Subgroups — #26107773

Order 12032.

1030 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
2	#20243368	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688871	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1797	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₉₄	C₄	1	2	yes	—
6	C₂×#13094024	—	2	8	no	—
7	C₂×#13094024	—	2	1	no	—
8	C₂×#13094024	—	2	1	no	—
9	C₂×#13094024	C₄	1	2	yes	—
10	C₂×#13094024	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
12	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
15	#15688878	—	2	1	no	—
16	#15688871	—	2	8	no	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
20	C₄₇×#336	—	2	1	no	—
21	C₂²×Dic₉₄	—	2	2	no	—
22	C₂²×Dic₉₄	—	2	8	no	—
23	C₂²×Dic₉₄	—	4	4	no	—
24	C₂²×Dic₉₄	—	4	4	no	—
25	C₂²×Dic₉₄	C₂×C₄	1	4	yes	—
26	C₂³×Dic₄₇	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	#13094024	—	2	8	no	—
29	#13094024	—	2	8	no	—
30	#13094024	—	2	8	no	—
31	#13094024	—	2	4	no	—
32	#13094024	D₄	1	8	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	8	no	—
34	C₂×C₄⋊Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	2	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
40	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	—	2	4	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₄₇	—	2	4	no	—
51	C₂²⋊C₈×C₄₇	—	4	2	no	—
52	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	8	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	4	4	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂×Dic₉₄	—	2	8	no	—
75	C₂×Dic₉₄	—	2	4	no	—
76	C₂×Dic₉₄	—	4	4	no	—
77	C₂×Dic₉₄	—	4	32	no	—
78	C₂×Dic₉₄	—	4	4	no	—
79	C₂×Dic₉₄	—	4	4	no	—
80	C₂×Dic₉₄	—	4	4	no	—
81	C₂×Dic₉₄	C₂²⋊C₄	1	8	yes	—
82	C₂×C₈×C₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	—	4	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	C₂×D₄	1	4	yes	—
89	C₄⋊Dic₄₇	—	2	8	no	—
90	C₄⋊Dic₄₇	—	2	8	no	—
91	C₄⋊Dic₄₇	—	2	8	no	—
92	C₄⋊Dic₄₇	—	2	8	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂³×C₄₇	—	2	1	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	#100	1	2	yes	—
101	C₂³×C₄₇	#103	1	2	yes	—
102	C₂³×C₄₇	#103	1	2	yes	—
103	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	#100	1	2	yes	—
118	C₂×C₄×C₄₇	#103	1	2	yes	—
119	C₂×C₄×C₄₇	#103	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	#120	1	2	yes	—
122	C₂×C₄×C₄₇	#120	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#137	1	2	yes	—
128	C₂×C₄×C₄₇	#137	1	2	yes	—
129	C₂×Dic₄₇	—	2	8	no	—
130	C₂×Dic₄₇	—	4	16	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	Dic₉₄	—	4	16	no	—
134	Dic₉₄	—	4	8	no	—
135	Dic₉₄	—	4	8	no	—
136	C₈×C₄₇	—	8	2	no	—
137	C₄₇⋊C₈	—	8	2	no	—
138	#8342	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	#333	1	4	yes	complemented
146	C₂²×C₄₇	#379	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
151	C₂²×C₄₇	#419	1	1	yes	characteristic
152	C₂²×C₄₇	#419	1	1	yes	characteristic
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	#448	1	2	yes	—
158	C₄×C₄₇	#448	1	2	yes	—
159	C₄×C₄₇	#451	1	2	yes	—
160	C₄×C₄₇	#451	1	2	yes	—
161	Dic₄₇	—	4	8	no	—
162	C₂²×C₄⋊Q₈	—	47	1	no	—
163	#1797	—	47	1	no	—
164	#1797	D₄₇	1	1	yes	characteristic
165	C₂×#333	—	47	4	no	—
166	C₂×C₄₇	—	2	4	no	—
167	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
168	C₂×C₄₇	#1548	1	1	yes	characteristic
169	C₂×C₄₇	#1548	1	1	yes	characteristic
170	C₂×C₄₇	#1558	1	1	yes	characteristic
171	C₂×C₄₇	#1783	1	1	yes	characteristic
172	C₂×C₄₇	#1787	1	1	yes	characteristic
173	C₂×C₄₇	#1787	1	1	yes	characteristic
174	C₂³×Q₈	—	47	2	no	—
175	C₂×C₄⋊Q₈	—	47	2	no	—
176	C₂×C₄⋊Q₈	—	47	2	no	—
177	C₂×C₄⋊Q₈	—	94	8	no	—
178	C₂×C₄⋊Q₈	—	94	1	no	—
179	C₂×C₄⋊Q₈	—	94	1	no	—
180	C₂²×C₄⋊C₄	—	94	1	no	—
181	C₂²×C₄⋊C₄	—	94	1	no	—
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	47	2	no	—
184	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
185	C₂×C₂²⋊C₈	—	2	1	no	—
186	C₂×C₂²⋊C₈	—	94	1	no	—
187	#336	—	2	1	no	—
188	#336	—	94	1	no	—
189	#333	—	94	8	no	—
190	C₄₇	#8342	1	1	yes	characteristic, complemented, Sylow
191	C₂²×Q₈	—	47	4	no	—
192	C₂²×Q₈	—	94	8	no	—
193	C₂²×Q₈	—	94	2	no	—
194	C₂²×Q₈	—	188	4	no	—
195	C₂²×Q₈	—	188	4	no	—
196	C₂³×C₄	—	94	2	no	—
197	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
198	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
199	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	94	2	no	—
202	C₄⋊Q₈	—	47	8	no	—
203	C₄⋊Q₈	—	94	8	no	—
204	C₄⋊Q₈	—	94	8	no	—
205	C₄⋊Q₈	—	94	8	no	—
206	C₄⋊Q₈	—	94	4	no	—
207	C₂×C₄⋊C₄	—	94	8	no	—
208	C₂×C₄⋊C₄	—	94	8	no	—
209	C₂×C₄⋊C₄	—	94	2	no	—
210	C₂×C₄⋊C₄	—	94	2	no	—
211	C₂×C₄⋊C₄	—	94	2	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄²	—	2	4	no	—
214	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
215	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
218	C₄⋊C₈	—	2	4	no	—
219	C₄⋊C₈	—	94	4	no	—
220	C₂²⋊C₈	—	4	2	no	—
221	C₂²⋊C₈	—	188	2	no	—
222	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
223	C₂×Q₈	—	47	8	no	—
224	C₂×Q₈	—	94	8	no	—
225	C₂×Q₈	—	94	4	no	—
226	C₂×Q₈	—	188	32	no	—
227	C₂×Q₈	—	188	4	no	—
228	C₂×Q₈	—	188	4	no	—
229	C₂×Q₈	—	188	4	no	—
230	C₂×Q₈	—	188	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	94	8	no	—
239	C₂²×C₄	—	94	4	no	—
240	C₂²×C₄	—	188	4	no	—
241	C₂²×C₄	—	188	4	no	—
242	C₂²×C₄	#10625325	1	2	yes	—
243	C₂²×C₄	Dic₁₈₈	1	2	yes	—
244	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	#10625335	1	2	yes	—
248	C₂²×C₄	#10625337	1	2	yes	—
249	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	188	2	no	—
256	C₂×C₈	—	188	2	no	—
257	C₂×C₈	—	188	2	no	—
258	C₄⋊C₄	—	94	8	no	—
259	C₄⋊C₄	—	94	8	no	—
260	C₄⋊C₄	—	94	8	no	—
261	C₄⋊C₄	—	94	8	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	D₄×D₄₇	1	4	yes	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	#13093940	1	2	yes	—
271	C₂³	#13093947	1	2	yes	—
272	C₂³	#13093949	1	2	yes	—
273	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
274	Q₈	—	188	8	no	—
275	Q₈	—	188	16	no	—
276	Q₈	—	188	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	94	8	no	—
291	C₂×C₄	—	188	16	no	—
292	C₂×C₄	—	188	4	no	—
293	C₂×C₄	—	188	4	no	—
294	C₂×C₄	#13093940	1	2	yes	—
295	C₂×C₄	#13093947	1	2	yes	—
296	C₂×C₄	#13093949	1	2	yes	—
297	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
298	C₂×C₄	#13093992	1	2	yes	—
299	C₂×C₄	#13093994	1	2	yes	—
300	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
301	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
302	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
303	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
304	C₂×C₄	#13094054	1	2	yes	—
305	C₂×C₄	#13094057	1	2	yes	—
306	C₈	—	8	2	no	—
307	C₈	—	376	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	#15688871	1	4	yes	complemented
315	C₂²	#15688978	1	1	yes	characteristic
316	C₂²	C₂×#13093940	1	1	yes	characteristic
317	C₂²	C₂×#13093940	1	1	yes	characteristic
318	C₂²	C₂×#13093947	1	1	yes	characteristic
319	C₂²	C₂×#13093949	1	1	yes	characteristic
320	C₂²	#15689131	1	1	yes	characteristic
321	C₂²	#15689133	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	188	8	no	—
327	C₄	#15689261	1	2	yes	—
328	C₄	#15689264	1	2	yes	—
329	C₄	#15689278	1	2	yes	—
330	C₄	#15689281	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	C₂×#15688871	1	1	yes	characteristic
333	C₂	#20242068	1	1	yes	characteristic
334	C₂	#20242070	1	1	yes	characteristic
335	C₂	#20242113	1	1	yes	characteristic
336	C₂	#20243308	1	1	yes	characteristic
337	C₂	#20243322	1	1	yes	characteristic
338	C₂	#20243324	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1797	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1797	D₄₇
hypocenter	C₄₇	#8342
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#15688871	C₂	semidirect
145. C₂²×C₄₇	#333	semidirect
190. C₄₇	#8342	semidirect
314. C₂²	#15688871	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13094024	2
#20243368	2
C₂×#15688871	2
C₄₇×#1797	2
#8342	47