Subgroups — #26186227

Order 12032.

678 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	#20253415	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689247	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252100	C₂	1	1	yes	maximal, characteristic
4	#20242269	C₂	1	4	yes	maximal, complemented
5	#20242197	C₂	1	4	yes	maximal, complemented
6	#20242162	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241818	C₂	1	1	yes	maximal, characteristic, complemented
8	#20241681	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1480	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094031	—	2	1	no	—
11	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
12	#15689439	C₄	1	2	yes	—
13	#15689405	C₄	1	2	yes	—
14	#15689320	C₄	1	2	yes	—
15	#15689261	C₄	1	2	yes	—
16	#15689247	—	2	1	no	—
17	#15689247	—	2	1	no	—
18	#15689247	C₂²	1	2	yes	—
19	#15689247	C₂²	1	2	yes	—
20	C₄×#10625335	C₂²	1	4	yes	—
21	#15689201	—	2	1	no	—
22	#15689201	C₂²	1	2	yes	—
23	D₄×C₄₇⋊C₈	—	2	1	no	—
24	D₄×C₄₇⋊C₈	C₂²	1	2	yes	—
25	#15689176	C₂²	1	1	yes	characteristic
26	#15689147	C₂²	1	1	yes	characteristic
27	C₂×Dic₄₇⋊C₈	—	2	1	no	—
28	#15689090	—	2	1	no	—
29	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₇	—	2	1	no	—
31	#15688875	C₂²	1	4	yes	—
32	#15688855	C₂²	1	4	yes	—
33	#15688845	C₂²	1	2	yes	—
34	#15688844	C₂²	1	2	yes	—
35	C₄₇×#532	C₂²	1	1	yes	characteristic
36	C₄₇×#431	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×D₄⋊C₈	C₂²	1	4	yes	—
39	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
40	C₂×#10625335	C₂×C₄	1	4	yes	—
41	#13094031	—	4	4	no	—
42	C₂×#10625329	—	2	1	no	—
43	C₂×#10625329	—	2	2	no	—
44	C₂²×C₄₇⋊C₈	—	2	2	no	—
45	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
46	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
47	C₄×Dic₉₄	—	2	1	no	—
48	C₄×Dic₉₄	—	2	1	no	—
49	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
50	C₄×Dic₉₄	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×Dic₄₇⋊C₄	—	2	1	no	—
53	C₂×C₄×Dic₄₇	—	2	1	no	—
54	#13093955	C₂×C₄	1	4	yes	—
55	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
56	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
57	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
58	Dic₄₇⋊C₈	—	2	2	no	—
59	Dic₄₇⋊C₈	—	2	2	no	—
60	#13093947	C₂×C₄	1	4	yes	—
61	#13093945	C₂×C₄	1	4	yes	—
62	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
63	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
64	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
65	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
66	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
67	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
68	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₈×Dic₄₇	—	2	2	no	—
71	C₈×Dic₄₇	—	2	2	no	—
72	C₂×#41×C₄₇	—	2	1	no	—
73	C₂²×C₈×C₄₇	—	2	1	no	—
74	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
75	#127×C₄₇	C₂×C₄	1	1	yes	characteristic
76	#124×C₄₇	—	2	1	no	—
77	#121×C₄₇	—	2	1	no	—
78	#121×C₄₇	C₂×C₄	1	2	yes	—
79	C₄×D₄×C₄₇	—	2	1	no	—
80	C₄×D₄×C₄₇	C₂³	1	2	yes	—
81	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₄₇	—	2	1	no	—
83	C₄⋊C₈×C₄₇	—	2	1	no	—
84	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
85	C₂²⋊C₈×C₄₇	—	2	1	no	—
86	C₄×C₈×C₄₇	C₂³	1	2	yes	—
87	#41×C₄₇	—	4	2	no	—
88	#41×C₄₇	—	4	2	no	—
89	C₂×Q₈×C₄₇	—	2	1	no	—
90	C₂×D₄×C₄₇	—	2	2	no	—
91	C₂×D₄×C₄₇	—	2	1	no	—
92	C₂×D₄×C₄₇	—	2	1	no	—
93	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₄₇	—	2	2	no	—
95	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
98	C₂²×Dic₄₇	—	2	1	no	—
99	C₂²×Dic₄₇	—	2	1	no	—
100	C₂×Dic₉₄	—	2	1	no	—
101	C₂×Dic₉₄	—	2	4	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	—	4	2	no	—
106	C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
108	C₂²⋊C₄×C₄₇	—	2	2	no	—
109	C₂²⋊C₄×C₄₇	—	2	2	no	—
110	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₇	#41	1	1	yes	characteristic
113	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
114	#10625335	—	2	8	no	—
115	C₄⋊Dic₄₇	—	2	1	no	—
116	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
117	Dic₄₇⋊C₄	—	2	2	no	—
118	Dic₄₇⋊C₄	—	2	2	no	—
119	C₄×Dic₄₇	—	2	2	no	—
120	C₄×Dic₄₇	—	2	2	no	—
121	#10625329	—	4	4	no	—
122	#10625329	—	4	2	no	—
123	C₂×C₄₇⋊C₈	—	2	2	no	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	4	4	no	—
127	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
128	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂³×C₄₇	—	2	2	no	—
133	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₄₇	—	4	2	no	—
135	D₄×C₄₇	—	2	4	no	—
136	D₄×C₄₇	—	4	2	no	—
137	D₄×C₄₇	—	4	4	no	—
138	D₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₄₇	#137	1	2	yes	—
149	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₄₇	—	2	4	no	—
153	C₂×Dic₄₇	—	2	2	no	—
154	C₂×Dic₄₇	—	2	2	no	—
155	C₂×Dic₄₇	—	4	2	no	—
156	Dic₉₄	—	2	4	no	—
157	Dic₉₄	—	2	2	no	—
158	Dic₉₄	#131	1	4	yes	complemented
159	C₈×C₄₇	—	4	2	no	—
160	C₈×C₄₇	—	4	2	no	—
161	C₄₇⋊C₈	—	2	4	no	—
162	C₄₇⋊C₈	—	2	1	no	—
163	C₄₇⋊C₈	—	2	1	no	—
164	C₄₇⋊C₈	—	2	1	no	—
165	C₄₇⋊C₈	—	2	1	no	—
166	C₄₇⋊C₈	—	4	2	no	—
167	#17908	—	47	1	no	maximal, Sylow
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	—	2	1	no	—
170	C₂²×C₄₇	—	4	2	no	—
171	C₂²×C₄₇	—	4	2	no	—
172	C₂²×C₄₇	—	4	2	no	—
173	C₂²×C₄₇	#440	1	2	yes	—
174	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	4	2	no	—
179	C₄×C₄₇	#440	1	2	yes	—
180	C₄×C₄₇	C₂×#131	1	2	yes	—
181	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
183	Dic₄₇	—	2	4	no	—
184	Dic₄₇	—	4	2	no	—
185	#3103	—	47	1	no	—
186	C₂×C₈×Q₈	—	47	1	no	—
187	#2931	—	47	1	no	—
188	#1588	—	47	4	no	—
189	SD₁₆⋊C₈	—	47	4	no	—
190	#1571	—	47	1	no	—
191	#1487	—	47	1	no	—
192	#1480	D₄₇	1	1	yes	characteristic
193	#1452	—	47	1	no	—
194	C₂×C₄₇	—	2	2	no	—
195	C₂×C₄₇	—	8	2	no	—
196	C₂×C₄₇	#2929	1	1	yes	characteristic
197	C₂×C₄₇	C₂×#440	1	1	yes	characteristic
198	C₂×C₄₇	#2955	1	1	yes	characteristic
199	C₂×#131	—	94	1	no	—
200	#532	C₂×D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
216	#431	C₂×D₄₇	1	1	yes	characteristic
217	#423	—	47	1	no	—
218	C₂×C₄⋊C₈	—	94	1	no	—
219	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
220	#408	—	94	1	no	—
221	C₂×C₄×C₈	—	47	1	no	—
222	C₂×C₄×C₈	—	94	1	no	—
223	#335	—	47	4	no	—
224	Q₈⋊C₈	—	47	4	no	—
225	D₄⋊C₈	C₂×D₄₇	1	4	yes	—
226	#322	—	47	2	no	—
227	#322	—	47	2	no	—
228	C₄₇	#17908	1	1	yes	characteristic, complemented, Sylow
229	C₂×#41	—	2	1	no	—
230	C₂²×Q₈	—	47	1	no	—
231	C₂×SD₁₆	—	47	4	no	—
232	#131	—	188	4	no	—
233	C₂×M₁₆	—	94	2	no	—
234	C₂×M₁₆	—	94	1	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	47	1	no	—
237	C₂²×C₈	—	47	1	no	—
238	C₂²×C₈	—	94	2	no	—
239	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
240	#127	C₄×D₄₇	1	1	yes	characteristic
241	#124	—	2	1	no	—
242	#121	—	2	1	no	—
243	#121	C₄×D₄₇	1	2	yes	—
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	—	2	1	no	—
249	C₄×D₄	C₂²×D₄₇	1	2	yes	—
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	—	2	1	no	—
257	C₄⋊C₈	—	47	2	no	—
258	C₄⋊C₈	—	47	2	no	—
259	C₄⋊C₈	—	94	2	no	—
260	C₄⋊C₈	—	94	2	no	—
261	C₄⋊C₈	—	94	1	no	—
262	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
263	#103	—	47	4	no	—
264	#102	—	47	4	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	47	2	no	—
267	C₂²⋊C₈	—	94	2	no	—
268	C₂²⋊C₈	—	94	2	no	—
269	C₄×C₈	—	47	1	no	—
270	C₄×C₈	—	47	1	no	—
271	C₄×C₈	—	47	2	no	—
272	C₄×C₈	—	47	1	no	—
273	C₄×C₈	—	47	1	no	—
274	C₄×C₈	—	94	2	no	—
275	C₄×C₈	—	94	2	no	—
276	C₄×C₈	C₂²×D₄₇	1	2	yes	—
277	#41	—	4	2	no	—
278	#41	—	4	2	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	47	2	no	—
281	C₂×Q₈	—	94	4	no	—
282	C₂×Q₈	—	94	1	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	94	1	no	—
289	C₂²×C₄	—	94	1	no	—
290	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
292	C₂²×C₄	#10625352	1	1	yes	characteristic
293	SD₁₆	—	94	8	no	—
294	M₁₆	—	188	4	no	—
295	M₁₆	—	188	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	4	2	no	—
299	C₂×C₈	—	47	4	no	—
300	C₂×C₈	—	47	1	no	—
301	C₂×C₈	—	47	1	no	—
302	C₂×C₈	—	47	1	no	—
303	C₂×C₈	—	47	1	no	—
304	C₂×C₈	—	94	2	no	—
305	C₂×C₈	—	94	2	no	—
306	C₂×C₈	—	94	2	no	—
307	C₂×C₈	—	188	4	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	47	2	no	—
310	C₄⋊C₄	—	94	2	no	—
311	C₄⋊C₄	—	94	1	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₄²	—	94	2	no	—
317	C₄²	—	94	2	no	—
318	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
319	C₄²	D₄×D₄₇	1	1	yes	characteristic
320	C₄²	#10625352	1	1	yes	characteristic
321	C₄²	C₂³×D₄₇	1	1	yes	characteristic
322	C₂³	—	2	2	no	—
323	C₂³	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
324	Q₈	—	4	2	no	—
325	Q₈	—	47	4	no	—
326	Q₈	—	94	4	no	—
327	Q₈	—	94	2	no	—
328	D₄	—	2	4	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	94	4	no	—
338	C₂×C₄	—	94	2	no	—
339	C₂×C₄	—	94	2	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
342	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
343	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
344	C₂×C₄	#13094048	1	2	yes	—
345	C₂×C₄	#13094053	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
348	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	94	4	no	—
352	C₈	—	94	1	no	—
353	C₈	—	94	1	no	—
354	C₈	—	94	1	no	—
355	C₈	—	94	1	no	—
356	C₈	—	188	2	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	#15689220	1	2	yes	—
363	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	4	2	no	—
368	C₄	—	94	4	no	—
369	C₄	—	188	2	no	—
370	C₄	#15689220	1	2	yes	—
371	C₄	#131×D₄₇	1	2	yes	—
372	C₄	C₂×#13094048	1	1	yes	characteristic
373	C₄	C₂×#13094053	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	8	2	no	—
376	C₂	#20252085	1	1	yes	characteristic
377	C₂	C₂×#15689220	1	1	yes	characteristic
378	C₂	#20252287	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1480	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₄₇×#532	C₂²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1480	D₄₇
hypocenter	C₄₇	#17908
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1480

Complements

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

N	H ≅ G/N	type
2. C₂×#15689247	C₂	semidirect
4. #20242269	C₂	semidirect
5. #20242197	C₂	semidirect
6. #20242162	C₂	semidirect
7. #20241818	C₂	semidirect
8. #20241681	C₂	semidirect
158. Dic₉₄	#131	semidirect
228. C₄₇	#17908	semidirect

Maximal subgroups

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

subgroup	index
#20253415	2
C₂×#15689247	2
#20252100	2
#20242269	2
#20242197	2
#20242162	2
#20241818	2
#20241681	2
C₄₇×#1480	2
#17908	47