Subgroups — #26369556

Order 12032.

690 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×#15689553	C₂	1	1	yes	maximal, characteristic
2	#135×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252159	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245244	C₂	1	2	yes	maximal, complemented
5	#20245243	C₂	1	2	yes	maximal, complemented
6	#20245237	C₂	1	2	yes	maximal, complemented
7	#20245232	C₂	1	2	yes	maximal, complemented
8	#20245170	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245096	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2138	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	#41×Dic₄₇	—	2	1	no	—
14	#15689591	—	2	1	no	—
15	#15689553	C₄	1	8	yes	complemented
16	C₂×#13093970	C₂²	1	1	yes	characteristic
17	C₂×#13093968	C₂²	1	1	yes	characteristic
18	C₂×#13093967	C₂²	1	1	yes	characteristic
19	C₂×#13093965	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
21	Q₁₆×Dic₄₇	C₂²	1	2	yes	—
22	#15689217	C₂²	1	2	yes	—
23	SD₁₆×Dic₄₇	—	2	2	no	—
24	#15689210	—	2	2	no	—
25	D₈×Dic₄₇	C₂²	1	2	yes	—
26	#15689205	C₂²	1	2	yes	—
27	#15689161	C₂²	1	1	yes	characteristic
28	#15689119	—	2	1	no	—
29	#15689117	—	2	1	no	—
30	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
31	#15688950	C₂²	1	2	yes	—
32	#15688948	C₂²	1	2	yes	—
33	#15688928	C₂²	1	2	yes	—
34	#15688925	C₂²	1	2	yes	—
35	C₁₆×Dic₄₇	C₂²	1	2	yes	—
36	C₁₆×Dic₄₇	C₂²	1	2	yes	—
37	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#427	C₂²	1	1	yes	characteristic
40	C₄₇×#358	C₂²	1	2	yes	—
41	C₄₇×#357	C₂²	1	2	yes	—
42	C₂×#10625350	—	2	1	no	—
43	#13094046	—	2	2	no	—
44	#13094046	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
46	C₂×#10625325	—	2	1	no	—
47	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
48	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
49	Q₈×Dic₄₇	—	2	2	no	—
50	#13093989	—	2	2	no	—
51	D₄×Dic₄₇	—	2	2	no	—
52	D₄×Dic₄₇	—	4	2	no	—
53	#13093986	—	2	2	no	—
54	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
55	#13093983	—	2	1	no	—
56	#13093983	—	4	1	no	—
57	#13093981	—	4	1	no	—
58	C₂×C₄×Dic₄₇	—	4	1	no	—
59	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
60	C₄²×D₄₇	—	4	1	no	—
61	#13093970	C₂×C₄	1	4	yes	complemented
62	#13093968	C₂×C₄	1	4	yes	complemented
63	#13093967	C₂×C₄	1	4	yes	complemented
64	#13093965	C₂×C₄	1	4	yes	complemented
65	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
66	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
68	#13093958	C₂³	1	2	yes	—
69	#13093956	—	2	1	no	—
70	#13093948	—	2	2	no	—
71	#13093947	—	2	2	no	—
72	#13093945	—	2	2	no	—
73	#13093944	—	2	2	no	—
74	C₈×Dic₄₇	D₄	1	2	yes	—
75	C₈×Dic₄₇	C₂³	1	2	yes	—
76	C₂×#41×C₄₇	—	2	1	no	—
77	#135×C₄₇	—	2	2	no	—
78	#135×C₄₇	C₂×C₄	1	4	yes	complemented
79	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
80	C₂×SD₁₆×C₄₇	—	2	1	no	—
81	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
83	#117×C₄₇	—	2	1	no	—
84	C₂×C₁₆×C₄₇	—	2	2	no	—
85	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
86	#107×C₄₇	C₂³	1	2	yes	—
87	#106×C₄₇	—	2	1	no	—
88	#41×C₄₇	—	2	4	no	—
89	#41×C₄₇	—	4	2	no	—
90	C₂×Q₈×C₄₇	—	2	1	no	—
91	C₂×D₄×C₄₇	—	2	1	no	—
92	C₂×D₄×C₄₇	—	4	1	no	—
93	C₂²×C₄×C₄₇	—	4	1	no	—
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₂²⋊D₄₇	—	4	1	no	—
96	C₂²×Dic₄₇	—	4	2	no	—
97	C₂²×Dic₄₇	D₈	1	2	yes	—
98	#10625350	—	2	4	no	—
99	#10625350	—	4	2	no	—
100	C₂×D₁₈₈	—	2	1	no	—
101	C₂×C₄×D₄₇	—	4	1	no	—
102	C₂×C₄×D₄₇	—	4	2	no	—
103	C₂×Dic₉₄	—	2	1	no	—
104	Q₁₆×C₄₇	—	2	1	no	—
105	Q₁₆×C₄₇	C₂²×C₄	1	2	yes	—
106	SD₁₆×C₄₇	—	2	4	no	—
107	D₈×C₄₇	—	2	1	no	—
108	D₈×C₄₇	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₇	#41	1	2	yes	—
112	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₄₇	—	2	2	no	—
114	C₂²⋊C₄×C₄₇	—	4	1	no	—
115	C₄²×C₄₇	—	4	1	no	—
116	C₂²⋊Dic₄₇	—	4	1	no	—
117	C₂²⋊Dic₄₇	—	4	2	no	—
118	(C₂×D₄₇)⋊C₄	—	4	2	no	—
119	C₄⋊Dic₄₇	—	2	2	no	—
120	C₄⋊Dic₄₇	—	4	2	no	—
121	Dic₄₇⋊C₄	—	4	2	no	—
122	C₄×Dic₄₇	—	4	1	no	—
123	C₄×Dic₄₇	—	4	2	no	—
124	C₄×Dic₄₇	—	4	2	no	—
125	C₄×Dic₄₇	D₈	1	1	yes	characteristic
126	C₄×Dic₄₇	D₈	1	1	yes	characteristic
127	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
128	C₂×C₄₇⋊C₈	—	2	2	no	—
129	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
130	Dic₁₈₈	—	2	1	no	—
131	Dic₁₈₈	C₂²×C₄	1	2	yes	—
132	D₃₇₆	—	2	1	no	—
133	D₃₇₆	C₂²×C₄	1	2	yes	—
134	#10625325	—	2	4	no	—
135	C₁₆×C₄₇	—	2	2	no	—
136	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
137	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
138	C₂³×C₄₇	—	4	1	no	—
139	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
140	C₂²×D₄₇	—	4	1	no	—
141	Q₈×C₄₇	—	2	2	no	—
142	Q₈×C₄₇	—	4	1	no	—
143	D₄×C₄₇	—	2	2	no	—
144	D₄×C₄₇	—	4	4	no	—
145	D₄×C₄₇	—	4	1	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	—	4	4	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₄₇	#135	1	2	yes	—
152	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₂²⋊D₄₇	—	4	4	no	—
154	C₂×Dic₄₇	—	2	4	no	—
155	C₂×Dic₄₇	—	4	1	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	C₂×Dic₄₇	—	8	4	no	—
158	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
159	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
160	D₁₈₈	—	2	2	no	—
161	D₁₈₈	—	4	1	no	—
162	C₄×D₄₇	—	4	4	no	—
163	C₄×D₄₇	—	8	4	no	—
164	Dic₉₄	—	2	2	no	—
165	Dic₉₄	—	4	1	no	—
166	C₈×C₄₇	C₄×D₄	1	2	yes	—
167	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
168	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
169	C₄₇⋊C₈	—	2	2	no	—
170	C₄×#508	—	47	1	no	maximal, Sylow
171	C₂²×C₄₇	—	2	2	no	—
172	C₂²×C₄₇	—	4	2	no	—
173	C₂²×C₄₇	—	8	2	no	—
174	C₂²×C₄₇	C₄×D₈	1	2	yes	—
175	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₄₇	—	4	2	no	—
177	C₂×D₄₇	—	8	2	no	—
178	C₄×C₄₇	—	4	2	no	—
179	C₄×C₄₇	—	8	2	no	—
180	C₄×C₄₇	C₄×D₈	1	2	yes	—
181	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
182	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
183	Dic₄₇	—	2	2	no	—
184	Dic₄₇	—	4	2	no	—
185	Dic₄₇	—	8	2	no	—
186	Dic₄₇	#508	1	4	yes	complemented
187	C₂×#508	—	47	1	no	—
188	C₄×#135	—	47	1	no	—
189	C₄×#135	—	47	1	no	—
190	C₄×Q₃₂	—	47	2	no	—
191	C₄×SD₃₂	—	47	2	no	—
192	C₄×SD₃₂	—	47	2	no	—
193	C₄×D₁₆	—	47	2	no	—
194	#2158	—	47	1	no	—
195	#2138	—	47	1	no	—
196	#2138	D₄₇	1	1	yes	characteristic, complemented
197	C₂×C₄×C₁₆	—	47	1	no	—
198	C₂×C₄₇	—	2	2	no	—
199	C₂×C₄₇	—	8	2	no	—
200	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
201	C₂×C₄₇	C₂×#508	1	2	yes	—
202	D₄₇	—	8	2	no	—
203	C₂×#135	—	47	1	no	—
204	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
205	C₄×#41	—	94	1	no	—
206	C₄×#41	—	94	1	no	—
207	#508	—	47	8	no	—
208	C₂×Q₃₂	—	47	1	no	—
209	C₂×SD₃₂	—	47	1	no	—
210	C₂×SD₃₂	—	47	1	no	—
211	C₂×D₁₆	—	47	1	no	—
212	C₂²×C₁₆	—	47	1	no	—
213	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
214	C₄×Q₁₆	—	47	2	no	—
215	C₄×Q₁₆	—	47	2	no	—
216	C₄×SD₁₆	—	94	2	no	—
217	C₄×SD₁₆	—	94	2	no	—
218	C₄×D₈	—	47	2	no	—
219	C₄×D₈	—	47	2	no	—
220	#427	—	47	1	no	—
221	#427	C₂×D₄₇	1	1	yes	characteristic
222	#416	—	94	1	no	—
223	#416	—	94	1	no	—
224	C₂×C₄×C₈	—	47	1	no	—
225	#367	—	47	2	no	—
226	#366	—	47	2	no	—
227	#358	—	47	2	no	—
228	#358	C₂×D₄₇	1	2	yes	—
229	#357	—	47	2	no	—
230	#357	C₂×D₄₇	1	2	yes	—
231	C₄×C₁₆	—	47	2	no	—
232	C₄×C₁₆	—	47	2	no	—
233	C₄₇	C₄×#508	1	1	yes	characteristic, complemented, Sylow
234	C₂×#41	—	2	1	no	—
235	C₂×#41	—	94	1	no	—
236	#135	—	2	2	no	—
237	#135	—	47	4	no	—
238	#135	—	94	2	no	—
239	#135	C₄×D₄₇	1	4	yes	complemented
240	C₂×Q₁₆	—	47	1	no	—
241	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
242	C₂×SD₁₆	—	2	1	no	—
243	C₂×SD₁₆	—	94	1	no	—
244	C₂×D₈	—	47	1	no	—
245	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
246	C₂²×C₈	—	47	1	no	—
247	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
248	C₄×Q₈	—	94	2	no	—
249	C₄×Q₈	—	94	2	no	—
250	C₄×D₄	—	94	2	no	—
251	C₄×D₄	—	94	2	no	—
252	C₄×D₄	—	188	2	no	—
253	C₄×D₄	—	188	2	no	—
254	#117	—	2	1	no	—
255	#117	—	94	1	no	—
256	#117	—	188	1	no	—
257	#117	—	188	1	no	—
258	C₂×C₄²	—	47	1	no	—
259	C₂×C₄²	—	188	1	no	—
260	C₂×C₄²	—	188	1	no	—
261	Q₃₂	—	47	4	no	—
262	SD₃₂	—	47	4	no	—
263	SD₃₂	—	47	4	no	—
264	D₁₆	—	47	4	no	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	—	47	4	no	—
267	C₂×C₁₆	—	47	1	no	—
268	C₂×C₁₆	—	47	1	no	—
269	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
270	#107	—	47	2	no	—
271	#107	C₂²×D₄₇	1	2	yes	—
272	#106	—	2	1	no	—
273	#106	—	94	1	no	—
274	#103	—	94	2	no	—
275	#103	—	94	2	no	—
276	#102	—	94	2	no	—
277	#102	—	94	2	no	—
278	C₄×C₈	—	47	2	no	—
279	C₄×C₈	—	47	2	no	—
280	#41	—	2	4	no	—
281	#41	—	4	2	no	—
282	#41	—	94	4	no	—
283	#41	—	188	2	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	—	94	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	94	1	no	—
289	C₂×D₄	—	188	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	47	2	no	—
292	C₂²×C₄	—	188	2	no	—
293	C₂²×C₄	—	188	2	no	—
294	C₂²×C₄	—	188	1	no	—
295	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
296	Q₁₆	—	2	1	no	—
297	Q₁₆	—	47	2	no	—
298	Q₁₆	—	94	1	no	—
299	Q₁₆	C₂×C₄×D₄₇	1	2	yes	—
300	SD₁₆	—	2	4	no	—
301	SD₁₆	—	94	4	no	—
302	D₈	—	2	1	no	—
303	D₈	—	47	2	no	—
304	D₈	—	94	1	no	—
305	D₈	C₂×C₄×D₄₇	1	2	yes	—
306	C₂×C₈	—	47	2	no	—
307	C₂×C₈	—	94	2	no	—
308	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
309	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
310	C₂×C₈	#10625352	1	2	yes	—
311	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	94	2	no	—
314	C₄⋊C₄	—	188	2	no	—
315	C₄⋊C₄	—	188	2	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	188	2	no	—
318	C₂²⋊C₄	—	188	2	no	—
319	C₂²⋊C₄	—	188	1	no	—
320	C₄²	—	4	1	no	—
321	C₄²	—	47	1	no	—
322	C₄²	—	47	1	no	—
323	C₄²	—	47	2	no	—
324	C₄²	—	188	2	no	—
325	C₄²	—	188	2	no	—
326	C₄²	—	188	1	no	—
327	C₁₆	—	2	2	no	—
328	C₁₆	—	47	2	no	—
329	C₁₆	—	47	2	no	—
330	C₂³	—	4	1	no	—
331	C₂³	—	188	1	no	—
332	C₂³	D₈×D₄₇	1	1	yes	characteristic
333	Q₈	—	2	2	no	—
334	Q₈	—	4	1	no	—
335	Q₈	—	94	2	no	—
336	Q₈	—	188	1	no	—
337	D₄	—	2	2	no	—
338	D₄	—	4	4	no	—
339	D₄	—	4	1	no	—
340	D₄	—	94	2	no	—
341	D₄	—	188	4	no	—
342	D₄	—	188	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	47	2	no	—
347	C₂×C₄	—	47	2	no	—
348	C₂×C₄	—	94	4	no	—
349	C₂×C₄	—	376	4	no	—
350	C₂×C₄	—	376	4	no	—
351	C₂×C₄	—	188	2	no	—
352	C₂×C₄	—	188	1	no	—
353	C₂×C₄	—	188	4	no	—
354	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
355	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
356	C₂×C₄	#13094045	1	2	yes	—
357	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
358	C₈	—	94	2	no	—
359	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
360	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
361	C₈	C₂×#10625352	1	1	yes	characteristic
362	C₂²	—	2	2	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	376	2	no	—
366	C₂²	—	188	2	no	—
367	C₂²	#15689203	1	2	yes	—
368	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
369	C₄	—	4	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	47	4	no	—
372	C₄	—	94	2	no	—
373	C₄	—	376	2	no	—
374	C₄	—	188	2	no	—
375	C₄	#15689203	1	2	yes	—
376	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
377	C₄	C₂×#13094045	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	8	2	no	—
380	C₂	—	376	2	no	—
381	C₂	C₂×#15689203	1	1	yes	characteristic
382	C₂	#508×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2138	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#15689553	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2138	D₄₇
hypocenter	C₄₇	C₄×#508
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

Complements

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

N	H ≅ G/N	type
2. #135×Dic₄₇	C₂	semidirect
3. #20252159	C₂	semidirect
4. #20245244	C₂	semidirect
5. #20245243	C₂	semidirect
6. #20245237	C₂	semidirect
7. #20245232	C₂	semidirect
8. #20245170	C₂	semidirect
9. #20245096	C₂	semidirect
10. C₂×C₁₆×Dic₄₇	C₂	semidirect
11. C₄₇×#2138	C₂	semidirect
15. #15689553	C₄	semidirect
44. #13094046	C₂×C₄	semidirect
61. #13093970	C₂×C₄	semidirect
62. #13093968	C₂×C₄	semidirect
63. #13093967	C₂×C₄	semidirect
64. #13093965	C₂×C₄	semidirect
65. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
78. #135×C₄₇	C₂×C₄	semidirect
186. Dic₄₇	#508	semidirect
196. #2138	D₄₇	semidirect
233. C₄₇	C₄×#508	semidirect
239. #135	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689553	2
#135×Dic₄₇	2
#20252159	2
#20245244	2
#20245243	2
#20245237	2
#20245232	2
#20245170	2
#20245096	2
C₂×C₁₆×Dic₄₇	2
C₄₇×#2138	2
C₄×#508	47