Subgroups — #26373150

Order 12032.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255735	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245617	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245587	C₂	1	2	yes	maximal, complemented
5	#20245485	C₂	1	2	yes	maximal, complemented
6	#20245434	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245306	C₂	1	2	yes	maximal, complemented
8	C₄×#13093967	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245207	C₂	1	1	yes	maximal, characteristic, complemented
10	#358×D₄₇	C₂	1	2	yes	maximal, complemented
11	C₄₇×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₇	C₂²	1	2	yes	—
13	Q₈×D₁₈₈	—	2	1	no	—
14	#15689776	—	2	1	no	—
15	C₂×#13093969	C₂²	1	2	yes	—
16	C₂×#13093967	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
18	#15689503	C₂²	1	1	yes	characteristic
19	#15689438	—	2	2	no	—
20	#15689400	—	2	1	no	—
21	#15689365	C₂²	1	2	yes	—
22	#15689360	C₂²	1	2	yes	—
23	#15689333	—	2	1	no	—
24	#15689280	—	2	2	no	—
25	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
26	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
27	#107×D₄₇	C₂²	1	2	yes	—
28	#15688950	C₂²	1	1	yes	characteristic
29	#15688928	C₂²	1	2	yes	—
30	#15688927	C₂²	1	1	yes	characteristic
31	#15688926	C₂²	1	2	yes	—
32	#15688926	C₂²	1	2	yes	—
33	#15688923	C₂²	1	1	yes	characteristic
34	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₄₇×#500	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#367	C₂²	1	2	yes	—
39	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄₇×#358	C₂²	1	2	yes	—
41	C₂×#10625354	—	4	1	no	—
42	C₂×Q₈×D₄₇	—	2	2	no	—
43	C₂×Dic₁₈₈	—	2	1	no	—
44	C₂×Dic₁₈₈	C₂³	1	2	yes	—
45	C₂×#10625337	—	2	2	no	—
46	Q₁₆×D₄₇	—	2	4	no	—
47	Q₁₆×D₄₇	—	4	2	no	—
48	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₄₇	C₂³	1	2	yes	—
50	#13094024	—	2	1	no	—
51	#13094024	—	4	1	no	—
52	#13094009	—	4	2	no	—
53	#13094004	—	2	2	no	—
54	#13094004	—	4	2	no	—
55	#13094000	—	4	1	no	—
56	#13093999	—	2	2	no	—
57	C₄×D₁₈₈	—	2	1	no	—
58	C₄×D₁₈₈	—	4	1	no	—
59	C₄×D₁₈₈	D₄	1	1	yes	characteristic
60	C₄⋊C₄×D₄₇	—	2	2	no	—
61	#13093969	—	4	2	no	—
62	#13093967	—	2	2	no	—
63	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
65	C₁₆×D₄₇	—	4	2	no	—
66	#13093958	C₂³	1	1	yes	characteristic
67	#13093957	—	2	2	no	—
68	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
69	#13093947	—	2	2	no	—
70	#13093946	—	2	1	no	—
71	#13093944	—	2	2	no	—
72	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
73	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₄₇	—	2	1	no	—
75	C₄×Q₈×C₄₇	—	2	1	no	—
76	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
77	#107×C₄₇	—	2	1	no	—
78	#107×C₄₇	C₂³	1	2	yes	—
79	#107×C₄₇	C₂³	1	1	yes	characteristic
80	#103×C₄₇	—	2	1	no	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₇	—	2	1	no	—
83	C₂×Q₈×C₄₇	—	4	1	no	—
84	#10625354	—	8	4	no	—
85	Q₈×D₄₇	—	4	4	no	—
86	Q₈×D₄₇	—	8	2	no	—
87	C₂×D₁₈₈	—	2	1	no	—
88	C₂×D₁₈₈	—	4	2	no	—
89	C₂×D₁₈₈	D₈	1	1	yes	characteristic
90	C₂×C₄×D₄₇	—	4	2	no	—
91	C₂×C₄×D₄₇	—	4	2	no	—
92	C₂×C₄×D₄₇	—	4	1	no	—
93	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
94	C₂×Dic₉₄	—	2	2	no	—
95	C₂×Dic₉₄	—	4	2	no	—
96	Q₁₆×C₄₇	—	2	1	no	—
97	Q₁₆×C₄₇	#41	1	2	yes	complemented
98	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₄₇	—	2	1	no	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄⋊C₄×C₄₇	—	4	1	no	—
103	C₄⋊C₄×C₄₇	—	4	1	no	—
104	C₄²×C₄₇	—	4	1	no	—
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	#10625337	—	4	4	no	—
107	(C₂×D₄₇)⋊C₄	—	4	2	no	—
108	(C₂×D₄₇)⋊C₄	—	4	1	no	—
109	(C₂×D₄₇)⋊C₄	—	4	2	no	—
110	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
111	C₄⋊Dic₄₇	—	2	1	no	—
112	C₄⋊Dic₄₇	—	4	1	no	—
113	C₄⋊Dic₄₇	—	4	1	no	—
114	C₄⋊Dic₄₇	—	4	2	no	—
115	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
116	Dic₄₇⋊C₄	—	4	2	no	—
117	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
118	Dic₁₈₈	—	2	2	no	—
119	Dic₁₈₈	—	2	2	no	—
120	Dic₁₈₈	—	8	1	no	—
121	D₃₇₆	—	2	1	no	—
122	D₃₇₆	—	2	1	no	—
123	C₈×D₄₇	—	2	2	no	—
124	C₈×D₄₇	—	2	2	no	—
125	C₁₆×C₄₇	—	2	2	no	—
126	C₄₇⋊C₁₆	—	2	1	no	—
127	C₄₇⋊C₁₆	—	2	1	no	—
128	C₂²×D₄₇	—	4	1	no	—
129	C₂²×D₄₇	C₂×D₈	1	2	yes	—
130	Q₈×C₄₇	—	2	2	no	—
131	Q₈×C₄₇	—	4	1	no	—
132	Q₈×C₄₇	—	8	2	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₄₇	—	4	1	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
142	D₁₈₈	—	4	1	no	—
143	D₁₈₈	—	4	1	no	—
144	D₁₈₈	—	8	4	no	—
145	D₁₈₈	SD₃₂	1	4	yes	complemented
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	8	4	no	—
149	C₄×D₄₇	—	8	4	no	—
150	C₄×D₄₇	—	8	2	no	—
151	Dic₉₄	—	4	2	no	—
152	Dic₉₄	—	4	2	no	—
153	Dic₉₄	—	8	4	no	—
154	C₈×C₄₇	—	2	1	no	—
155	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
156	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
157	C₄₇⋊C₈	—	2	2	no	—
158	#33865	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
160	C₂×D₄₇	—	2	2	no	—
161	C₂×D₄₇	—	2	2	no	—
162	C₂×D₄₇	—	8	1	no	—
163	C₂×D₄₇	—	8	1	no	—
164	C₂×D₄₇	—	8	1	no	—
165	C₂×D₄₇	C₂×SD₃₂	1	4	yes	—
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	—	4	2	no	—
168	C₄×C₄₇	—	8	2	no	—
169	C₄×C₄₇	—	8	1	no	—
170	C₄×C₄₇	C₂×SD₃₂	1	2	yes	—
171	C₄×C₄₇	#546	1	1	yes	characteristic
172	C₄×C₄₇	#577	1	1	yes	characteristic
173	Dic₄₇	—	2	2	no	—
174	Dic₄₇	—	8	1	no	—
175	Dic₄₇	—	8	2	no	—
176	#3334	—	47	1	no	—
177	D₄×Q₁₆	—	47	1	no	—
178	#2241	—	47	1	no	—
179	#2235	—	47	2	no	—
180	#2225	D₄₇	1	1	yes	characteristic, complemented
181	#2217	—	47	2	no	—
182	#2208	—	47	1	no	—
183	#2185	—	47	2	no	—
184	C₄×SD₃₂	—	47	1	no	—
185	D₄×C₁₆	—	47	1	no	—
186	C₂×#358	—	47	2	no	—
187	C₂×C₄₇	#3294	1	1	yes	characteristic
188	C₂×C₄₇	C₂²×SD₃₂	1	1	yes	characteristic
189	C₂×C₄₇	#3417	1	1	yes	characteristic
190	D₄₇	—	2	4	no	—
191	D₄₇	—	16	1	no	—
192	C₂²×Q₁₆	—	47	2	no	—
193	D₄×Q₈	—	94	1	no	—
194	#547	—	94	1	no	—
195	C₂×SD₃₂	—	47	1	no	—
196	C₂×SD₃₂	—	47	2	no	—
197	C₂²×C₁₆	—	47	2	no	—
198	#500	C₂×D₄₇	1	1	yes	characteristic
199	#494	—	47	1	no	—
200	#480	—	94	2	no	—
201	#474	—	94	1	no	—
202	#467	—	47	2	no	—
203	#466	—	47	2	no	—
204	#462	—	94	1	no	—
205	#451	—	94	2	no	—
206	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
207	C₄×D₈	—	47	1	no	—
208	C₈×D₄	—	47	1	no	—
209	C₂×#107	—	47	2	no	—
210	#367	—	47	1	no	—
211	#367	C₂×D₄₇	1	2	yes	—
212	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
213	#358	—	47	1	no	—
214	#358	—	47	2	no	—
215	#358	—	47	2	no	—
216	#358	—	47	2	no	—
217	#358	C₂×D₄₇	1	2	yes	—
218	#357	—	47	1	no	—
219	C₂²⋊C₁₆	—	47	2	no	—
220	C₄×C₁₆	—	47	1	no	—
221	C₄₇	#33865	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	188	1	no	—
223	C₂²×Q₈	—	94	2	no	—
224	C₂×Q₁₆	—	47	2	no	—
225	C₂×Q₁₆	—	94	2	no	—
226	C₂×Q₁₆	—	94	4	no	—
227	C₂×Q₁₆	—	94	1	no	—
228	C₂×Q₁₆	—	188	2	no	—
229	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
230	C₂×D₈	—	47	1	no	—
231	C₂²×C₈	—	47	2	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	94	1	no	—
234	C₄⋊Q₈	—	188	1	no	—
235	#123	—	188	2	no	—
236	C₂²⋊Q₈	—	94	2	no	—
237	C₂²⋊Q₈	—	188	2	no	—
238	C₂²⋊Q₈	—	188	1	no	—
239	#121	—	94	2	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	47	1	no	—
242	C₄×D₄	—	94	1	no	—
243	C₄×D₄	—	188	1	no	—
244	C₂×C₄⋊C₄	—	94	2	no	—
245	SD₃₂	—	94	2	no	—
246	SD₃₂	—	188	2	no	—
247	C₂×C₁₆	—	47	1	no	—
248	C₂×C₁₆	—	47	1	no	—
249	C₂×C₁₆	—	188	2	no	—
250	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
251	#107	—	2	1	no	—
252	#107	—	47	1	no	—
253	#107	—	94	2	no	—
254	#107	C₂²×D₄₇	1	1	yes	characteristic
255	#107	C₂²×D₄₇	1	2	yes	—
256	C₄⋊C₈	—	47	1	no	—
257	#103	—	2	1	no	—
258	#103	—	94	2	no	—
259	#102	—	94	2	no	—
260	#102	—	94	1	no	—
261	C₂²⋊C₈	—	47	2	no	—
262	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
263	#41	—	376	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	94	2	no	—
267	C₂×Q₈	—	376	2	no	—
268	C₂×Q₈	—	188	2	no	—
269	C₂×Q₈	—	188	4	no	—
270	C₂×D₄	—	47	1	no	—
271	C₂×D₄	—	94	1	no	—
272	C₂×D₄	—	188	2	no	—
273	C₂²×C₄	—	47	2	no	—
274	C₂²×C₄	—	188	2	no	—
275	C₂²×C₄	—	188	2	no	—
276	C₂²×C₄	—	188	1	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	—	94	2	no	—
279	Q₁₆	—	94	2	no	—
280	Q₁₆	—	376	1	no	—
281	Q₁₆	—	188	4	no	—
282	Q₁₆	#10625354	1	2	yes	complemented
283	D₈	—	94	1	no	—
284	D₈	—	94	1	no	—
285	C₂×C₈	—	47	2	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	47	1	no	—
295	C₄⋊C₄	—	94	1	no	—
296	C₄⋊C₄	—	188	2	no	—
297	C₄⋊C₄	—	188	2	no	—
298	C₄⋊C₄	—	188	1	no	—
299	C₄⋊C₄	—	188	1	no	—
300	C₂²⋊C₄	—	47	2	no	—
301	C₂²⋊C₄	—	188	2	no	—
302	C₂²⋊C₄	—	188	2	no	—
303	C₂²⋊C₄	—	188	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	D₄×D₄₇	1	1	yes	characteristic
306	C₁₆	—	2	2	no	—
307	C₁₆	—	94	1	no	—
308	C₁₆	—	94	1	no	—
309	C₂³	—	47	2	no	—
310	C₂³	—	188	1	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	8	2	no	—
314	Q₈	—	376	4	no	—
315	Q₈	—	188	2	no	—
316	Q₈	—	188	2	no	—
317	D₄	—	47	4	no	—
318	D₄	—	376	4	no	—
319	D₄	—	188	1	no	—
320	D₄	—	188	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	47	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	376	4	no	—
328	C₂×C₄	—	376	4	no	—
329	C₂×C₄	—	376	2	no	—
330	C₂×C₄	—	188	1	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
333	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	94	2	no	—
337	C₈	C₂×#10625354	1	1	yes	characteristic
338	C₈	#13094070	1	1	yes	characteristic
339	C₂²	—	47	4	no	—
340	C₂²	—	94	2	no	—
341	C₂²	—	94	2	no	—
342	C₂²	—	376	1	no	—
343	C₂²	—	376	1	no	—
344	C₂²	—	376	1	no	—
345	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	94	2	no	—
351	C₄	—	376	2	no	—
352	C₄	—	376	1	no	—
353	C₄	SD₃₂×D₄₇	1	2	yes	—
354	C₄	#15689764	1	1	yes	characteristic
355	C₄	#15689932	1	1	yes	characteristic
356	C₂	—	94	4	no	—
357	C₂	—	752	1	no	—
358	C₂	#20255198	1	1	yes	characteristic
359	C₂	C₂×SD₃₂×D₄₇	1	1	yes	characteristic
360	C₂	#20256540	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2225	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20255735	C₂
hypercenter	#2225	D₄₇
hypocenter	C₄₇	#33865
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₁₈₈	C₂	semidirect
3. #20245617	C₂	semidirect
4. #20245587	C₂	semidirect
5. #20245485	C₂	semidirect
6. #20245434	C₂	semidirect
7. #20245306	C₂	semidirect
8. C₄×#13093967	C₂	semidirect
9. #20245207	C₂	semidirect
10. #358×D₄₇	C₂	semidirect
11. C₄₇×#2225	C₂	semidirect
97. Q₁₆×C₄₇	#41	semidirect
145. D₁₈₈	SD₃₂	semidirect
180. #2225	D₄₇	semidirect
221. C₄₇	#33865	semidirect
282. Q₁₆	#10625354	semidirect

Maximal subgroups

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

subgroup	index
#20255735	2
Q₁₆×D₁₈₈	2
#20245617	2
#20245587	2
#20245485	2
#20245434	2
#20245306	2
C₄×#13093967	2
#20245207	2
#358×D₄₇	2
C₄₇×#2225	2
#33865	47