Subgroups — #26115188

Order 12032.

534 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625325	C₂	1	1	yes	maximal, characteristic
2	#20244323	C₂	1	1	yes	maximal, characteristic, complemented
3	#20244181	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688853	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688850	C₂	1	1	yes	maximal, characteristic
6	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625325	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	C₄×#10625325	—	2	2	no	—
12	C₄×#10625325	—	2	2	no	—
13	C₂×#13093956	C₄	1	1	yes	characteristic
14	C₂×#13093949	C₄	1	1	yes	characteristic
15	C₂×#13093946	C₄	1	1	yes	characteristic
16	C₂×(C₂×D₄₇)⋊C₈	—	2	2	no	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	#15688877	—	2	2	no	—
20	#15688853	—	2	2	no	—
21	#15688850	—	2	2	no	—
22	#15688845	—	2	2	no	—
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#335	—	2	2	no	—
27	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₄₇	—	2	1	no	—
30	C₂×#10625325	—	2	2	no	—
31	C₂×#10625325	—	2	2	no	—
32	C₂×#10625325	—	2	2	no	—
33	C₂²×C₄₇⋊C₈	—	2	1	no	—
34	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
35	C₄×Dic₉₄	—	2	1	no	—
36	C₄×Dic₉₄	—	2	1	no	—
37	C₄×Dic₉₄	—	2	1	no	—
38	C₄×Dic₉₄	—	2	1	no	—
39	C₄×D₁₈₈	—	2	1	no	—
40	C₄×D₁₈₈	—	2	1	no	—
41	C₄×D₁₈₈	—	2	1	no	—
42	C₄×D₁₈₈	—	2	1	no	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×Dic₄₇⋊C₄	—	2	1	no	—
46	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
47	C₂×C₄×Dic₄₇	—	2	1	no	—
48	#13093956	—	2	2	no	—
49	#13093949	—	2	2	no	—
50	#13093946	—	2	2	no	—
51	(C₂×D₄₇)⋊C₈	—	4	4	no	—
52	C₄×C₄₇⋊C₈	—	2	1	no	—
53	C₄×C₄₇⋊C₈	—	2	1	no	—
54	C₄×C₄₇⋊C₈	—	2	1	no	—
55	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
56	C₂²×C₈×C₄₇	—	2	1	no	—
57	C₂²×C₈×C₄₇	—	2	1	no	—
58	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₇	—	2	1	no	—
62	C₄⋊C₈×C₄₇	—	2	1	no	—
63	C₄⋊C₈×C₄₇	—	2	1	no	—
64	C₄⋊C₈×C₄₇	—	2	1	no	—
65	C₄×C₈×C₄₇	D₄	1	2	yes	—
66	C₄×C₈×C₄₇	D₄	1	2	yes	—
67	C₂³×D₄₇	—	2	1	no	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	1	no	—
72	C₂²×Dic₄₇	—	2	1	no	—
73	C₂²×Dic₄₇	—	2	1	no	—
74	C₂×D₁₈₈	—	2	1	no	—
75	C₂×D₁₈₈	—	2	1	no	—
76	C₂×D₁₈₈	—	2	1	no	—
77	C₂×D₁₈₈	—	2	1	no	—
78	C₂×D₁₈₈	—	2	1	no	—
79	C₂×D₁₈₈	M₁₆	1	2	yes	—
80	C₂×C₄×D₄₇	—	4	1	no	—
81	C₂×C₄×D₄₇	—	4	1	no	—
82	C₂×C₄×D₄₇	—	4	2	no	—
83	C₂×C₄×D₄₇	—	4	2	no	—
84	C₂×Dic₉₄	—	2	1	no	—
85	C₂×Dic₉₄	—	2	1	no	—
86	C₂×Dic₉₄	—	2	1	no	—
87	C₂×Dic₉₄	—	2	1	no	—
88	C₂×Dic₉₄	—	2	1	no	—
89	C₂×Dic₉₄	M₁₆	1	2	yes	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	—	2	1	no	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	—	4	1	no	—
95	C₂×C₈×C₄₇	—	4	1	no	—
96	C₂×C₈×C₄₇	—	4	2	no	—
97	C₂×C₈×C₄₇	—	4	2	no	—
98	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₇	#41	1	1	yes	characteristic
104	(C₂×D₄₇)⋊C₄	—	4	1	no	—
105	(C₂×D₄₇)⋊C₄	—	4	1	no	—
106	C₄⋊Dic₄₇	—	2	1	no	—
107	C₄⋊Dic₄₇	—	2	1	no	—
108	C₄⋊Dic₄₇	—	2	1	no	—
109	C₄⋊Dic₄₇	—	2	1	no	—
110	Dic₄₇⋊C₄	—	4	1	no	—
111	Dic₄₇⋊C₄	—	4	1	no	—
112	C₄×Dic₄₇	—	4	1	no	—
113	C₄×Dic₄₇	—	4	1	no	—
114	C₂×C₄₇⋊C₈	—	2	4	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	—	2	2	no	—
119	#10625325	—	2	8	no	—
120	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
121	C₂²×D₄₇	—	2	2	no	—
122	C₂²×D₄₇	—	4	1	no	—
123	C₂²×D₄₇	—	4	1	no	—
124	C₂²×D₄₇	—	4	2	no	—
125	C₂²×D₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
137	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₄₇	#137	1	2	yes	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	4	1	no	—
141	C₂×Dic₄₇	—	4	1	no	—
142	C₂×Dic₄₇	—	4	1	no	—
143	C₂×Dic₄₇	—	4	1	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	D₁₈₈	—	2	2	no	—
147	D₁₈₈	—	2	2	no	—
148	D₁₈₈	—	4	1	no	—
149	D₁₈₈	—	4	1	no	—
150	C₄×D₄₇	—	8	2	no	—
151	C₄×D₄₇	—	8	1	no	—
152	C₄×D₄₇	—	8	1	no	—
153	Dic₉₄	—	2	2	no	—
154	Dic₉₄	—	2	2	no	—
155	Dic₉₄	—	4	1	no	—
156	Dic₉₄	—	4	1	no	—
157	C₈×C₄₇	—	2	2	no	—
158	C₈×C₄₇	—	2	2	no	—
159	C₈×C₄₇	—	8	1	no	—
160	C₈×C₄₇	—	8	1	no	—
161	C₄₇⋊C₈	—	2	4	no	—
162	C₄₇⋊C₈	—	4	2	no	—
163	#9666	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	#390	1	1	yes	characteristic
165	C₂²×C₄₇	#435	1	1	yes	characteristic
166	C₂²×C₄₇	#435	1	1	yes	characteristic
167	C₂²×C₄₇	#440	1	2	yes	—
168	C₂²×C₄₇	#443	1	2	yes	—
169	C₂×D₄₇	—	4	4	no	—
170	C₂×D₄₇	—	4	2	no	—
171	C₂×D₄₇	—	8	2	no	—
172	C₂×D₄₇	—	8	1	no	—
173	C₂×D₄₇	—	8	1	no	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	#407	1	1	yes	characteristic
177	C₄×C₄₇	#408	1	1	yes	characteristic
178	C₄×C₄₇	#435	1	1	yes	characteristic
179	C₄×C₄₇	#435	1	1	yes	characteristic
180	C₄×C₄₇	#468	1	2	yes	—
181	C₄×C₄₇	#496	1	1	yes	characteristic
182	C₄×C₄₇	#497	1	1	yes	characteristic
183	Dic₄₇	—	4	2	no	—
184	Dic₄₇	—	8	1	no	—
185	Dic₄₇	—	8	1	no	—
186	C₂×C₄×SD₁₆	—	47	1	no	—
187	#1984	—	47	1	no	—
188	#1954	—	47	1	no	—
189	C₂×#335	D₄₇	1	1	yes	characteristic, complemented
190	C₂×Q₈⋊C₈	—	47	1	no	—
191	C₂×D₄⋊C₈	—	47	1	no	—
192	C₂×#322	—	47	1	no	—
193	C₂×C₄₇	#1585	1	2	yes	—
194	C₂×C₄₇	#1588	1	2	yes	—
195	C₂×C₄₇	#1955	1	1	yes	characteristic
196	C₂×C₄₇	#1973	1	1	yes	characteristic
197	C₂×C₄₇	#1974	1	1	yes	characteristic
198	D₄₇	—	8	2	no	—
199	C₂²×SD₁₆	—	47	1	no	—
200	C₂×C₄×Q₈	—	47	1	no	—
201	C₂×C₄×D₄	—	47	1	no	—
202	C₄×SD₁₆	—	94	2	no	—
203	C₄×SD₁₆	—	94	2	no	—
204	C₂×#106	—	47	1	no	—
205	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
206	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
207	C₂×#103	—	47	1	no	—
208	C₂×#102	—	47	1	no	—
209	C₂×C₂²⋊C₈	—	94	2	no	—
210	C₂×C₄×C₈	—	47	1	no	—
211	C₂×C₄×C₈	—	47	1	no	—
212	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
213	#336	—	94	2	no	—
214	#335	—	2	2	no	—
215	Q₈⋊C₈	—	94	2	no	—
216	D₄⋊C₈	—	94	2	no	—
217	#322	—	94	2	no	—
218	C₄₇	#9666	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	—	47	1	no	—
220	C₂²×D₄	—	47	1	no	—
221	C₂³×C₄	—	94	1	no	—
222	C₂×SD₁₆	—	94	2	no	—
223	C₂×SD₁₆	—	94	2	no	—
224	C₂×SD₁₆	—	94	2	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	—	47	2	no	—
228	C₂²×C₈	—	94	1	no	—
229	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
230	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
231	C₄×Q₈	—	94	1	no	—
232	C₄×Q₈	—	94	1	no	—
233	C₄×Q₈	—	94	1	no	—
234	C₄×Q₈	—	94	1	no	—
235	C₄×D₄	—	94	1	no	—
236	C₄×D₄	—	94	1	no	—
237	C₄×D₄	—	94	1	no	—
238	C₄×D₄	—	94	1	no	—
239	C₂×C₄⋊C₄	—	47	1	no	—
240	C₂×C₄⋊C₄	—	47	1	no	—
241	C₂×C₄⋊C₄	—	94	1	no	—
242	C₂×C₂²⋊C₄	—	94	1	no	—
243	C₂×C₄²	—	94	1	no	—
244	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
245	#106	—	94	2	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	#103	—	94	2	no	—
251	#102	—	94	2	no	—
252	C₂²⋊C₈	—	188	4	no	—
253	C₄×C₈	—	47	2	no	—
254	C₄×C₈	—	94	1	no	—
255	C₄×C₈	—	94	1	no	—
256	C₄×C₈	—	94	1	no	—
257	C₄×C₈	D₁₈₈	1	2	yes	—
258	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
259	C₂⁴	—	94	1	no	—
260	C₂×Q₈	—	47	2	no	—
261	C₂×Q₈	—	94	1	no	—
262	C₂×Q₈	—	94	1	no	—
263	C₂×Q₈	—	94	1	no	—
264	C₂×Q₈	—	94	1	no	—
265	C₂×Q₈	—	94	1	no	—
266	C₂×D₄	—	47	2	no	—
267	C₂×D₄	—	94	1	no	—
268	C₂×D₄	—	94	1	no	—
269	C₂×D₄	—	94	1	no	—
270	C₂×D₄	—	94	1	no	—
271	C₂×D₄	—	94	1	no	—
272	C₂²×C₄	—	94	1	no	—
273	C₂²×C₄	—	94	1	no	—
274	C₂²×C₄	—	94	1	no	—
275	C₂²×C₄	—	188	1	no	—
276	C₂²×C₄	—	188	1	no	—
277	C₂²×C₄	—	188	2	no	—
278	C₂²×C₄	—	188	2	no	—
279	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
281	C₂²×C₄	#10625354	1	1	yes	characteristic
282	SD₁₆	—	94	8	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	—	94	4	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	—	94	2	no	—
294	C₂×C₈	—	94	2	no	—
295	C₂×C₈	—	94	2	no	—
296	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
297	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
298	C₄⋊C₄	—	94	1	no	—
299	C₄⋊C₄	—	94	1	no	—
300	C₄⋊C₄	—	94	1	no	—
301	C₄⋊C₄	—	94	1	no	—
302	C₄⋊C₄	—	188	1	no	—
303	C₄⋊C₄	—	188	1	no	—
304	C₂²⋊C₄	—	188	1	no	—
305	C₂²⋊C₄	—	188	1	no	—
306	C₄²	—	188	1	no	—
307	C₄²	—	188	1	no	—
308	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
309	C₄²	D₄×D₄₇	1	1	yes	characteristic
310	C₄²	#10625354	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
312	C₂³	—	94	2	no	—
313	C₂³	—	188	2	no	—
314	C₂³	—	188	2	no	—
315	C₂³	—	188	1	no	—
316	C₂³	—	188	1	no	—
317	C₂³	#13093986	1	1	yes	characteristic
318	Q₈	—	94	2	no	—
319	Q₈	—	94	2	no	—
320	Q₈	—	188	1	no	—
321	Q₈	—	188	1	no	—
322	D₄	—	94	2	no	—
323	D₄	—	94	2	no	—
324	D₄	—	188	1	no	—
325	D₄	—	188	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	376	1	no	—
329	C₂×C₄	—	376	1	no	—
330	C₂×C₄	—	376	2	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	—	188	2	no	—
333	C₂×C₄	—	188	1	no	—
334	C₂×C₄	—	188	1	no	—
335	C₂×C₄	—	188	1	no	—
336	C₂×C₄	—	188	1	no	—
337	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
338	C₂×C₄	#13093986	1	1	yes	characteristic
339	C₂×C₄	#13093999	1	1	yes	characteristic
340	C₂×C₄	#13093999	1	1	yes	characteristic
341	C₂×C₄	#13094013	1	1	yes	characteristic
342	C₂×C₄	#13094020	1	1	yes	characteristic
343	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
344	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
345	C₂×C₄	#13094030	1	1	yes	characteristic
346	C₂×C₄	#13094030	1	1	yes	characteristic
347	C₂×C₄	#13094048	1	2	yes	—
348	C₂×C₄	#13094056	1	2	yes	—
349	C₈	—	2	2	no	—
350	C₈	—	2	2	no	—
351	C₈	—	8	1	no	—
352	C₈	—	8	1	no	—
353	C₈	—	94	4	no	—
354	C₈	—	188	2	no	—
355	C₂²	—	376	1	no	—
356	C₂²	—	376	1	no	—
357	C₂²	—	376	2	no	—
358	C₂²	—	188	4	no	—
359	C₂²	—	188	2	no	—
360	C₂²	#15689024	1	1	yes	characteristic
361	C₂²	#15689193	1	1	yes	characteristic
362	C₂²	#15689193	1	1	yes	characteristic
363	C₂²	#15689222	1	2	yes	—
364	C₂²	#15689239	1	2	yes	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	376	1	no	—
368	C₄	—	376	1	no	—
369	C₄	—	188	2	no	—
370	C₄	#15689086	1	1	yes	characteristic
371	C₄	#15689089	1	1	yes	characteristic
372	C₄	#15689193	1	1	yes	characteristic
373	C₄	#15689193	1	1	yes	characteristic
374	C₄	#15689370	1	2	yes	—
375	C₄	#15689511	1	1	yes	characteristic
376	C₄	#15689516	1	1	yes	characteristic
377	C₂	—	376	2	no	—
378	C₂	#20242244	1	2	yes	—
379	C₂	#20242265	1	2	yes	—
380	C₂	#20244185	1	1	yes	characteristic
381	C₂	#20244267	1	1	yes	characteristic
382	C₂	#20244274	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	#9666
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. #20244323	C₂	semidirect
4. C₂×#15688853	C₂	semidirect
6. C₂×#15688845	C₂	semidirect
7. C₂×C₄₇×#335	C₂	semidirect
189. C₂×#335	D₄₇	semidirect
218. C₄₇	#9666	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625325	2
#20244323	2
#20244181	2
C₂×#15688853	2
C₂×#15688850	2
C₂×#15688845	2
C₂×C₄₇×#335	2
#9666	47