Subgroups — #26200094

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253967	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253479	C₂	1	2	yes	maximal, complemented
3	#20242797	C₂	1	4	yes	maximal, complemented
4	#20242787	C₂	1	2	yes	maximal, complemented
5	#20242743	C₂	1	2	yes	maximal, complemented
6	#20242151	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242081	C₂	1	2	yes	maximal, complemented
8	C₂×C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689673	C₂²	1	2	yes	—
10	#15689528	C₂²	1	1	yes	characteristic, complemented
11	#15689500	C₂²	1	1	yes	characteristic, complemented
12	#15689470	C₂²	1	2	yes	complemented
13	#15689441	—	2	2	no	—
14	#15689413	C₂²	1	4	yes	—
15	#15689359	—	2	1	no	—
16	#15689359	—	2	1	no	—
17	#15689318	C₂²	1	4	yes	—
18	#15689257	—	2	2	no	—
19	#15689233	—	2	4	no	—
20	C₄×D₃₇₆	—	2	1	no	—
21	C₄×D₃₇₆	—	2	1	no	—
22	#15689180	C₂²	1	2	yes	—
23	#15688876	C₂²	1	2	yes	complemented
24	#15688874	C₂²	1	2	yes	complemented
25	#15688871	C₂²	1	4	yes	complemented
26	#15688866	C₂²	1	4	yes	complemented
27	C₂³×C₈×C₄₇	—	2	1	no	—
28	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
29	C₈×D₄×C₄₇	—	2	1	no	—
30	C₈×D₄×C₄₇	—	2	1	no	—
31	C₈×D₄×C₄₇	C₂²	1	2	yes	—
32	C₈×D₄×C₄₇	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#10625350	—	2	2	no	—
37	C₂×D₃₇₆	—	2	1	no	—
38	C₂×D₃₇₆	—	2	1	no	—
39	C₂×#10625334	—	2	4	no	—
40	#13094024	C₂³	1	2	yes	—
41	#13094021	C₂³	1	2	yes	—
42	#13094012	—	2	2	no	—
43	#13093998	—	2	2	no	—
44	#13093998	—	2	2	no	—
45	C₄×D₁₈₈	—	2	2	no	—
46	C₄×D₁₈₈	—	2	2	no	—
47	#13093958	—	2	1	no	—
48	#13093958	—	2	1	no	—
49	#13093955	—	2	4	no	—
50	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
51	#13093946	—	2	2	no	—
52	#13093946	—	2	2	no	—
53	#13093945	—	2	4	no	—
54	#13093944	—	2	4	no	—
55	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
56	#13093935	—	2	2	no	—
57	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₈×C₄₇	—	2	1	no	—
60	C₂²×C₈×C₄₇	—	2	2	no	—
61	C₂²×C₈×C₄₇	—	2	2	no	—
62	C₂²×C₈×C₄₇	—	4	4	no	—
63	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
65	C₄×D₄×C₄₇	—	2	1	no	—
66	C₄×D₄×C₄₇	—	2	1	no	—
67	C₄×D₄×C₄₇	D₄	1	2	yes	—
68	C₄×D₄×C₄₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₄₇	—	2	1	no	—
73	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
74	C₂²⋊C₈×C₄₇	—	2	2	no	—
75	C₂²⋊C₈×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	C₂⁴×C₄₇	—	2	1	no	—
81	C₂×D₄×C₄₇	—	2	4	no	—
82	C₂×D₄×C₄₇	—	2	1	no	—
83	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	—	2	1	no	—
85	C₂²×C₄×C₄₇	—	2	2	no	—
86	C₂²×C₄×C₄₇	—	2	2	no	—
87	C₂²×C₄×C₄₇	—	4	2	no	—
88	C₂²×C₄×C₄₇	—	4	2	no	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₇	—	4	2	no	—
93	#10625350	—	8	4	no	—
94	C₂×D₁₈₈	—	2	2	no	—
95	C₂×D₁₈₈	—	2	2	no	—
96	C₂×D₁₈₈	—	2	2	no	—
97	C₂×C₄×D₄₇	—	4	2	no	—
98	C₂×Dic₉₄	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	4	1	no	—
102	C₂×C₈×C₄₇	—	4	4	no	—
103	C₂×C₈×C₄₇	—	4	2	no	—
104	C₂×C₈×C₄₇	—	4	4	no	—
105	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₄⋊C₄×C₄₇	—	2	1	no	—
110	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₄₇	—	2	2	no	—
112	C₂²⋊C₄×C₄₇	—	2	2	no	—
113	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	#41	1	1	yes	characteristic
116	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
117	#10625334	—	4	8	no	—
118	(C₂×D₄₇)⋊C₄	—	4	2	no	—
119	(C₂×D₄₇)⋊C₄	—	4	2	no	—
120	C₄⋊Dic₄₇	—	2	2	no	—
121	C₄⋊Dic₄₇	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	2	4	no	—
123	D₃₇₆	—	4	2	no	—
124	D₃₇₆	—	4	2	no	—
125	C₂³×C₄₇	—	2	4	no	—
126	C₂³×C₄₇	—	2	2	no	—
127	C₂³×C₄₇	—	4	2	no	—
128	C₂³×C₄₇	—	4	2	no	—
129	C₂³×C₄₇	#121	1	1	yes	characteristic
130	C₂²×D₄₇	—	4	2	no	—
131	D₄×C₄₇	—	2	4	no	—
132	D₄×C₄₇	—	2	2	no	—
133	D₄×C₄₇	#135	1	4	yes	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	1	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	4	4	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
149	C₂²⋊D₄₇	—	8	4	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	D₁₈₈	—	4	4	no	—
152	D₁₈₈	—	4	2	no	—
153	D₁₈₈	—	4	2	no	—
154	D₁₈₈	—	8	2	no	—
155	C₄×D₄₇	—	8	4	no	—
156	Dic₉₄	—	8	2	no	—
157	C₈×C₄₇	—	2	1	no	—
158	C₈×C₄₇	—	2	1	no	—
159	C₈×C₄₇	—	2	1	no	—
160	C₈×C₄₇	—	4	2	no	—
161	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
162	C₄₇⋊C₈	—	4	4	no	—
163	#19258	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	—	2	1	no	—
166	C₂²×C₄₇	—	2	4	no	—
167	C₂²×C₄₇	—	4	8	no	—
168	C₂²×C₄₇	—	4	2	no	—
169	C₂²×C₄₇	—	4	2	no	—
170	C₂²×C₄₇	#472	1	2	yes	—
171	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
172	C₂×D₄₇	—	8	2	no	—
173	C₂×D₄₇	—	8	2	no	—
174	C₂×D₄₇	—	8	2	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	—	4	2	no	—
180	C₄×C₄₇	C₂×#135	1	2	yes	—
181	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₄₇	#575	1	1	yes	characteristic
183	Dic₄₇	—	8	2	no	—
184	#3169	—	47	1	no	—
185	#3109	—	47	2	no	—
186	C₂×C₈×D₄	D₄₇	1	1	yes	characteristic, complemented
187	#1677	—	47	4	no	—
188	#1675	—	47	2	no	—
189	#1667	—	47	2	no	—
190	#1568	—	47	1	no	—
191	#1551	—	47	2	no	—
192	C₂×C₄₇	—	2	2	no	—
193	C₂×C₄₇	—	4	4	no	—
194	C₂×C₄₇	#3036	1	1	yes	characteristic
195	C₂×C₄₇	#3050	1	1	yes	characteristic
196	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
197	D₄₇	—	16	2	no	—
198	C₂³×C₈	—	2	1	no	—
199	#532	—	47	2	no	—
200	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
201	#500	—	47	1	no	—
202	#493	—	47	1	no	—
203	#486	—	47	2	no	—
204	#481	—	94	2	no	—
205	#476	—	47	4	no	—
206	#466	—	94	1	no	—
207	#466	—	94	1	no	—
208	#459	—	47	4	no	—
209	#447	—	94	2	no	—
210	#442	—	94	4	no	—
211	C₄×D₈	—	94	1	no	—
212	C₄×D₈	—	94	1	no	—
213	C₈×D₄	—	2	1	no	—
214	C₈×D₄	—	2	1	no	—
215	C₈×D₄	C₂×D₄₇	1	2	yes	—
216	C₈×D₄	C₂×D₄₇	1	2	yes	—
217	#432	—	47	2	no	—
218	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
219	C₂×C₂²⋊C₈	—	2	1	no	—
220	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
221	#335	—	47	2	no	—
222	#334	—	47	2	no	—
223	#333	—	47	4	no	—
224	#331	—	47	4	no	—
225	C₄₇	#19258	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	94	2	no	—
227	C₂²×D₄	D₁₈₈	1	1	yes	characteristic
228	C₂³×C₄	—	2	1	no	—
229	C₂×D₈	—	94	4	no	—
230	C₂×D₈	—	94	1	no	—
231	C₂×D₈	—	94	1	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	2	2	no	—
234	C₂²×C₈	—	2	2	no	—
235	C₂²×C₈	—	4	4	no	—
236	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
237	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
238	C₄⋊Q₈	—	47	2	no	—
239	#127	—	47	2	no	—
240	#124	—	94	2	no	—
241	#121	—	94	2	no	—
242	#121	—	94	2	no	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	94	2	no	—
246	C₄×D₄	—	94	2	no	—
247	C₄×D₄	D₁₈₈	1	2	yes	—
248	C₄×D₄	C₂²×D₄₇	1	2	yes	—
249	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
250	C₂×C₂²⋊C₄	—	2	1	no	—
251	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
252	#107	—	94	1	no	—
253	#107	—	94	1	no	—
254	#106	—	94	4	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	47	4	no	—
257	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
258	#102	—	94	4	no	—
259	#102	—	94	2	no	—
260	#102	—	94	2	no	—
261	#102	—	94	4	no	—
262	C₂²⋊C₈	—	2	2	no	—
263	C₂²⋊C₈	—	2	2	no	—
264	C₂²⋊C₈	—	94	2	no	—
265	#97	—	94	2	no	—
266	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
267	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
270	C₂⁴	—	2	1	no	—
271	#41	—	376	4	no	—
272	C₂×Q₈	—	94	2	no	—
273	C₂×D₄	—	2	4	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	94	2	no	—
276	C₂×D₄	—	94	2	no	—
277	C₂×D₄	—	94	2	no	—
278	C₂×D₄	—	188	2	no	—
279	C₂×D₄	C₂×D₁₈₈	1	2	yes	—
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	—	188	2	no	—
286	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
287	C₂²×C₄	#10625350	1	1	yes	characteristic
288	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
289	D₈	—	188	2	no	—
290	D₈	—	188	8	no	—
291	D₈	—	188	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	4	4	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	4	4	no	—
298	C₂×C₈	—	94	4	no	—
299	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
301	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
302	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	94	2	no	—
305	C₄⋊C₄	—	94	2	no	—
306	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	188	2	no	—
310	C₂²⋊C₄	—	188	2	no	—
311	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
312	C₄²	#10625350	1	1	yes	characteristic
313	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
314	C₄²	C₂³×D₄₇	1	1	yes	characteristic
315	C₂³	—	2	4	no	—
316	C₂³	—	2	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	188	2	no	—
320	C₂³	#13093998	1	1	yes	characteristic
321	Q₈	—	376	2	no	—
322	D₄	—	2	4	no	—
323	D₄	—	2	2	no	—
324	D₄	—	376	2	no	—
325	D₄	—	376	4	no	—
326	D₄	—	188	4	no	—
327	D₄	—	188	2	no	—
328	D₄	—	188	2	no	—
329	D₄	#13094046	1	4	yes	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	4	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	376	4	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	#13093998	1	1	yes	characteristic
342	C₂×C₄	#13093998	1	1	yes	characteristic
343	C₂×C₄	#13093998	1	1	yes	characteristic
344	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
345	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
346	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₈	—	2	1	no	—
350	C₈	—	4	2	no	—
351	C₈	—	188	4	no	—
352	C₈	#13094050	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	4	no	—
356	C₂²	—	4	8	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	376	2	no	—
360	C₂²	—	376	2	no	—
361	C₂²	—	376	2	no	—
362	C₂²	#15689393	1	2	yes	—
363	C₂²	C₂×#13093998	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	376	2	no	—
370	C₄	C₂×#13094046	1	2	yes	—
371	C₄	C₂×#13094050	1	1	yes	characteristic
372	C₄	#15689921	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	4	4	no	—
375	C₂	—	752	2	no	—
376	C₂	#20252852	1	1	yes	characteristic
377	C₂	#20252969	1	1	yes	characteristic
378	C₂	C₂×#15689393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093998
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₈×D₄×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#121
hypercenter	C₂×C₈×D₄	D₄₇
hypocenter	C₄₇	#19258
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20253967	C₂	semidirect
2. #20253479	C₂	semidirect
3. #20242797	C₂	semidirect
4. #20242787	C₂	semidirect
5. #20242743	C₂	semidirect
6. #20242151	C₂	semidirect
7. #20242081	C₂	semidirect
8. C₂×C₈×D₄×C₄₇	C₂	semidirect
10. #15689528	C₂²	semidirect
11. #15689500	C₂²	semidirect
12. #15689470	C₂²	semidirect
23. #15688876	C₂²	semidirect
24. #15688874	C₂²	semidirect
25. #15688871	C₂²	semidirect
26. #15688866	C₂²	semidirect
186. C₂×C₈×D₄	D₄₇	semidirect
225. C₄₇	#19258	semidirect

Maximal subgroups

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

subgroup	index
#20253967	2
#20253479	2
#20242797	2
#20242787	2
#20242743	2
#20242151	2
#20242081	2
C₂×C₈×D₄×C₄₇	2
#19258	47