Subgroups — #26218401

Order 12032.

906 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
3	#20245871	C₂	1	1	yes	maximal, characteristic
4	#20243955	C₂	1	4	yes	maximal, complemented
5	#20243948	C₂	1	4	yes	maximal, complemented
6	#20243084	C₂	1	1	yes	maximal, characteristic
7	#20243084	C₂	1	1	yes	maximal, characteristic
8	#20243084	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂²×#10625329	C₂²	1	1	yes	characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	C₂×#13093983	C₂²	1	1	yes	characteristic
14	#15689147	—	2	4	no	—
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
16	#15689078	—	2	4	no	—
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
19	#15689021	C₄	1	2	yes	—
20	#15689021	C₄	1	2	yes	—
21	#15688996	C₄	1	4	yes	—
22	C₄×Dic₄₇⋊C₄	C₂²	1	4	yes	—
23	#15688877	C₂²	1	2	yes	—
24	#15688877	C₂²	1	4	yes	—
25	#15688877	C₂²	1	4	yes	—
26	#15688877	C₂²	1	2	yes	—
27	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#336	C₂²	1	2	yes	—
31	C₄₇×#336	C₂²	1	2	yes	—
32	C₂×#10625329	—	2	2	no	—
33	C₂×#10625329	—	2	2	no	—
34	C₂×#10625329	—	2	2	no	—
35	C₂×#10625329	—	2	2	no	—
36	C₂×#10625329	—	2	2	no	—
37	C₂×#10625329	D₄	1	4	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	#13093983	—	2	2	no	—
41	#13093983	—	2	2	no	—
42	#13093983	—	2	2	no	—
43	#13093983	—	2	2	no	—
44	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
46	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
47	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
52	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
53	#13093934	—	2	4	no	—
54	C₈×Dic₄₇	—	2	4	no	—
55	C₄₇⋊#95	C₂×C₄	1	4	yes	—
56	C₄₇⋊#95	C₂×C₄	1	4	yes	—
57	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
58	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	—	2	2	no	—
60	C₂²×C₈×C₄₇	—	2	2	no	—
61	C₂²×C₈×C₄₇	—	2	2	no	—
62	C₂²×C₈×C₄₇	—	2	2	no	—
63	C₂²×C₈×C₄₇	—	2	2	no	—
64	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
65	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	—	2	2	no	—
69	C₂×C₄²×C₄₇	—	2	2	no	—
70	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
73	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	—	2	4	no	—
75	C₂²×C₄×C₄₇	—	2	4	no	—
76	C₂²×C₄×C₄₇	—	4	2	no	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₄₇	#41	1	2	yes	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×C₄×C₄₇	#41	1	2	yes	—
85	C₂²×C₄×C₄₇	#41	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
88	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₄₇	—	2	8	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	—	2	1	no	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	4	4	no	—
98	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
100	C₄²×C₄₇	—	2	2	no	—
101	C₄²×C₄₇	—	2	2	no	—
102	C₄²×C₄₇	—	2	2	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₂²⋊Dic₄₇	—	2	4	no	—
105	C₂²⋊Dic₄₇	—	2	4	no	—
106	C₄⋊Dic₄₇	—	2	4	no	—
107	C₄⋊Dic₄₇	—	2	4	no	—
108	Dic₄₇⋊C₄	—	2	8	no	—
109	Dic₄₇⋊C₄	—	4	4	no	—
110	C₄×Dic₄₇	—	2	2	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	2	2	no	—
114	#10625329	—	2	8	no	—
115	#10625329	—	4	4	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	—	2	2	no	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
122	C₂³×C₄₇	—	2	2	no	—
123	C₂³×C₄₇	—	2	2	no	—
124	C₂³×C₄₇	#117	1	2	yes	—
125	C₂³×C₄₇	C₄×D₄	1	2	yes	—
126	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
127	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	2	no	—
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	4	8	no	—
134	C₂×C₄×C₄₇	#117	1	2	yes	—
135	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
138	C₂×C₄×C₄₇	#123	1	4	yes	—
139	C₂×C₄×C₄₇	#125	1	4	yes	—
140	C₂×C₄×C₄₇	#126	1	4	yes	—
141	C₂×C₄×C₄₇	#131	1	4	yes	—
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₄₇	—	2	4	no	—
149	C₂×Dic₄₇	—	2	4	no	—
150	C₂×Dic₄₇	—	2	4	no	—
151	C₂×Dic₄₇	—	2	4	no	—
152	C₂×Dic₄₇	—	2	4	no	—
153	C₂×Dic₄₇	—	2	4	no	—
154	C₈×C₄₇	—	2	2	no	—
155	C₈×C₄₇	—	2	2	no	—
156	C₈×C₄₇	—	4	2	no	—
157	C₄₇⋊C₈	—	2	4	no	—
158	C₄₇⋊C₈	—	4	2	no	—
159	#21027	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	2	4	no	—
161	C₂²×C₄₇	—	2	4	no	—
162	C₂²×C₄₇	—	2	2	no	—
163	C₂²×C₄₇	—	2	2	no	—
164	C₂²×C₄₇	#387	1	4	yes	—
165	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
166	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂²×C₄₇	C₂×#131	1	2	yes	—
169	C₂²×C₄₇	#568	1	2	yes	—
170	C₄×C₄₇	—	2	4	no	—
171	C₄×C₄₇	—	4	2	no	—
172	C₄×C₄₇	#387	1	4	yes	—
173	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
174	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
175	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
176	C₄×C₄₇	C₂×#126	1	1	yes	characteristic
177	Dic₄₇	—	4	4	no	—
178	Dic₄₇	—	4	4	no	—
179	C₂×#423	—	47	1	no	—
180	C₂×#405	—	47	1	no	—
181	#2296	—	47	1	no	—
182	#1917	—	47	4	no	—
183	#1916	—	47	4	no	—
184	#1729	—	47	1	no	—
185	#1729	—	47	1	no	—
186	#1729	—	47	1	no	—
187	C₂×#336	D₄₇	1	1	yes	characteristic
188	C₂×C₄₇	—	2	4	no	—
189	C₂×C₄₇	C₂×#387	1	1	yes	characteristic
190	C₂×C₄₇	#2925	1	2	yes	—
191	C₂×C₄₇	#2933	1	2	yes	—
192	C₂×C₄₇	#2960	1	1	yes	characteristic
193	C₂×C₄₇	#2961	1	1	yes	characteristic
194	C₂²×M₁₆	—	47	1	no	—
195	C₂²×M₁₆	—	47	1	no	—
196	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
197	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
198	C₂×#117	—	47	1	no	—
199	C₂×#117	—	47	1	no	—
200	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
201	#423	—	94	4	no	—
202	C₂×C₄⋊C₈	—	47	4	no	—
203	#405	—	94	4	no	—
204	C₂×#97	—	47	2	no	—
205	C₂×C₄×C₈	—	47	2	no	—
206	#389	—	47	2	no	—
207	#389	—	47	2	no	—
208	#385	—	47	4	no	—
209	C₄×C₄⋊C₄	—	47	4	no	—
210	#336	—	47	2	no	—
211	#336	—	47	4	no	—
212	#336	—	47	4	no	—
213	#336	—	47	2	no	—
214	#336	C₂×D₄₇	1	2	yes	—
215	#336	C₂×D₄₇	1	2	yes	—
216	C₄₇	#21027	1	1	yes	characteristic, complemented, Sylow
217	C₂³×C₄	C₄×D₄₇	1	2	yes	—
218	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
219	C₂×M₁₆	—	47	4	no	—
220	C₂×M₁₆	—	94	2	no	—
221	C₂×M₁₆	—	94	2	no	—
222	C₂×M₁₆	—	94	2	no	—
223	C₂×M₁₆	—	94	2	no	—
224	C₂×M₁₆	—	94	2	no	—
225	C₂²×C₈	—	2	2	no	—
226	C₂²×C₈	—	2	2	no	—
227	C₂²×C₈	—	2	2	no	—
228	C₂²×C₈	—	2	2	no	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	—	47	2	no	—
231	C₂²×C₈	—	47	2	no	—
232	C₂²×C₈	Dic₉₄	1	4	yes	—
233	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
234	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	#117	—	94	2	no	—
237	#117	—	94	2	no	—
238	#117	—	94	2	no	—
239	#117	—	94	2	no	—
240	C₂×C₄⋊C₄	—	47	2	no	—
241	C₂×C₄⋊C₄	—	47	2	no	—
242	C₂×C₄⋊C₄	—	47	4	no	—
243	C₂×C₄⋊C₄	—	47	4	no	—
244	C₂×C₂²⋊C₄	—	47	2	no	—
245	C₂×C₂²⋊C₄	—	47	2	no	—
246	C₂×C₄²	—	2	2	no	—
247	C₂×C₄²	—	2	2	no	—
248	C₂×C₄²	—	47	2	no	—
249	C₂×C₄²	—	47	2	no	—
250	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
251	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
253	C₄⋊C₈	—	94	8	no	—
254	#97	—	94	4	no	—
255	C₄×C₈	—	94	4	no	—
256	#95	—	47	4	no	—
257	#95	—	47	4	no	—
258	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	47	4	no	—
263	C₂²×C₄	—	47	4	no	—
264	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
265	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
266	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
267	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
268	C₂²×C₄	#10625350	1	2	yes	—
269	C₂²×C₄	D₄×D₄₇	1	2	yes	—
270	C₂²×C₄	#10625352	1	2	yes	—
271	C₂²×C₄	#10625352	1	2	yes	—
272	C₂²×C₄	#10625354	1	2	yes	—
273	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
274	M₁₆	—	94	8	no	—
275	M₁₆	—	188	4	no	—
276	C₂×C₈	—	2	8	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	4	no	—
285	C₂×C₈	—	47	4	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	—	94	2	no	—
289	C₂×C₈	—	94	2	no	—
290	C₂×C₈	—	94	2	no	—
291	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
292	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
293	C₄⋊C₄	—	94	4	no	—
294	C₄⋊C₄	—	94	4	no	—
295	C₄⋊C₄	—	94	8	no	—
296	C₄⋊C₄	—	188	4	no	—
297	C₂²⋊C₄	—	94	4	no	—
298	C₂²⋊C₄	—	94	4	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	94	2	no	—
304	C₄²	—	94	2	no	—
305	C₄²	—	94	2	no	—
306	C₄²	—	94	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	#13093982	1	2	yes	—
310	C₂³	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
311	C₂³	C₄×Dic₉₄	1	2	yes	—
312	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	8	no	—
319	C₂×C₄	—	94	4	no	—
320	C₂×C₄	—	94	4	no	—
321	C₂×C₄	—	94	4	no	—
322	C₂×C₄	—	94	4	no	—
323	C₂×C₄	—	94	4	no	—
324	C₂×C₄	—	94	4	no	—
325	C₂×C₄	#13093982	1	2	yes	—
326	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
327	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
328	C₂×C₄	#13094001	1	4	yes	—
329	C₂×C₄	#13094009	1	4	yes	—
330	C₂×C₄	#13094015	1	4	yes	—
331	C₂×C₄	#13094017	1	4	yes	—
332	C₂×C₄	#13094030	1	4	yes	—
333	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
334	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
335	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
336	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
337	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
338	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	94	4	no	—
343	C₈	—	188	2	no	—
344	C₂²	—	2	4	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	4	no	—
348	C₂²	#15689010	1	4	yes	—
349	C₂²	C₂×#13093982	1	1	yes	characteristic
350	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
351	C₂²	C₂×C₄×Dic₉₄	1	1	yes	characteristic
352	C₂²	C₂×#13094030	1	2	yes	—
353	C₂²	#15689889	1	2	yes	—
354	C₄	—	2	4	no	—
355	C₄	—	4	2	no	—
356	C₄	—	188	4	no	—
357	C₄	—	188	4	no	—
358	C₄	#15689010	1	4	yes	—
359	C₄	C₂×#13094001	1	1	yes	characteristic
360	C₄	C₂×#13094009	1	1	yes	characteristic
361	C₄	C₂×#13094015	1	1	yes	characteristic
362	C₄	C₂×#13094017	1	1	yes	characteristic
363	C₂	—	2	4	no	—
364	C₂	C₂×#15689010	1	1	yes	characteristic
365	C₂	#20252053	1	2	yes	—
366	C₂	#20252109	1	2	yes	—
367	C₂	#20252322	1	1	yes	characteristic
368	C₂	#20252328	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#336	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#21027
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
4. #20243955	C₂	semidirect
5. #20243948	C₂	semidirect
216. C₄₇	#21027	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×#15689078	2
#20245871	2
#20243955	2
#20243948	2
#20243084	2
C₂×C₄₇×#336	2
#21027	47