Subgroups — #26185557

Order 12032.

588 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252354	C₂	1	1	yes	maximal, characteristic
2	#20252049	C₂	1	2	yes	maximal
3	#20242175	C₂	1	2	yes	maximal, complemented
4	#20242170	C₂	1	2	yes	maximal, complemented
5	#20242166	C₂	1	2	yes	maximal, complemented
6	#20242154	C₂	1	2	yes	maximal, complemented
7	#20242151	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242139	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#10625350	C₂²	1	1	yes	characteristic
11	#15689253	C₄	1	1	yes	characteristic
12	#15689253	C₄	1	1	yes	characteristic
13	#15689202	C₄	1	1	yes	characteristic
14	#15689202	C₄	1	1	yes	characteristic
15	#15689194	C₄	1	2	yes	—
16	#15689194	C₄	1	2	yes	—
17	#15689182	—	2	2	no	—
18	#15689181	—	2	2	no	—
19	#15689180	C₂²	1	2	yes	—
20	#15689179	C₂²	1	2	yes	—
21	#15689176	C₂²	1	1	yes	characteristic
22	#15689175	C₂²	1	2	yes	—
23	#15689174	C₂²	1	2	yes	—
24	#15689145	C₂²	1	2	yes	—
25	C₄⋊C₈×D₄₇	C₂²	1	2	yes	—
26	#15689088	—	2	2	no	—
27	#15689077	—	2	2	no	—
28	C₄×#10625324	C₂²	1	2	yes	—
29	#15688876	C₂²	1	1	yes	characteristic
30	#15688876	C₂²	1	1	yes	characteristic
31	#15688875	C₂²	1	2	yes	—
32	#15688875	C₂²	1	2	yes	—
33	#15688874	C₂²	1	2	yes	—
34	#15688873	C₂²	1	2	yes	—
35	#15688873	C₂²	1	2	yes	—
36	C₄₇×#431	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#335	C₂²	1	1	yes	characteristic
39	C₄₇×#335	C₂²	1	1	yes	characteristic
40	C₄₇×#334	C₂²	1	2	yes	—
41	C₂×#10625350	D₄	1	1	yes	characteristic
42	C₂×#10625324	—	2	2	no	—
43	C₂×#10625324	C₂×C₄	1	2	yes	—
44	C₂×#10625324	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₄₇	—	2	2	no	—
46	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₉₄	Q₈	1	1	yes	characteristic
48	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
49	C₄×D₁₈₈	Q₈	1	1	yes	characteristic
50	C₄×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
51	C₄×C₂²⋊D₄₇	Q₈	1	2	yes	—
52	#13093983	D₄	1	1	yes	characteristic
53	#13093981	D₄	1	2	yes	—
54	C₄²×D₄₇	C₂³	1	2	yes	—
55	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
56	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
57	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
58	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
59	Dic₄₇⋊C₈	—	2	2	no	—
60	Dic₄₇⋊C₈	—	2	2	no	—
61	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
62	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
63	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
64	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
65	(C₂×D₄₇)⋊C₈	—	2	2	no	—
66	(C₂×D₄₇)⋊C₈	—	2	2	no	—
67	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
68	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
69	#13093935	—	2	2	no	—
70	#13093934	—	2	2	no	—
71	#13093934	C₂×C₄	1	2	yes	—
72	#13093934	C₂×C₄	1	2	yes	—
73	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
74	C₈×Dic₄₇	—	2	2	no	—
75	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
80	C₂²⋊C₈×C₄₇	—	2	2	no	—
81	C₄×C₈×C₄₇	—	2	2	no	—
82	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₂²⋊D₄₇	C₄⋊C₄	1	2	yes	—
90	#10625350	—	4	1	no	—
91	#10625350	—	4	1	no	—
92	C₂×D₁₈₈	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
96	C₂×Dic₉₄	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	4	1	no	—
101	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
108	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
109	C₂²⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
110	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	2	yes	—
111	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	2	yes	—
112	C₄⋊Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
113	C₄⋊Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
114	Dic₄₇⋊C₄	C₄⋊C₄	1	2	yes	—
115	Dic₄₇⋊C₄	C₄⋊C₄	1	2	yes	—
116	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
117	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
118	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
122	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
123	#10625324	—	2	4	no	—
124	#10625324	—	4	2	no	—
125	#10625324	—	4	4	no	—
126	C₈×D₄₇	—	4	4	no	—
127	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂²⋊D₄₇	—	4	2	no	—
141	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
144	D₁₈₈	—	4	1	no	—
145	C₄×D₄₇	—	2	2	no	—
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	2	2	no	—
149	C₄×D₄₇	—	4	2	no	—
150	Dic₉₄	—	4	1	no	—
151	C₈×C₄₇	—	2	1	no	—
152	C₈×C₄₇	—	2	1	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	—	4	2	no	—
155	C₈×C₄₇	—	4	2	no	—
156	C₈×C₄₇	#131	1	2	yes	—
157	C₄₇⋊C₈	—	2	2	no	—
158	C₄₇⋊C₈	—	2	2	no	—
159	C₄₇⋊C₈	—	4	2	no	—
160	C₄₇⋊C₈	—	4	2	no	—
161	#17833	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₂×D₄₇	—	2	2	no	—
167	C₂×D₄₇	—	2	2	no	—
168	C₂×D₄₇	—	2	2	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
174	C₄×C₄₇	#568	1	1	yes	characteristic
175	C₄×C₄₇	#575	1	1	yes	characteristic
176	C₄×C₄₇	#575	1	1	yes	characteristic
177	Dic₄₇	—	2	2	no	—
178	Dic₄₇	—	2	2	no	—
179	Dic₄₇	—	2	2	no	—
180	#2965	—	47	1	no	—
181	#2925	—	47	2	no	—
182	#1574	—	47	2	no	—
183	#1573	—	47	2	no	—
184	#1572	—	47	2	no	—
185	#1569	—	47	2	no	—
186	#1568	—	47	1	no	—
187	#1565	—	47	2	no	—
188	#1564	D₄₇	1	1	yes	characteristic, complemented
189	C₂×C₄₇	—	4	1	no	—
190	C₂×C₄₇	#2906	1	1	yes	characteristic
191	C₂×C₄₇	#2911	1	1	yes	characteristic
192	C₂×C₄₇	#2916	1	1	yes	characteristic
193	D₄₇	—	4	2	no	—
194	C₄×#41	—	47	1	no	—
195	#446	—	47	1	no	—
196	#446	—	47	1	no	—
197	#436	—	47	1	no	—
198	#436	—	47	1	no	—
199	#435	—	47	2	no	—
200	#435	—	47	2	no	—
201	#433	—	94	2	no	—
202	#433	—	94	2	no	—
203	#432	—	47	2	no	—
204	#432	—	47	2	no	—
205	#431	—	47	1	no	—
206	#431	—	47	2	no	—
207	#431	—	47	2	no	—
208	#431	C₂×D₄₇	1	2	yes	—
209	#423	—	47	2	no	—
210	C₂×C₄⋊C₈	—	47	2	no	—
211	#408	—	94	2	no	—
212	#405	—	94	2	no	—
213	C₄×M₁₆	—	47	2	no	—
214	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
215	#335	—	47	2	no	—
216	#335	—	47	2	no	—
217	#335	—	47	1	no	—
218	#335	—	47	1	no	—
219	#335	C₂×D₄₇	1	1	yes	characteristic
220	#335	C₂×D₄₇	1	1	yes	characteristic
221	#334	—	47	2	no	—
222	#334	—	47	2	no	—
223	#334	—	47	2	no	—
224	#334	C₂×D₄₇	1	2	yes	—
225	C₄₇	#17833	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	47	1	no	—
227	C₂×M₁₆	—	47	2	no	—
228	C₂×M₁₆	—	47	2	no	—
229	C₂×M₁₆	—	94	2	no	—
230	C₂²×C₈	—	94	2	no	—
231	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
232	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
233	C₄×Q₈	—	47	1	no	—
234	C₄×Q₈	—	47	1	no	—
235	C₄×D₄	—	47	2	no	—
236	C₄×D₄	—	47	2	no	—
237	C₄×D₄	—	47	1	no	—
238	C₄×D₄	—	47	1	no	—
239	#117	—	47	1	no	—
240	#117	—	47	2	no	—
241	C₂×C₄²	—	47	2	no	—
242	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
243	C₄⋊C₈	—	47	2	no	—
244	C₄⋊C₈	—	47	2	no	—
245	C₄⋊C₈	—	47	1	no	—
246	C₄⋊C₈	—	47	1	no	—
247	C₄⋊C₈	—	47	2	no	—
248	C₄⋊C₈	—	47	2	no	—
249	C₄⋊C₈	—	94	2	no	—
250	C₄⋊C₈	—	94	2	no	—
251	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
252	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	47	2	no	—
255	C₂²⋊C₈	—	47	2	no	—
256	C₂²⋊C₈	—	47	2	no	—
257	C₂²⋊C₈	—	94	2	no	—
258	C₂²⋊C₈	—	94	2	no	—
259	C₂²⋊C₈	—	94	2	no	—
260	#97	—	47	2	no	—
261	#97	—	47	2	no	—
262	#97	—	94	2	no	—
263	#97	—	94	2	no	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	47	2	no	—
266	C₄×C₈	—	94	2	no	—
267	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
268	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
271	#41	—	188	1	no	—
272	#41	—	188	1	no	—
273	C₂×Q₈	—	47	1	no	—
274	C₂×D₄	—	47	2	no	—
275	C₂×D₄	—	47	1	no	—
276	C₂²×C₄	—	47	2	no	—
277	C₂²×C₄	—	47	2	no	—
278	C₂²×C₄	—	47	2	no	—
279	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
281	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
282	M₁₆	—	94	4	no	—
283	M₁₆	—	188	2	no	—
284	M₁₆	—	188	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	47	2	no	—
290	C₂×C₈	—	47	2	no	—
291	C₂×C₈	—	94	2	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	—	188	4	no	—
294	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
298	C₄⋊C₄	—	47	2	no	—
299	C₄⋊C₄	—	47	2	no	—
300	C₄⋊C₄	—	47	1	no	—
301	C₄⋊C₄	—	47	1	no	—
302	C₂²⋊C₄	—	47	2	no	—
303	C₂²⋊C₄	—	47	2	no	—
304	C₂²⋊C₄	—	47	2	no	—
305	C₄²	—	47	2	no	—
306	C₄²	—	47	2	no	—
307	C₄²	—	47	2	no	—
308	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
309	C₄²	D₄×D₄₇	1	1	yes	characteristic
310	C₄²	Q₈×D₄₇	1	1	yes	characteristic
311	C₄²	C₂³×D₄₇	1	1	yes	characteristic
312	C₂³	—	47	2	no	—
313	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
314	Q₈	—	188	1	no	—
315	D₄	—	188	1	no	—
316	D₄	—	188	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	47	2	no	—
323	C₂×C₄	—	47	2	no	—
324	C₂×C₄	—	47	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
331	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
332	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
335	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	94	2	no	—
342	C₈	—	94	2	no	—
343	C₈	—	188	2	no	—
344	C₈	—	188	2	no	—
345	C₈	#13094030	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	94	2	no	—
350	C₂²	—	94	2	no	—
351	C₂²	—	94	2	no	—
352	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	94	2	no	—
358	C₄	—	94	2	no	—
359	C₄	—	94	2	no	—
360	C₄	C₂×#13094030	1	1	yes	characteristic
361	C₄	#15689889	1	1	yes	characteristic
362	C₄	#15689920	1	1	yes	characteristic
363	C₄	#15689920	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	188	2	no	—
366	C₂	#20251953	1	1	yes	characteristic
367	C₂	#20251976	1	1	yes	characteristic
368	C₂	#20251996	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂×#10625350	D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#17833
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #20242175	C₂	semidirect
4. #20242170	C₂	semidirect
5. #20242166	C₂	semidirect
6. #20242154	C₂	semidirect
7. #20242151	C₂	semidirect
8. #20242139	C₂	semidirect
9. C₄₇×#1564	C₂	semidirect
188. #1564	D₄₇	semidirect
225. C₄₇	#17833	semidirect

Maximal subgroups

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

subgroup	index
#20252354	2
#20252049	2
#20242175	2
#20242170	2
#20242166	2
#20242154	2
#20242151	2
#20242139	2
C₄₇×#1564	2
#17833	47