Subgroups — #26234413

Order 12032.

982 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	C₂×C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689158	C₂	1	2	yes	maximal, complemented
3	#20246339	C₂	1	2	yes	maximal
4	#20244144	C₂	1	2	yes	maximal, complemented
5	#20243586	C₂	1	4	yes	maximal, complemented
6	#20243410	C₂	1	2	yes	maximal, complemented
7	C₄×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	C₂×#13093982	C₂²	1	4	yes	—
12	C₈×D₁₈₈	C₄	1	8	yes	complemented
13	#15689158	—	2	8	no	—
14	C₂×#13093957	C₂²	1	4	yes	—
15	C₂×#13093956	C₂²	1	1	yes	characteristic
16	C₂×#13093956	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
19	#15689024	—	2	2	no	—
20	#15689013	—	2	4	no	—
21	#15688980	C₂²	1	4	yes	—
22	C₄⋊C₄×Dic₄₇	—	2	2	no	—
23	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
24	#15688886	C₂²	1	2	yes	—
25	#15688885	C₂²	1	4	yes	—
26	C₂×#106×C₄₇	C₂²	1	2	yes	—
27	C₂×#106×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#384	C₂²	1	2	yes	—
30	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₇	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂×C₈×D₄₇	—	2	4	no	—
34	C₂×C₈×D₄₇	—	2	4	no	—
35	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
36	C₄×D₁₈₈	C₂×C₄	1	4	yes	—
37	C₄×D₁₈₈	Q₈	1	4	yes	—
38	#13093982	—	2	8	no	—
39	#13093982	—	4	4	no	—
40	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
44	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
45	C₂×(C₂×D₄₇)⋊C₄	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₇	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	4	no	—
48	#13093957	—	2	8	no	—
49	#13093956	—	2	2	no	—
50	#13093956	D₄	1	4	yes	complemented
51	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
52	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
53	C₄₇⋊#95	—	2	2	no	—
54	C₄₇⋊#95	—	2	4	no	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#106×C₄₇	—	2	4	no	—
62	#106×C₄₇	—	4	2	no	—
63	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
64	C₂³×D₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
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	2	no	—
71	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
72	C₂×D₁₈₈	—	2	4	no	—
73	C₂×D₁₈₈	C₄⋊C₄	1	4	yes	—
74	C₂×C₄×D₄₇	—	2	2	no	—
75	C₂×C₄×D₄₇	—	2	2	no	—
76	C₂×C₄×D₄₇	—	2	2	no	—
77	C₂×C₄×D₄₇	—	2	2	no	—
78	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₄₇	C₂×Q₈	1	4	yes	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	—	2	1	no	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	#41	1	2	yes	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	—	2	4	no	—
88	C₄⋊C₄×C₄₇	—	4	4	no	—
89	C₄⋊C₄×C₄₇	—	8	2	no	—
90	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
92	(C₂×D₄₇)⋊C₄	—	4	8	no	—
93	(C₂×D₄₇)⋊C₄	—	8	4	no	—
94	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	8	yes	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
99	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
100	C₄×Dic₄₇	—	4	8	no	—
101	C₄×Dic₄₇	—	4	4	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
105	C₈×D₄₇	—	2	16	no	—
106	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂²×D₄₇	—	2	2	no	—
108	C₂²×D₄₇	—	2	2	no	—
109	C₂²×D₄₇	—	2	4	no	—
110	C₂²×D₄₇	—	2	4	no	—
111	C₂²×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	#135	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
134	D₁₈₈	—	2	8	no	—
135	C₄×D₄₇	—	2	4	no	—
136	C₄×D₄₇	—	2	4	no	—
137	C₄×D₄₇	—	2	4	no	—
138	C₄×D₄₇	—	2	4	no	—
139	C₈×C₄₇	—	2	2	no	—
140	C₈×C₄₇	C₄×D₄	1	4	yes	—
141	C₄₇⋊C₈	—	2	4	no	—
142	#22657	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₄₇	D₄²	1	1	yes	characteristic
145	C₂²×C₄₇	#547	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#135	1	2	yes	—
147	C₂²×C₄₇	#577	1	2	yes	—
148	C₂×D₄₇	—	2	4	no	—
149	C₂×D₄₇	—	2	4	no	—
150	C₂×D₄₇	—	2	4	no	—
151	C₂×D₄₇	—	4	4	no	—
152	C₂×D₄₇	—	4	4	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	8	4	no	—
155	C₄×C₄₇	#427	1	4	yes	complemented
156	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₄₇	#520	1	1	yes	characteristic
158	C₄×C₄₇	D₄×Q₈	1	1	yes	characteristic
159	C₄×C₄₇	#554	1	1	yes	characteristic
160	Dic₄₇	—	2	4	no	—
161	Dic₄₇	—	4	4	no	—
162	Dic₄₇	—	8	2	no	—
163	C₂×C₈×D₄	—	47	1	no	—
164	C₂×#427	—	47	2	no	—
165	#2354	—	47	2	no	—
166	#1947	—	47	2	no	—
167	#1854	—	47	4	no	—
168	#1844	D₄₇	1	1	yes	characteristic, complemented
169	#1807	—	47	2	no	—
170	C₄×#106	—	47	1	no	—
171	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
173	C₂×C₄₇	#2912	1	1	yes	characteristic
174	C₂×C₄₇	#3284	1	2	yes	—
175	C₂×C₄₇	#3340	1	2	yes	—
176	D₄₇	—	4	4	no	—
177	C₂³×C₈	—	47	2	no	—
178	C₂×C₄×D₄	—	47	1	no	—
179	C₂×#117	—	47	4	no	—
180	C₈×D₄	—	47	8	no	—
181	#427	—	94	8	no	—
182	C₂×#107	—	47	4	no	—
183	C₂×#106	—	47	1	no	—
184	C₂×#106	—	47	1	no	—
185	C₂×#106	C₂×D₄₇	1	2	yes	—
186	C₂×#106	C₂×D₄₇	1	2	yes	—
187	C₂×C₄⋊C₈	—	47	1	no	—
188	C₂×C₂²⋊C₈	—	47	2	no	—
189	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
190	#390	—	94	2	no	—
191	#388	—	94	4	no	—
192	#384	C₂×D₄₇	1	2	yes	—
193	#380	—	47	4	no	—
194	C₄×C₄⋊C₄	—	47	2	no	—
195	C₄×C₄⋊C₄	—	94	2	no	—
196	#340	—	47	2	no	—
197	#340	—	47	4	no	—
198	C₄₇	#22657	1	1	yes	characteristic, complemented, Sylow
199	C₂²×D₄	—	47	1	no	—
200	C₂³×C₄	—	47	2	no	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	—	47	8	no	—
203	C₂²×C₈	—	94	4	no	—
204	C₂²×C₈	—	94	4	no	—
205	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₄×D₄	—	47	4	no	—
208	C₄×D₄	—	47	4	no	—
209	#117	—	94	8	no	—
210	#117	—	188	4	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	47	1	no	—
214	C₂×C₄⋊C₄	—	47	2	no	—
215	C₂×C₄⋊C₄	—	47	2	no	—
216	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
217	C₂×C₂²⋊C₄	—	47	2	no	—
218	C₂×C₂²⋊C₄	—	94	4	no	—
219	C₂×C₂²⋊C₄	—	94	4	no	—
220	C₂×C₄²	—	94	4	no	—
221	C₂×C₄²	—	94	2	no	—
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	#107	—	94	8	no	—
224	#106	—	2	4	no	—
225	#106	—	4	2	no	—
226	#106	—	47	4	no	—
227	#106	—	94	2	no	—
228	C₄⋊C₈	—	47	4	no	—
229	C₂²⋊C₈	—	47	8	no	—
230	C₄×C₈	C₄×D₄₇	1	4	yes	—
231	#95	—	94	4	no	—
232	#95	—	94	2	no	—
233	C₂⁴	—	47	2	no	—
234	C₂×D₄	—	47	4	no	—
235	C₂×D₄	—	94	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	47	4	no	—
239	C₂²×C₄	—	47	4	no	—
240	C₂²×C₄	—	94	2	no	—
241	C₂²×C₄	—	94	2	no	—
242	C₂²×C₄	—	94	2	no	—
243	C₂²×C₄	—	94	2	no	—
244	C₂²×C₄	—	94	2	no	—
245	C₂²×C₄	—	94	2	no	—
246	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	47	4	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	—	94	16	no	—
255	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
257	C₂×C₈	D₄×D₄₇	1	2	yes	—
258	C₂×C₈	#10625354	1	2	yes	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	8	2	no	—
263	C₄⋊C₄	—	47	4	no	—
264	C₄⋊C₄	—	47	4	no	—
265	C₄⋊C₄	—	94	2	no	—
266	C₄⋊C₄	—	94	2	no	—
267	C₄⋊C₄	—	94	2	no	—
268	C₂²⋊C₄	—	47	8	no	—
269	C₂²⋊C₄	—	376	4	no	—
270	C₂²⋊C₄	—	188	8	no	—
271	C₄²	—	188	8	no	—
272	C₄²	—	188	4	no	—
273	C₄²	C₂×C₄×D₄₇	1	2	yes	—
274	C₄²	Q₈×D₄₇	1	2	yes	—
275	C₂³	—	47	4	no	—
276	C₂³	—	94	2	no	—
277	C₂³	—	94	2	no	—
278	C₂³	—	94	4	no	—
279	C₂³	—	94	4	no	—
280	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
281	D₄	—	94	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	47	4	no	—
287	C₂×C₄	—	94	2	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	94	4	no	—
290	C₂×C₄	—	94	4	no	—
291	C₂×C₄	—	94	4	no	—
292	C₂×C₄	—	94	4	no	—
293	C₂×C₄	—	94	4	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	C₄⋊C₄×D₄₇	1	2	yes	—
300	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
301	C₂×C₄	#13094045	1	4	yes	—
302	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
306	C₂×C₄	#13094069	1	1	yes	characteristic
307	C₂×C₄	#13094072	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	94	4	no	—
310	C₈	#13093986	1	4	yes	—
311	C₂²	—	94	4	no	—
312	C₂²	—	94	4	no	—
313	C₂²	—	94	4	no	—
314	C₂²	—	188	4	no	—
315	C₂²	—	188	4	no	—
316	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
317	C₂²	#15689758	1	1	yes	characteristic
318	C₂²	#15689776	1	1	yes	characteristic
319	C₂²	C₂×#13094045	1	2	yes	—
320	C₂²	#15689929	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	8	4	no	—
323	C₄	—	94	4	no	—
324	C₄	—	376	2	no	—
325	C₄	—	188	4	no	—
326	C₄	#15689158	1	4	yes	complemented
327	C₄	C₂×#13093986	1	1	yes	characteristic
328	C₄	#15689604	1	1	yes	characteristic
329	C₄	#15689788	1	1	yes	characteristic
330	C₄	#15689834	1	1	yes	characteristic
331	C₂	—	188	4	no	—
332	C₂	#20246288	1	1	yes	characteristic
333	C₂	C₂×#15689158	1	1	yes	characteristic
334	C₂	#20251979	1	1	yes	characteristic
335	C₂	#20255117	1	2	yes	—
336	C₂	#20255800	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₁₈₈	D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#22657
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
2. C₂×#15689158	C₂	semidirect
4. #20244144	C₂	semidirect
5. #20243586	C₂	semidirect
6. #20243410	C₂	semidirect
7. C₄×#13093956	C₂	semidirect
8. C₄₇×#1844	C₂	semidirect
12. C₈×D₁₈₈	C₄	semidirect
50. #13093956	D₄	semidirect
155. C₄×C₄₇	#427	semidirect
168. #1844	D₄₇	semidirect
198. C₄₇	#22657	semidirect
326. C₄	#15689158	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₁₈₈	2
C₂×#15689158	2
#20246339	2
#20244144	2
#20243586	2
#20243410	2
C₄×#13093956	2
C₄₇×#1844	2
#22657	47