Subgroups — #26123222

Order 12032.

1010 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251952	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093941	C₂	1	2	yes	maximal
3	C₂×#15688893	C₂	1	2	yes	maximal, complemented
4	#20243127	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2309	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15689150	—	2	4	no	—
9	#15689147	—	2	2	no	—
10	C₂×#13093941	—	2	4	no	—
11	C₂×#13093941	—	2	4	no	—
12	C₂×#13093941	—	2	4	no	—
13	#15688893	—	2	4	no	—
14	#15688892	—	2	2	no	—
15	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#390	—	2	2	no	—
19	C₄₇×#390	—	2	2	no	—
20	C₄₇×#379	C₄	1	4	yes	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	2	no	—
31	C₂×#10625329	D₄	1	4	yes	—
32	C₂×#10625329	D₄	1	4	yes	—
33	C₂²×C₄₇⋊C₈	—	2	2	no	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
36	#13093941	—	2	16	no	—
37	C₂⁵×C₄₇	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₄₇	—	2	2	no	—
39	C₂²×D₄×C₄₇	—	2	2	no	—
40	C₂²×D₄×C₄₇	—	2	1	no	—
41	C₂²×D₄×C₄₇	—	2	1	no	—
42	C₂²×D₄×C₄₇	—	2	1	no	—
43	C₂²×D₄×C₄₇	—	2	2	no	—
44	C₂²×D₄×C₄₇	—	2	1	no	—
45	C₂²×D₄×C₄₇	—	2	2	no	—
46	C₂²×D₄×C₄₇	D₄	1	2	yes	—
47	C₂²×D₄×C₄₇	Q₈	1	2	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	—	2	2	no	—
52	#117×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
56	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₄₇	—	2	1	no	—
59	C₂×C₄²×C₄₇	D₄	1	2	yes	—
60	C₂⁴×C₄₇	—	2	2	no	—
61	C₂⁴×C₄₇	—	2	2	no	—
62	C₂⁴×C₄₇	—	2	4	no	—
63	C₂⁴×C₄₇	—	2	4	no	—
64	C₂⁴×C₄₇	—	8	4	no	—
65	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×D₄×C₄₇	—	2	4	no	—
68	C₂×D₄×C₄₇	—	2	2	no	—
69	C₂×D₄×C₄₇	—	2	2	no	—
70	C₂×D₄×C₄₇	—	2	4	no	—
71	C₂×D₄×C₄₇	—	2	4	no	—
72	C₂×D₄×C₄₇	—	2	2	no	—
73	C₂×D₄×C₄₇	—	2	2	no	—
74	C₂×D₄×C₄₇	—	2	4	no	—
75	C₂×D₄×C₄₇	—	4	1	no	—
76	C₂×D₄×C₄₇	—	4	2	no	—
77	C₂×D₄×C₄₇	—	4	2	no	—
78	C₂×D₄×C₄₇	—	4	2	no	—
79	C₂×D₄×C₄₇	—	4	1	no	—
80	C₂×D₄×C₄₇	—	4	2	no	—
81	C₂×D₄×C₄₇	—	4	2	no	—
82	C₂×D₄×C₄₇	—	4	1	no	—
83	C₂×D₄×C₄₇	—	4	2	no	—
84	C₂×D₄×C₄₇	—	4	1	no	—
85	C₂²×C₄×C₄₇	—	2	2	no	—
86	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	#41	1	2	yes	—
95	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
96	C₂²×C₄×C₄₇	#41	1	2	yes	—
97	C₄⋊C₄×C₄₇	—	2	4	no	—
98	C₄⋊C₄×C₄₇	—	4	1	no	—
99	C₄⋊C₄×C₄₇	—	4	1	no	—
100	C₂²⋊C₄×C₄₇	—	2	4	no	—
101	C₂²⋊C₄×C₄₇	—	2	4	no	—
102	C₂²⋊C₄×C₄₇	—	8	8	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₄²×C₄₇	—	2	2	no	—
105	C₄²×C₄₇	—	4	2	no	—
106	#10625329	—	2	4	no	—
107	#10625329	—	2	4	no	—
108	#10625329	—	2	4	no	—
109	#10625329	—	2	4	no	—
110	#10625329	—	4	2	no	—
111	#10625329	—	4	2	no	—
112	#10625329	—	4	2	no	—
113	#10625329	—	4	2	no	—
114	C₂×C₄₇⋊C₈	—	2	4	no	—
115	C₂×C₄₇⋊C₈	—	2	4	no	—
116	C₂×C₄₇⋊C₈	—	4	2	no	—
117	C₂×C₄₇⋊C₈	—	4	2	no	—
118	C₂×C₄₇⋊C₈	—	4	2	no	—
119	C₂×C₄₇⋊C₈	—	4	2	no	—
120	C₂³×C₄₇	—	2	4	no	—
121	C₂³×C₄₇	—	2	8	no	—
122	C₂³×C₄₇	—	4	2	no	—
123	C₂³×C₄₇	—	4	8	no	—
124	C₂³×C₄₇	—	4	2	no	—
125	C₂³×C₄₇	—	4	4	no	—
126	C₂³×C₄₇	—	4	2	no	—
127	C₂³×C₄₇	—	8	8	no	—
128	C₂³×C₄₇	—	8	4	no	—
129	C₂³×C₄₇	—	8	4	no	—
130	C₂³×C₄₇	—	8	8	no	—
131	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂³×C₄₇	#117	1	1	yes	characteristic
134	C₂³×C₄₇	C₄×D₄	1	2	yes	—
135	C₂³×C₄₇	C₄×D₄	1	2	yes	—
136	D₄×C₄₇	—	4	2	no	—
137	D₄×C₄₇	—	4	2	no	—
138	D₄×C₄₇	—	4	4	no	—
139	D₄×C₄₇	—	4	4	no	—
140	D₄×C₄₇	—	4	2	no	—
141	D₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	2	4	no	—
143	C₂×C₄×C₄₇	—	4	4	no	—
144	C₂×C₄×C₄₇	—	4	4	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
147	C₂×C₄×C₄₇	#120	1	2	yes	—
148	C₂×C₄×C₄₇	#121	1	2	yes	—
149	C₂×C₄×C₄₇	#121	1	2	yes	—
150	C₂×C₄×C₄₇	#121	1	2	yes	—
151	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
154	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
158	C₂×C₄×C₄₇	#123	1	2	yes	—
159	C₂×C₄×C₄₇	#123	1	2	yes	—
160	C₂×C₄×C₄₇	#124	1	2	yes	—
161	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
162	C₂×C₄×C₄₇	#126	1	2	yes	—
163	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
164	C₄₇⋊C₈	—	4	4	no	—
165	C₄₇⋊C₈	—	4	4	no	—
166	#11047	—	47	1	no	maximal, Sylow
167	C₂²×C₄₇	—	2	2	no	—
168	C₂²×C₄₇	—	2	4	no	—
169	C₂²×C₄₇	—	4	1	no	—
170	C₂²×C₄₇	—	4	1	no	—
171	C₂²×C₄₇	—	4	2	no	—
172	C₂²×C₄₇	—	4	8	no	—
173	C₂²×C₄₇	—	4	4	no	—
174	C₂²×C₄₇	—	8	8	no	—
175	C₂²×C₄₇	—	8	4	no	—
176	C₂²×C₄₇	—	8	8	no	—
177	C₂²×C₄₇	—	8	4	no	—
178	C₂²×C₄₇	#380	1	1	yes	characteristic
179	C₂²×C₄₇	#385	1	2	yes	—
180	C₂²×C₄₇	#386	1	1	yes	characteristic
181	C₂²×C₄₇	#388	1	2	yes	—
182	C₂²×C₄₇	#390	1	1	yes	characteristic
183	C₄×C₄₇	—	4	4	no	—
184	C₄×C₄₇	—	8	2	no	—
185	C₄×C₄₇	#392	1	1	yes	characteristic
186	C₄×C₄₇	#393	1	1	yes	characteristic
187	C₄×C₄₇	#394	1	2	yes	—
188	C₄×C₄₇	#396	1	2	yes	—
189	C₄×C₄₇	#397	1	1	yes	characteristic
190	C₄×C₄₇	#399	1	1	yes	characteristic
191	#2905	—	47	1	no	—
192	C₂²×#100	—	47	2	no	—
193	#2309	D₄₇	1	1	yes	characteristic
194	C₂×#344	—	47	2	no	—
195	#1742	—	47	1	no	—
196	C₂×C₄₇	—	2	1	no	—
197	C₂×C₄₇	—	2	1	no	—
198	C₂×C₄₇	—	2	2	no	—
199	C₂×C₄₇	—	2	2	no	—
200	C₂×C₄₇	—	8	4	no	—
201	C₂×C₄₇	#1439	1	1	yes	characteristic
202	C₂×C₄₇	#1914	1	2	yes	—
203	C₂³×D₄	C₂×D₄₇	1	1	yes	characteristic
204	C₂²×M₁₆	—	47	2	no	—
205	C₂²×M₁₆	—	47	2	no	—
206	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
207	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
208	#424	—	94	4	no	—
209	#423	—	94	2	no	—
210	C₂×#100	—	94	4	no	—
211	C₂×#100	—	94	4	no	—
212	C₂×#100	—	94	4	no	—
213	#390	—	2	2	no	—
214	#390	—	2	2	no	—
215	#379	Dic₄₇	1	4	yes	—
216	#344	—	94	4	no	—
217	#343	—	94	2	no	—
218	C₄₇	#11047	1	1	yes	characteristic, complemented, Sylow
219	C₂⁵	C₂×Dic₄₇	1	2	yes	—
220	C₂²×D₄	—	2	1	no	—
221	C₂²×D₄	—	2	1	no	—
222	C₂²×D₄	—	2	2	no	—
223	C₂²×D₄	—	2	2	no	—
224	C₂²×D₄	—	2	2	no	—
225	C₂²×D₄	—	2	2	no	—
226	C₂²×D₄	—	2	1	no	—
227	C₂²×D₄	—	2	1	no	—
228	C₂²×D₄	Dic₉₄	1	2	yes	—
229	C₂²×D₄	D₁₈₈	1	2	yes	—
230	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
231	C₂×M₁₆	—	47	4	no	—
232	C₂×M₁₆	—	47	4	no	—
233	C₂×M₁₆	—	94	2	no	—
234	C₂×M₁₆	—	94	2	no	—
235	C₂×M₁₆	—	94	2	no	—
236	C₂×M₁₆	—	94	2	no	—
237	C₂×M₁₆	—	94	2	no	—
238	C₂×M₁₆	—	94	2	no	—
239	C₂×M₁₆	—	94	2	no	—
240	C₂×M₁₆	—	94	2	no	—
241	C₂×M₁₆	—	94	2	no	—
242	C₂×M₁₆	—	94	2	no	—
243	C₂²×C₈	—	94	2	no	—
244	C₂²×C₈	—	94	2	no	—
245	#117	—	2	2	no	—
246	#117	—	2	2	no	—
247	#117	—	2	2	no	—
248	#117	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
251	C₂×C₂²⋊C₄	—	2	8	no	—
252	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
253	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
254	C₂×C₄²	—	2	1	no	—
255	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
256	C₄⋊C₈	—	94	8	no	—
257	#100	—	94	16	no	—
258	C₂⁴	—	2	2	no	—
259	C₂⁴	—	2	2	no	—
260	C₂⁴	—	2	4	no	—
261	C₂⁴	—	2	4	no	—
262	C₂⁴	—	8	4	no	—
263	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
264	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
265	C₂×D₄	—	2	4	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	4	no	—
269	C₂×D₄	—	2	4	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	4	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
285	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
287	C₂²×C₄	#10625350	1	1	yes	characteristic
288	C₂²×C₄	#10625350	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₄₇	1	2	yes	—
290	C₂²×C₄	D₄×D₄₇	1	2	yes	—
291	C₂²×C₄	#10625352	1	2	yes	—
292	C₂²×C₄	#10625352	1	2	yes	—
293	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
295	M₁₆	—	94	4	no	—
296	M₁₆	—	94	4	no	—
297	M₁₆	—	94	4	no	—
298	M₁₆	—	94	4	no	—
299	M₁₆	—	188	2	no	—
300	M₁₆	—	188	2	no	—
301	M₁₆	—	188	2	no	—
302	M₁₆	—	188	2	no	—
303	C₂×C₈	—	94	4	no	—
304	C₂×C₈	—	94	4	no	—
305	C₂×C₈	—	188	2	no	—
306	C₂×C₈	—	188	2	no	—
307	C₂×C₈	—	188	2	no	—
308	C₂×C₈	—	188	2	no	—
309	C₄⋊C₄	—	2	4	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₂²⋊C₄	—	2	4	no	—
313	C₂²⋊C₄	—	2	4	no	—
314	C₂²⋊C₄	—	8	8	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	4	2	no	—
318	C₂³	—	2	4	no	—
319	C₂³	—	2	8	no	—
320	C₂³	—	4	2	no	—
321	C₂³	—	4	8	no	—
322	C₂³	—	4	2	no	—
323	C₂³	—	4	4	no	—
324	C₂³	—	4	2	no	—
325	C₂³	—	8	8	no	—
326	C₂³	—	8	4	no	—
327	C₂³	—	8	4	no	—
328	C₂³	—	8	8	no	—
329	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
330	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
331	C₂³	#13093983	1	1	yes	characteristic
332	C₂³	D₄×Dic₄₇	1	2	yes	—
333	C₂³	D₄×Dic₄₇	1	2	yes	—
334	D₄	—	4	2	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	4	no	—
337	D₄	—	4	4	no	—
338	D₄	—	4	2	no	—
339	D₄	—	4	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	4	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	#13093992	1	1	yes	characteristic
345	C₂×C₄	#13093994	1	2	yes	—
346	C₂×C₄	#13093995	1	2	yes	—
347	C₂×C₄	#13093996	1	2	yes	—
348	C₂×C₄	#13093997	1	2	yes	—
349	C₂×C₄	#13093998	1	1	yes	characteristic
350	C₂×C₄	#13093998	1	1	yes	characteristic
351	C₂×C₄	#13094001	1	2	yes	—
352	C₂×C₄	#13094001	1	2	yes	—
353	C₂×C₄	#13094003	1	1	yes	characteristic
354	C₂×C₄	#13094003	1	1	yes	characteristic
355	C₂×C₄	#13094005	1	1	yes	characteristic
356	C₂×C₄	#13094006	1	2	yes	—
357	C₂×C₄	#13094008	1	2	yes	—
358	C₂×C₄	#13094010	1	2	yes	—
359	C₂×C₄	#13094011	1	1	yes	characteristic
360	C₂×C₄	#13094017	1	2	yes	—
361	C₂×C₄	#13094020	1	1	yes	characteristic
362	C₈	—	188	4	no	—
363	C₈	—	188	4	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	4	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	8	no	—
370	C₂²	—	4	4	no	—
371	C₂²	—	8	8	no	—
372	C₂²	—	8	4	no	—
373	C₂²	—	8	8	no	—
374	C₂²	—	8	4	no	—
375	C₂²	#15688983	1	1	yes	characteristic
376	C₂²	#15688998	1	2	yes	—
377	C₂²	#15689003	1	1	yes	characteristic
378	C₂²	#15689016	1	2	yes	—
379	C₂²	#15689026	1	1	yes	characteristic
380	C₄	—	4	4	no	—
381	C₄	—	8	2	no	—
382	C₄	#15689032	1	1	yes	characteristic
383	C₄	#15689035	1	1	yes	characteristic
384	C₄	#15689039	1	2	yes	—
385	C₄	#15689046	1	2	yes	—
386	C₄	#15689048	1	1	yes	characteristic
387	C₄	#15689059	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	2	1	no	—
390	C₂	—	2	2	no	—
391	C₂	—	2	2	no	—
392	C₂	—	8	4	no	—
393	C₂	#20241633	1	1	yes	characteristic
394	C₂	#20243936	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689003
commutator	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2309	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂³×D₄×C₄₇	C₂²
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2309	D₄₇
hypocenter	C₄₇	#11047
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2309

Complements

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

N	H ≅ G/N	type
1. #20251952	C₂	semidirect
3. C₂×#15688893	C₂	semidirect
4. #20243127	C₂	semidirect
218. C₄₇	#11047	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
C₂²×#13093941	2
C₂×#15688893	2
#20243127	2
C₄₇×#2309	2
#11047	47