Subgroups — #26186028

Order 12032.

682 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	#20252355	C₂	1	2	yes	maximal, complemented
2	#20242285	C₂	1	2	yes	maximal, complemented
3	#20242279	C₂	1	4	yes	maximal, complemented
4	#20242273	C₂	1	2	yes	maximal, complemented
5	#20242136	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241770	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689254	C₂²	1	4	yes	—
10	#15689201	C₂²	1	4	yes	—
11	#15689195	—	2	4	no	—
12	#15689176	—	2	2	no	—
13	#15689176	C₂²	1	2	yes	—
14	C₄×#10625329	—	2	2	no	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15688876	C₂²	1	2	yes	—
18	#15688874	C₂²	1	2	yes	—
19	Q₈⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
20	D₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
21	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
22	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
23	C₂×#135×C₄₇	C₄	1	1	yes	characteristic
24	C₄×#41×C₄₇	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₄₇	C₄	1	2	yes	—
28	C₄×SD₁₆×C₄₇	C₂²	1	2	yes	—
29	C₄×D₈×C₄₇	C₄	1	1	yes	characteristic
30	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#427	C₄	1	1	yes	characteristic
32	C₄₇×#416	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#10625329	—	2	4	no	—
35	C₂²×C₄₇⋊C₈	—	2	1	no	—
36	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
37	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
38	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
39	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
40	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
41	#13093935	—	2	4	no	—
42	C₄×C₄₇⋊C₈	—	2	2	no	—
43	C₄×C₄₇⋊C₈	—	2	1	no	—
44	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
45	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
46	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
47	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
48	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
49	#135×C₄₇	—	2	4	no	—
50	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
57	C₄×D₄×C₄₇	—	2	2	no	—
58	C₄×D₄×C₄₇	—	2	2	no	—
59	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₄₇	C₂³	1	2	yes	—
61	#117×C₄₇	—	2	2	no	—
62	#117×C₄₇	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₄₇	—	2	2	no	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
66	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
67	#106×C₄₇	C₂×C₄	1	2	yes	—
68	#103×C₄₇	C₂×C₄	1	2	yes	—
69	#103×C₄₇	C₂×C₄	1	2	yes	—
70	#102×C₄₇	C₂×C₄	1	2	yes	—
71	#102×C₄₇	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
76	#41×C₄₇	—	2	4	no	—
77	#41×C₄₇	—	2	4	no	—
78	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₄₇	—	2	2	no	—
80	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	—	2	2	no	—
84	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
87	Q₁₆×C₄₇	—	2	2	no	—
88	SD₁₆×C₄₇	—	2	4	no	—
89	D₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	M₁₆	1	4	yes	complemented
92	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
99	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
100	C₂²⋊C₄×C₄₇	—	2	2	no	—
101	C₂²⋊C₄×C₄₇	—	2	2	no	—
102	C₂²⋊C₄×C₄₇	—	2	2	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₄²×C₄₇	—	2	2	no	—
105	C₄²×C₄₇	—	2	2	no	—
106	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₇	#41	1	1	yes	characteristic
109	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
110	#10625329	—	4	8	no	—
111	C₂×C₄₇⋊C₈	—	2	4	no	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	—	2	4	no	—
114	C₂×C₄₇⋊C₈	—	2	4	no	—
115	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
116	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
117	C₂³×C₄₇	—	2	2	no	—
118	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₄₇	—	2	2	no	—
120	Q₈×C₄₇	—	2	2	no	—
121	D₄×C₄₇	—	2	2	no	—
122	D₄×C₄₇	—	2	2	no	—
123	D₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	—	4	4	no	—
132	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₄₇	#131	1	2	yes	—
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₈×C₄₇	—	2	1	no	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
143	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
144	C₄₇⋊C₈	—	2	4	no	—
145	C₄₇⋊C₈	—	2	2	no	—
146	C₄₇⋊C₈	—	4	4	no	—
147	C₄₇⋊C₈	#135	1	4	yes	complemented
148	#17886	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	4	2	no	—
152	C₂²×C₄₇	—	4	2	no	—
153	C₂²×C₄₇	—	4	2	no	—
154	C₂²×C₄₇	#436	1	2	yes	—
155	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	4	2	no	—
159	C₄×C₄₇	—	4	2	no	—
160	C₄×C₄₇	—	4	2	no	—
161	C₄×C₄₇	#436	1	2	yes	—
162	C₄×C₄₇	#443	1	2	yes	—
163	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
164	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
165	C₄×C₄₇	C₂×#135	1	2	yes	—
166	#2965	—	47	2	no	—
167	C₄×#135	D₄₇	1	1	yes	characteristic
168	#1591	—	47	2	no	—
169	#1590	—	47	4	no	—
170	#1589	—	47	2	no	—
171	#1564	—	47	1	no	—
172	#1476	—	47	2	no	—
173	C₂×C₈²	—	47	1	no	—
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	—	8	2	no	—
176	C₂×C₄₇	C₂×#436	1	1	yes	characteristic
177	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#443	1	1	yes	characteristic
179	C₂×#135	Dic₄₇	1	1	yes	characteristic
180	C₄×#41	C₂×D₄₇	1	2	yes	—
181	#446	—	47	4	no	—
182	C₄×Q₁₆	Dic₄₇	1	1	yes	characteristic
183	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
184	C₄×SD₁₆	Dic₄₇	1	2	yes	—
185	C₄×SD₁₆	C₂×D₄₇	1	2	yes	—
186	C₄×D₈	Dic₄₇	1	1	yes	characteristic
187	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
188	#436	—	47	4	no	—
189	#435	—	94	4	no	—
190	#431	—	47	2	no	—
191	#431	—	94	2	no	—
192	#427	Dic₄₇	1	1	yes	characteristic
193	#416	Dic₄₇	1	2	yes	—
194	C₄×M₁₆	—	94	2	no	—
195	C₂×C₄×C₈	—	47	1	no	—
196	C₂×C₄×C₈	—	47	1	no	—
197	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
198	#335	—	47	2	no	—
199	#334	—	47	2	no	—
200	Q₈⋊C₈	—	47	4	no	—
201	D₄⋊C₈	—	47	4	no	—
202	C₈²	—	47	2	no	—
203	C₈²	—	47	2	no	—
204	C₄₇	#17886	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	C₂×Dic₄₇	1	2	yes	—
206	#135	—	2	4	no	—
207	C₂×Q₁₆	C₂×Dic₄₇	1	1	yes	characteristic
208	C₂×SD₁₆	C₂×Dic₄₇	1	2	yes	—
209	C₂×D₈	C₂×Dic₄₇	1	1	yes	characteristic
210	C₂×M₁₆	—	94	4	no	—
211	C₂²×C₈	—	47	2	no	—
212	C₂²×C₈	—	94	1	no	—
213	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
214	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
215	C₄×Q₈	C₂×Dic₄₇	1	2	yes	—
216	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	C₂×Dic₄₇	1	2	yes	—
220	C₄×D₄	C₂²×D₄₇	1	2	yes	—
221	#117	—	2	2	no	—
222	#117	C₂×Dic₄₇	1	2	yes	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
225	#107	C₂×Dic₄₇	1	1	yes	characteristic
226	#107	C₂×Dic₄₇	1	1	yes	characteristic
227	#106	C₂×Dic₄₇	1	2	yes	—
228	C₄⋊C₈	—	47	4	no	—
229	C₄⋊C₈	—	94	4	no	—
230	#103	C₂×Dic₄₇	1	2	yes	—
231	#103	C₂×Dic₄₇	1	2	yes	—
232	#102	C₂×Dic₄₇	1	2	yes	—
233	#102	C₂×Dic₄₇	1	2	yes	—
234	C₂²⋊C₈	—	94	4	no	—
235	C₂²⋊C₈	—	94	2	no	—
236	#97	—	94	4	no	—
237	C₄×C₈	—	47	1	no	—
238	C₄×C₈	—	47	1	no	—
239	C₄×C₈	—	47	2	no	—
240	C₄×C₈	—	47	2	no	—
241	C₄×C₈	—	94	2	no	—
242	C₄×C₈	—	94	1	no	—
243	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
244	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
247	#41	—	2	4	no	—
248	#41	—	2	4	no	—
249	C₂×Q₈	C₂²×Dic₄₇	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	C₂²×Dic₄₇	1	2	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
256	C₂²×C₄	#10625352	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
258	Q₁₆	—	2	2	no	—
259	SD₁₆	—	2	4	no	—
260	D₈	—	2	2	no	—
261	M₁₆	—	188	8	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	47	2	no	—
264	C₂×C₈	—	47	2	no	—
265	C₂×C₈	—	94	4	no	—
266	C₂×C₈	—	94	4	no	—
267	C₂×C₈	—	94	2	no	—
268	C₂×C₈	—	94	4	no	—
269	C₂×C₈	#10625329	1	4	yes	complemented
270	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
273	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂²×Dic₄₇	1	2	yes	—
277	C₄⋊C₄	C₂²×Dic₄₇	1	2	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	D₄×D₄₇	1	1	yes	characteristic
285	C₄²	#10625352	1	1	yes	characteristic
286	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
287	C₄²	C₂³×D₄₇	1	1	yes	characteristic
288	C₂³	—	2	2	no	—
289	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	—	2	2	no	—
292	D₄	—	2	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
304	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×#10625329	1	2	yes	—
307	C₂×C₄	#13094031	1	2	yes	—
308	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	47	4	no	—
314	C₈	—	94	4	no	—
315	C₈	—	94	2	no	—
316	C₈	—	188	4	no	—
317	C₈	C₂×#10625329	1	2	yes	—
318	C₈	C₂×#10625329	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	#15689201	1	2	yes	—
325	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	#15689201	1	2	yes	—
332	C₄	#15689240	1	2	yes	—
333	C₄	C₂²×#10625329	1	1	yes	characteristic
334	C₄	C₂×#13094031	1	1	yes	characteristic
335	C₄	#135×D₄₇	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	8	2	no	—
338	C₂	C₂×#15689201	1	1	yes	characteristic
339	C₂	#135×Dic₄₇	1	1	yes	characteristic
340	C₂	C₂×#15689240	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄×#135×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#135×C₄₇	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₄×#135	D₄₇
hypocenter	C₄₇	#17886
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #20252355	C₂	semidirect
2. #20242285	C₂	semidirect
3. #20242279	C₂	semidirect
4. #20242273	C₂	semidirect
5. #20242136	C₂	semidirect
6. #20241770	C₂	semidirect
7. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
91. C₂×C₈×C₄₇	M₁₆	semidirect
147. C₄₇⋊C₈	#135	semidirect
204. C₄₇	#17886	semidirect
269. C₂×C₈	#10625329	semidirect

Maximal subgroups

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

subgroup	index
#20252355	2
#20242285	2
#20242279	2
#20242273	2
#20242136	2
#20241770	2
C₂×C₈×C₄₇⋊C₈	2
C₄×#135×C₄₇	2
#17886	47