Subgroups — #26234559

Order 12032.

1122 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689236	C₂	1	2	yes	maximal
2	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×D₁₈₈	C₂	1	2	yes	maximal, complemented
4	#20244126	C₂	1	2	yes	maximal, complemented
5	#20243652	C₂	1	4	yes	maximal, complemented
6	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243389	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	C₂²	1	4	yes	—
11	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×D₄₇	—	2	2	no	—
13	#15689236	—	2	8	no	—
14	C₄×D₃₇₆	C₄	1	8	yes	complemented
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093956	C₂²	1	2	yes	—
17	C₂×#13093956	C₂²	1	2	yes	—
18	C₂×#13093946	C₂²	1	4	yes	—
19	C₂×#13093946	C₂²	1	2	yes	—
20	#15689021	C₂²	1	4	yes	—
21	#15688997	—	2	4	no	—
22	#15688993	C₂²	1	2	yes	—
23	#15688980	—	2	2	no	—
24	#15688886	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#336	C₂²	1	2	yes	—
30	C₂²×D₁₈₈	C₂³	1	2	yes	—
31	C₂²×C₄×D₄₇	—	2	2	no	—
32	C₂²×C₄×D₄₇	—	2	4	no	—
33	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
34	C₂×D₃₇₆	D₄	1	4	yes	—
35	C₂×D₃₇₆	Q₈	1	4	yes	—
36	C₄×D₁₈₈	—	2	4	no	—
37	C₄×D₁₈₈	—	2	8	no	—
38	C₄×D₁₈₈	—	4	4	no	—
39	C₄×D₁₈₈	C₂×C₄	1	8	yes	complemented
40	C₂×C₄⋊Dic₄₇	—	2	2	no	—
41	C₂×C₄⋊Dic₄₇	—	2	2	no	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
43	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
44	C₂×Dic₄₇⋊C₄	—	2	4	no	—
45	C₄⋊C₄×D₄₇	—	4	8	no	—
46	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
47	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
48	#13093958	C₂×C₄	1	4	yes	complemented
49	#13093956	—	2	4	no	—
50	#13093956	—	4	2	no	—
51	#13093946	—	2	8	no	—
52	#13093946	C₂×C₄	1	8	yes	complemented
53	C₄₇⋊#95	—	2	4	no	—
54	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	—	2	2	no	—
61	#97×C₄₇	—	2	4	no	—
62	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
63	#95×C₄₇	D₄	1	2	yes	—
64	C₂³×D₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂×D₁₈₈	—	2	2	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	—	2	2	no	—
74	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
75	C₂×D₁₈₈	C₂×D₄	1	4	yes	—
76	C₂×D₁₈₈	C₂×Q₈	1	4	yes	—
77	C₂×C₄×D₄₇	—	2	8	no	—
78	C₂×C₄×D₄₇	—	4	4	no	—
79	C₂×C₄×D₄₇	—	4	4	no	—
80	C₂×C₄×D₄₇	—	4	8	no	—
81	C₂×C₄×D₄₇	—	4	8	no	—
82	C₂×C₄×D₄₇	—	8	4	no	—
83	C₂×C₈×C₄₇	—	2	1	no	—
84	C₂×C₈×C₄₇	—	2	1	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	2	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
92	C₄⋊C₄×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₇	#41	1	2	yes	—
97	(C₂×D₄₇)⋊C₄	—	2	8	no	—
98	(C₂×D₄₇)⋊C₄	—	4	8	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	2	4	no	—
102	C₄⋊Dic₄₇	—	4	4	no	—
103	C₄⋊Dic₄₇	—	8	2	no	—
104	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
105	Dic₄₇⋊C₄	—	4	8	no	—
106	D₃₇₆	C₄⋊C₄	1	16	yes	complemented
107	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂²×D₄₇	—	2	4	no	—
109	C₂²×D₄₇	—	2	4	no	—
110	C₂²×D₄₇	—	2	4	no	—
111	C₂²×D₄₇	—	4	2	no	—
112	C₂²×D₄₇	—	4	2	no	—
113	C₂²×D₄₇	—	4	2	no	—
114	C₂²×D₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	D₁₈₈	—	2	4	no	—
135	D₁₈₈	—	2	4	no	—
136	D₁₈₈	—	2	4	no	—
137	D₁₈₈	C₂×C₄⋊C₄	1	8	yes	complemented
138	C₄×D₄₇	—	4	16	no	—
139	C₄×D₄₇	—	8	16	no	—
140	C₈×C₄₇	—	2	2	no	—
141	C₈×C₄₇	—	4	2	no	—
142	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	complemented
143	#22670	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂²×C₄₇	D₄²	1	1	yes	characteristic
146	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
148	C₂²×C₄₇	#577	1	2	yes	—
149	C₂×D₄₇	—	2	8	no	—
150	C₂×D₄₇	—	4	4	no	—
151	C₂×D₄₇	—	4	4	no	—
152	C₂×D₄₇	—	4	4	no	—
153	C₂×D₄₇	—	4	4	no	—
154	C₂×D₄₇	—	4	4	no	—
155	C₂×D₄₇	—	4	4	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	4	4	no	—
158	C₄×C₄₇	#442	1	4	yes	complemented
159	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₄₇	#520	1	1	yes	characteristic
161	C₄×C₄₇	#547	1	1	yes	characteristic
162	C₄×C₄₇	#554	1	1	yes	characteristic
163	Dic₄₇	—	4	4	no	—
164	Dic₄₇	—	8	4	no	—
165	C₂×#442	—	47	2	no	—
166	C₂×C₄×D₈	—	47	1	no	—
167	D₄×C₄⋊C₄	—	47	2	no	—
168	#1943	—	47	2	no	—
169	#1864	—	47	4	no	—
170	#1844	—	47	1	no	—
171	#1802	—	47	2	no	—
172	#1770	D₄₇	1	1	yes	characteristic, complemented
173	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
175	C₂×C₄₇	#2952	1	1	yes	characteristic
176	C₂×C₄₇	#3272	1	2	yes	—
177	C₂×C₄₇	#3384	1	2	yes	—
178	D₄₇	—	4	8	no	—
179	C₂²×D₈	—	47	1	no	—
180	C₂×C₄×D₄	—	47	2	no	—
181	C₂×C₄×D₄	—	47	4	no	—
182	C₂²×C₄⋊C₄	—	94	2	no	—
183	#442	—	94	8	no	—
184	C₄×D₈	—	47	8	no	—
185	C₂×#107	—	47	1	no	—
186	C₂×#106	—	47	2	no	—
187	C₂×#106	—	47	2	no	—
188	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
189	C₂×#102	—	47	2	no	—
190	C₂×#102	—	47	4	no	—
191	C₂×#97	C₂×D₄₇	1	2	yes	—
192	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
193	#389	—	47	4	no	—
194	#385	—	94	4	no	—
195	#384	—	47	2	no	—
196	#380	—	94	2	no	—
197	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
198	#340	—	47	4	no	—
199	#336	C₂×D₄₇	1	2	yes	—
200	C₄₇	#22670	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₄	—	47	2	no	—
202	C₂³×C₄	—	94	2	no	—
203	C₂³×C₄	—	94	4	no	—
204	C₂×D₈	—	47	4	no	—
205	C₂×D₈	—	47	4	no	—
206	C₂×D₈	—	47	4	no	—
207	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
208	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
210	C₄×D₄	—	47	8	no	—
211	C₄×D₄	—	94	8	no	—
212	C₄×D₄	—	94	4	no	—
213	C₄×D₄	—	188	4	no	—
214	C₂×C₄⋊C₄	—	47	2	no	—
215	C₂×C₄⋊C₄	—	47	4	no	—
216	C₂×C₄⋊C₄	—	94	2	no	—
217	C₂×C₄⋊C₄	—	94	4	no	—
218	C₂×C₄⋊C₄	—	94	2	no	—
219	C₂×C₄⋊C₄	—	188	8	no	—
220	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
221	C₂×C₂²⋊C₄	—	94	2	no	—
222	C₂×C₂²⋊C₄	—	94	4	no	—
223	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
224	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
225	#107	—	47	4	no	—
226	#106	—	94	4	no	—
227	#106	—	188	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	#102	—	47	8	no	—
230	#102	—	94	8	no	—
231	#97	—	2	4	no	—
232	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
233	#95	—	94	4	no	—
234	#95	D₁₈₈	1	2	yes	—
235	C₂⁴	—	94	2	no	—
236	C₂×D₄	—	47	4	no	—
237	C₂×D₄	—	47	4	no	—
238	C₂×D₄	—	47	4	no	—
239	C₂×D₄	—	94	2	no	—
240	C₂×D₄	—	94	2	no	—
241	C₂×D₄	—	94	2	no	—
242	C₂²×C₄	—	94	2	no	—
243	C₂²×C₄	—	94	8	no	—
244	C₂²×C₄	—	94	4	no	—
245	C₂²×C₄	—	376	4	no	—
246	C₂²×C₄	—	188	8	no	—
247	C₂²×C₄	—	188	8	no	—
248	C₂²×C₄	—	188	4	no	—
249	C₂²×C₄	—	188	4	no	—
250	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
251	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
254	D₈	—	47	16	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
261	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
262	C₂×C₈	D₄×D₄₇	1	2	yes	—
263	C₂×C₈	Q₈×D₄₇	1	2	yes	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	47	4	no	—
266	C₄⋊C₄	—	94	2	no	—
267	C₄⋊C₄	—	94	4	no	—
268	C₄⋊C₄	—	94	4	no	—
269	C₄⋊C₄	—	376	2	no	—
270	C₄⋊C₄	—	188	8	no	—
271	C₄⋊C₄	—	188	4	no	—
272	C₂²⋊C₄	—	94	8	no	—
273	C₂²⋊C₄	—	188	8	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	C₂×C₄×D₄₇	1	2	yes	—
277	C₄²	#10625350	1	2	yes	—
278	C₂³	—	94	4	no	—
279	C₂³	—	94	4	no	—
280	C₂³	—	94	4	no	—
281	C₂³	—	188	2	no	—
282	C₂³	—	188	2	no	—
283	C₂³	—	188	2	no	—
284	C₂³	—	188	2	no	—
285	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
286	D₄	—	47	8	no	—
287	D₄	—	94	4	no	—
288	D₄	—	94	4	no	—
289	D₄	—	94	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	94	4	no	—
295	C₂×C₄	—	376	16	no	—
296	C₂×C₄	—	188	4	no	—
297	C₂×C₄	—	188	4	no	—
298	C₂×C₄	—	188	4	no	—
299	C₂×C₄	—	188	2	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	188	16	no	—
302	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
303	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
304	C₂×C₄	#13094052	1	4	yes	—
305	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
309	C₂×C₄	#13094070	1	1	yes	characteristic
310	C₂×C₄	#13094071	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	4	2	no	—
313	C₈	C₄⋊C₄×D₄₇	1	4	yes	complemented
314	C₂²	—	94	8	no	—
315	C₂²	—	188	4	no	—
316	C₂²	—	188	4	no	—
317	C₂²	—	188	4	no	—
318	C₂²	—	188	4	no	—
319	C₂²	—	188	4	no	—
320	C₂²	—	188	4	no	—
321	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
322	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
323	C₂²	Q₈×D₁₈₈	1	1	yes	characteristic
324	C₂²	C₂×#13094052	1	2	yes	—
325	C₂²	#15689935	1	2	yes	—
326	C₄	—	2	2	no	—
327	C₄	—	4	4	no	—
328	C₄	—	376	4	no	—
329	C₄	—	188	4	no	—
330	C₄	#15689236	1	4	yes	complemented
331	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
332	C₄	#15689603	1	1	yes	characteristic
333	C₄	#15689778	1	1	yes	characteristic
334	C₄	#15689833	1	1	yes	characteristic
335	C₂	—	188	8	no	—
336	C₂	C₄⋊C₄×D₁₈₈	1	1	yes	characteristic
337	C₂	C₂×#15689236	1	1	yes	characteristic
338	C₂	#20252273	1	1	yes	characteristic
339	C₂	#20254990	1	2	yes	—
340	C₂	#20256289	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1770	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1770	D₄₇
hypocenter	C₄₇	#22670
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×D₁₈₈	C₂	semidirect
4. #20244126	C₂	semidirect
5. #20243652	C₂	semidirect
6. #20243545	C₂	semidirect
7. #20243389	C₂	semidirect
8. C₄₇×#1770	C₂	semidirect
14. C₄×D₃₇₆	C₄	semidirect
39. C₄×D₁₈₈	C₂×C₄	semidirect
48. #13093958	C₂×C₄	semidirect
52. #13093946	C₂×C₄	semidirect
62. C₄×C₈×C₄₇	C₂×C₄	semidirect
106. D₃₇₆	C₄⋊C₄	semidirect
137. D₁₈₈	C₂×C₄⋊C₄	semidirect
142. C₈×C₄₇	C₂×C₄⋊C₄	semidirect
158. C₄×C₄₇	#442	semidirect
172. #1770	D₄₇	semidirect
200. C₄₇	#22670	semidirect
232. C₄×C₈	C₄×D₄₇	semidirect
313. C₈	C₄⋊C₄×D₄₇	semidirect
330. C₄	#15689236	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689236	2
C₂×C₄×D₃₇₆	2
C₄⋊C₄×D₁₈₈	2
#20244126	2
#20243652	2
#20243545	2
#20243389	2
C₄₇×#1770	2
#22670	47