Subgroups — #26215045

Order 12032.

906 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689184	C₂	1	1	yes	maximal, characteristic
2	#20245872	C₂	1	1	yes	maximal, characteristic
3	#20243265	C₂	1	2	yes	maximal, complemented
4	#20243260	C₂	1	2	yes	maximal, complemented
5	#20243253	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊Dic₄₇	C₂	1	2	yes	maximal, complemented
7	#20243156	C₂	1	2	yes	maximal
8	C₂²⋊C₈×Dic₄₇	C₂	1	2	yes	maximal
9	C₂×C₄₇×#433	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	#15689184	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	1	yes	characteristic
16	C₂×#13093934	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
19	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
20	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
21	#15689022	C₄	1	4	yes	—
22	#15688998	C₄	1	4	yes	—
23	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
25	#15688877	C₂²	1	2	yes	—
26	#15688877	C₂²	1	2	yes	—
27	#15688877	C₂²	1	2	yes	—
28	#15688877	C₂²	1	2	yes	—
29	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#433	C₄	1	8	yes	complemented
31	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
33	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	#13093983	—	4	2	no	—
39	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₄₇	—	4	2	no	—
44	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
46	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
47	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
48	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
49	#13093935	C₂×C₄	1	4	yes	complemented
50	#13093934	—	4	2	no	—
51	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
52	C₈×Dic₄₇	—	4	2	no	—
53	C₈×Dic₄₇	—	4	2	no	—
54	C₈×Dic₄₇	—	4	2	no	—
55	C₄₇⋊#95	C₂×C₄	1	4	yes	—
56	C₄₇⋊#95	C₂×C₄	1	4	yes	—
57	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
59	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
60	#117×C₄₇	C₂×C₄	1	4	yes	—
61	#117×C₄₇	C₂×C₄	1	4	yes	—
62	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	complemented
67	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
68	#97×C₄₇	C₂×C₄	1	4	yes	complemented
69	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
70	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	8	no	—
80	C₂²×Dic₄₇	—	4	2	no	—
81	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
82	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₄₇	—	2	2	no	—
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	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
90	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	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	C₂²⋊Dic₄₇	—	4	2	no	—
96	C₄⋊Dic₄₇	—	2	8	no	—
97	C₄⋊Dic₄₇	—	2	8	no	—
98	C₄⋊Dic₄₇	—	4	2	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	C₄×Dic₄₇	—	2	2	no	—
102	C₄×Dic₄₇	—	2	2	no	—
103	C₄×Dic₄₇	—	4	2	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	—	2	2	no	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
109	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
110	C₂³×C₄₇	—	2	1	no	—
111	C₂³×C₄₇	—	2	1	no	—
112	C₂³×C₄₇	—	2	4	no	—
113	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
114	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	—	4	8	no	—
134	C₂×Dic₄₇	#131	1	8	yes	—
135	C₈×C₄₇	—	2	4	no	—
136	C₈×C₄₇	—	2	4	no	—
137	C₄₇⋊C₈	—	2	4	no	—
138	C₄₇⋊C₈	—	2	4	no	—
139	#20540	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	4	2	no	—
143	C₂²×C₄₇	—	4	2	no	—
144	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
145	C₂²×C₄₇	#518	1	1	yes	characteristic
146	C₂²×C₄₇	#519	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#131	1	4	yes	—
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	#520	1	2	yes	—
152	C₄×C₄₇	#520	1	2	yes	—
153	Dic₄₇	—	2	8	no	—
154	Dic₄₇	—	4	4	no	—
155	C₂×#433	—	47	1	no	—
156	C₂×#433	D₄₇	1	1	yes	characteristic
157	#2296	—	47	1	no	—
158	#1771	—	47	2	no	—
159	#1771	—	47	2	no	—
160	C₈×C₄⋊C₄	—	47	2	no	—
161	C₈×C₄⋊C₄	—	47	2	no	—
162	#1750	—	47	2	no	—
163	C₄×C₂²⋊C₈	—	47	2	no	—
164	C₂×C₄₇	—	4	2	no	—
165	C₂×C₄₇	#2274	1	1	yes	characteristic
166	C₂×C₄₇	C₄×#131	1	2	yes	—
167	C₂×C₄₇	#2977	1	2	yes	—
168	C₂×C₄₇	#2986	1	2	yes	—
169	C₂×#117	—	47	1	no	—
170	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
171	C₂²×C₄²	—	47	1	no	—
172	#433	—	47	8	no	—
173	#433	Dic₄₇	1	8	yes	complemented
174	C₂×C₄⋊C₈	—	47	2	no	—
175	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
176	C₂×C₂²⋊C₈	—	47	2	no	—
177	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
178	C₂×#97	—	47	1	no	—
179	C₂×#97	—	47	2	no	—
180	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
181	C₂×C₄×C₈	—	47	1	no	—
182	C₂×C₄×C₈	—	47	2	no	—
183	C₂×C₄×C₈	—	47	2	no	—
184	C₂×C₄×C₈	—	47	2	no	—
185	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
186	#389	—	47	4	no	—
187	#385	—	47	4	no	—
188	C₄×C₄⋊C₄	—	47	2	no	—
189	C₄×C₄⋊C₄	—	47	2	no	—
190	#336	—	47	2	no	—
191	#336	—	47	2	no	—
192	#336	—	47	2	no	—
193	#336	—	47	2	no	—
194	C₄₇	#20540	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	47	2	no	—
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂²×C₈	—	47	2	no	—
198	C₂²×C₈	—	47	2	no	—
199	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
200	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
201	#117	—	188	2	no	—
202	#117	C₄×D₄₇	1	4	yes	—
203	#117	C₂×Dic₄₇	1	4	yes	—
204	C₂×C₄⋊C₄	—	47	4	no	—
205	C₂×C₄⋊C₄	—	47	4	no	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
208	C₂×C₂²⋊C₄	—	47	2	no	—
209	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
210	C₂×C₄²	—	47	2	no	—
211	C₂×C₄²	—	47	2	no	—
212	C₂×C₄²	—	47	2	no	—
213	C₂×C₄²	—	188	2	no	—
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₄⋊C₈	—	47	8	no	—
217	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	complemented
218	C₂²⋊C₈	—	47	8	no	—
219	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
220	#97	—	47	4	no	—
221	#97	—	188	2	no	—
222	#97	C₂×Dic₄₇	1	4	yes	complemented
223	C₄×C₈	—	47	4	no	—
224	C₄×C₈	—	188	2	no	—
225	C₄×C₈	—	188	2	no	—
226	C₄×C₈	—	188	2	no	—
227	C₄×C₈	C₂×Dic₄₇	1	4	yes	complemented
228	#95	—	47	4	no	—
229	#95	—	47	4	no	—
230	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	47	4	no	—
234	C₂²×C₄	—	47	4	no	—
235	C₂²×C₄	—	94	8	no	—
236	C₂²×C₄	—	188	2	no	—
237	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
238	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
242	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	47	4	no	—
248	C₂×C₈	—	47	4	no	—
249	C₂×C₈	—	94	2	no	—
250	C₂×C₈	—	94	2	no	—
251	C₂×C₈	—	94	2	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
254	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
255	C₄⋊C₄	—	94	8	no	—
256	C₄⋊C₄	—	94	8	no	—
257	C₄⋊C₄	—	188	2	no	—
258	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	—
259	C₂²⋊C₄	—	188	2	no	—
260	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
261	C₄²	—	94	2	no	—
262	C₄²	—	94	2	no	—
263	C₄²	—	94	2	no	—
264	C₄²	—	94	2	no	—
265	C₄²	—	188	2	no	—
266	C₄²	C₂×C₄×D₄₇	1	2	yes	—
267	C₄²	C₂×C₄×D₄₇	1	2	yes	—
268	C₄²	C₂²×Dic₄₇	1	2	yes	—
269	C₄²	C₂²×Dic₄₇	1	2	yes	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	4	no	—
273	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
274	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	47	8	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	188	8	no	—
288	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
289	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
290	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
291	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
293	C₂×C₄	#13094070	1	2	yes	—
294	C₂×C₄	#13094071	1	2	yes	—
295	C₈	—	2	4	no	—
296	C₈	—	2	4	no	—
297	C₈	—	94	4	no	—
298	C₈	—	94	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
304	C₂²	#15689597	1	1	yes	characteristic
305	C₂²	#15689600	1	1	yes	characteristic
306	C₂²	#131×D₄₇	1	4	yes	—
307	C₄	—	2	4	no	—
308	C₄	—	2	4	no	—
309	C₄	—	2	2	no	—
310	C₄	—	94	8	no	—
311	C₄	—	188	4	no	—
312	C₄	#15689603	1	2	yes	—
313	C₄	#15689606	1	2	yes	—
314	C₂	—	4	2	no	—
315	C₂	#20245744	1	1	yes	characteristic
316	C₂	#131×Dic₄₇	1	2	yes	—
317	C₂	#20252443	1	2	yes	—
318	C₂	#20252487	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#433	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#433	D₄₇
hypocenter	C₄₇	#20540
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
3. #20243265	C₂	semidirect
4. #20243260	C₂	semidirect
5. #20243253	C₂	semidirect
6. C₈×C₄⋊Dic₄₇	C₂	semidirect
12. #15689184	C₄	semidirect
30. C₄₇×#433	C₄	semidirect
47. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
49. #13093935	C₂×C₄	semidirect
51. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
66. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
68. #97×C₄₇	C₂×C₄	semidirect
69. C₄×C₈×C₄₇	C₂×C₄	semidirect
173. #433	Dic₄₇	semidirect
194. C₄₇	#20540	semidirect
217. C₄⋊C₈	C₂×Dic₄₇	semidirect
222. #97	C₂×Dic₄₇	semidirect
227. 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×#15689184	2
#20245872	2
#20243265	2
#20243260	2
#20243253	2
C₈×C₄⋊Dic₄₇	2
#20243156	2
C₂²⋊C₈×Dic₄₇	2
C₂×C₄₇×#433	2
#20540	47