Subgroups — #26199554

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254536	C₂	1	1	yes	maximal, characteristic, complemented
2	#20254326	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689470	C₂	1	1	yes	maximal, characteristic
4	#20242928	C₂	1	4	yes	maximal, complemented
5	#20242804	C₂	1	4	yes	maximal, complemented
6	#20242136	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242070	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242057	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×#13094021	C₂²	1	1	yes	characteristic
12	#15689526	—	2	1	no	—
13	#15689497	—	2	1	no	—
14	#15689473	—	2	1	no	—
15	#15689471	—	2	1	no	—
16	#15689470	—	2	1	no	—
17	#15689470	—	2	1	no	—
18	#15689470	C₂²	1	2	yes	—
19	#15689470	C₂²	1	2	yes	—
20	#15689456	—	2	2	no	—
21	#15689418	C₂²	1	4	yes	complemented
22	#15689323	C₂²	1	4	yes	—
23	#15689273	—	2	2	no	—
24	#15689176	C₂²	1	1	yes	characteristic
25	#15689176	C₂²	1	1	yes	characteristic
26	C₂×#13093946	—	2	1	no	—
27	C₂×#13093946	—	2	1	no	—
28	#15688876	C₂²	1	2	yes	complemented
29	#15688874	C₂²	1	2	yes	complemented
30	#15688871	C₂²	1	4	yes	complemented
31	#15688866	C₂²	1	4	yes	—
32	C₄₇×#533	C₂²	1	1	yes	characteristic
33	C₄₇×#446	—	2	1	no	—
34	C₄₇×#446	C₂²	1	2	yes	complemented
35	C₈×Q₈×C₄₇	—	2	1	no	—
36	C₈×Q₈×C₄₇	C₂²	1	2	yes	complemented
37	C₄₇×#424	—	2	1	no	—
38	C₄₇×#423	C₂²	1	1	yes	characteristic
39	C₄₇×#405	—	2	1	no	—
40	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂²×Dic₉₄	—	2	1	no	—
42	C₂²×D₁₈₈	—	2	1	no	—
43	C₂²×D₁₈₈	—	2	1	no	—
44	C₂²×D₁₈₈	—	2	1	no	—
45	C₂×#10625336	—	2	4	no	—
46	#13094024	—	2	1	no	—
47	#13094024	—	2	1	no	—
48	#13094024	—	2	1	no	—
49	#13094024	C₂³	1	2	yes	—
50	#13094021	—	2	1	no	—
51	#13094021	—	2	1	no	—
52	#13094021	—	2	1	no	—
53	#13094021	C₂³	1	2	yes	—
54	C₂×C₄⋊Dic₄₇	—	2	1	no	—
55	C₂×C₄⋊Dic₄₇	—	2	1	no	—
56	#13093957	—	2	4	no	—
57	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
58	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
59	#13093948	—	2	4	no	—
60	#13093946	—	2	2	no	—
61	#13093946	—	2	2	no	—
62	#13093946	—	2	2	no	—
63	#13093946	—	2	2	no	—
64	#13093944	—	2	4	no	—
65	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
66	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
67	C₄×C₄₇⋊C₈	—	2	1	no	—
68	C₄×C₄₇⋊C₈	—	2	1	no	—
69	C₂×M₁₆×C₄₇	—	2	1	no	—
70	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
72	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
73	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
74	#125×C₄₇	—	2	1	no	—
75	C₂²⋊Q₈×C₄₇	—	2	1	no	—
76	C₂²⋊Q₈×C₄₇	D₄	1	2	yes	—
77	C₄×Q₈×C₄₇	—	2	1	no	—
78	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
79	#117×C₄₇	—	2	1	no	—
80	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₄₇	—	2	2	no	—
82	C₄⋊C₈×C₄₇	—	2	2	no	—
83	C₄⋊C₈×C₄₇	—	2	1	no	—
84	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
85	#97×C₄₇	—	2	2	no	—
86	C₄×C₈×C₄₇	—	2	2	no	—
87	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
88	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
89	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₂³×D₄₇	—	4	1	no	—
92	C₂×Q₈×C₄₇	—	2	1	no	—
93	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
97	C₂²×Dic₄₇	—	4	1	no	—
98	C₂×D₁₈₈	—	2	2	no	—
99	C₂×D₁₈₈	—	2	2	no	—
100	C₂×D₁₈₈	—	2	2	no	—
101	C₂×D₁₈₈	—	4	2	no	—
102	C₂×D₁₈₈	—	4	2	no	—
103	C₂×D₁₈₈	—	4	1	no	—
104	C₂×D₁₈₈	—	4	1	no	—
105	C₂×D₁₈₈	—	4	1	no	—
106	C₂×D₁₈₈	—	8	2	no	—
107	C₂×D₁₈₈	—	8	2	no	—
108	C₂×Dic₉₄	—	2	2	no	—
109	C₂×Dic₉₄	—	4	1	no	—
110	C₂×Dic₉₄	—	8	2	no	—
111	M₁₆×C₄₇	—	4	2	no	—
112	C₂×C₈×C₄₇	—	2	2	no	—
113	C₂×C₈×C₄₇	—	2	2	no	—
114	C₂×C₈×C₄₇	—	2	2	no	—
115	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₄₇	—	2	1	no	—
120	C₄⋊C₄×C₄₇	—	2	1	no	—
121	C₄⋊C₄×C₄₇	—	2	1	no	—
122	C₄⋊C₄×C₄₇	—	2	2	no	—
123	C₄⋊C₄×C₄₇	—	2	2	no	—
124	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
125	C₂²⋊C₄×C₄₇	—	2	2	no	—
126	C₄²×C₄₇	—	2	2	no	—
127	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₄₇	#41	1	1	yes	characteristic
130	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
131	#10625336	—	8	4	no	—
132	C₄⋊Dic₄₇	—	2	2	no	—
133	C₄⋊Dic₄₇	—	2	2	no	—
134	C₄⋊Dic₄₇	—	4	1	no	—
135	C₄⋊Dic₄₇	—	4	1	no	—
136	C₂×C₄₇⋊C₈	—	2	2	no	—
137	C₂×C₄₇⋊C₈	—	2	2	no	—
138	C₂³×C₄₇	#121	1	1	yes	characteristic
139	C₂²×D₄₇	—	4	2	no	—
140	C₂²×D₄₇	—	8	2	no	—
141	C₂²×D₄₇	—	8	2	no	—
142	C₂²×D₄₇	—	8	2	no	—
143	Q₈×C₄₇	—	2	4	no	—
144	Q₈×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	—	2	2	no	—
146	C₂×C₄×C₄₇	—	2	2	no	—
147	C₂×C₄×C₄₇	—	2	2	no	—
148	C₂×C₄×C₄₇	—	2	2	no	—
149	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
150	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
151	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₄₇	#135	1	2	yes	—
153	C₂×C₄×C₄₇	#137	1	2	yes	—
154	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
157	C₂×Dic₄₇	—	4	2	no	—
158	C₂×Dic₄₇	—	8	2	no	—
159	D₁₈₈	—	4	2	no	—
160	D₁₈₈	—	4	2	no	—
161	D₁₈₈	—	4	4	no	—
162	D₁₈₈	—	8	2	no	—
163	D₁₈₈	—	8	2	no	—
164	D₁₈₈	—	8	2	no	—
165	D₁₈₈	—	8	2	no	—
166	Dic₉₄	—	8	2	no	—
167	Dic₉₄	—	8	2	no	—
168	C₈×C₄₇	—	2	1	no	—
169	C₈×C₄₇	—	2	1	no	—
170	C₈×C₄₇	—	2	1	no	—
171	C₈×C₄₇	—	2	1	no	—
172	C₈×C₄₇	—	4	2	no	—
173	C₄₇⋊C₈	—	4	2	no	—
174	C₄₇⋊C₈	—	4	2	no	—
175	#19212	—	47	1	no	maximal, Sylow
176	C₂²×C₄₇	—	2	1	no	—
177	C₂²×C₄₇	—	2	1	no	—
178	C₂²×C₄₇	#464	1	2	yes	—
179	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
180	C₂×D₄₇	—	8	2	no	—
181	C₂×D₄₇	—	8	2	no	—
182	C₂×D₄₇	—	8	2	no	—
183	C₂×D₄₇	—	16	4	no	—
184	C₄×C₄₇	—	2	1	no	—
185	C₄×C₄₇	—	2	1	no	—
186	C₄×C₄₇	—	4	2	no	—
187	C₄×C₄₇	—	4	2	no	—
188	C₄×C₄₇	#464	1	2	yes	—
189	C₄×C₄₇	#472	1	2	yes	—
190	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
191	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
192	C₄×C₄₇	#577	1	2	yes	—
193	Dic₄₇	—	8	2	no	—
194	#3230	—	47	1	no	—
195	#3208	—	47	1	no	—
196	C₂×#486	—	47	1	no	—
197	#2960	D₄₇	1	1	yes	characteristic, complemented
198	#1698	—	47	4	no	—
199	#1678	—	47	4	no	—
200	#1564	—	47	1	no	—
201	#1548	—	47	1	no	—
202	#1545	—	47	1	no	—
203	C₂×C₄₇	—	2	2	no	—
204	C₂×C₄₇	C₂×#464	1	1	yes	characteristic
205	C₂×C₄₇	#3057	1	1	yes	characteristic
206	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
207	D₄₇	—	16	2	no	—
208	#533	C₂×D₄₇	1	1	yes	characteristic
209	C₂×C₄⋊Q₈	—	47	1	no	—
210	C₂×#127	—	47	1	no	—
211	#500	—	94	1	no	—
212	#492	—	94	1	no	—
213	#487	—	94	1	no	—
214	#486	—	47	2	no	—
215	#486	—	47	2	no	—
216	#486	—	94	1	no	—
217	#486	—	94	1	no	—
218	#486	—	94	1	no	—
219	#484	—	94	2	no	—
220	#477	—	47	4	no	—
221	#460	—	47	4	no	—
222	#450	—	94	2	no	—
223	#446	—	2	1	no	—
224	#446	C₂×D₄₇	1	2	yes	complemented
225	C₈×Q₈	—	2	1	no	—
226	C₈×Q₈	C₂×D₄₇	1	2	yes	complemented
227	#431	—	47	1	no	—
228	#431	—	47	1	no	—
229	#424	—	2	1	no	—
230	#423	C₂×D₄₇	1	1	yes	characteristic
231	C₂×#102	—	94	1	no	—
232	C₂×#102	—	94	1	no	—
233	#405	—	2	1	no	—
234	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
235	#335	—	47	2	no	—
236	#334	—	47	2	no	—
237	#333	—	47	4	no	—
238	#331	—	47	4	no	—
239	C₄₇	#19212	1	1	yes	characteristic, complemented, Sylow
240	C₂²×Q₈	—	94	1	no	—
241	C₂²×D₄	—	94	1	no	—
242	C₂²×D₄	—	94	1	no	—
243	C₂²×D₄	—	94	1	no	—
244	C₂×SD₁₆	—	94	4	no	—
245	C₂×M₁₆	—	2	1	no	—
246	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
248	C₄⋊Q₈	—	47	2	no	—
249	C₄⋊Q₈	—	94	1	no	—
250	C₄⋊Q₈	—	94	1	no	—
251	C₄⋊Q₈	—	94	1	no	—
252	C₄⋊Q₈	D₁₈₈	1	1	yes	characteristic
253	C₄⋊Q₈	D₁₈₈	1	1	yes	characteristic
254	#127	—	47	2	no	—
255	#127	—	94	1	no	—
256	#127	—	94	1	no	—
257	#127	—	94	1	no	—
258	#125	—	2	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	C₂²⋊Q₈	D₁₈₈	1	2	yes	—
261	C₄×Q₈	—	2	1	no	—
262	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
263	#117	—	2	1	no	—
264	C₂×C₄⋊C₄	—	94	1	no	—
265	C₂×C₄⋊C₄	—	94	1	no	—
266	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
267	#107	—	94	4	no	—
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	—	2	2	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	47	2	no	—
272	C₄⋊C₈	—	47	2	no	—
273	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
274	#103	—	94	4	no	—
275	#102	—	94	2	no	—
276	#102	—	94	2	no	—
277	#102	—	94	2	no	—
278	#102	—	94	2	no	—
279	#102	—	94	4	no	—
280	C₂²⋊C₈	—	94	1	no	—
281	C₂²⋊C₈	—	94	1	no	—
282	#97	—	2	2	no	—
283	C₄×C₈	—	2	2	no	—
284	C₄×C₈	—	94	1	no	—
285	C₄×C₈	—	94	1	no	—
286	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
287	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
290	C₂⁴	—	188	1	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×Q₈	—	94	2	no	—
293	C₂×Q₈	—	376	2	no	—
294	C₂×Q₈	—	188	1	no	—
295	C₂×Q₈	C₂×D₁₈₈	1	2	yes	—
296	C₂×D₄	—	94	2	no	—
297	C₂×D₄	—	94	2	no	—
298	C₂×D₄	—	94	2	no	—
299	C₂×D₄	—	376	2	no	—
300	C₂×D₄	—	376	2	no	—
301	C₂×D₄	—	188	1	no	—
302	C₂×D₄	—	188	1	no	—
303	C₂×D₄	—	188	1	no	—
304	C₂×D₄	—	188	2	no	—
305	C₂×D₄	—	188	2	no	—
306	C₂²×C₄	—	188	1	no	—
307	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
308	C₂²×C₄	#10625350	1	1	yes	characteristic
309	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
310	SD₁₆	—	376	4	no	—
311	M₁₆	—	4	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	94	2	no	—
316	C₂×C₈	—	94	2	no	—
317	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
318	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
319	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
320	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	94	2	no	—
327	C₄⋊C₄	—	94	2	no	—
328	C₄⋊C₄	—	188	1	no	—
329	C₄⋊C₄	—	188	1	no	—
330	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
334	C₄²	#10625350	1	1	yes	characteristic
335	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
336	C₄²	C₂³×D₄₇	1	1	yes	characteristic
337	C₂³	—	376	2	no	—
338	C₂³	—	376	2	no	—
339	C₂³	—	376	2	no	—
340	C₂³	—	188	2	no	—
341	C₂³	#13093998	1	1	yes	characteristic
342	Q₈	—	2	4	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	376	2	no	—
345	Q₈	—	376	2	no	—
346	D₄	—	376	2	no	—
347	D₄	—	376	2	no	—
348	D₄	—	376	2	no	—
349	D₄	—	376	2	no	—
350	D₄	—	188	2	no	—
351	D₄	—	188	2	no	—
352	D₄	—	188	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	376	2	no	—
358	C₂×C₄	—	188	2	no	—
359	C₂×C₄	#13093998	1	1	yes	characteristic
360	C₂×C₄	#13093998	1	1	yes	characteristic
361	C₂×C₄	#13093998	1	1	yes	characteristic
362	C₂×C₄	#13094044	1	2	yes	—
363	C₂×C₄	#13094055	1	2	yes	—
364	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
365	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
366	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	4	2	no	—
372	C₈	—	188	2	no	—
373	C₈	—	188	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	376	2	no	—
377	C₂²	—	376	2	no	—
378	C₂²	—	376	2	no	—
379	C₂²	—	752	4	no	—
380	C₂²	#15689345	1	2	yes	—
381	C₂²	C₂×#13093998	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	2	no	—
386	C₄	—	376	2	no	—
387	C₄	#15689345	1	2	yes	—
388	C₄	#15689393	1	2	yes	—
389	C₄	C₂×#13094044	1	1	yes	characteristic
390	C₄	C₂×#13094055	1	1	yes	characteristic
391	C₄	#15689935	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	752	2	no	—
394	C₂	C₂×#15689345	1	1	yes	characteristic
395	C₂	#20253027	1	1	yes	characteristic
396	C₂	C₂×#15689393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093998
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#2960	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂×#13094021	C₂²
hypercenter	#2960	D₄₇
hypocenter	C₄₇	#19212
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #20254536	C₂	semidirect
4. #20242928	C₂	semidirect
5. #20242804	C₂	semidirect
6. #20242136	C₂	semidirect
7. #20242070	C₂	semidirect
9. C₄₇×#2960	C₂	semidirect
21. #15689418	C₂²	semidirect
28. #15688876	C₂²	semidirect
29. #15688874	C₂²	semidirect
30. #15688871	C₂²	semidirect
34. C₄₇×#446	C₂²	semidirect
36. C₈×Q₈×C₄₇	C₂²	semidirect
197. #2960	D₄₇	semidirect
224. #446	C₂×D₄₇	semidirect
226. C₈×Q₈	C₂×D₄₇	semidirect
239. C₄₇	#19212	semidirect

Maximal subgroups

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

subgroup	index
#20254536	2
#20254326	2
C₂×#15689470	2
#20242928	2
#20242804	2
#20242136	2
#20242070	2
#20242057	2
C₄₇×#2960	2
#19212	47