Subgroups — #26200071

Order 12032.

690 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253181	C₂	1	2	yes	maximal
2	C₂×C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#20242836	C₂	1	4	yes	maximal, complemented
4	#20242784	C₂	1	2	yes	maximal, complemented
5	#20242750	C₂	1	2	yes	maximal, complemented
6	#20242147	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242078	C₂	1	2	yes	maximal, complemented
8	C₄₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	—	2	1	no	—
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#15689420	—	2	4	no	—
12	#15689410	C₂²	1	4	yes	—
13	#15689400	C₂²	1	4	yes	—
14	C₈×D₁₈₈	—	2	1	no	—
15	C₈×D₁₈₈	—	2	1	no	—
16	C₈×D₁₈₈	C₂²	1	2	yes	—
17	C₈×D₁₈₈	C₂²	1	2	yes	—
18	#15689166	—	2	2	no	—
19	#15689147	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
21	#15689127	—	2	2	no	—
22	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
23	#15688876	C₂²	1	2	yes	complemented
24	#15688876	C₂²	1	2	yes	complemented
25	#15688872	C₂²	1	4	yes	complemented
26	#15688869	C₂²	1	4	yes	complemented
27	C₄₇×#533	C₂²	1	2	yes	—
28	C₄₇×#500	C₂²	1	1	yes	characteristic, complemented
29	C₄₇×#494	C₂²	1	1	yes	characteristic, complemented
30	C₄₇×#487	C₂²	1	2	yes	complemented
31	C₄₇×#467	—	2	1	no	—
32	C₄₇×#467	—	2	1	no	—
33	C₄×Q₁₆×C₄₇	—	2	1	no	—
34	C₄×Q₁₆×C₄₇	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₄₇	—	2	1	no	—
38	C₂×#10625329	—	2	2	no	—
39	C₂²×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₈×D₄₇	—	2	2	no	—
41	C₂×C₈×D₄₇	—	2	2	no	—
42	C₂×C₈×D₄₇	—	4	4	no	—
43	C₄×D₁₈₈	—	2	1	no	—
44	C₄×D₁₈₈	—	2	1	no	—
45	C₄×D₁₈₈	D₄	1	2	yes	—
46	C₄×D₁₈₈	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
48	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
49	#13093957	—	2	4	no	—
50	#13093955	—	2	4	no	—
51	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
52	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
54	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
55	#13093944	—	2	4	no	—
56	#13093944	—	2	4	no	—
57	(C₂×D₄₇)⋊C₈	—	2	2	no	—
58	(C₂×D₄₇)⋊C₈	—	2	2	no	—
59	C₂×Q₁₆×C₄₇	—	2	1	no	—
60	C₂×Q₁₆×C₄₇	—	2	1	no	—
61	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
65	#125×C₄₇	—	2	2	no	—
66	C₂²⋊Q₈×C₄₇	—	2	2	no	—
67	C₂²⋊Q₈×C₄₇	—	2	2	no	—
68	C₄×Q₈×C₄₇	—	2	2	no	—
69	C₄×Q₈×C₄₇	—	2	2	no	—
70	#117×C₄₇	—	2	2	no	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	#107×C₄₇	—	2	1	no	—
73	#107×C₄₇	—	2	1	no	—
74	#103×C₄₇	—	2	2	no	—
75	#103×C₄₇	—	2	2	no	—
76	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₂³×D₄₇	—	2	1	no	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
86	C₂²×Dic₄₇	—	2	1	no	—
87	C₂×D₁₈₈	—	2	1	no	—
88	C₂×D₁₈₈	—	2	2	no	—
89	C₂×D₁₈₈	Q₁₆	1	4	yes	complemented
90	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
91	C₂×C₄×D₄₇	—	2	2	no	—
92	C₂×C₄×D₄₇	—	2	2	no	—
93	C₂×C₄×D₄₇	—	4	2	no	—
94	C₂×C₄×D₄₇	—	4	2	no	—
95	Q₁₆×C₄₇	—	4	2	no	—
96	Q₁₆×C₄₇	—	4	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	4	1	no	—
99	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	—	4	2	no	—
108	C₄⋊C₄×C₄₇	—	4	2	no	—
109	C₂²⋊C₄×C₄₇	—	4	2	no	—
110	C₄²×C₄₇	—	4	2	no	—
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	#41	1	1	yes	characteristic
114	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
115	(C₂×D₄₇)⋊C₄	—	2	2	no	—
116	(C₂×D₄₇)⋊C₄	—	2	2	no	—
117	C₄⋊Dic₄₇	—	2	1	no	—
118	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
119	#10625329	—	4	4	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	—	2	4	no	—
122	C₂×C₄₇⋊C₈	—	4	2	no	—
123	C₈×D₄₇	—	4	4	no	—
124	C₈×D₄₇	—	4	4	no	—
125	C₂³×C₄₇	#121	1	1	yes	characteristic
126	C₂²×D₄₇	—	2	2	no	—
127	C₂²×D₄₇	—	2	4	no	—
128	C₂²×D₄₇	—	4	2	no	—
129	C₂²×D₄₇	—	4	2	no	—
130	Q₈×C₄₇	—	4	2	no	—
131	Q₈×C₄₇	—	4	2	no	—
132	Q₈×C₄₇	—	8	2	no	—
133	C₂×C₄×C₄₇	—	2	1	no	—
134	C₂×C₄×C₄₇	—	2	1	no	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
142	C₂×C₄×C₄₇	#135	1	2	yes	—
143	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	D₁₈₈	—	2	2	no	—
149	D₁₈₈	—	2	4	no	—
150	D₁₈₈	C₂×Q₁₆	1	4	yes	—
151	C₄×D₄₇	—	4	4	no	—
152	C₄×D₄₇	—	4	2	no	—
153	C₄×D₄₇	—	4	2	no	—
154	C₈×C₄₇	—	2	1	no	—
155	C₈×C₄₇	—	2	1	no	—
156	C₈×C₄₇	—	2	1	no	—
157	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
158	C₄₇⋊C₈	—	4	4	no	—
159	C₄₇⋊C₈	—	4	2	no	—
160	#19256	—	47	1	no	maximal, Sylow
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	#467	1	2	yes	—
164	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
165	C₂×D₄₇	—	2	4	no	—
166	C₂×D₄₇	—	4	8	no	—
167	C₂×D₄₇	—	4	2	no	—
168	C₂×D₄₇	—	4	2	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	#467	1	2	yes	—
175	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
176	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
177	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₄×C₄₇	#575	1	1	yes	characteristic
179	Dic₄₇	—	4	2	no	—
180	#3170	D₄₇	1	1	yes	characteristic, complemented
181	#3077	—	47	2	no	—
182	C₂×C₈×D₄	—	47	1	no	—
183	#1683	—	47	4	no	—
184	#1675	—	47	2	no	—
185	#1669	—	47	2	no	—
186	#1567	—	47	1	no	—
187	#1550	—	47	2	no	—
188	C₂×C₄₇	—	2	2	no	—
189	C₂×C₄₇	#3036	1	1	yes	characteristic
190	C₂×C₄₇	C₂×#467	1	1	yes	characteristic
191	C₂×C₄₇	#3067	1	1	yes	characteristic
192	D₄₇	—	4	4	no	—
193	C₂³×C₈	—	94	1	no	—
194	#533	C₂×D₄₇	1	2	yes	—
195	C₂×C₄×D₄	—	47	1	no	—
196	#500	C₂×D₄₇	1	1	yes	characteristic, complemented
197	#494	C₂×D₄₇	1	1	yes	characteristic, complemented
198	#487	C₂×D₄₇	1	2	yes	complemented
199	#478	—	94	4	no	—
200	#476	—	47	4	no	—
201	#474	—	47	4	no	—
202	#467	—	2	1	no	—
203	#467	—	2	1	no	—
204	C₄×Q₁₆	—	2	1	no	—
205	C₄×Q₁₆	—	2	1	no	—
206	C₈×D₄	—	47	2	no	—
207	C₈×D₄	—	47	2	no	—
208	C₈×D₄	—	94	1	no	—
209	C₈×D₄	—	94	1	no	—
210	#428	—	94	2	no	—
211	#423	—	47	2	no	—
212	C₂×C₄⋊C₈	—	47	1	no	—
213	#418	—	94	2	no	—
214	C₂×C₂²⋊C₈	—	94	1	no	—
215	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
216	#335	—	47	2	no	—
217	#335	—	47	2	no	—
218	#333	—	47	4	no	—
219	#332	—	47	4	no	—
220	C₄₇	#19256	1	1	yes	characteristic, complemented, Sylow
221	C₂²×D₄	—	47	1	no	—
222	C₂³×C₄	—	94	1	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	2	1	no	—
225	C₂×M₁₆	—	94	2	no	—
226	C₂²×C₈	—	94	1	no	—
227	C₂²×C₈	—	94	2	no	—
228	C₂²×C₈	—	94	2	no	—
229	C₂²×C₈	—	188	4	no	—
230	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
232	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
233	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
234	#125	—	2	2	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	C₄×D₄	—	47	2	no	—
240	C₄×D₄	—	47	2	no	—
241	C₄×D₄	—	94	1	no	—
242	C₄×D₄	—	94	1	no	—
243	#117	—	2	2	no	—
244	C₂×C₄⋊C₄	—	47	1	no	—
245	C₂×C₂²⋊C₄	—	94	1	no	—
246	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
247	#107	—	2	1	no	—
248	#107	—	2	1	no	—
249	#107	—	94	4	no	—
250	#106	—	94	4	no	—
251	C₄⋊C₈	—	47	2	no	—
252	C₄⋊C₈	—	47	4	no	—
253	C₄⋊C₈	—	94	2	no	—
254	C₄⋊C₈	—	94	1	no	—
255	#103	—	2	2	no	—
256	#103	—	2	2	no	—
257	#102	—	94	4	no	—
258	#102	—	94	4	no	—
259	C₂²⋊C₈	—	94	2	no	—
260	C₂²⋊C₈	—	94	2	no	—
261	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
262	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
265	C₂⁴	—	94	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×D₄	—	47	4	no	—
269	C₂×D₄	—	47	2	no	—
270	C₂×D₄	—	94	1	no	—
271	C₂×D₄	—	94	2	no	—
272	C₂²×C₄	—	94	1	no	—
273	C₂²×C₄	—	94	2	no	—
274	C₂²×C₄	—	94	2	no	—
275	C₂²×C₄	—	188	2	no	—
276	C₂²×C₄	—	188	2	no	—
277	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
278	C₂²×C₄	#10625352	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
280	Q₁₆	—	4	2	no	—
281	Q₁₆	—	4	2	no	—
282	M₁₆	—	188	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	94	4	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	—	188	4	no	—
288	C₂×C₈	—	188	2	no	—
289	C₂×C₈	—	188	4	no	—
290	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄⋊C₄	—	4	2	no	—
300	C₄⋊C₄	—	47	2	no	—
301	C₄⋊C₄	—	94	1	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	94	2	no	—
304	C₂²⋊C₄	—	94	2	no	—
305	C₄²	—	4	2	no	—
306	C₄²	D₄×D₄₇	1	1	yes	characteristic
307	C₄²	#10625352	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
309	C₄²	C₂³×D₄₇	1	1	yes	characteristic
310	C₂³	—	94	2	no	—
311	C₂³	—	94	4	no	—
312	C₂³	—	188	2	no	—
313	C₂³	—	188	2	no	—
314	C₂³	#13093995	1	1	yes	characteristic
315	Q₈	—	4	2	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	8	2	no	—
318	D₄	—	47	4	no	—
319	D₄	—	94	4	no	—
320	D₄	—	94	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	188	4	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	#13093995	1	1	yes	characteristic
332	C₂×C₄	#13093995	1	1	yes	characteristic
333	C₂×C₄	#13093995	1	1	yes	characteristic
334	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
335	C₂×C₄	#13094047	1	2	yes	—
336	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
337	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
338	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	188	4	no	—
343	C₈	—	188	2	no	—
344	C₈	#13094049	1	2	yes	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	94	4	no	—
348	C₂²	—	188	8	no	—
349	C₂²	—	188	2	no	—
350	C₂²	—	188	2	no	—
351	C₂²	#15689361	1	2	yes	—
352	C₂²	C₂×#13093995	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	—	188	2	no	—
359	C₄	#15689361	1	2	yes	—
360	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
361	C₄	C₂×#13094047	1	1	yes	characteristic
362	C₄	C₂×#13094049	1	1	yes	characteristic
363	C₄	#15689921	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	188	4	no	—
366	C₂	#20252848	1	1	yes	characteristic
367	C₂	C₂×#15689361	1	1	yes	characteristic
368	C₂	#20253104	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3170	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	#3170	D₄₇
hypocenter	C₄₇	#19256
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #20242836	C₂	semidirect
4. #20242784	C₂	semidirect
5. #20242750	C₂	semidirect
6. #20242147	C₂	semidirect
7. #20242078	C₂	semidirect
8. C₄₇×#3170	C₂	semidirect
23. #15688876	C₂²	semidirect
24. #15688876	C₂²	semidirect
25. #15688872	C₂²	semidirect
26. #15688869	C₂²	semidirect
28. C₄₇×#500	C₂²	semidirect
29. C₄₇×#494	C₂²	semidirect
30. C₄₇×#487	C₂²	semidirect
89. C₂×D₁₈₈	Q₁₆	semidirect
180. #3170	D₄₇	semidirect
196. #500	C₂×D₄₇	semidirect
197. #494	C₂×D₄₇	semidirect
198. #487	C₂×D₄₇	semidirect
220. C₄₇	#19256	semidirect

Maximal subgroups

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

subgroup	index
#20253181	2
C₂×C₈×D₁₈₈	2
#20242836	2
#20242784	2
#20242750	2
#20242147	2
#20242078	2
C₄₇×#3170	2
#19256	47