄⋊C₄	—	172	4	no	—
259	C₄⋊C₄	—	172	2	no	—
260	C₄⋊C₄	—	172	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	172	4	no	—
265	C₄²	#10621873	1	4	yes	—
266	C₄²	Dic₁₇₂	1	4	yes	—
267	C₄²	#10621898	1	4	yes	—
268	C₄²	#10621900	1	2	yes	—
269	C₄²	Q₈×D₄₃	1	2	yes	—
270	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
271	Q₈	—	86	8	no	—
272	Q₈	—	344	8	no	—
273	Q₈	—	172	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	86	4	no	—
281	C₂×C₄	—	172	2	no	—
282	C₂×C₄	—	172	4	no	—
283	C₂×C₄	—	172	4	no	—
284	C₂×C₄	—	172	2	no	—
285	C₂×C₄	—	172	2	no	—
286	C₂×C₄	—	172	4	no	—
287	C₂×C₄	—	172	2	no	—
288	C₂×C₄	—	172	2	no	—
289	C₂×C₄	#13063305	1	2	yes	—
290	C₂×C₄	#13063305	1	2	yes	—
291	C₂×C₄	#13063313	1	2	yes	—
292	C₂×C₄	#13063313	1	2	yes	—
293	C₂×C₄	C₂×#10621873	1	2	yes	—
294	C₂×C₄	C₂×#10621873	1	2	yes	—
295	C₂×C₄	C₂×#10621873	1	2	yes	—
296	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
297	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
298	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
299	C₂×C₄	#13063353	1	2	yes	—
300	C₂×C₄	#13063358	1	2	yes	—
301	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
302	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
303	C₂×C₄	C₂×#10621898	1	2	yes	—
304	C₂×C₄	#13063371	1	2	yes	—
305	C₈	—	4	4	no	—
306	C₂²	C₂×#13063305	1	1	yes	characteristic
307	C₂²	C₂×#13063313	1	1	yes	characteristic
308	C₂²	#14583468	1	1	yes	characteristic
309	C₂²	C₂²×#10621873	1	1	yes	characteristic
310	C₂²	C₂²×Dic₁₇₂	1	1	yes	characteristic
311	C₂²	C₂×#13063353	1	1	yes	characteristic
312	C₂²	C₂×#13063358	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	344	2	no	—
317	C₄	—	344	4	no	—
318	C₄	—	172	4	no	—
319	C₄	#14583185	1	4	yes	—
320	C₄	#14583195	1	4	yes	—
321	C₄	#14583250	1	4	yes	—
322	C₄	#14583609	1	2	yes	—
323	C₄	#14583614	1	2	yes	—
324	C₂	#19139764	1	1	yes	characteristic
325	C₂	C₂×#14583185	1	1	yes	characteristic
326	C₂	C₂×#14583195	1	1	yes	characteristic
327	C₂	C₂×#14583250	1	1	yes	characteristic
328	C₂	#19144989	1	1	yes	characteristic
329	C₂	#19145538	1	1	yes	characteristic
330	C₂	#19145545	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	#29141
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
35. #13063325	Q₈	semidirect
194. C₄₃	#29141	semidirect

Maximal subgroups

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

subgroup	index
#19142993	2
C₄×#13063325	2
#19135737	2
#19135397	2
#19134723	2
#19134601	2
C₄×C₄⋊C₈×C₄₃	2
#29141	43