Subgroups — #26107072

Order 12032.

570 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×C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093944	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688850	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688844	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241229	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241222	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	—	2	1	no	—
9	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
10	C₈×D₁₈₈	—	2	2	no	—
11	C₈×D₁₈₈	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#13093944	C₄	1	2	yes	—
14	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15688885	C₄	1	2	yes	—
18	#15688850	C₄	1	4	yes	complemented
19	#15688844	—	2	2	no	—
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₄₇	—	2	1	no	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#336	—	2	1	no	—
25	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₄₇	—	2	1	no	—
27	C₂²×C₄₇⋊C₈	—	2	1	no	—
28	C₂²×C₄₇⋊C₈	—	2	1	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₄₇	—	2	4	no	—
31	C₂×C₈×D₄₇	—	4	2	no	—
32	C₂×C₈×D₄₇	—	4	2	no	—
33	C₄×D₁₈₈	—	2	1	no	—
34	C₄×D₁₈₈	—	2	1	no	—
35	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
36	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
40	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
41	#13093944	—	2	4	no	—
42	(C₂×D₄₇)⋊C₈	—	2	4	no	—
43	C₄×C₄₇⋊C₈	—	2	1	no	—
44	C₄×C₄₇⋊C₈	—	2	1	no	—
45	C₄×C₄₇⋊C₈	—	2	1	no	—
46	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	—	2	1	no	—
48	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	—	2	1	no	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₇	—	2	1	no	—
54	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
55	#97×C₄₇	—	4	2	no	—
56	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₄₇	D₄	1	2	yes	—
58	#95×C₄₇	—	2	1	no	—
59	C₂³×D₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	1	no	—
65	C₂×D₁₈₈	—	2	1	no	—
66	C₂×D₁₈₈	—	2	2	no	—
67	C₂×D₁₈₈	C₄²	1	2	yes	—
68	C₂×D₁₈₈	M₁₆	1	4	yes	—
69	C₂×C₄×D₄₇	—	2	2	no	—
70	C₂×C₄×D₄₇	—	2	2	no	—
71	C₂×C₄×D₄₇	—	4	1	no	—
72	C₂×C₄×D₄₇	—	4	1	no	—
73	C₂×C₄×D₄₇	—	4	1	no	—
74	C₂×C₄×D₄₇	—	4	1	no	—
75	C₂×C₈×C₄₇	—	2	1	no	—
76	C₂×C₈×C₄₇	—	2	1	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	4	1	no	—
79	C₂×C₈×C₄₇	—	4	1	no	—
80	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₄₇	D₈	1	2	yes	—
83	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	4	1	no	—
87	C₄²×C₄₇	—	4	1	no	—
88	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	#41	1	1	yes	characteristic
92	(C₂×D₄₇)⋊C₄	—	2	2	no	—
93	(C₂×D₄₇)⋊C₄	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	1	no	—
95	C₄⋊Dic₄₇	C₄²	1	2	yes	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	4	2	no	—
102	C₂×C₄₇⋊C₈	—	4	1	no	—
103	C₂×C₄₇⋊C₈	—	4	1	no	—
104	C₈×D₄₇	—	4	8	no	—
105	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₄₇	—	2	2	no	—
107	C₂²×D₄₇	—	2	4	no	—
108	C₂²×D₄₇	—	4	1	no	—
109	C₂²×D₄₇	—	4	1	no	—
110	C₂²×D₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₄₇	#131	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
127	C₂×Dic₄₇	—	2	2	no	—
128	C₂×Dic₄₇	—	4	1	no	—
129	C₂×Dic₄₇	—	4	1	no	—
130	D₁₈₈	—	2	2	no	—
131	D₁₈₈	—	2	4	no	—
132	D₁₈₈	#97	1	4	yes	complemented
133	C₄×D₄₇	—	4	2	no	—
134	C₄×D₄₇	—	4	2	no	—
135	C₄×D₄₇	—	4	2	no	—
136	C₄×D₄₇	—	4	2	no	—
137	C₈×C₄₇	—	2	2	no	—
138	C₈×C₄₇	—	4	2	no	—
139	C₈×C₄₇	#102	1	4	yes	complemented
140	C₄₇⋊C₈	—	4	2	no	—
141	C₄₇⋊C₈	—	4	2	no	—
142	C₄₇⋊C₈	—	4	2	no	—
143	#8202	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₇	#435	1	2	yes	—
146	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
147	C₂²×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
148	C₂²×C₄₇	#444	1	2	yes	—
149	C₂×D₄₇	—	2	4	no	—
150	C₂×D₄₇	—	4	4	no	—
151	C₂×D₄₇	—	4	4	no	—
152	C₂×D₄₇	—	4	2	no	—
153	C₂×D₄₇	—	4	2	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	8	2	no	—
157	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
158	C₄×C₄₇	#405	1	1	yes	characteristic
159	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
160	C₄×C₄₇	#416	1	1	yes	characteristic
161	C₄×C₄₇	#435	1	2	yes	—
162	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
163	C₄×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
164	Dic₄₇	—	4	2	no	—
165	C₂×C₈×D₄	—	47	1	no	—
166	#1770	D₄₇	1	1	yes	characteristic, complemented
167	C₄×#102	—	47	1	no	—
168	C₂×D₄⋊C₈	—	47	1	no	—
169	C₂×#322	—	47	1	no	—
170	#1279	—	47	1	no	—
171	#1277	—	47	1	no	—
172	C₂×C₄₇	#1581	1	2	yes	—
173	C₂×C₄₇	#1582	1	2	yes	—
174	C₂×C₄₇	#1752	1	1	yes	characteristic
175	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
176	C₂×C₄₇	#1765	1	1	yes	characteristic
177	D₄₇	—	4	4	no	—
178	C₂³×C₈	—	94	1	no	—
179	C₂×C₄×D₄	—	47	1	no	—
180	C₈×D₄	—	47	4	no	—
181	C₈×D₄	—	94	2	no	—
182	C₂×C₄⋊C₈	—	47	1	no	—
183	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂×#102	—	47	2	no	—
185	C₂×C₂²⋊C₈	—	94	1	no	—
186	C₂×#97	—	2	1	no	—
187	C₂×C₄×C₈	—	47	1	no	—
188	C₂×C₄×C₈	—	47	1	no	—
189	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
191	#340	—	47	2	no	—
192	#336	—	2	1	no	—
193	D₄⋊C₈	—	47	4	no	—
194	#322	—	94	2	no	—
195	C₄₇	#8202	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	47	1	no	—
197	C₂³×C₄	—	94	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	47	2	no	—
200	C₂²×C₈	—	94	1	no	—
201	C₂²×C₈	—	94	1	no	—
202	C₂²×C₈	—	94	4	no	—
203	C₂²×C₈	—	188	2	no	—
204	C₂²×C₈	—	188	2	no	—
205	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
207	C₄×D₄	—	47	2	no	—
208	C₄×D₄	—	47	2	no	—
209	C₄×D₄	—	94	1	no	—
210	C₄×D₄	—	94	1	no	—
211	C₂×C₄⋊C₄	—	47	1	no	—
212	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
213	C₂×C₂²⋊C₄	—	94	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	47	2	no	—
218	C₄⋊C₈	—	94	1	no	—
219	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
220	#102	—	94	4	no	—
221	C₂²⋊C₈	—	94	4	no	—
222	#97	—	4	2	no	—
223	C₄×C₈	—	47	2	no	—
224	C₄×C₈	—	94	1	no	—
225	C₄×C₈	—	94	1	no	—
226	C₄×C₈	—	94	1	no	—
227	C₄×C₈	C₄×D₄₇	1	2	yes	—
228	C₄×C₈	D₁₈₈	1	2	yes	—
229	#95	—	2	1	no	—
230	C₂⁴	—	94	1	no	—
231	C₂×D₄	—	47	2	no	—
232	C₂×D₄	—	47	4	no	—
233	C₂×D₄	—	94	1	no	—
234	C₂×D₄	—	94	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	94	1	no	—
237	C₂²×C₄	—	94	2	no	—
238	C₂²×C₄	—	94	2	no	—
239	C₂²×C₄	—	188	1	no	—
240	C₂²×C₄	—	188	1	no	—
241	C₂²×C₄	—	188	1	no	—
242	C₂²×C₄	—	188	1	no	—
243	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂²×C₄	#10625350	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	94	2	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	—	94	2	no	—
255	C₂×C₈	—	94	2	no	—
256	C₂×C₈	—	188	2	no	—
257	C₂×C₈	—	188	1	no	—
258	C₂×C₈	—	188	1	no	—
259	C₂×C₈	—	188	8	no	—
260	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
261	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
262	C₂×C₈	#10625334	1	2	yes	—
263	C₂×C₈	#10625336	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	47	2	no	—
267	C₄⋊C₄	—	94	1	no	—
268	C₂²⋊C₄	—	94	2	no	—
269	C₂²⋊C₄	—	94	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
274	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
275	C₄²	#10625350	1	1	yes	characteristic
276	C₂³	—	94	2	no	—
277	C₂³	—	94	4	no	—
278	C₂³	—	188	2	no	—
279	C₂³	—	188	1	no	—
280	C₂³	—	188	1	no	—
281	C₂³	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
282	D₄	—	47	4	no	—
283	D₄	—	94	2	no	—
284	D₄	—	94	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	188	1	no	—
291	C₂×C₄	—	188	1	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	—	188	2	no	—
296	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
298	C₂×C₄	#13093981	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
301	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
302	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
303	C₂×C₄	#13094032	1	2	yes	—
304	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
305	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
306	C₂×C₄	#13094044	1	1	yes	characteristic
307	C₂×C₄	#13094044	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	188	2	no	—
311	C₈	—	188	2	no	—
312	C₈	—	188	2	no	—
313	C₈	#13093944	1	4	yes	complemented
314	C₂²	—	94	4	no	—
315	C₂²	—	188	2	no	—
316	C₂²	—	188	2	no	—
317	C₂²	—	188	4	no	—
318	C₂²	—	188	4	no	—
319	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
320	C₂²	#15689192	1	2	yes	—
321	C₂²	C₄×#10625334	1	1	yes	characteristic
322	C₂²	C₄×#10625336	1	1	yes	characteristic
323	C₂²	#15689243	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	188	2	no	—
328	C₄	#97×D₄₇	1	1	yes	characteristic
329	C₄	#15689080	1	1	yes	characteristic
330	C₄	C₂×#13093944	1	1	yes	characteristic
331	C₄	#15689117	1	1	yes	characteristic
332	C₄	#15689196	1	2	yes	—
333	C₄	C₄×#10625334	1	1	yes	characteristic
334	C₄	C₄×#10625336	1	1	yes	characteristic
335	C₂	—	188	4	no	—
336	C₂	#20242215	1	2	yes	—
337	C₂	#20242222	1	2	yes	—
338	C₂	#20243165	1	1	yes	characteristic
339	C₂	C₄×#13093944	1	1	yes	characteristic
340	C₂	#20243234	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1770	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1770	D₄₇
hypocenter	C₄₇	#8202
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
4. C₂×#15688844	C₂	semidirect
5. #20241229	C₂	semidirect
6. #20241222	C₂	semidirect
7. C₄₇×#1770	C₂	semidirect
11. C₈×D₁₈₈	C₄	semidirect
18. #15688850	C₄	semidirect
132. D₁₈₈	#97	semidirect
139. C₈×C₄₇	#102	semidirect
166. #1770	D₄₇	semidirect
195. C₄₇	#8202	semidirect
313. C₈	#13093944	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
C₄×#13093944	2
C₂×#15688850	2
C₂×#15688844	2
#20241229	2
#20241222	2
C₄₇×#1770	2
#8202	47