Subgroups — #26221141

Order 12032.

958 conjugacy classes of subgroups.

The subgroup lattice has 350 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	C₄×#13093958	C₂	1	2	yes	maximal, complemented
7	C₄×#13093956	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₄₇	—	2	1	no	—
11	C₂×#13093958	C₂²	1	4	yes	—
12	C₂×#13093956	C₂²	1	4	yes	—
13	#15688991	—	2	1	no	—
14	#15688989	C₂²	1	2	yes	—
15	#15688986	—	2	1	no	—
16	#15688983	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₄₇	—	2	1	no	—
18	C₄×C₄⋊Dic₄₇	—	2	1	no	—
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	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
24	#15688969	—	2	1	no	—
25	C₂×C₄₇⋊#95	—	2	1	no	—
26	#15688886	C₂²	1	4	yes	—
27	#15688886	C₂²	1	4	yes	—
28	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#431	C₄	1	8	yes	complemented
30	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
34	C₂³×Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	—	2	8	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
41	C₂×C₄×Dic₄₇	—	2	1	no	—
42	C₂×C₄×Dic₄₇	—	2	1	no	—
43	C₂×C₄×Dic₄₇	—	2	2	no	—
44	C₂×C₄×Dic₄₇	—	2	2	no	—
45	C₂×C₄×Dic₄₇	—	4	4	no	—
46	#13093958	—	2	8	no	—
47	#13093956	—	2	8	no	—
48	C₄₇⋊#95	—	2	2	no	—
49	C₄₇⋊#95	—	2	2	no	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	2	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₄₇⋊#95	—	2	1	no	—
54	C₄₇⋊#95	—	2	1	no	—
55	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
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	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	Q₈	1	4	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	4	yes	—
68	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
70	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	2	4	no	—
75	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	2	8	no	—
86	C₂²×Dic₄₇	—	2	4	no	—
87	C₂²×Dic₄₇	—	2	2	no	—
88	C₂²×Dic₄₇	—	2	2	no	—
89	C₂²×Dic₄₇	—	4	4	no	—
90	C₂²×Dic₄₇	—	4	4	no	—
91	C₂×C₈×C₄₇	—	2	4	no	—
92	C₂×C₈×C₄₇	—	2	4	no	—
93	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
95	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
96	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
97	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
98	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
100	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
101	C₂²⋊Dic₄₇	—	4	4	no	—
102	C₄⋊Dic₄₇	—	2	8	no	—
103	C₄⋊Dic₄₇	—	2	4	no	—
104	C₄⋊Dic₄₇	—	2	4	no	—
105	C₄⋊Dic₄₇	—	2	4	no	—
106	C₄⋊Dic₄₇	#41	1	8	yes	—
107	C₄×Dic₄₇	—	4	4	no	—
108	C₄×Dic₄₇	—	4	2	no	—
109	C₄×Dic₄₇	—	4	4	no	—
110	C₄×Dic₄₇	—	4	2	no	—
111	C₂³×C₄₇	—	2	2	no	—
112	C₂³×C₄₇	—	2	1	no	—
113	C₂³×C₄₇	—	2	1	no	—
114	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂³×C₄₇	#135	1	4	yes	—
116	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	8	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	2	8	no	—
126	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₄₇	#137	1	2	yes	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
136	C₂×Dic₄₇	—	2	8	no	—
137	C₂×Dic₄₇	—	4	16	no	—
138	C₂×Dic₄₇	—	4	4	no	—
139	C₂×Dic₄₇	—	4	4	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₈×C₄₇	—	2	4	no	—
147	C₈×C₄₇	—	2	4	no	—
148	#21361	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	2	4	no	—
150	C₂²×C₄₇	—	2	4	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	#427	1	4	yes	—
154	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₄₇	#525	1	2	yes	—
156	C₂²×C₄₇	C₂×#135	1	2	yes	—
157	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	#428	1	4	yes	—
164	C₄×C₄₇	C₄×#41	1	2	yes	—
165	C₄×C₄₇	#533	1	2	yes	—
166	Dic₄₇	—	4	8	no	—
167	Dic₄₇	—	8	2	no	—
168	Dic₄₇	—	8	2	no	—
169	C₂×#431	D₄₇	1	1	yes	characteristic
170	#2283	—	47	1	no	—
171	C₂×C₄×C₄⋊C₄	—	47	1	no	—
172	#1949	—	47	4	no	—
173	#1853	—	47	2	no	—
174	#1852	—	47	2	no	—
175	C₄×#107	—	47	2	no	—
176	C₄×#106	—	47	2	no	—
177	C₂×C₄₇	—	2	4	no	—
178	C₂×C₄₇	#2296	1	1	yes	characteristic
179	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
180	C₂×C₄₇	C₂×#428	1	1	yes	characteristic
181	C₂×C₄₇	#3105	1	2	yes	—
182	C₂×C₄₇	#3108	1	2	yes	—
183	C₂²×C₄⋊C₄	—	47	2	no	—
184	C₂²×C₄²	—	94	1	no	—
185	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
186	#431	Dic₄₇	1	8	yes	complemented
187	C₂×#107	—	47	4	no	—
188	C₂×#106	—	47	4	no	—
189	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
190	C₂×C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
191	C₂×C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
193	#383	—	94	1	no	—
194	#382	—	47	2	no	—
195	#381	—	94	1	no	—
196	#380	—	47	2	no	—
197	C₄×C₄⋊C₄	—	47	2	no	—
198	C₄×C₄⋊C₄	—	47	2	no	—
199	C₄×C₄⋊C₄	—	47	1	no	—
200	C₄×C₄⋊C₄	—	47	1	no	—
201	C₄×C₄⋊C₄	—	94	1	no	—
202	C₄×C₄⋊C₄	—	94	1	no	—
203	C₄×C₂²⋊C₄	—	94	1	no	—
204	#376	—	94	1	no	—
205	C₂×#95	—	94	1	no	—
206	#340	—	47	4	no	—
207	#340	—	47	4	no	—
208	C₄₇	#21361	1	1	yes	characteristic, complemented, Sylow
209	C₂³×C₄	—	94	2	no	—
210	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
211	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
212	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
214	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
215	C₂×C₄⋊C₄	—	47	4	no	—
216	C₂×C₄⋊C₄	—	47	2	no	—
217	C₂×C₄⋊C₄	—	47	2	no	—
218	C₂×C₄⋊C₄	—	94	8	no	—
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₂²⋊C₄	—	94	2	no	—
221	C₂×C₄²	—	94	2	no	—
222	C₂×C₄²	—	94	1	no	—
223	C₂×C₄²	—	94	1	no	—
224	C₂×C₄²	—	94	2	no	—
225	C₂×C₄²	—	188	4	no	—
226	C₂×C₄²	Dic₉₄	1	4	yes	—
227	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
228	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
230	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	#107	—	94	8	no	—
233	#106	—	94	8	no	—
234	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	complemented
235	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
236	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
237	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
238	#95	—	94	2	no	—
239	#95	—	94	2	no	—
240	#95	—	94	2	no	—
241	#95	—	94	2	no	—
242	#95	—	94	2	no	—
243	#95	—	94	1	no	—
244	#95	—	94	1	no	—
245	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	94	8	no	—
251	C₂²×C₄	—	94	4	no	—
252	C₂²×C₄	—	94	2	no	—
253	C₂²×C₄	—	94	2	no	—
254	C₂²×C₄	—	188	4	no	—
255	C₂²×C₄	—	188	4	no	—
256	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
257	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
258	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
259	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
265	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
269	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
270	C₄⋊C₄	—	47	8	no	—
271	C₄⋊C₄	—	94	8	no	—
272	C₄⋊C₄	—	94	4	no	—
273	C₄⋊C₄	—	94	4	no	—
274	C₄⋊C₄	—	94	4	no	—
275	C₂²⋊C₄	—	188	4	no	—
276	C₄²	—	188	4	no	—
277	C₄²	—	188	2	no	—
278	C₄²	—	188	4	no	—
279	C₄²	—	188	2	no	—
280	C₄²	C₄⋊Dic₄₇	1	4	yes	—
281	C₄²	C₄⋊Dic₄₇	1	4	yes	—
282	C₄²	C₂×Dic₉₄	1	2	yes	—
283	C₄²	C₂×Dic₉₄	1	2	yes	—
284	C₄²	C₂²×Dic₄₇	1	2	yes	—
285	C₄²	C₂²×Dic₄₇	1	2	yes	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
290	C₂³	#13094046	1	4	yes	—
291	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	94	8	no	—
302	C₂×C₄	—	188	2	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	—	188	2	no	—
306	C₂×C₄	—	188	2	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	—	188	16	no	—
309	C₂×C₄	—	188	4	no	—
310	C₂×C₄	—	188	4	no	—
311	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
314	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
315	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
316	C₂×C₄	#13094052	1	2	yes	—
317	C₂×C₄	#13094057	1	2	yes	—
318	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
319	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
320	C₂×C₄	#41×D₄₇	1	4	yes	—
321	C₈	—	2	4	no	—
322	C₈	—	2	4	no	—
323	C₂²	—	2	4	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	4	no	—
327	C₂²	#15689161	1	4	yes	—
328	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
329	C₂²	#15689635	1	2	yes	—
330	C₂²	C₂×#13094046	1	2	yes	—
331	C₂²	C₂×#13094052	1	1	yes	characteristic
332	C₂²	C₂×#13094057	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	376	2	no	—
338	C₄	—	376	2	no	—
339	C₄	—	188	8	no	—
340	C₄	#15689165	1	4	yes	—
341	C₄	#41×Dic₄₇	1	2	yes	—
342	C₄	#15689676	1	2	yes	—
343	C₂	—	2	4	no	—
344	C₂	#20245872	1	1	yes	characteristic
345	C₂	C₂×#15689161	1	1	yes	characteristic
346	C₂	C₂×#15689165	1	1	yes	characteristic
347	C₂	#20253437	1	2	yes	—
348	C₂	#20253470	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#21361
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20244158	C₂	semidirect
6. C₄×#13093958	C₂	semidirect
7. C₄×#13093956	C₂	semidirect
29. C₄₇×#431	C₄	semidirect
66. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
69. C₄×C₈×C₄₇	C₂×C₄	semidirect
186. #431	Dic₄₇	semidirect
208. C₄₇	#21361	semidirect
234. C₄⋊C₈	C₂×Dic₄₇	semidirect
237. 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
#20245786	2
C₂×C₄×C₄⋊Dic₄₇	2
#20244158	2
#20243583	2
#20243578	2
C₄×#13093958	2
C₄×#13093956	2
C₂×C₄₇×#431	2
#21361	47