Subgroups — #26188022

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252422	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
3	#20242303	C₂	1	2	yes	maximal, complemented
4	#20242219	C₂	1	4	yes	maximal, complemented
5	#20242213	C₂	1	2	yes	maximal, complemented
6	#20242151	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241832	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	C₄	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
11	C₄×D₃₇₆	C₄	1	1	yes	characteristic
12	C₄×D₃₇₆	C₄	1	2	yes	—
13	C₄×D₃₇₆	C₄	1	1	yes	characteristic
14	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
15	C₄×D₃₇₆	C₂²	1	2	yes	—
16	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
17	#15689202	C₂²	1	4	yes	complemented
18	#15689193	—	2	4	no	—
19	C₈×D₁₈₈	C₂²	1	4	yes	complemented
20	#15689180	C₂²	1	2	yes	—
21	C₂×#13093958	C₄	1	1	yes	characteristic
22	C₂×#13093946	C₄	1	2	yes	—
23	#15689085	—	2	2	no	—
24	C₂²⋊C₈×D₄₇	—	2	2	no	—
25	#15688876	C₂²	1	2	yes	complemented
26	#15688874	C₂²	1	2	yes	complemented
27	#15688852	C₂²	1	4	yes	complemented
28	#15688852	C₂²	1	4	yes	complemented
29	C₄₇×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#322	C₂²	1	2	yes	complemented
33	C₄₇×#322	C₂²	1	2	yes	complemented
34	C₂²×D₁₈₈	C₂×C₄	1	2	yes	—
35	C₂²×C₄×D₄₇	—	2	2	no	—
36	C₂×D₃₇₆	—	2	4	no	—
37	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
38	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
39	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
40	C₂×#10625324	—	2	4	no	—
41	C₂×C₈×D₄₇	—	2	4	no	—
42	C₄×D₁₈₈	—	2	2	no	—
43	C₄×D₁₈₈	—	2	2	no	—
44	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
45	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
46	C₄×D₁₈₈	C₂³	1	2	yes	—
47	C₄×D₁₈₈	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
49	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
50	#13093958	C₂×C₄	1	2	yes	—
51	#13093958	C₂×C₄	1	1	yes	characteristic
52	#13093958	C₂×C₄	1	1	yes	characteristic
53	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
54	#13093946	C₂×C₄	1	2	yes	—
55	#13093946	C₂×C₄	1	2	yes	—
56	#13093946	C₂×C₄	1	2	yes	—
57	#13093946	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
59	(C₂×D₄₇)⋊C₈	—	2	4	no	—
60	(C₂×D₄₇)⋊C₈	—	2	4	no	—
61	#13093935	—	2	2	no	—
62	C₂×M₁₆×C₄₇	—	2	1	no	—
63	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
67	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
68	#97×C₄₇	—	2	1	no	—
69	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₇	C₂³	1	2	yes	—
72	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	2	yes	—
75	C₂³×D₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	2	no	—
80	C₂×D₁₈₈	—	2	4	no	—
81	C₂×D₁₈₈	—	2	2	no	—
82	C₂×D₁₈₈	—	4	2	no	—
83	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
84	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₄₇	—	2	2	no	—
86	C₂×C₄×D₄₇	—	2	2	no	—
87	C₂×C₄×D₄₇	—	2	2	no	—
88	C₂×C₄×D₄₇	—	2	2	no	—
89	C₂×C₄×D₄₇	—	8	4	no	—
90	M₁₆×C₄₇	—	4	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	4	1	no	—
94	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₇	#41	1	1	yes	characteristic
102	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
103	(C₂×D₄₇)⋊C₄	—	2	2	no	—
104	(C₂×D₄₇)⋊C₄	—	2	2	no	—
105	(C₂×D₄₇)⋊C₄	—	2	2	no	—
106	(C₂×D₄₇)⋊C₄	—	2	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
109	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄₇⋊C₈	—	2	4	no	—
111	D₃₇₆	—	2	4	no	—
112	D₃₇₆	—	2	2	no	—
113	D₃₇₆	M₁₆	1	4	yes	complemented
114	#10625324	—	8	4	no	—
115	C₈×D₄₇	—	8	4	no	—
116	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₄₇	—	2	2	no	—
118	C₂²×D₄₇	—	2	2	no	—
119	C₂²×D₄₇	—	4	4	no	—
120	C₂²×D₄₇	—	4	4	no	—
121	C₂²×D₄₇	—	8	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	8	2	no	—
135	D₁₈₈	—	2	2	no	—
136	D₁₈₈	—	2	2	no	—
137	D₁₈₈	—	2	2	no	—
138	D₁₈₈	—	4	4	no	—
139	D₁₈₈	C₂×M₁₆	1	4	yes	—
140	C₄×D₄₇	—	4	4	no	—
141	C₄×D₄₇	—	4	4	no	—
142	C₄×D₄₇	—	8	2	no	—
143	C₄×D₄₇	—	8	2	no	—
144	C₈×C₄₇	—	2	1	no	—
145	C₈×C₄₇	—	2	1	no	—
146	C₈×C₄₇	—	2	1	no	—
147	C₈×C₄₇	—	2	4	no	—
148	C₈×C₄₇	—	4	2	no	—
149	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
150	C₄₇⋊C₈	—	4	4	no	—
151	#18064	—	47	1	no	maximal, Sylow
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	1	no	—
154	C₂²×C₄₇	#444	1	2	yes	—
155	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂×D₄₇	—	2	4	no	—
157	C₂×D₄₇	—	4	2	no	—
158	C₂×D₄₇	—	4	2	no	—
159	C₂×D₄₇	—	4	2	no	—
160	C₂×D₄₇	—	4	2	no	—
161	C₂×D₄₇	—	4	2	no	—
162	C₂×D₄₇	—	8	8	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
168	C₄×C₄₇	#568	1	1	yes	characteristic
169	C₄×C₄₇	#576	1	2	yes	—
170	Dic₄₇	—	4	2	no	—
171	Dic₄₇	—	4	2	no	—
172	#2974	—	47	2	no	—
173	C₂×C₄×D₈	—	47	1	no	—
174	#1594	—	47	2	no	—
175	#1581	—	47	4	no	—
176	#1580	—	47	2	no	—
177	#1568	—	47	1	no	—
178	#1490	—	47	2	no	—
179	#1451	D₄₇	1	1	yes	characteristic, complemented
180	C₂×C₄₇	—	2	2	no	—
181	C₂×C₄₇	D₄×M₁₆	1	1	yes	characteristic
182	C₂×C₄₇	#2948	1	1	yes	characteristic
183	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
184	D₄₇	—	4	4	no	—
185	D₄₇	—	8	2	no	—
186	C₂²×D₈	—	47	1	no	—
187	C₂×C₄×D₄	—	47	2	no	—
188	C₄×D₈	—	47	1	no	—
189	C₄×D₈	—	47	2	no	—
190	C₄×D₈	—	47	1	no	—
191	C₄×D₈	—	47	1	no	—
192	C₄×D₈	—	47	2	no	—
193	C₄×D₈	—	47	1	no	—
194	#436	—	47	4	no	—
195	#435	—	94	4	no	—
196	C₈×D₄	—	47	4	no	—
197	#432	—	47	2	no	—
198	#431	C₂×D₄₇	1	1	yes	characteristic
199	C₂×#107	—	47	1	no	—
200	C₂×#102	—	47	2	no	—
201	#407	—	94	2	no	—
202	C₂×C₂²⋊C₈	—	94	2	no	—
203	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
205	#335	—	47	2	no	—
206	#334	—	47	2	no	—
207	D₄⋊C₈	—	47	4	no	—
208	D₄⋊C₈	—	47	4	no	—
209	#322	C₂×D₄₇	1	2	yes	complemented
210	#322	C₂×D₄₇	1	2	yes	complemented
211	C₄₇	#18064	1	1	yes	characteristic, complemented, Sylow
212	C₂²×D₄	—	47	2	no	—
213	C₂³×C₄	—	94	2	no	—
214	C₂×D₈	—	47	1	no	—
215	C₂×D₈	—	47	2	no	—
216	C₂×D₈	—	47	1	no	—
217	C₂×D₈	—	94	4	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	94	4	no	—
220	C₂²×C₈	—	94	4	no	—
221	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
222	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
223	C₄×D₄	—	47	2	no	—
224	C₄×D₄	—	47	2	no	—
225	C₄×D₄	—	47	2	no	—
226	C₄×D₄	—	47	2	no	—
227	C₄×D₄	—	94	2	no	—
228	C₄×D₄	—	94	2	no	—
229	C₂×C₄⋊C₄	—	47	2	no	—
230	C₂×C₂²⋊C₄	—	94	2	no	—
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	#107	—	47	2	no	—
233	#107	—	47	1	no	—
234	#107	—	47	1	no	—
235	C₄⋊C₈	—	47	4	no	—
236	C₄⋊C₈	D₁₈₈	1	2	yes	—
237	#102	—	47	2	no	—
238	#102	—	47	2	no	—
239	#102	—	47	2	no	—
240	#102	—	47	2	no	—
241	C₂²⋊C₈	—	94	2	no	—
242	C₂²⋊C₈	—	94	4	no	—
243	C₂²⋊C₈	—	94	4	no	—
244	C₂²⋊C₈	D₁₈₈	1	2	yes	—
245	#97	—	2	1	no	—
246	#97	—	94	2	no	—
247	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
248	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₄₇	1	2	yes	—
250	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₄₇	1	2	yes	—
253	C₂⁴	—	94	2	no	—
254	C₂×D₄	—	47	2	no	—
255	C₂×D₄	—	47	2	no	—
256	C₂×D₄	—	94	4	no	—
257	C₂×D₄	—	94	2	no	—
258	C₂×D₄	—	188	2	no	—
259	C₂²×C₄	—	94	2	no	—
260	C₂²×C₄	—	94	2	no	—
261	C₂²×C₄	—	94	2	no	—
262	C₂²×C₄	—	94	2	no	—
263	C₂²×C₄	—	94	2	no	—
264	C₂²×C₄	—	376	4	no	—
265	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
267	C₂²×C₄	#10625350	1	1	yes	characteristic
268	D₈	—	47	4	no	—
269	D₈	—	94	4	no	—
270	D₈	—	94	2	no	—
271	M₁₆	—	4	2	no	—
272	M₁₆	—	376	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	94	4	no	—
277	C₂×C₈	—	376	4	no	—
278	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
280	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
281	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
282	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
283	C₄⋊C₄	—	47	2	no	—
284	C₄⋊C₄	—	47	2	no	—
285	C₄⋊C₄	—	94	2	no	—
286	C₂²⋊C₄	—	94	2	no	—
287	C₂²⋊C₄	—	94	2	no	—
288	C₂²⋊C₄	—	94	2	no	—
289	C₂²⋊C₄	—	94	2	no	—
290	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
291	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
292	C₄²	#10625350	1	1	yes	characteristic
293	C₄²	C₂³×D₄₇	1	1	yes	characteristic
294	C₂³	—	94	2	no	—
295	C₂³	—	94	2	no	—
296	C₂³	—	376	2	no	—
297	C₂³	—	188	4	no	—
298	C₂³	—	188	4	no	—
299	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
300	D₄	—	47	4	no	—
301	D₄	—	94	2	no	—
302	D₄	—	94	2	no	—
303	D₄	—	94	2	no	—
304	D₄	—	188	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	94	2	no	—
310	C₂×C₄	—	94	2	no	—
311	C₂×C₄	—	376	2	no	—
312	C₂×C₄	—	376	2	no	—
313	C₂×C₄	—	376	2	no	—
314	C₂×C₄	—	188	4	no	—
315	C₂×C₄	—	188	4	no	—
316	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
317	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
318	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
321	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	4	no	—
326	C₈	—	4	2	no	—
327	C₈	—	188	4	no	—
328	C₈	M₁₆×D₄₇	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	94	4	no	—
332	C₂²	—	376	8	no	—
333	C₂²	—	188	2	no	—
334	C₂²	—	188	2	no	—
335	C₂²	—	188	2	no	—
336	C₂²	—	188	2	no	—
337	C₂²	—	188	2	no	—
338	C₂²	#15689246	1	2	yes	—
339	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	188	2	no	—
345	C₄	—	188	2	no	—
346	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
347	C₄	#15689889	1	1	yes	characteristic
348	C₄	#15689928	1	2	yes	—
349	C₂	—	2	2	no	—
350	C₂	—	376	2	no	—
351	C₂	—	188	4	no	—
352	C₂	M₁₆×D₁₈₈	1	1	yes	characteristic
353	C₂	#20252241	1	1	yes	characteristic
354	C₂	C₂×#15689246	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1451	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂²×D₃₇₆	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1451	D₄₇
hypocenter	C₄₇	#18064
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
1. #20252422	C₂	semidirect
3. #20242303	C₂	semidirect
4. #20242219	C₂	semidirect
5. #20242213	C₂	semidirect
6. #20242151	C₂	semidirect
7. #20241832	C₂	semidirect
8. C₄₇×#1451	C₂	semidirect
17. #15689202	C₂²	semidirect
19. C₈×D₁₈₈	C₂²	semidirect
25. #15688876	C₂²	semidirect
26. #15688874	C₂²	semidirect
27. #15688852	C₂²	semidirect
28. #15688852	C₂²	semidirect
32. C₄₇×#322	C₂²	semidirect
33. C₄₇×#322	C₂²	semidirect
113. D₃₇₆	M₁₆	semidirect
179. #1451	D₄₇	semidirect
209. #322	C₂×D₄₇	semidirect
210. #322	C₂×D₄₇	semidirect
211. C₄₇	#18064	semidirect

Maximal subgroups

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

subgroup	index
#20252422	2
C₂×C₄×D₃₇₆	2
#20242303	2
#20242219	2
#20242213	2
#20242151	2
#20241832	2
C₄₇×#1451	2
#18064	47