Subgroups — #26199345

Order 12032.

842 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254306	C₂	1	2	yes	maximal, complemented
2	#20252101	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242864	C₂	1	4	yes	maximal, complemented
4	#20242740	C₂	1	4	yes	maximal, complemented
5	#20242136	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242056	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094030	—	2	1	no	—
9	#15689673	C₂²	1	1	yes	characteristic
10	#15689499	—	2	2	no	—
11	#15689469	—	2	2	no	—
12	#15689315	C₂²	1	8	yes	complemented
13	#15689257	—	2	4	no	—
14	#15689202	—	2	1	no	—
15	#15689202	C₂²	1	2	yes	complemented
16	C₈×D₁₈₈	—	2	1	no	—
17	C₈×D₁₈₈	C₂²	1	2	yes	complemented
18	#15689176	C₂²	1	2	yes	—
19	#15689147	C₂²	1	1	yes	characteristic
20	#15689089	—	2	1	no	—
21	#15688876	C₂²	1	2	yes	complemented
22	#15688874	C₂²	1	2	yes	complemented
23	#15688867	C₂²	1	8	yes	complemented
24	C₂²×D₈×C₄₇	—	2	1	no	—
25	C₂²×D₈×C₄₇	—	2	1	no	—
26	C₂×#127×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#493	—	2	1	no	—
28	C₄₇×#493	—	2	1	no	—
29	C₄₇×#493	C₂²	1	2	yes	—
30	C₄₇×#493	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×#10625350	—	2	1	no	—
33	C₂×#10625334	—	2	8	no	—
34	#13094030	—	4	4	no	—
35	C₂×#10625329	—	2	1	no	—
36	C₂×#10625324	—	2	2	no	—
37	C₂×C₈×D₄₇	—	2	2	no	—
38	#13094024	D₄	1	1	yes	characteristic
39	#13094021	D₄	1	1	yes	characteristic
40	#13094012	—	2	1	no	—
41	#13093998	—	2	1	no	—
42	#13093998	D₄	1	2	yes	—
43	C₄×D₁₈₈	—	2	1	no	—
44	C₄×D₁₈₈	C₂³	1	2	yes	—
45	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
47	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
48	#13093945	—	2	8	no	—
49	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
50	(C₂×D₄₇)⋊C₈	—	2	2	no	—
51	(C₂×D₄₇)⋊C₈	—	2	2	no	—
52	C₄×C₄₇⋊C₈	—	2	2	no	—
53	C₂²×D₄×C₄₇	—	2	2	no	—
54	C₂²×D₄×C₄₇	—	2	2	no	—
55	C₂²×D₄×C₄₇	—	2	2	no	—
56	C₂×D₈×C₄₇	—	2	2	no	—
57	C₂×D₈×C₄₇	—	2	2	no	—
58	C₂×D₈×C₄₇	—	2	2	no	—
59	C₂×D₈×C₄₇	—	2	2	no	—
60	C₂×D₈×C₄₇	—	4	4	no	—
61	C₂×D₈×C₄₇	—	4	4	no	—
62	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
64	#127×C₄₇	—	2	2	no	—
65	#127×C₄₇	—	2	2	no	—
66	#127×C₄₇	—	2	2	no	—
67	#127×C₄₇	C₂³	1	4	yes	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
71	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂⁴×C₄₇	—	4	2	no	—
74	C₂×D₄×C₄₇	—	2	4	no	—
75	C₂×D₄×C₄₇	—	2	4	no	—
76	C₂×D₄×C₄₇	—	2	4	no	—
77	C₂×D₄×C₄₇	—	4	2	no	—
78	C₂×D₄×C₄₇	—	4	2	no	—
79	C₂×D₄×C₄₇	—	4	2	no	—
80	C₂×D₄×C₄₇	—	4	4	no	—
81	C₂×D₄×C₄₇	—	4	4	no	—
82	C₂×D₄×C₄₇	—	8	4	no	—
83	C₂×D₄×C₄₇	—	8	4	no	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
87	C₂×C₂²⋊D₄₇	—	2	2	no	—
88	#10625350	—	4	2	no	—
89	#10625350	—	4	2	no	—
90	C₂×D₁₈₈	—	2	1	no	—
91	C₂×D₁₈₈	—	2	1	no	—
92	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
93	C₂×C₄×D₄₇	—	2	2	no	—
94	C₂×Dic₉₄	—	2	1	no	—
95	D₈×C₄₇	—	4	4	no	—
96	D₈×C₄₇	—	4	4	no	—
97	D₈×C₄₇	—	4	4	no	—
98	D₈×C₄₇	—	4	4	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	D₈	1	4	yes	complemented
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₇	#41	1	1	yes	characteristic
108	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
109	#10625334	—	8	8	no	—
110	(C₂×D₄₇)⋊C₄	—	2	2	no	—
111	(C₂×D₄₇)⋊C₄	—	2	2	no	—
112	C₄⋊Dic₄₇	—	2	1	no	—
113	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
114	#10625329	—	4	2	no	—
115	C₂×C₄₇⋊C₈	—	2	4	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	#10625324	—	4	4	no	—
118	C₈×D₄₇	—	4	4	no	—
119	C₂³×C₄₇	—	4	4	no	—
120	C₂³×C₄₇	—	8	4	no	—
121	C₂³×C₄₇	—	8	4	no	—
122	C₂³×C₄₇	—	8	4	no	—
123	C₂³×C₄₇	#121	1	1	yes	characteristic
124	C₂²×D₄₇	—	2	2	no	—
125	D₄×C₄₇	—	4	4	no	—
126	D₄×C₄₇	—	4	4	no	—
127	D₄×C₄₇	—	4	8	no	—
128	D₄×C₄₇	—	8	4	no	—
129	D₄×C₄₇	—	8	4	no	—
130	D₄×C₄₇	—	8	4	no	—
131	D₄×C₄₇	—	8	4	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂²⋊D₄₇	—	4	4	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	D₁₈₈	—	2	4	no	—
145	D₁₈₈	—	4	2	no	—
146	D₁₈₈	—	4	2	no	—
147	C₄×D₄₇	—	4	2	no	—
148	C₄×D₄₇	—	4	2	no	—
149	Dic₉₄	—	4	2	no	—
150	C₈×C₄₇	—	2	1	no	—
151	C₈×C₄₇	—	2	1	no	—
152	C₈×C₄₇	C₂×D₈	1	2	yes	—
153	C₈×C₄₇	C₂×D₈	1	2	yes	—
154	C₄₇⋊C₈	—	4	2	no	—
155	C₄₇⋊C₈	—	4	4	no	—
156	#19187	—	47	1	no	maximal, Sylow
157	C₂²×C₄₇	—	2	1	no	—
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	8	4	no	—
160	C₂²×C₄₇	—	8	4	no	—
161	C₂²×C₄₇	—	8	4	no	—
162	C₂²×C₄₇	—	16	8	no	—
163	C₂²×C₄₇	#459	1	2	yes	—
164	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
165	C₂×D₄₇	—	4	2	no	—
166	C₂×D₄₇	—	4	2	no	—
167	C₂×D₄₇	—	4	2	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	#459	1	2	yes	—
171	C₄×C₄₇	#472	1	2	yes	—
172	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₄×C₄₇	#576	1	2	yes	—
175	Dic₄₇	—	4	2	no	—
176	#3205	—	47	2	no	—
177	C₂×#493	D₄₇	1	1	yes	characteristic, complemented
178	#2931	—	47	1	no	—
179	#1687	—	47	4	no	—
180	#1667	—	47	4	no	—
181	#1564	—	47	1	no	—
182	#1545	—	47	2	no	—
183	C₂×C₄₇	—	2	2	no	—
184	C₂×C₄₇	—	16	4	no	—
185	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
186	C₂×C₄₇	#3058	1	1	yes	characteristic
187	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
188	D₄₇	—	8	2	no	—
189	C₂²×D₈	—	2	1	no	—
190	C₂²×D₈	—	2	1	no	—
191	C₂×#131	—	94	1	no	—
192	#532	—	47	1	no	—
193	C₂×#127	C₂×D₄₇	1	2	yes	—
194	#493	—	2	1	no	—
195	#493	—	2	1	no	—
196	#493	—	94	2	no	—
197	#493	C₂×D₄₇	1	2	yes	—
198	#493	C₂×D₄₇	1	2	yes	—
199	#486	—	94	2	no	—
200	#459	—	47	8	no	—
201	#447	—	94	4	no	—
202	#436	—	47	2	no	—
203	#436	—	94	1	no	—
204	C₈×D₄	—	47	2	no	—
205	C₈×D₄	—	94	1	no	—
206	#431	—	47	2	no	—
207	#423	—	47	1	no	—
208	#408	—	94	1	no	—
209	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
210	#335	—	47	2	no	—
211	#334	—	47	2	no	—
212	#331	—	47	8	no	—
213	C₄₇	#19187	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	94	1	no	—
215	C₂²×D₄	—	2	2	no	—
216	C₂²×D₄	—	2	2	no	—
217	C₂²×D₄	—	2	2	no	—
218	C₂×D₈	—	2	2	no	—
219	C₂×D₈	—	2	2	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	—	4	4	no	—
223	C₂×D₈	—	4	4	no	—
224	C₂×D₈	—	94	8	no	—
225	#131	—	188	4	no	—
226	C₂×M₁₆	—	94	1	no	—
227	C₂×M₁₆	—	94	2	no	—
228	C₂²×C₈	—	94	2	no	—
229	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
231	C₄⋊Q₈	—	47	1	no	—
232	#127	—	2	2	no	—
233	#127	—	2	2	no	—
234	#127	—	2	2	no	—
235	#127	—	47	1	no	—
236	#127	C₂²×D₄₇	1	4	yes	—
237	#124	—	94	1	no	—
238	#121	—	47	2	no	—
239	#121	—	94	1	no	—
240	C₄×D₄	—	47	2	no	—
241	C₄×D₄	—	94	1	no	—
242	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
243	C₄⋊C₈	—	47	2	no	—
244	C₄⋊C₈	—	47	4	no	—
245	C₄⋊C₈	—	94	1	no	—
246	#102	—	94	8	no	—
247	C₂²⋊C₈	—	94	2	no	—
248	C₂²⋊C₈	—	94	2	no	—
249	C₂²⋊C₈	—	94	2	no	—
250	C₄×C₈	—	94	2	no	—
251	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
252	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
255	C₂⁴	—	4	2	no	—
256	#41	—	188	2	no	—
257	#41	—	188	2	no	—
258	C₂×Q₈	—	94	1	no	—
259	C₂×D₄	—	2	4	no	—
260	C₂×D₄	—	2	4	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	4	4	no	—
267	C₂×D₄	—	8	4	no	—
268	C₂×D₄	—	8	4	no	—
269	C₂×D₄	—	47	2	no	—
270	C₂×D₄	—	94	2	no	—
271	C₂×D₄	—	94	1	no	—
272	C₂×D₄	—	94	1	no	—
273	C₂²×C₄	—	94	2	no	—
274	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
275	C₂²×C₄	#10625352	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
277	D₈	—	4	4	no	—
278	D₈	—	4	4	no	—
279	D₈	—	4	4	no	—
280	D₈	—	4	4	no	—
281	D₈	—	376	8	no	—
282	M₁₆	—	188	2	no	—
283	M₁₆	—	188	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	94	4	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	—	188	4	no	—
288	C₂×C₈	#10625334	1	4	yes	complemented
289	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
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	C₄⋊C₄	—	47	2	no	—
294	C₄⋊C₄	—	94	1	no	—
295	C₂²⋊C₄	—	94	2	no	—
296	C₂²⋊C₄	—	94	2	no	—
297	C₄²	D₄×D₄₇	1	1	yes	characteristic
298	C₄²	#10625352	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₄²	C₂³×D₄₇	1	1	yes	characteristic
301	C₂³	—	4	4	no	—
302	C₂³	—	8	4	no	—
303	C₂³	—	8	4	no	—
304	C₂³	—	8	4	no	—
305	C₂³	—	94	2	no	—
306	C₂³	#13093995	1	1	yes	characteristic
307	Q₈	—	188	2	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	8	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	4	no	—
314	D₄	—	8	4	no	—
315	D₄	—	94	4	no	—
316	D₄	—	188	4	no	—
317	D₄	—	188	2	no	—
318	D₄	—	188	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	188	2	no	—
323	C₂×C₄	—	188	2	no	—
324	C₂×C₄	#13093995	1	1	yes	characteristic
325	C₂×C₄	#13093995	1	1	yes	characteristic
326	C₂×C₄	#13093995	1	1	yes	characteristic
327	C₂×C₄	C₂×#10625334	1	2	yes	—
328	C₂×C₄	#13094050	1	2	yes	—
329	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
331	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	188	2	no	—
335	C₈	—	188	4	no	—
336	C₈	C₂×#10625334	1	2	yes	—
337	C₈	C₂×#10625334	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	8	4	no	—
341	C₂²	—	8	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	16	8	no	—
344	C₂²	—	188	2	no	—
345	C₂²	—	188	2	no	—
346	C₂²	—	188	2	no	—
347	C₂²	#15689315	1	2	yes	—
348	C₂²	C₂×#13093995	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	188	2	no	—
352	C₄	#15689315	1	2	yes	—
353	C₄	#15689390	1	2	yes	—
354	C₄	C₂²×#10625334	1	1	yes	characteristic
355	C₄	C₂×#13094050	1	1	yes	characteristic
356	C₄	#15689924	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	—	16	4	no	—
359	C₂	—	376	2	no	—
360	C₂	C₂×#15689315	1	1	yes	characteristic
361	C₂	#20253032	1	1	yes	characteristic
362	C₂	C₂×#15689390	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#493	C₂
socle	C₂²×C₄₇	C₂×#121
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	#19187
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #20254306	C₂	semidirect
2. #20252101	C₂	semidirect
3. #20242864	C₂	semidirect
4. #20242740	C₂	semidirect
5. #20242136	C₂	semidirect
6. #20242056	C₂	semidirect
7. C₂×C₄₇×#493	C₂	semidirect
12. #15689315	C₂²	semidirect
15. #15689202	C₂²	semidirect
17. C₈×D₁₈₈	C₂²	semidirect
21. #15688876	C₂²	semidirect
22. #15688874	C₂²	semidirect
23. #15688867	C₂²	semidirect
100. C₂×C₈×C₄₇	D₈	semidirect
177. C₂×#493	D₄₇	semidirect
213. C₄₇	#19187	semidirect
288. C₂×C₈	#10625334	semidirect

Maximal subgroups

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

subgroup	index
#20254306	2
#20252101	2
#20242864	2
#20242740	2
#20242136	2
#20242056	2
C₂×C₄₇×#493	2
#19187	47