Subgroups — #26186157

Order 12032.

686 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	#20253420	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252101	C₂	1	1	yes	maximal, characteristic
4	#20242233	C₂	1	4	yes	maximal, complemented
5	C₈×#10625325	C₂	1	4	yes	maximal, complemented
6	#20242159	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241818	C₂	1	1	yes	maximal, characteristic, complemented
8	#20241788	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094030	—	2	1	no	—
11	#15689673	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
13	#15689442	C₄	1	2	yes	—
14	#15689409	C₄	1	2	yes	—
15	#15689324	C₄	1	2	yes	—
16	#15689264	C₄	1	2	yes	—
17	C₈×Dic₉₄	—	2	1	no	—
18	C₈×Dic₉₄	—	2	1	no	—
19	C₈×Dic₉₄	C₂²	1	2	yes	—
20	C₈×Dic₉₄	C₂²	1	2	yes	—
21	C₄×#10625325	C₂²	1	4	yes	—
22	#15689202	—	2	1	no	—
23	#15689202	C₂²	1	2	yes	—
24	C₈×D₁₈₈	—	2	1	no	—
25	C₈×D₁₈₈	C₂²	1	2	yes	—
26	#15689147	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×Dic₄₇⋊C₈	—	2	1	no	—
29	#15689089	—	2	1	no	—
30	C₂×C₈×Dic₄₇	—	2	1	no	—
31	#15688874	C₂²	1	4	yes	—
32	#15688855	C₂²	1	4	yes	—
33	#15688852	C₂²	1	4	yes	—
34	C₄₇×#431	C₂²	1	1	yes	characteristic
35	C₄₇×#431	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×#322	C₂²	1	2	yes	—
38	C₈²×C₄₇	C₂²	1	2	yes	—
39	C₂×#10625350	—	2	1	no	—
40	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
41	C₂×#10625325	C₂×C₄	1	4	yes	—
42	#13094030	—	4	4	no	—
43	C₂×#10625329	—	2	1	no	—
44	C₂×#10625324	—	2	2	no	—
45	C₂²×C₄₇⋊C₈	—	2	1	no	—
46	C₂×C₈×D₄₇	—	2	2	no	—
47	#13094024	C₂×C₄	1	1	yes	characteristic
48	#13094021	C₂×C₄	1	1	yes	characteristic
49	#13094012	—	2	1	no	—
50	#13093998	—	2	1	no	—
51	#13093998	C₂×C₄	1	2	yes	—
52	C₄×Dic₉₄	—	2	1	no	—
53	C₄×Dic₉₄	—	2	1	no	—
54	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₉₄	C₂³	1	2	yes	—
56	C₄×D₁₈₈	—	2	1	no	—
57	C₄×D₁₈₈	C₂³	1	2	yes	—
58	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×Dic₄₇⋊C₄	—	2	1	no	—
60	C₂×C₄×Dic₄₇	—	2	1	no	—
61	#13093956	C₂×C₄	1	4	yes	—
62	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
63	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
64	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
65	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
66	Dic₄₇⋊C₈	—	2	2	no	—
67	Dic₄₇⋊C₈	—	2	2	no	—
68	#13093949	C₂×C₄	1	4	yes	—
69	#13093946	C₂×C₄	1	4	yes	—
70	(C₂×D₄₇)⋊C₈	—	2	2	no	—
71	(C₂×D₄₇)⋊C₈	—	2	2	no	—
72	C₈×Dic₄₇	—	2	2	no	—
73	C₈×Dic₄₇	—	2	2	no	—
74	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₄₇	—	2	1	no	—
78	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
79	C₂²⋊C₈×C₄₇	—	2	1	no	—
80	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
82	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
83	C₄×C₈×C₄₇	C₂³	1	2	yes	—
84	C₄×C₈×C₄₇	C₂³	1	2	yes	—
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₇	—	2	2	no	—
91	C₂²×Dic₄₇	—	2	1	no	—
92	C₂²×Dic₄₇	—	2	1	no	—
93	#10625350	—	4	2	no	—
94	#10625350	—	4	2	no	—
95	C₂×D₁₈₈	—	2	1	no	—
96	C₂×D₁₈₈	—	2	1	no	—
97	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₄₇	—	2	2	no	—
99	C₂×Dic₉₄	—	2	1	no	—
100	C₂×Dic₉₄	—	2	4	no	—
101	C₂×Dic₉₄	—	2	1	no	—
102	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	SD₁₆	1	4	yes	complemented
106	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₄₇	C₂²×C₄	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	—	2	1	no	—
119	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
120	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
121	Dic₄₇⋊C₄	—	2	2	no	—
122	Dic₄₇⋊C₄	—	2	2	no	—
123	C₄×Dic₄₇	—	2	2	no	—
124	C₄×Dic₄₇	—	2	2	no	—
125	#10625329	—	4	2	no	—
126	C₂×C₄₇⋊C₈	—	2	2	no	—
127	C₂×C₄₇⋊C₈	—	2	2	no	—
128	C₂×C₄₇⋊C₈	—	4	2	no	—
129	#10625325	—	2	8	no	—
130	#10625324	—	4	4	no	—
131	C₈×D₄₇	—	4	4	no	—
132	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
133	C₂²×D₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
140	C₂×C₄×C₄₇	#135	1	2	yes	—
141	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₄₇	—	4	4	no	—
145	C₂×Dic₄₇	—	2	4	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	D₁₈₈	—	2	4	no	—
151	D₁₈₈	—	4	2	no	—
152	D₁₈₈	—	4	2	no	—
153	C₄×D₄₇	—	4	2	no	—
154	C₄×D₄₇	—	4	2	no	—
155	Dic₉₄	—	2	4	no	—
156	Dic₉₄	—	2	2	no	—
157	Dic₉₄	—	4	2	no	—
158	Dic₉₄	#131	1	4	yes	complemented
159	C₈×C₄₇	—	2	4	no	—
160	C₈×C₄₇	—	2	1	no	—
161	C₈×C₄₇	—	2	1	no	—
162	C₈×C₄₇	—	4	2	no	—
163	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
164	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
165	C₄₇⋊C₈	—	4	2	no	—
166	C₄₇⋊C₈	—	4	2	no	—
167	#17901	—	47	1	no	maximal, Sylow
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	—	2	1	no	—
170	C₂²×C₄₇	C₄×SD₁₆	1	2	yes	—
171	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₄₇	—	4	2	no	—
173	C₂×D₄₇	—	4	2	no	—
174	C₂×D₄₇	—	4	2	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
179	C₄×C₄₇	C₂×#131	1	2	yes	—
180	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
181	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
182	Dic₄₇	—	2	4	no	—
183	Dic₄₇	—	4	2	no	—
184	Dic₄₇	—	4	2	no	—
185	#3103	—	47	1	no	—
186	C₂×C₈×Q₈	—	47	1	no	—
187	#2931	—	47	1	no	—
188	#1583	—	47	4	no	—
189	C₈×SD₁₆	—	47	4	no	—
190	#1570	—	47	1	no	—
191	#1487	—	47	1	no	—
192	#1480	—	47	1	no	—
193	#1450	D₄₇	1	1	yes	characteristic, complemented
194	C₂×C₄₇	—	2	2	no	—
195	C₂×C₄₇	#2929	1	1	yes	characteristic
196	C₂×C₄₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
197	C₂×C₄₇	#2955	1	1	yes	characteristic
198	D₄₇	—	8	2	no	—
199	C₂×#131	—	94	1	no	—
200	#532	—	47	1	no	—
201	C₂×C₄×Q₈	—	47	1	no	—
202	#481	—	47	2	no	—
203	#475	—	47	2	no	—
204	#460	—	47	2	no	—
205	#448	—	47	2	no	—
206	C₈×Q₈	—	47	2	no	—
207	C₈×Q₈	—	47	2	no	—
208	C₈×Q₈	—	94	1	no	—
209	C₈×Q₈	—	94	1	no	—
210	C₄×SD₁₆	—	47	4	no	—
211	#436	—	47	2	no	—
212	#436	—	94	1	no	—
213	C₈×D₄	—	47	2	no	—
214	C₈×D₄	—	94	1	no	—
215	#431	C₂×D₄₇	1	1	yes	characteristic
216	#431	C₂×D₄₇	1	1	yes	characteristic
217	#423	—	47	1	no	—
218	C₂×C₄⋊C₈	—	47	1	no	—
219	C₂×C₄⋊C₈	—	94	1	no	—
220	#408	—	94	1	no	—
221	C₂×C₄×C₈	—	94	1	no	—
222	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
223	#334	—	47	4	no	—
224	Q₈⋊C₈	—	47	4	no	—
225	D₄⋊C₈	—	47	4	no	—
226	#322	C₂×D₄₇	1	2	yes	—
227	C₈²	C₂×D₄₇	1	2	yes	—
228	C₄₇	#17901	1	1	yes	characteristic, complemented, Sylow
229	C₂×#41	—	94	1	no	—
230	C₂²×Q₈	—	47	1	no	—
231	C₂×SD₁₆	—	47	4	no	—
232	#131	—	188	4	no	—
233	C₂×M₁₆	—	94	1	no	—
234	C₂×M₁₆	—	94	2	no	—
235	C₂²×C₈	—	94	1	no	—
236	C₂²×C₈	—	94	2	no	—
237	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
238	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
239	C₄⋊Q₈	—	47	1	no	—
240	#127	—	47	1	no	—
241	#124	—	94	1	no	—
242	#121	—	47	2	no	—
243	#121	—	94	1	no	—
244	C₄×Q₈	—	47	2	no	—
245	C₄×Q₈	—	47	2	no	—
246	C₄×Q₈	—	94	1	no	—
247	C₄×Q₈	—	94	1	no	—
248	C₄×D₄	—	47	2	no	—
249	C₄×D₄	—	94	1	no	—
250	C₂×C₄⋊C₄	—	47	1	no	—
251	C₂×C₄⋊C₄	—	94	1	no	—
252	C₂×C₄²	—	94	1	no	—
253	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
254	#106	—	47	4	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	47	2	no	—
257	C₄⋊C₈	—	47	2	no	—
258	C₄⋊C₈	—	94	2	no	—
259	C₄⋊C₈	—	94	2	no	—
260	C₄⋊C₈	—	94	1	no	—
261	C₄⋊C₈	—	94	1	no	—
262	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
263	#103	—	47	4	no	—
264	#102	—	47	4	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	94	2	no	—
267	C₂²⋊C₈	—	94	2	no	—
268	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
269	C₄×C₈	—	94	2	no	—
270	C₄×C₈	—	94	2	no	—
271	C₄×C₈	D₁₈₈	1	1	yes	characteristic
272	C₄×C₈	D₁₈₈	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₄₇	1	2	yes	—
274	C₄×C₈	C₂²×D₄₇	1	2	yes	—
275	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
277	#41	—	188	2	no	—
278	#41	—	188	2	no	—
279	C₂×Q₈	—	47	2	no	—
280	C₂×Q₈	—	94	4	no	—
281	C₂×Q₈	—	94	1	no	—
282	C₂×Q₈	—	94	1	no	—
283	C₂×D₄	—	47	2	no	—
284	C₂×D₄	—	94	2	no	—
285	C₂×D₄	—	94	1	no	—
286	C₂×D₄	—	94	1	no	—
287	C₂²×C₄	—	94	1	no	—
288	C₂²×C₄	—	94	1	no	—
289	C₂²×C₄	—	94	2	no	—
290	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
292	C₂²×C₄	#10625350	1	1	yes	characteristic
293	SD₁₆	—	94	8	no	—
294	M₁₆	—	188	2	no	—
295	M₁₆	—	188	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	94	2	no	—
299	C₂×C₈	—	94	2	no	—
300	C₂×C₈	—	188	2	no	—
301	C₂×C₈	—	188	4	no	—
302	C₂×C₈	#10625325	1	4	yes	complemented
303	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
304	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
305	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
306	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
307	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
308	C₄⋊C₄	—	47	2	no	—
309	C₄⋊C₄	—	47	2	no	—
310	C₄⋊C₄	—	94	2	no	—
311	C₄⋊C₄	—	94	1	no	—
312	C₄⋊C₄	—	94	1	no	—
313	C₄⋊C₄	—	94	2	no	—
314	C₂²⋊C₄	—	94	2	no	—
315	C₂²⋊C₄	—	94	2	no	—
316	C₄²	—	94	2	no	—
317	C₄²	—	94	2	no	—
318	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
319	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
320	C₄²	#10625350	1	1	yes	characteristic
321	C₄²	C₂³×D₄₇	1	1	yes	characteristic
322	C₂³	—	94	2	no	—
323	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
324	Q₈	—	47	4	no	—
325	Q₈	—	94	4	no	—
326	Q₈	—	94	2	no	—
327	Q₈	—	188	2	no	—
328	D₄	—	94	4	no	—
329	D₄	—	188	2	no	—
330	D₄	—	188	4	no	—
331	D₄	—	188	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	94	4	no	—
335	C₂×C₄	—	94	2	no	—
336	C₂×C₄	—	94	2	no	—
337	C₂×C₄	—	94	2	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	—	188	2	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
342	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
343	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
344	C₂×C₄	C₂×#10625325	1	2	yes	—
345	C₂×C₄	#13094046	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
347	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
348	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	188	2	no	—
354	C₈	—	188	2	no	—
355	C₈	C₂×#10625325	1	2	yes	—
356	C₈	C₂×#10625325	1	2	yes	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	188	2	no	—
360	C₂²	—	188	2	no	—
361	C₂²	—	188	2	no	—
362	C₂²	C₄×#10625325	1	2	yes	—
363	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	94	4	no	—
368	C₄	—	188	2	no	—
369	C₄	—	188	2	no	—
370	C₄	C₄×#10625325	1	2	yes	—
371	C₄	#131×D₄₇	1	2	yes	—
372	C₄	C₂²×#10625325	1	1	yes	characteristic
373	C₄	C₂×#13094046	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	376	2	no	—
376	C₂	#20252084	1	1	yes	characteristic
377	C₂	C₂×C₄×#10625325	1	1	yes	characteristic
378	C₂	#20252292	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689673	C₂²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#17901
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Dic₉₄	C₂	semidirect
4. #20242233	C₂	semidirect
5. C₈×#10625325	C₂	semidirect
6. #20242159	C₂	semidirect
7. #20241818	C₂	semidirect
9. C₄₇×#1450	C₂	semidirect
105. C₂×C₈×C₄₇	SD₁₆	semidirect
158. Dic₉₄	#131	semidirect
193. #1450	D₄₇	semidirect
228. C₄₇	#17901	semidirect
302. C₂×C₈	#10625325	semidirect

Maximal subgroups

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

subgroup	index
#20253420	2
C₂×C₈×Dic₉₄	2
#20252101	2
#20242233	2
C₈×#10625325	2
#20242159	2
#20241818	2
#20241788	2
C₄₇×#1450	2
#17901	47