Subgroups — #26371129

Order 12032.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253039	C₂	1	1	yes	maximal, characteristic
2	#20251984	C₂	1	1	yes	maximal, characteristic
3	#20245595	C₂	1	2	yes	maximal, complemented
4	#20245594	C₂	1	2	yes	maximal, complemented
5	#20245589	C₂	1	2	yes	maximal, complemented
6	#20245584	C₂	1	2	yes	maximal, complemented
7	#20245174	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245108	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245001	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄₇×#358	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094030	C₂²	1	1	yes	characteristic
13	#15689697	—	2	1	no	—
14	#15689604	—	2	1	no	—
15	C₂×#13093964	C₂²	1	1	yes	characteristic
16	#15689378	—	2	2	no	—
17	#15689361	C₂²	1	2	yes	—
18	#15689361	C₂²	1	2	yes	—
19	#15689164	—	2	2	no	—
20	#15689163	—	2	1	no	—
21	#15689161	C₂²	1	1	yes	characteristic
22	#15689161	C₂²	1	1	yes	characteristic
23	#15689160	C₂²	1	2	yes	—
24	#107×D₄₇	C₂²	1	2	yes	—
25	#15689132	—	2	1	no	—
26	#15688951	C₂²	1	1	yes	characteristic
27	#15688951	C₂²	1	1	yes	characteristic
28	#15688950	C₂²	1	2	yes	—
29	#15688949	C₂²	1	1	yes	characteristic
30	#15688949	C₂²	1	1	yes	characteristic
31	#15688948	C₂²	1	2	yes	—
32	#15688927	C₂²	1	2	yes	—
33	#15688927	C₂²	1	2	yes	—
34	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
35	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
36	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#358	C₂²	1	2	yes	—
40	C₄₇×#358	C₂²	1	2	yes	—
41	C₂×#10625350	D₄	1	1	yes	characteristic
42	#13094030	—	2	2	no	—
43	#13094030	—	2	2	no	—
44	C₂×#10625329	D₄	1	1	yes	characteristic
45	C₂×#10625324	D₄	1	2	yes	—
46	C₂×C₈×D₄₇	C₂³	1	2	yes	—
47	#13094022	—	4	1	no	—
48	#13094013	—	4	1	no	—
49	#13094007	—	4	2	no	—
50	#13094000	—	2	2	no	—
51	#13094000	—	2	2	no	—
52	#13093989	—	4	1	no	—
53	#13093986	—	4	1	no	—
54	C₄×C₂²⋊D₄₇	—	4	2	no	—
55	#13093983	—	2	1	no	—
56	#13093983	—	2	1	no	—
57	#13093982	—	2	2	no	—
58	C₄⋊C₄×D₄₇	—	2	2	no	—
59	#13093964	—	2	2	no	—
60	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
61	#13093958	C₂³	1	1	yes	characteristic
62	#13093958	C₂³	1	1	yes	characteristic
63	#13093958	C₂³	1	1	yes	characteristic
64	#13093958	C₂³	1	1	yes	characteristic
65	#13093957	—	2	2	no	—
66	#13093956	—	2	1	no	—
67	#13093956	—	2	1	no	—
68	#13093955	—	2	2	no	—
69	#13093948	—	2	2	no	—
70	#13093948	—	2	2	no	—
71	C₂²×Q₈×C₄₇	—	2	1	no	—
72	C₂×Q₁₆×C₄₇	—	2	1	no	—
73	C₂×Q₁₆×C₄₇	—	2	2	no	—
74	C₂×Q₁₆×C₄₇	—	2	2	no	—
75	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
78	C₂×C₁₆×C₄₇	—	2	2	no	—
79	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
81	#107×C₄₇	—	2	1	no	—
82	#107×C₄₇	C₂³	1	2	yes	—
83	C₂×Q₈×C₄₇	—	2	2	no	—
84	C₂×Q₈×C₄₇	—	4	1	no	—
85	C₂×Q₈×C₄₇	—	4	2	no	—
86	C₂×Q₈×C₄₇	—	4	2	no	—
87	C₂²×C₄×C₄₇	—	4	1	no	—
88	C₂²×C₄×C₄₇	—	4	1	no	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₇	D₈	1	2	yes	—
91	#10625350	—	2	2	no	—
92	#10625350	—	2	2	no	—
93	C₂×D₁₈₈	D₈	1	1	yes	characteristic
94	C₂×C₄×D₄₇	—	4	2	no	—
95	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
96	C₂×Dic₉₄	D₈	1	1	yes	characteristic
97	Q₁₆×C₄₇	—	2	2	no	—
98	Q₁₆×C₄₇	—	2	2	no	—
99	Q₁₆×C₄₇	—	4	2	no	—
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	#41	1	2	yes	—
102	C₂×C₈×C₄₇	#41	1	2	yes	—
103	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	4	1	no	—
106	C₂²⋊Dic₄₇	—	4	1	no	—
107	C₂²⋊Dic₄₇	—	4	1	no	—
108	(C₂×D₄₇)⋊C₄	—	4	2	no	—
109	(C₂×D₄₇)⋊C₄	—	4	2	no	—
110	(C₂×D₄₇)⋊C₄	—	4	2	no	—
111	C₄⋊Dic₄₇	—	2	1	no	—
112	C₄⋊Dic₄₇	—	2	1	no	—
113	C₄⋊Dic₄₇	—	2	1	no	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	Dic₄₇⋊C₄	—	4	2	no	—
116	Dic₄₇⋊C₄	—	4	2	no	—
117	Dic₄₇⋊C₄	—	4	2	no	—
118	C₄×Dic₄₇	—	4	1	no	—
119	C₄×Dic₄₇	—	4	1	no	—
120	C₄×Dic₄₇	—	4	2	no	—
121	#10625329	—	2	2	no	—
122	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
123	#10625324	—	2	4	no	—
124	C₈×D₄₇	—	2	2	no	—
125	C₈×D₄₇	—	2	2	no	—
126	C₁₆×C₄₇	—	2	1	no	—
127	C₁₆×C₄₇	—	2	1	no	—
128	C₄₇⋊C₁₆	—	2	2	no	—
129	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂²×D₄₇	C₂×D₈	1	2	yes	—
131	Q₈×C₄₇	—	4	2	no	—
132	Q₈×C₄₇	—	4	2	no	—
133	Q₈×C₄₇	—	8	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	8	2	no	—
137	C₂×C₄×C₄₇	—	16	1	no	—
138	C₂×C₄×C₄₇	#123	1	2	yes	—
139	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#137	1	2	yes	—
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₄₇	—	2	4	no	—
143	C₂×Dic₄₇	—	4	1	no	—
144	C₂×Dic₄₇	—	4	1	no	—
145	C₂×Dic₄₇	—	4	1	no	—
146	C₂×Dic₄₇	—	4	1	no	—
147	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
148	D₁₈₈	—	2	2	no	—
149	C₄×D₄₇	—	2	2	no	—
150	C₄×D₄₇	—	2	2	no	—
151	C₄×D₄₇	—	16	2	no	—
152	Dic₉₄	—	2	2	no	—
153	C₈×C₄₇	#123	1	2	yes	—
154	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
156	C₄₇⋊C₈	—	2	2	no	—
157	#33651	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	#480	1	2	yes	—
161	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₄₇	—	2	2	no	—
163	C₂×D₄₇	—	2	2	no	—
164	C₂×D₄₇	—	2	2	no	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	—	8	2	no	—
167	C₄×C₄₇	#480	1	2	yes	—
168	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
169	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
170	Dic₄₇	—	2	2	no	—
171	Dic₄₇	—	8	1	no	—
172	Dic₄₇	—	8	1	no	—
173	Dic₄₇	—	8	1	no	—
174	Dic₄₇	—	8	1	no	—
175	#3059	—	47	1	no	—
176	#2913	—	47	1	no	—
177	#2237	—	47	2	no	—
178	#2237	—	47	2	no	—
179	#2236	—	47	2	no	—
180	#2235	—	47	2	no	—
181	#2159	—	47	1	no	—
182	#2158	—	47	1	no	—
183	#2141	—	47	1	no	—
184	C₂×#358	D₄₇	1	1	yes	characteristic, complemented
185	#2113	—	47	1	no	—
186	C₂×C₄₇	—	2	2	no	—
187	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
188	C₂×C₄₇	#3416	1	1	yes	characteristic
189	C₂×C₄₇	#3417	1	1	yes	characteristic
190	D₄₇	—	4	2	no	—
191	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
192	C₂×#131	—	47	1	no	—
193	#536	—	94	1	no	—
194	#520	—	94	1	no	—
195	C₂×M₃₂	—	47	1	no	—
196	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
197	#470	—	94	2	no	—
198	#467	—	47	2	no	—
199	#467	—	47	2	no	—
200	#428	—	94	2	no	—
201	#428	—	94	1	no	—
202	#427	—	47	1	no	—
203	#427	—	47	1	no	—
204	#427	—	47	2	no	—
205	C₂×#107	—	47	2	no	—
206	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
207	#419	—	94	1	no	—
208	#367	—	47	1	no	—
209	#367	—	47	1	no	—
210	#367	—	47	2	no	—
211	#366	—	47	1	no	—
212	#366	—	47	1	no	—
213	#366	—	47	2	no	—
214	#358	—	47	2	no	—
215	#358	—	47	2	no	—
216	#358	C₂×D₄₇	1	2	yes	—
217	#358	C₂×D₄₇	1	2	yes	—
218	C₂²⋊C₁₆	—	47	2	no	—
219	C₂²⋊C₁₆	—	47	2	no	—
220	C₄₇	#33651	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	47	1	no	—
222	C₂²×Q₈	—	2	1	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	—	2	1	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
227	#131	—	94	2	no	—
228	#131	—	94	2	no	—
229	C₂×M₁₆	—	47	1	no	—
230	C₂×M₁₆	—	47	2	no	—
231	C₂²×C₈	—	47	2	no	—
232	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₄⋊Q₈	—	188	1	no	—
234	#124	—	188	1	no	—
235	#123	—	188	2	no	—
236	C₂²⋊Q₈	—	94	2	no	—
237	C₂²⋊Q₈	—	94	2	no	—
238	C₄×Q₈	—	188	1	no	—
239	C₄×D₄	—	188	1	no	—
240	C₄×D₄	—	188	2	no	—
241	#117	—	94	2	no	—
242	#117	—	94	1	no	—
243	#117	—	94	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	94	2	no	—
246	M₃₂	—	94	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	47	2	no	—
249	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
251	#107	—	2	1	no	—
252	#107	—	47	1	no	—
253	#107	—	47	1	no	—
254	#107	—	47	1	no	—
255	#107	—	47	1	no	—
256	#107	—	94	2	no	—
257	#107	C₂²×D₄₇	1	2	yes	—
258	#106	—	94	2	no	—
259	#106	—	94	1	no	—
260	#106	—	94	1	no	—
261	#103	—	94	2	no	—
262	#103	—	94	2	no	—
263	#41	—	94	2	no	—
264	#41	—	94	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	47	1	no	—
270	C₂×D₄	—	47	2	no	—
271	C₂×D₄	—	47	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	47	2	no	—
275	C₂²×C₄	—	188	2	no	—
276	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	4	2	no	—
280	M₁₆	—	94	2	no	—
281	M₁₆	—	94	4	no	—
282	C₂×C₈	—	47	2	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	#10625350	1	2	yes	—
286	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
287	C₂×C₈	#10625352	1	2	yes	—
288	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	94	1	no	—
292	C₄⋊C₄	—	94	1	no	—
293	C₄⋊C₄	—	94	1	no	—
294	C₄⋊C₄	—	94	1	no	—
295	C₄⋊C₄	—	188	2	no	—
296	C₄⋊C₄	—	188	2	no	—
297	C₄⋊C₄	—	188	2	no	—
298	C₂²⋊C₄	—	188	2	no	—
299	C₂²⋊C₄	—	188	2	no	—
300	C₂²⋊C₄	—	188	2	no	—
301	C₂²⋊C₄	—	188	1	no	—
302	C₂²⋊C₄	—	188	1	no	—
303	C₄²	—	188	2	no	—
304	C₄²	—	188	1	no	—
305	C₄²	—	188	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	94	2	no	—
309	C₂³	—	47	2	no	—
310	C₂³	D₈×D₄₇	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	2	no	—
314	Q₈	—	94	2	no	—
315	D₄	—	94	4	no	—
316	D₄	—	94	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	16	1	no	—
321	C₂×C₄	—	47	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	188	1	no	—
325	C₂×C₄	—	188	1	no	—
326	C₂×C₄	—	188	1	no	—
327	C₂×C₄	—	188	1	no	—
328	C₂×C₄	—	752	2	no	—
329	C₂×C₄	#13094006	1	2	yes	—
330	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
331	C₂×C₄	#13094053	1	2	yes	—
332	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
333	C₈	—	94	2	no	—
334	C₈	#13094006	1	2	yes	—
335	C₈	C₂×#10625352	1	1	yes	characteristic
336	C₈	C₂×#10625350	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	94	2	no	—
340	C₂²	—	94	2	no	—
341	C₂²	—	94	2	no	—
342	C₂²	#15689435	1	2	yes	—
343	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	94	2	no	—
347	C₄	—	376	1	no	—
348	C₄	—	376	1	no	—
349	C₄	—	376	1	no	—
350	C₄	—	376	1	no	—
351	C₄	#15689435	1	2	yes	—
352	C₄	C₂×#13094006	1	1	yes	characteristic
353	C₄	C₂×#13094053	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	188	2	no	—
356	C₂	C₂×#15689435	1	1	yes	characteristic
357	C₂	#20256530	1	1	yes	characteristic
358	C₂	#20256537	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#358	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#10625350	D₄
hypercenter	C₂×#358	D₄₇
hypocenter	C₄₇	#33651
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
3. #20245595	C₂	semidirect
4. #20245594	C₂	semidirect
5. #20245589	C₂	semidirect
6. #20245584	C₂	semidirect
7. #20245174	C₂	semidirect
8. #20245172	C₂	semidirect
9. #20245108	C₂	semidirect
11. C₂×C₄₇×#358	C₂	semidirect
184. C₂×#358	D₄₇	semidirect
220. C₄₇	#33651	semidirect

Maximal subgroups

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

subgroup	index
#20253039	2
#20251984	2
#20245595	2
#20245594	2
#20245589	2
#20245584	2
#20245174	2
#20245172	2
#20245108	2
#20245001	2
C₂×C₄₇×#358	2
#33651	47