Subgroups — #26373147

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	D₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245617	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245585	C₂	1	2	yes	maximal, complemented
4	#20245527	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245482	C₂	1	2	yes	maximal, complemented
6	#20245433	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245304	C₂	1	2	yes	maximal, complemented
8	C₄×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15688926	C₂	1	2	yes	maximal, complemented
11	C₄₇×#3334	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
13	D₄×Dic₉₄	—	2	1	no	—
14	C₂×#13093968	C₂²	1	1	yes	characteristic
15	C₂×#13093968	C₂²	1	2	yes	—
16	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
17	#15689503	C₂²	1	1	yes	characteristic
18	#15689364	C₂²	1	2	yes	—
19	#15689334	—	2	1	no	—
20	#15689281	—	2	2	no	—
21	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
22	#15688950	C₂²	1	2	yes	—
23	#15688950	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#15688926	C₂²	1	2	yes	—
26	#15688926	C₂²	1	2	yes	—
27	#15688926	C₂²	1	2	yes	—
28	#15688926	C₂²	1	1	yes	characteristic
29	#15688926	C₂²	1	2	yes	—
30	#15688924	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₇×#547	—	2	1	no	—
34	C₄₇×#500	C₂²	1	1	yes	characteristic
35	C₄₇×#480	—	2	2	no	—
36	C₄₇×#474	—	2	1	no	—
37	C₄₇×#466	C₂²	1	2	yes	—
38	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
40	C₂×#107×C₄₇	C₂²	1	2	yes	—
41	C₂²×Dic₉₄	—	2	2	no	—
42	C₂×Dic₁₈₈	—	2	4	no	—
43	C₂×Dic₁₈₈	—	2	2	no	—
44	C₂×Dic₁₈₈	—	2	1	no	—
45	C₂×Dic₁₈₈	—	4	2	no	—
46	C₂×Dic₁₈₈	C₂³	1	2	yes	—
47	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
48	#13094024	—	2	1	no	—
49	#13094023	—	4	1	no	—
50	#13094003	—	2	2	no	—
51	#13094001	—	4	2	no	—
52	C₄×Dic₉₄	—	2	1	no	—
53	D₄×Dic₄₇	—	4	1	no	—
54	#13093968	—	2	2	no	—
55	#13093968	—	4	2	no	—
56	C₂×C₄₇⋊C₁₆	—	4	2	no	—
57	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	#13093958	C₂³	1	1	yes	characteristic
61	#13093949	—	2	1	no	—
62	#13093949	—	2	2	no	—
63	C₂×#41×C₄₇	—	4	1	no	—
64	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₄₇	—	2	1	no	—
67	#123×C₄₇	—	4	2	no	—
68	C₂²⋊Q₈×C₄₇	—	4	1	no	—
69	#121×C₄₇	—	2	2	no	—
70	C₄×D₄×C₄₇	—	2	1	no	—
71	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
73	#107×C₄₇	—	2	2	no	—
74	#107×C₄₇	—	2	1	no	—
75	#107×C₄₇	C₂³	1	2	yes	—
76	#107×C₄₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
78	#102×C₄₇	—	2	2	no	—
79	#102×C₄₇	—	2	1	no	—
80	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	#41×C₄₇	—	8	4	no	—
83	C₂×Q₈×C₄₇	—	4	1	no	—
84	C₂×D₄×C₄₇	—	2	1	no	—
85	C₂×D₄×C₄₇	—	4	2	no	—
86	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	—	4	2	no	—
88	C₂²×C₄×C₄₇	—	4	1	no	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×Dic₄₇	—	4	2	no	—
91	C₂×Dic₉₄	—	2	1	no	—
92	C₂×Dic₉₄	—	2	2	no	—
93	C₂×Dic₉₄	—	4	4	no	—
94	C₂×Dic₉₄	—	4	2	no	—
95	C₂×Dic₉₄	—	8	2	no	—
96	D₈×C₄₇	—	2	1	no	—
97	D₈×C₄₇	—	2	1	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₄₇	—	2	1	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	4	2	no	—
106	C₄⋊C₄×C₄₇	—	4	1	no	—
107	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₄₇	—	4	1	no	—
109	C₂²⋊C₄×C₄₇	—	4	2	no	—
110	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₂²⋊Dic₄₇	—	4	2	no	—
113	C₄⋊Dic₄₇	—	2	1	no	—
114	C₄⋊Dic₄₇	—	4	1	no	—
115	C₄⋊Dic₄₇	—	4	1	no	—
116	Dic₄₇⋊C₄	—	4	1	no	—
117	Dic₄₇⋊C₄	—	4	2	no	—
118	C₄×Dic₄₇	—	4	1	no	—
119	Dic₁₈₈	—	2	2	no	—
120	Dic₁₈₈	—	2	1	no	—
121	Dic₁₈₈	—	2	2	no	—
122	Dic₁₈₈	—	4	4	no	—
123	Dic₁₈₈	—	8	1	no	—
124	Dic₁₈₈	#41	1	2	yes	complemented
125	C₄₇⋊C₁₆	—	2	1	no	—
126	C₄₇⋊C₁₆	—	2	1	no	—
127	C₄₇⋊C₁₆	—	2	2	no	—
128	C₂³×C₄₇	—	4	1	no	—
129	C₂³×C₄₇	C₂×D₈	1	2	yes	—
130	Q₈×C₄₇	—	8	2	no	—
131	D₄×C₄₇	—	4	1	no	—
132	D₄×C₄₇	—	4	1	no	—
133	D₄×C₄₇	—	8	4	no	—
134	D₄×C₄₇	SD₃₂	1	4	yes	complemented
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	8	4	no	—
140	C₂×C₄×C₄₇	—	8	2	no	—
141	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×Dic₄₇	—	4	1	no	—
146	C₂×Dic₄₇	—	4	1	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	8	4	no	—
149	Dic₉₄	—	2	2	no	—
150	Dic₉₄	—	4	1	no	—
151	Dic₉₄	—	4	2	no	—
152	Dic₉₄	—	4	2	no	—
153	Dic₉₄	—	8	4	no	—
154	C₈×C₄₇	—	2	2	no	—
155	C₈×C₄₇	—	2	1	no	—
156	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
157	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
158	#33865	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	2	2	no	—
161	C₂²×C₄₇	—	8	1	no	—
162	C₂²×C₄₇	—	8	1	no	—
163	C₂²×C₄₇	—	8	1	no	—
164	C₂²×C₄₇	C₂×SD₃₂	1	4	yes	—
165	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	—	2	2	no	—
168	C₄×C₄₇	—	8	1	no	—
169	C₄×C₄₇	—	8	2	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	—	4	2	no	—
174	Dic₄₇	—	8	1	no	—
175	Dic₄₇	—	8	2	no	—
176	#3334	D₄₇	1	1	yes	characteristic
177	D₄×Q₁₆	—	47	1	no	—
178	#2241	—	47	1	no	—
179	#2235	—	47	2	no	—
180	#2225	—	47	1	no	—
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	—	2	4	no	—
188	C₂×C₄₇	—	16	1	no	—
189	C₂×C₄₇	#3294	1	1	yes	characteristic
190	C₂×C₄₇	C₂²×SD₃₂	1	1	yes	characteristic
191	C₂×C₄₇	#3417	1	1	yes	characteristic
192	C₂²×Q₁₆	—	47	2	no	—
193	D₄×Q₈	—	94	1	no	—
194	#547	—	2	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	—	2	2	no	—
201	#474	—	2	1	no	—
202	#467	—	47	2	no	—
203	#466	C₂×D₄₇	1	2	yes	—
204	#462	—	94	1	no	—
205	#451	—	94	2	no	—
206	C₄×Q₁₆	—	47	1	no	—
207	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
208	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
209	C₂×#107	C₂×D₄₇	1	2	yes	—
210	#367	—	47	1	no	—
211	#367	—	47	2	no	—
212	C₄⋊C₁₆	—	47	1	no	—
213	#358	—	47	1	no	—
214	#358	—	47	2	no	—
215	#358	—	47	2	no	—
216	#358	—	47	2	no	—
217	#358	—	47	2	no	—
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	—	4	1	no	—
223	C₂²×Q₈	—	94	2	no	—
224	C₂×Q₁₆	—	47	1	no	—
225	C₂×Q₁₆	—	47	2	no	—
226	C₂×Q₁₆	—	94	4	no	—
227	C₂×Q₁₆	—	94	1	no	—
228	C₂×Q₁₆	—	94	2	no	—
229	C₂×Q₁₆	—	188	2	no	—
230	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
231	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	94	1	no	—
234	C₄⋊Q₈	—	188	1	no	—
235	#123	—	4	2	no	—
236	C₂²⋊Q₈	—	4	1	no	—
237	C₂²⋊Q₈	—	94	2	no	—
238	C₂²⋊Q₈	—	188	2	no	—
239	#121	—	2	2	no	—
240	C₄×Q₈	—	94	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	188	1	no	—
243	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	2	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	—	47	2	no	—
250	C₂×C₁₆	—	188	2	no	—
251	#107	—	2	2	no	—
252	#107	—	2	1	no	—
253	#107	—	47	1	no	—
254	#107	C₂²×D₄₇	1	1	yes	characteristic
255	#107	C₂²×D₄₇	1	2	yes	—
256	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
257	#103	—	94	1	no	—
258	#103	—	94	2	no	—
259	#102	—	2	1	no	—
260	#102	—	2	2	no	—
261	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
262	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
263	#41	—	8	4	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	94	1	no	—
266	C₂×Q₈	—	94	2	no	—
267	C₂×Q₈	—	376	2	no	—
268	C₂×Q₈	—	188	4	no	—
269	C₂×Q₈	—	188	2	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	#10625334	1	1	yes	characteristic
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	188	2	no	—
276	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
277	Q₁₆	—	47	2	no	—
278	Q₁₆	—	94	2	no	—
279	Q₁₆	—	94	1	no	—
280	Q₁₆	—	94	2	no	—
281	Q₁₆	—	376	1	no	—
282	Q₁₆	—	188	4	no	—
283	D₈	—	2	1	no	—
284	D₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
289	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	94	1	no	—
295	C₄⋊C₄	—	188	1	no	—
296	C₄⋊C₄	—	188	2	no	—
297	C₄⋊C₄	—	188	1	no	—
298	C₄⋊C₄	—	188	1	no	—
299	C₄⋊C₄	#10625334	1	1	yes	characteristic
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	188	2	no	—
303	C₂²⋊C₄	#10625334	1	2	yes	—
304	C₄²	—	188	1	no	—
305	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
306	C₁₆	—	94	1	no	—
307	C₁₆	—	94	1	no	—
308	C₁₆	—	94	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	C₂×#10625334	1	2	yes	—
311	Q₈	—	8	2	no	—
312	Q₈	—	94	2	no	—
313	Q₈	—	376	4	no	—
314	Q₈	—	188	2	no	—
315	Q₈	—	188	2	no	—
316	Q₈	—	188	1	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	8	4	no	—
320	D₄	#13093968	1	4	yes	complemented
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	—	376	4	no	—
328	C₂×C₄	—	188	1	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	1	no	—
331	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
332	C₂×C₄	C₂×#10625334	1	2	yes	—
333	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	1	no	—
337	C₈	#41×D₄₇	1	1	yes	characteristic
338	C₈	#13094070	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	1	no	—
343	C₂²	—	8	1	no	—
344	C₂²	C₂×#13093968	1	4	yes	—
345	C₂²	C₂²×#10625334	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	376	1	no	—
351	C₄	—	376	2	no	—
352	C₄	—	188	2	no	—
353	C₄	C₂×#13093968	1	2	yes	—
354	C₄	#15689761	1	1	yes	characteristic
355	C₄	#15689931	1	1	yes	characteristic
356	C₂	—	2	4	no	—
357	C₂	—	16	1	no	—
358	C₂	#20255195	1	1	yes	characteristic
359	C₂	C₂²×#13093968	1	1	yes	characteristic
360	C₂	#20256539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3334	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3334	D₄₇
hypocenter	C₄₇	#33865
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3334

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₁₈₈	C₂	semidirect
2. #20245617	C₂	semidirect
3. #20245585	C₂	semidirect
4. #20245527	C₂	semidirect
5. #20245482	C₂	semidirect
6. #20245433	C₂	semidirect
7. #20245304	C₂	semidirect
8. C₄×#13093968	C₂	semidirect
9. D₄×C₄₇⋊C₁₆	C₂	semidirect
10. C₂×#15688926	C₂	semidirect
124. Dic₁₈₈	#41	semidirect
134. D₄×C₄₇	SD₃₂	semidirect
221. C₄₇	#33865	semidirect
320. D₄	#13093968	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₁₈₈	2
#20245617	2
#20245585	2
#20245527	2
#20245482	2
#20245433	2
#20245304	2
C₄×#13093968	2
D₄×C₄₇⋊C₁₆	2
C₂×#15688926	2
C₄₇×#3334	2
#33865	47