Subgroups — #26109117

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	2	yes	maximal
2	C₂×#15688876	C₂	1	2	yes	maximal, complemented
3	#20241223	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
7	#15688878	—	2	2	no	—
8	#15688876	—	2	4	no	—
9	C₂³×C₈×C₄₇	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₄₇	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₇	—	2	4	no	—
13	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	4	4	no	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	—	2	2	no	—
25	C₂²×C₈×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	C₈	1	4	yes	complemented
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	—	2	2	no	—
35	C₂×C₄²×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₄₇	—	2	2	no	—
41	C₄×C₈×C₄₇	—	2	2	no	—
42	C₄×C₈×C₄₇	—	2	2	no	—
43	C₄×C₈×C₄₇	D₄	1	4	yes	—
44	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	1	no	—
48	C₂²×C₄×C₄₇	—	2	1	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	1	no	—
52	C₂²×C₄×C₄₇	—	2	1	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
59	C₂²×C₄×C₄₇	D₈	1	2	yes	—
60	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₄₇	—	2	8	no	—
65	C₂×C₈×C₄₇	—	2	1	no	—
66	C₂×C₈×C₄₇	—	2	1	no	—
67	C₂×C₈×C₄₇	—	2	1	no	—
68	C₂×C₈×C₄₇	—	2	1	no	—
69	C₂×C₈×C₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	—	2	1	no	—
71	C₂×C₈×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
77	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
78	C₄²×C₄₇	—	2	1	no	—
79	C₄²×C₄₇	—	2	1	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	C₂×D₄	1	2	yes	—
83	C₄²×C₄₇	#41	1	2	yes	—
84	C₂×C₄₇⋊C₈	—	4	4	no	—
85	C₂×C₄₇⋊C₈	—	4	4	no	—
86	C₂×C₄₇⋊C₈	—	4	4	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
92	C₂³×C₄₇	#107	1	2	yes	—
93	C₂³×C₄₇	#108	1	2	yes	—
94	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	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	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
111	C₂×C₄×C₄₇	#107	1	2	yes	—
112	C₂×C₄×C₄₇	#108	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#121	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#135	1	2	yes	—
124	C₈×C₄₇	—	2	2	no	—
125	C₈×C₄₇	—	2	2	no	—
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	C₂²⋊C₈	1	4	yes	complemented
128	C₄₇⋊C₈	—	8	4	no	—
129	#8596	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	#335	1	4	yes	complemented
137	C₂²×C₄₇	#380	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₂²×C₄₇	#423	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
141	C₂²×C₄₇	#427	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₄×C₄₇	#407	1	1	yes	characteristic
150	C₄×C₄₇	#431	1	1	yes	characteristic
151	C₄×C₄₇	#432	1	1	yes	characteristic
152	C₄×C₄₇	#466	1	2	yes	—
153	C₄×C₄₇	#467	1	2	yes	—
154	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
155	#1797	—	47	2	no	—
156	C₂×#335	—	47	2	no	—
157	#1277	—	47	2	no	—
158	C₂×C₄₇	—	2	4	no	—
159	C₂×C₄₇	C₂×#335	1	1	yes	characteristic
160	C₂×C₄₇	#1564	1	1	yes	characteristic
161	C₂×C₄₇	#1567	1	1	yes	characteristic
162	C₂×C₄₇	#1568	1	1	yes	characteristic
163	C₂×C₄₇	#1797	1	1	yes	characteristic
164	C₂×C₄₇	#1808	1	1	yes	characteristic
165	C₂×C₄₇	#1810	1	1	yes	characteristic
166	C₂³×C₈	Dic₄₇	1	1	yes	characteristic
167	C₂³×C₈	Dic₄₇	1	1	yes	characteristic
168	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	47	4	no	—
170	C₂×C₂²⋊C₈	—	94	2	no	—
171	C₂×C₄×C₈	—	2	4	no	—
172	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
173	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
174	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
176	#336	—	94	2	no	—
177	#335	—	94	4	no	—
178	C₄₇	#8596	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
180	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
181	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	94	4	no	—
188	C₂²×C₈	C₄₇⋊C₈	1	4	yes	complemented
189	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
190	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
191	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
196	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
199	C₄⋊C₈	—	94	4	no	—
200	C₄⋊C₈	—	94	4	no	—
201	C₂²⋊C₈	—	188	4	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	2	2	no	—
205	C₄×C₈	C₂²⋊D₄₇	1	4	yes	—
206	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	D₃₇₆	1	2	yes	—
217	C₂²×C₄	Dic₁₈₈	1	2	yes	—
218	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
219	C₂²×C₄	#10625329	1	2	yes	—
220	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
226	C₂×C₈	—	2	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	—	188	4	no	—
235	C₂×C₈	—	188	4	no	—
236	C₂×C₈	—	188	4	no	—
237	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
238	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
239	C₂×C₈	#10625329	1	2	yes	—
240	C₂×C₈	#10625329	1	2	yes	—
241	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	#10625350	1	2	yes	—
248	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
254	C₂³	#13093958	1	2	yes	—
255	C₂³	#13093960	1	2	yes	—
256	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
273	C₂×C₄	#13093958	1	2	yes	—
274	C₂×C₄	#13093960	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
276	C₂×C₄	#13093983	1	1	yes	characteristic
277	C₂×C₄	#13093998	1	2	yes	—
278	C₂×C₄	#13094003	1	2	yes	—
279	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
281	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
282	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
283	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
284	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
285	C₂×C₄	#13094046	1	2	yes	—
286	C₈	—	2	2	no	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	376	4	no	—
290	C₈	C₂²⋊(C₄₇⋊C₈)	1	4	yes	complemented
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	#15688876	1	4	yes	complemented
298	C₂²	#15688983	1	1	yes	characteristic
299	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
300	C₂²	#15689147	1	1	yes	characteristic
301	C₂²	C₂×#13093958	1	1	yes	characteristic
302	C₂²	#15689161	1	1	yes	characteristic
303	C₂²	C₂×#13093960	1	1	yes	characteristic
304	C₂²	C₂×#13093960	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
310	C₄	#15689087	1	1	yes	characteristic
311	C₄	#15689176	1	1	yes	characteristic
312	C₄	#15689180	1	1	yes	characteristic
313	C₄	#15689359	1	2	yes	—
314	C₄	#15689364	1	2	yes	—
315	C₂	—	2	4	no	—
316	C₂	C₂×#15688876	1	1	yes	characteristic
317	C₂	#20242136	1	1	yes	characteristic
318	C₂	#20242147	1	1	yes	characteristic
319	C₂	#20242151	1	1	yes	characteristic
320	C₂	#20243368	1	1	yes	characteristic
321	C₂	#20243417	1	1	yes	characteristic
322	C₂	#20243426	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#8596
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15688876	C₂	semidirect
3. #20241223	C₂	semidirect
29. C₂²×C₈×C₄₇	C₈	semidirect
127. C₈×C₄₇	C₂²⋊C₈	semidirect
136. C₂²×C₄₇	#335	semidirect
178. C₄₇	#8596	semidirect
188. C₂²×C₈	C₄₇⋊C₈	semidirect
290. C₈	C₂²⋊(C₄₇⋊C₈)	semidirect
297. C₂²	#15688876	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₂×#15688876	2
#20241223	2
C₂²×C₄×C₈×C₄₇	2
#8596	47