Subgroups — #26215018

Order 12032.

1038 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243265	C₂	1	4	yes	maximal, complemented
4	#20243255	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊Dic₄₇	C₂	1	2	yes	maximal, complemented
6	#20243152	C₂	1	2	yes	maximal
7	C₂²⋊C₈×Dic₄₇	C₂	1	2	yes	maximal
8	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄⋊Dic₄₇	C₄	1	2	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
11	#15689180	C₄	1	8	yes	complemented
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×#13093934	C₂²	1	4	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
18	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₄₇⋊#95	C₄	1	2	yes	—
23	#15688877	C₂²	1	4	yes	—
24	#15688877	C₂²	1	4	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#431	C₄	1	8	yes	complemented
27	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
31	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	8	no	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	—	4	4	no	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
42	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
43	#13093935	C₂×C₄	1	8	yes	complemented
44	#13093934	—	4	4	no	—
45	C₈×Dic₄₇	—	4	4	no	—
46	C₄₇⋊#95	C₂×C₄	1	2	yes	—
47	C₄₇⋊#95	C₂×C₄	1	4	yes	—
48	C₄₇⋊#95	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	complemented
61	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
62	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
63	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
64	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
70	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	8	no	—
80	C₂²×Dic₄₇	—	2	8	no	—
81	C₂²×Dic₄₇	—	4	4	no	—
82	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
83	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
90	C₄²×C₄₇	C₄²	1	4	yes	—
91	C₄²×C₄₇	C₄²	1	4	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
95	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
96	C₄⋊Dic₄₇	—	2	4	no	—
97	C₄⋊Dic₄₇	—	2	8	no	—
98	C₄⋊Dic₄₇	M₁₆	1	8	yes	—
99	C₄×Dic₄₇	—	2	4	no	—
100	C₄×Dic₄₇	—	2	4	no	—
101	C₄×Dic₄₇	—	4	4	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
105	C₂³×C₄₇	—	2	1	no	—
106	C₂³×C₄₇	—	2	1	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
109	C₂³×C₄₇	#131	1	4	yes	—
110	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
129	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	—	2	4	no	—
134	C₂×Dic₄₇	—	2	4	no	—
135	C₂×Dic₄₇	—	4	16	no	—
136	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
137	C₈×C₄₇	—	2	4	no	—
138	C₈×C₄₇	—	2	4	no	—
139	C₄₇⋊C₈	—	2	8	no	—
140	#20535	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	4	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	#405	1	4	yes	—
146	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
147	C₂²×C₄₇	#518	1	1	yes	characteristic
148	C₂²×C₄₇	#519	1	1	yes	characteristic
149	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
150	C₂²×C₄₇	C₂×#131	1	2	yes	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
156	C₄×C₄₇	#520	1	2	yes	—
157	C₄×C₄₇	#520	1	2	yes	—
158	Dic₄₇	—	2	8	no	—
159	Dic₄₇	—	4	4	no	—
160	C₂×#432	—	47	1	no	—
161	C₂×#431	D₄₇	1	1	yes	characteristic
162	C₂×C₄×C₄⋊C₄	—	47	1	no	—
163	#1771	—	47	4	no	—
164	#1770	—	47	2	no	—
165	C₈×C₄⋊C₄	—	47	2	no	—
166	#1749	—	47	2	no	—
167	C₄×C₂²⋊C₈	—	47	2	no	—
168	C₂×C₄₇	—	2	4	no	—
169	C₂×C₄₇	#2274	1	1	yes	characteristic
170	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
172	C₂×C₄₇	#2974	1	2	yes	—
173	C₂×C₄₇	#2984	1	2	yes	—
174	C₂²×C₄⋊C₄	—	47	2	no	—
175	C₂²×C₄²	—	47	2	no	—
176	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
177	#432	—	47	8	no	—
178	#431	Dic₄₇	1	8	yes	complemented
179	C₂×C₄⋊C₈	—	47	2	no	—
180	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
181	C₂×C₂²⋊C₈	—	47	2	no	—
182	C₂×C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
183	C₂×C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂×#97	—	47	2	no	—
185	C₂×#97	—	47	4	no	—
186	C₂×C₄×C₈	—	47	4	no	—
187	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
188	C₄×C₄⋊C₄	—	47	2	no	—
189	C₄×C₄⋊C₄	—	47	2	no	—
190	C₄×C₄⋊C₄	—	47	1	no	—
191	C₄×C₄⋊C₄	—	47	1	no	—
192	C₄×C₄⋊C₄	—	47	2	no	—
193	C₂×#95	—	47	2	no	—
194	#336	—	47	4	no	—
195	#336	—	47	4	no	—
196	C₄₇	#20535	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	—	47	4	no	—
198	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
199	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
200	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₈	—	47	4	no	—
202	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
203	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
204	C₂×C₄⋊C₄	—	47	2	no	—
205	C₂×C₄⋊C₄	—	47	4	no	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	—	94	8	no	—
208	C₂×C₄²	—	47	2	no	—
209	C₂×C₄²	—	47	2	no	—
210	C₂×C₄²	—	47	4	no	—
211	C₂×C₄²	—	188	4	no	—
212	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₄₇	1	4	yes	—
215	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
216	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
218	C₄⋊C₈	—	47	8	no	—
219	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	complemented
220	C₂²⋊C₈	—	47	8	no	—
221	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
222	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
223	#97	—	47	8	no	—
224	#97	—	188	4	no	—
225	C₄×C₈	—	188	4	no	—
226	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
227	#95	—	47	2	no	—
228	#95	—	47	4	no	—
229	#95	—	47	2	no	—
230	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	47	4	no	—
236	C₂²×C₄	—	47	4	no	—
237	C₂²×C₄	—	94	8	no	—
238	C₂²×C₄	—	94	8	no	—
239	C₂²×C₄	—	188	4	no	—
240	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
241	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
242	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
249	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	47	8	no	—
255	C₂×C₈	—	94	4	no	—
256	C₂×C₈	—	94	4	no	—
257	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
258	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
259	C₄⋊C₄	—	47	8	no	—
260	C₄⋊C₄	—	94	8	no	—
261	C₄⋊C₄	—	94	4	no	—
262	C₄²	—	94	4	no	—
263	C₄²	—	94	4	no	—
264	C₄²	—	188	4	no	—
265	C₄²	C₄×Dic₄₇	1	4	yes	—
266	C₄²	C₄×Dic₄₇	1	4	yes	—
267	C₄²	C₂×C₄×D₄₇	1	2	yes	—
268	C₄²	C₂×C₄×D₄₇	1	2	yes	—
269	C₄²	C₂²×Dic₄₇	1	2	yes	—
270	C₄²	C₂²×Dic₄₇	1	2	yes	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	2	no	—
274	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
275	C₂³	#13094030	1	4	yes	—
276	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	47	8	no	—
287	C₂×C₄	—	94	4	no	—
288	C₂×C₄	—	94	4	no	—
289	C₂×C₄	—	94	4	no	—
290	C₂×C₄	—	94	4	no	—
291	C₂×C₄	—	94	4	no	—
292	C₂×C₄	—	188	16	no	—
293	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
294	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
295	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
296	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
297	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
298	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
299	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
301	C₂×C₄	#13094070	1	2	yes	—
302	C₂×C₄	#13094071	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	2	4	no	—
305	C₈	—	94	8	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#15689078	1	4	yes	—
311	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
312	C₂²	#15689597	1	1	yes	characteristic
313	C₂²	#15689600	1	1	yes	characteristic
314	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
315	C₂²	C₂×#13094030	1	2	yes	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	94	8	no	—
321	C₄	—	188	4	no	—
322	C₄	M₁₆×Dic₄₇	1	4	yes	—
323	C₄	#15689603	1	2	yes	—
324	C₄	#15689606	1	2	yes	—
325	C₂	—	2	4	no	—
326	C₂	#20245744	1	1	yes	characteristic
327	C₂	C₂×M₁₆×Dic₄₇	1	1	yes	characteristic
328	C₂	C₂×#15689078	1	1	yes	characteristic
329	C₂	#20252422	1	2	yes	—
330	C₂	#20252475	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#20535
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20243265	C₂	semidirect
4. #20243255	C₂	semidirect
5. C₈×C₄⋊Dic₄₇	C₂	semidirect
11. #15689180	C₄	semidirect
26. C₄₇×#431	C₄	semidirect
41. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
43. #13093935	C₂×C₄	semidirect
60. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
63. C₄×C₈×C₄₇	C₂×C₄	semidirect
178. #431	Dic₄₇	semidirect
196. C₄₇	#20535	semidirect
219. C₄⋊C₈	C₂×Dic₄₇	semidirect
226. C₄×C₈	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₂×C₄×C₄⋊Dic₄₇	2
#20243265	2
#20243255	2
C₈×C₄⋊Dic₄₇	2
#20243152	2
C₂²⋊C₈×Dic₄₇	2
C₂×C₄₇×#431	2
#20535	47