Subgroups — #26221156

Order 12032.

938 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245786	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20244158	C₂	1	4	yes	maximal, complemented
4	#20243583	C₂	1	2	yes	maximal
5	#20243578	C₂	1	2	yes	maximal
6	#20243285	C₂	1	4	yes	maximal, complemented
7	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	—	2	1	no	—
10	C₂×#13093958	C₂²	1	4	yes	—
11	C₂×#13093956	C₂²	1	4	yes	—
12	#15688991	—	2	1	no	—
13	#15688989	C₂²	1	2	yes	—
14	#15688986	—	2	1	no	—
15	#15688983	C₂²	1	1	yes	characteristic
16	#15688983	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊Dic₄₇	—	2	1	no	—
18	C₄×C₄⋊Dic₄₇	—	2	1	no	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
23	#15688969	—	2	1	no	—
24	C₂×C₄₇⋊#95	—	2	1	no	—
25	#15688886	C₂²	1	4	yes	—
26	#15688886	C₂²	1	4	yes	—
27	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#432	C₄	1	8	yes	complemented
29	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₂×#97×C₄₇	C₂²	1	2	yes	—
32	C₂³×Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
39	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄×Dic₄₇	—	2	2	no	—
41	C₂×C₄×Dic₄₇	—	2	2	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	C₂×C₄×Dic₄₇	—	4	4	no	—
44	#13093958	—	2	8	no	—
45	#13093956	—	2	8	no	—
46	C₄₇⋊#95	—	2	2	no	—
47	C₄₇⋊#95	—	2	2	no	—
48	C₄₇⋊#95	—	2	1	no	—
49	C₄₇⋊#95	—	2	1	no	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	2	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	complemented
64	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
65	#97×C₄₇	C₂×C₄	1	8	yes	complemented
66	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	2	8	no	—
82	C₂²×Dic₄₇	—	2	4	no	—
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₂²×Dic₄₇	—	2	2	no	—
85	C₂²×Dic₄₇	—	4	4	no	—
86	C₂²×Dic₄₇	—	4	4	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	2	4	no	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
90	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
91	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
95	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
96	C₂²⋊Dic₄₇	—	4	2	no	—
97	C₂²⋊Dic₄₇	—	4	2	no	—
98	C₄⋊Dic₄₇	—	2	8	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	2	4	no	—
102	C₄⋊Dic₄₇	#41	1	8	yes	—
103	C₄×Dic₄₇	—	4	4	no	—
104	C₄×Dic₄₇	—	4	4	no	—
105	C₄×Dic₄₇	—	4	4	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
110	C₂³×C₄₇	#137	1	2	yes	—
111	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	—	2	2	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	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
131	C₂×Dic₄₇	—	2	8	no	—
132	C₂×Dic₄₇	—	4	16	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₈×C₄₇	—	2	8	no	—
142	#21363	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	#428	1	4	yes	—
148	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₇	#525	1	2	yes	—
150	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
152	C₂²×C₄₇	#575	1	2	yes	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	C₄×#41	1	2	yes	—
159	C₄×C₄₇	#533	1	2	yes	—
160	Dic₄₇	—	4	8	no	—
161	Dic₄₇	—	8	2	no	—
162	Dic₄₇	—	8	2	no	—
163	C₂×#432	D₄₇	1	1	yes	characteristic
164	#2283	—	47	1	no	—
165	C₂×C₄×C₄⋊C₄	—	47	1	no	—
166	#1949	—	47	4	no	—
167	#1853	—	47	2	no	—
168	#1852	—	47	2	no	—
169	#1776	—	47	4	no	—
170	C₂×C₄₇	—	2	4	no	—
171	C₂×C₄₇	#2296	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#428	1	1	yes	characteristic
173	C₂×C₄₇	#2911	1	1	yes	characteristic
174	C₂×C₄₇	#3114	1	2	yes	—
175	C₂×C₄₇	#3117	1	2	yes	—
176	C₂²×C₄⋊C₄	—	47	2	no	—
177	C₂²×C₄²	—	94	1	no	—
178	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
179	#432	Dic₄₇	1	8	yes	complemented
180	C₂×#107	—	47	4	no	—
181	C₂×#106	—	47	4	no	—
182	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
183	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
184	C₂×#97	C₂×D₄₇	1	2	yes	—
185	#383	—	94	1	no	—
186	#382	—	47	2	no	—
187	#381	—	94	1	no	—
188	#380	—	47	1	no	—
189	#380	—	47	1	no	—
190	C₄×C₄⋊C₄	—	47	2	no	—
191	C₄×C₄⋊C₄	—	47	2	no	—
192	C₄×C₄⋊C₄	—	47	2	no	—
193	C₄×C₄⋊C₄	—	94	1	no	—
194	C₄×C₄⋊C₄	—	94	1	no	—
195	C₄×C₂²⋊C₄	—	94	1	no	—
196	#376	—	94	1	no	—
197	C₂×#95	—	94	1	no	—
198	#340	—	47	4	no	—
199	#340	—	47	4	no	—
200	C₄₇	#21363	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₄	—	94	2	no	—
202	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
203	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
204	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
206	C₂×C₄⋊C₄	—	47	4	no	—
207	C₂×C₄⋊C₄	—	47	2	no	—
208	C₂×C₄⋊C₄	—	47	2	no	—
209	C₂×C₄⋊C₄	—	94	8	no	—
210	C₂×C₄⋊C₄	—	94	2	no	—
211	C₂×C₂²⋊C₄	—	94	1	no	—
212	C₂×C₂²⋊C₄	—	94	1	no	—
213	C₂×C₄²	—	94	2	no	—
214	C₂×C₄²	—	94	2	no	—
215	C₂×C₄²	—	94	2	no	—
216	C₂×C₄²	—	188	4	no	—
217	C₂×C₄²	Dic₉₄	1	4	yes	—
218	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
219	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	#107	—	94	8	no	—
224	#106	—	94	8	no	—
225	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	complemented
226	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
227	#97	C₂×Dic₄₇	1	8	yes	complemented
228	#95	—	94	2	no	—
229	#95	—	94	2	no	—
230	#95	—	94	1	no	—
231	#95	—	94	1	no	—
232	#95	—	94	2	no	—
233	#95	—	94	2	no	—
234	#95	—	94	2	no	—
235	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	94	8	no	—
241	C₂²×C₄	—	94	4	no	—
242	C₂²×C₄	—	94	2	no	—
243	C₂²×C₄	—	94	2	no	—
244	C₂²×C₄	—	188	4	no	—
245	C₂²×C₄	—	188	4	no	—
246	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
247	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
248	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
249	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
255	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
259	C₄⋊C₄	—	47	8	no	—
260	C₄⋊C₄	—	94	8	no	—
261	C₄⋊C₄	—	94	4	no	—
262	C₄⋊C₄	—	94	4	no	—
263	C₄⋊C₄	—	94	4	no	—
264	C₂²⋊C₄	—	188	2	no	—
265	C₂²⋊C₄	—	188	2	no	—
266	C₄²	—	188	4	no	—
267	C₄²	—	188	4	no	—
268	C₄²	—	188	4	no	—
269	C₄²	C₄⋊Dic₄₇	1	4	yes	—
270	C₄²	C₄⋊Dic₄₇	1	4	yes	—
271	C₄²	C₂×Dic₉₄	1	2	yes	—
272	C₄²	C₂×Dic₉₄	1	2	yes	—
273	C₄²	C₂²×Dic₄₇	1	2	yes	—
274	C₄²	C₂²×Dic₄₇	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
278	C₂³	#13094052	1	2	yes	—
279	C₂³	#13094057	1	2	yes	—
280	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	94	8	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	—	188	2	no	—
296	C₂×C₄	—	188	2	no	—
297	C₂×C₄	—	188	2	no	—
298	C₂×C₄	—	188	2	no	—
299	C₂×C₄	—	188	16	no	—
300	C₂×C₄	—	188	4	no	—
301	C₂×C₄	—	188	4	no	—
302	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
304	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
305	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
306	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
307	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
309	C₂×C₄	#41×D₄₇	1	4	yes	—
310	C₈	—	2	8	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	4	no	—
315	C₂²	#15689165	1	4	yes	—
316	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
317	C₂²	#15689635	1	2	yes	—
318	C₂²	C₂×#13094052	1	1	yes	characteristic
319	C₂²	C₂×#13094057	1	1	yes	characteristic
320	C₂²	#15689923	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	376	2	no	—
327	C₄	—	376	2	no	—
328	C₄	—	188	8	no	—
329	C₄	#41×Dic₄₇	1	2	yes	—
330	C₄	#15689676	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	#20245872	1	1	yes	characteristic
333	C₂	C₂×#15689165	1	1	yes	characteristic
334	C₂	#20251978	1	1	yes	characteristic
335	C₂	#20253528	1	2	yes	—
336	C₂	#20253561	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#21363
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20244158	C₂	semidirect
6. #20243285	C₂	semidirect
28. C₄₇×#432	C₄	semidirect
63. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
65. #97×C₄₇	C₂×C₄	semidirect
179. #432	Dic₄₇	semidirect
200. C₄₇	#21363	semidirect
225. C₄⋊C₈	C₂×Dic₄₇	semidirect
227. #97	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20245786	2
C₂×C₄×C₄⋊Dic₄₇	2
#20244158	2
#20243583	2
#20243578	2
#20243285	2
C₂×C₄₇×#432	2
#21363	47