Subgroups — #26102830

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
2	#20241618	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	—	2	1	no	—
5	C₂²×C₄⋊Dic₄₇	C₄	1	2	yes	—
6	C₂×C₄₇⋊#95	—	2	1	no	—
7	C₂×C₄₇⋊#95	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₄	1	2	yes	complemented
10	C₂³×C₈×C₄₇	—	2	1	no	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#336	—	2	1	no	—
13	C₄₇×#336	C₄	1	2	yes	complemented
14	C₂³×Dic₄₇	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	4	no	—
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	4	2	no	—
18	C₂²×C₄₇⋊C₈	—	4	1	no	—
19	C₂²×C₄₇⋊C₈	—	4	1	no	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	4	2	no	—
22	C₂×C₄⋊Dic₄₇	—	4	2	no	—
23	C₂×C₄⋊Dic₄₇	C₈	1	4	yes	complemented
24	C₄₇⋊#95	—	4	1	no	—
25	C₄₇⋊#95	—	4	1	no	—
26	C₄₇⋊#95	—	4	1	no	—
27	C₄₇⋊#95	—	4	1	no	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	—	2	4	no	—
32	C₂²×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₈×C₄₇	—	4	2	no	—
34	C₂²×C₈×C₄₇	—	4	1	no	—
35	C₂²×C₈×C₄₇	—	4	1	no	—
36	C₂×C₄²×C₄₇	—	4	2	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
41	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	—	2	2	no	—
45	C₂²×C₄×C₄₇	—	2	1	no	—
46	C₂²×C₄×C₄₇	—	2	1	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
57	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	4	no	—
66	C₂²×Dic₄₇	—	4	2	no	—
67	C₂²×Dic₄₇	—	4	2	no	—
68	C₂²×Dic₄₇	—	4	2	no	—
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂²×Dic₄₇	—	4	2	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	4	4	no	—
74	C₂×C₈×C₄₇	—	4	4	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₄²×C₄₇	—	4	1	no	—
78	C₄²×C₄₇	—	4	1	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₄²×C₄₇	—	4	1	no	—
81	C₄⋊Dic₄₇	—	4	4	no	—
82	C₄⋊Dic₄₇	—	4	4	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	4	4	no	—
85	C₂×C₄₇⋊C₈	—	4	4	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂³×C₄₇	#95	1	1	yes	characteristic
89	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
90	C₂³×C₄₇	#99	1	2	yes	—
91	C₂³×C₄₇	#100	1	1	yes	characteristic
92	C₂³×C₄₇	#101	1	1	yes	characteristic
93	C₂³×C₄₇	#102	1	1	yes	characteristic
94	C₂³×C₄₇	#102	1	1	yes	characteristic
95	C₂³×C₄₇	#103	1	1	yes	characteristic
96	C₂³×C₄₇	#103	1	1	yes	characteristic
97	C₂³×C₄₇	#104	1	2	yes	—
98	C₂³×C₄₇	#104	1	2	yes	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
130	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	8	2	no	—
138	C₂×Dic₄₇	—	8	2	no	—
139	C₂×Dic₄₇	—	8	2	no	—
140	C₂×Dic₄₇	—	8	2	no	—
141	C₈×C₄₇	—	4	4	no	—
142	C₄₇⋊C₈	—	4	4	no	—
143	#7025	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	#323	1	2	yes	—
145	C₂²×C₄₇	#324	1	2	yes	—
146	C₂²×C₄₇	D₄⋊C₈	1	2	yes	—
147	C₂²×C₄₇	D₄⋊C₈	1	2	yes	—
148	C₂²×C₄₇	Q₈⋊C₈	1	2	yes	—
149	C₂²×C₄₇	Q₈⋊C₈	1	2	yes	—
150	C₂²×C₄₇	#327	1	2	yes	—
151	C₂²×C₄₇	#327	1	2	yes	—
152	C₂²×C₄₇	#328	1	2	yes	—
153	C₂²×C₄₇	#328	1	2	yes	—
154	C₂²×C₄₇	#329	1	2	yes	—
155	C₂²×C₄₇	#330	1	2	yes	—
156	C₂²×C₄₇	#331	1	1	yes	characteristic
157	C₂²×C₄₇	#332	1	1	yes	characteristic
158	C₂²×C₄₇	#332	1	1	yes	characteristic
159	C₂²×C₄₇	#333	1	1	yes	characteristic
160	C₂²×C₄₇	#336	1	1	yes	characteristic
161	C₂²×C₄₇	#339	1	1	yes	characteristic
162	C₂²×C₄₇	#339	1	1	yes	characteristic
163	C₂²×C₄₇	#340	1	1	yes	characteristic
164	C₂²×C₄₇	#340	1	1	yes	characteristic
165	C₂²×C₄₇	#342	1	1	yes	characteristic
166	C₂²×C₄₇	#343	1	1	yes	characteristic
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	2	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	—	2	1	no	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	Dic₄₇	—	8	4	no	—
179	C₂×#336	—	47	1	no	—
180	C₂×#336	D₄₇	1	1	yes	characteristic
181	C₄⋊#95	—	47	1	no	—
182	C₂×C₄₇	#1270	1	2	yes	—
183	C₂×C₄₇	#1271	1	2	yes	—
184	C₂×C₄₇	#1271	1	2	yes	—
185	C₂×C₄₇	#1272	1	2	yes	—
186	C₂×C₄₇	#1279	1	1	yes	characteristic
187	C₂×C₄₇	#1279	1	1	yes	characteristic
188	C₂×C₄₇	#1280	1	1	yes	characteristic
189	C₂×C₄₇	#1294	1	1	yes	characteristic
190	C₂×C₄₇	#1294	1	1	yes	characteristic
191	C₂×C₄₇	#1295	1	1	yes	characteristic
192	C₂×C₄₇	#1296	1	1	yes	characteristic
193	C₂³×C₈	—	2	1	no	—
194	C₂³×C₈	—	94	1	no	—
195	C₂²×C₄⋊C₄	—	47	2	no	—
196	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
197	C₂×#95	—	94	1	no	—
198	C₂×#95	—	94	1	no	—
199	#336	—	2	1	no	—
200	#336	—	47	2	no	—
201	#336	—	94	1	no	—
202	#336	Dic₄₇	1	2	yes	complemented
203	C₄₇	#7025	1	1	yes	characteristic, complemented, Sylow
204	C₂³×C₄	—	94	2	no	—
205	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
206	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
207	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂²×C₈	—	2	4	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	4	2	no	—
211	C₂²×C₈	—	4	1	no	—
212	C₂²×C₈	—	4	1	no	—
213	C₂²×C₈	—	94	4	no	—
214	C₂²×C₈	—	94	2	no	—
215	C₂²×C₈	—	188	2	no	—
216	C₂²×C₈	—	188	1	no	—
217	C₂²×C₈	—	188	1	no	—
218	C₂×C₄⋊C₄	—	47	4	no	—
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₄⋊C₄	—	188	2	no	—
221	C₂×C₄⋊C₄	—	188	2	no	—
222	C₂×C₄²	—	4	2	no	—
223	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
227	#95	—	188	1	no	—
228	#95	—	188	1	no	—
229	#95	—	188	1	no	—
230	#95	—	188	1	no	—
231	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	94	4	no	—
240	C₂²×C₄	—	188	2	no	—
241	C₂²×C₄	—	188	2	no	—
242	C₂²×C₄	—	188	2	no	—
243	C₂²×C₄	—	188	2	no	—
244	C₂²×C₄	—	188	2	no	—
245	C₂²×C₄	—	188	2	no	—
246	C₂²×C₄	C₈×D₄₇	1	2	yes	—
247	C₂²×C₄	#10625324	1	2	yes	—
248	C₂²×C₄	#10625325	1	1	yes	characteristic
249	C₂²×C₄	#10625325	1	1	yes	characteristic
250	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
251	C₂²×C₄	Dic₁₈₈	1	1	yes	characteristic
252	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
256	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
257	C₂²×C₄	#10625334	1	1	yes	characteristic
258	C₂²×C₄	#10625335	1	1	yes	characteristic
259	C₂²×C₄	#10625336	1	1	yes	characteristic
260	C₂²×C₄	#10625337	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	4	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	94	4	no	—
268	C₂×C₈	—	188	4	no	—
269	C₂×C₈	—	188	4	no	—
270	C₂×C₈	—	188	2	no	—
271	C₂×C₈	—	188	2	no	—
272	C₄⋊C₄	—	188	4	no	—
273	C₄⋊C₄	—	188	4	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₂³	C₄₇⋊#95	1	1	yes	characteristic
279	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
280	C₂³	#13093938	1	2	yes	—
281	C₂³	#13093940	1	1	yes	characteristic
282	C₂³	#13093942	1	1	yes	characteristic
283	C₂³	#13093944	1	1	yes	characteristic
284	C₂³	#13093946	1	1	yes	characteristic
285	C₂³	#13093947	1	1	yes	characteristic
286	C₂³	#13093949	1	1	yes	characteristic
287	C₂³	#13093950	1	2	yes	—
288	C₂³	#13093952	1	2	yes	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	376	2	no	—
314	C₂×C₄	—	376	2	no	—
315	C₂×C₄	—	376	2	no	—
316	C₂×C₄	—	376	2	no	—
317	C₂×C₄	—	188	4	no	—
318	C₂×C₄	—	188	4	no	—
319	C₂×C₄	—	188	4	no	—
320	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
321	C₂×C₄	#13093934	1	1	yes	characteristic
322	C₂×C₄	#13093945	1	1	yes	characteristic
323	C₂×C₄	#13093946	1	1	yes	characteristic
324	C₂×C₄	#13093948	1	1	yes	characteristic
325	C₂×C₄	#13093949	1	1	yes	characteristic
326	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
327	C₂×C₄	#13093955	1	1	yes	characteristic
328	C₂×C₄	#13093956	1	1	yes	characteristic
329	C₂×C₄	#13093957	1	1	yes	characteristic
330	C₂×C₄	#13093958	1	1	yes	characteristic
331	C₈	—	4	4	no	—
332	C₈	—	188	4	no	—
333	C₂²	#15688846	1	2	yes	—
334	C₂²	#15688848	1	2	yes	—
335	C₂²	#15688850	1	2	yes	—
336	C₂²	#15688852	1	2	yes	—
337	C₂²	#15688853	1	2	yes	—
338	C₂²	#15688855	1	2	yes	—
339	C₂²	#15688856	1	2	yes	—
340	C₂²	#15688858	1	2	yes	—
341	C₂²	#15688859	1	2	yes	—
342	C₂²	#15688861	1	2	yes	—
343	C₂²	#15688862	1	2	yes	—
344	C₂²	#15688864	1	2	yes	—
345	C₂²	#15688866	1	1	yes	characteristic
346	C₂²	#15688868	1	1	yes	characteristic
347	C₂²	#15688870	1	1	yes	characteristic
348	C₂²	#15688871	1	1	yes	characteristic
349	C₂²	#15688877	1	1	yes	characteristic
350	C₂²	#15688883	1	1	yes	characteristic
351	C₂²	#15688884	1	1	yes	characteristic
352	C₂²	#15688885	1	1	yes	characteristic
353	C₂²	#15688886	1	1	yes	characteristic
354	C₂²	#15688889	1	1	yes	characteristic
355	C₂²	#15688891	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	376	4	no	—
368	C₂	#20241208	1	2	yes	—
369	C₂	#20241210	1	2	yes	—
370	C₂	#20241212	1	2	yes	—
371	C₂	#20241213	1	2	yes	—
372	C₂	#20241227	1	1	yes	characteristic
373	C₂	#20241228	1	1	yes	characteristic
374	C₂	#20241230	1	1	yes	characteristic
375	C₂	#20241267	1	1	yes	characteristic
376	C₂	#20241268	1	1	yes	characteristic
377	C₂	#20241270	1	1	yes	characteristic
378	C₂	#20241272	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₄₇
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂²×C₄²×C₄₇	C₂²
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#7025
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
9. #15688878	C₄	semidirect
13. C₄₇×#336	C₄	semidirect
23. C₂×C₄⋊Dic₄₇	C₈	semidirect
202. #336	Dic₄₇	semidirect
203. C₄₇	#7025	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688878	2
#20241618	2
C₂×C₄₇×#336	2
#7025	47