Subgroups — #26188066

Order 12032.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253956	C₂	1	1	yes	maximal, characteristic
2	#20252395	C₂	1	2	yes	maximal, complemented
3	#20242284	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242229	C₂	1	2	yes	maximal, complemented
5	#20242221	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242135	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241793	C₂	1	2	yes	maximal, complemented
10	C₈×#10625324	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689655	C₂²	1	2	yes	—
13	#15689524	C₄	1	1	yes	characteristic
14	#15689506	C₄	1	1	yes	characteristic
15	#15689476	C₄	1	2	yes	—
16	#15689365	C₄	1	1	yes	characteristic
17	#15689356	C₄	1	1	yes	characteristic
18	#15689351	C₄	1	2	yes	—
19	#15689250	C₂²	1	2	yes	—
20	C₈×Dic₉₄	C₂²	1	2	yes	—
21	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
22	C₄×#10625335	C₂²	1	2	yes	—
23	#15689199	—	2	2	no	—
24	#15689191	—	2	2	no	—
25	D₄×C₄₇⋊C₈	C₂²	1	2	yes	—
26	#15689181	—	2	2	no	—
27	#15689175	C₂²	1	2	yes	—
28	#15689174	C₂²	1	1	yes	characteristic
29	#15689081	—	2	2	no	—
30	C₄×#10625324	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
32	#15688876	C₂²	1	1	yes	characteristic
33	#15688873	C₂²	1	2	yes	—
34	#15688855	C₂²	1	2	yes	—
35	#15688853	C₂²	1	2	yes	—
36	D₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
37	#15688845	C₂²	1	1	yes	characteristic
38	#15688844	C₂²	1	1	yes	characteristic
39	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₄₇×#436	C₂²	1	2	yes	—
42	C₄₇×#335	C₂²	1	1	yes	characteristic
43	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₄₇	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#10625335	C₂×C₄	1	2	yes	—
47	C₂×#10625324	D₄	1	1	yes	characteristic
48	C₂×#10625324	D₄	1	1	yes	characteristic
49	C₂²×C₄₇⋊C₈	—	2	2	no	—
50	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
52	#13094017	—	2	2	no	—
53	#13094015	C₂×C₄	1	2	yes	—
54	#13094011	C₂×C₄	1	2	yes	—
55	#13094006	—	2	2	no	—
56	#13094004	C₂×C₄	1	2	yes	—
57	#13093995	C₂×C₄	1	2	yes	—
58	C₄×Dic₉₄	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₄₇	—	2	2	no	—
60	#13093983	—	2	2	no	—
61	C₄²×D₄₇	C₂³	1	1	yes	characteristic
62	#13093958	C₂×C₄	1	1	yes	characteristic
63	#13093955	C₂×C₄	1	2	yes	—
64	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
65	Dic₄₇⋊C₈	—	2	2	no	—
66	Dic₄₇⋊C₈	—	2	1	no	—
67	Dic₄₇⋊C₈	—	2	2	no	—
68	#13093949	C₂×C₄	1	2	yes	—
69	#13093947	C₂×C₄	1	2	yes	—
70	#13093945	C₂×C₄	1	2	yes	—
71	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
72	(C₂×D₄₇)⋊C₈	—	2	2	no	—
73	(C₂×D₄₇)⋊C₈	—	2	1	no	—
74	#13093934	—	2	2	no	—
75	#13093934	D₄	1	1	yes	characteristic
76	#13093934	D₄	1	1	yes	characteristic
77	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₈×Dic₄₇	—	2	2	no	—
81	C₈×Dic₄₇	—	2	2	no	—
82	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
84	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
85	C₂×M₁₆×C₄₇	—	2	2	no	—
86	C₄×D₄×C₄₇	C₂³	1	2	yes	—
87	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
89	#102×C₄₇	C₂×C₄	1	2	yes	—
90	C₂²⋊C₈×C₄₇	—	2	2	no	—
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₄₇	—	2	2	no	—
96	C₂×C₂²⋊D₄₇	—	2	2	no	—
97	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
100	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
101	D₈×C₄₇	—	2	2	no	—
102	M₁₆×C₄₇	—	4	4	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	2	1	no	—
105	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
110	C₂²⋊C₄×C₄₇	—	2	2	no	—
111	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
112	C₂²⋊Dic₄₇	—	2	2	no	—
113	C₂²⋊Dic₄₇	—	2	2	no	—
114	C₂²⋊Dic₄₇	—	2	2	no	—
115	#10625335	—	4	2	no	—
116	(C₂×D₄₇)⋊C₄	—	2	2	no	—
117	C₄⋊Dic₄₇	—	2	2	no	—
118	C₄⋊Dic₄₇	—	2	2	no	—
119	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
120	Dic₄₇⋊C₄	—	2	2	no	—
121	Dic₄₇⋊C₄	—	2	2	no	—
122	C₄×Dic₄₇	—	2	2	no	—
123	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₄₇	#41	1	1	yes	characteristic
126	C₂×C₄₇⋊C₈	—	2	2	no	—
127	C₂×C₄₇⋊C₈	—	2	1	no	—
128	C₂×C₄₇⋊C₈	—	4	4	no	—
129	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
133	Dic₁₈₈	—	2	2	no	—
134	#10625324	—	2	2	no	—
135	#10625324	—	2	2	no	—
136	C₈×D₄₇	—	2	2	no	—
137	C₈×D₄₇	—	4	1	no	—
138	C₂³×C₄₇	—	2	2	no	—
139	C₂²×D₄₇	C₄×D₄	1	1	yes	characteristic
140	D₄×C₄₇	—	2	2	no	—
141	D₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₄₇	—	8	2	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
154	C₄×D₄₇	—	2	1	no	—
155	C₄×D₄₇	—	2	1	no	—
156	C₄×D₄₇	—	2	1	no	—
157	C₄×D₄₇	—	2	1	no	—
158	C₄×D₄₇	—	4	1	no	—
159	Dic₉₄	—	2	2	no	—
160	Dic₉₄	—	2	2	no	—
161	C₈×C₄₇	—	2	1	no	—
162	C₈×C₄₇	—	2	1	no	—
163	C₈×C₄₇	—	4	1	no	—
164	C₈×C₄₇	—	4	2	no	—
165	C₈×C₄₇	#131	1	2	yes	—
166	C₈×C₄₇	#135	1	2	yes	—
167	C₄₇⋊C₈	—	2	1	no	—
168	C₄₇⋊C₈	—	2	1	no	—
169	C₄₇⋊C₈	—	2	1	no	—
170	C₄₇⋊C₈	—	4	1	no	—
171	C₄₇⋊C₈	—	4	2	no	—
172	C₄₇⋊C₈	#135	1	2	yes	—
173	#18069	—	47	1	no	maximal, Sylow
174	C₂²×C₄₇	—	4	2	no	—
175	C₂²×C₄₇	—	4	2	no	—
176	C₂²×C₄₇	—	4	2	no	—
177	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×D₄₇	—	2	1	no	—
179	C₂×D₄₇	—	2	1	no	—
180	C₂×D₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	—	4	2	no	—
183	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
184	C₄×C₄₇	#568	1	1	yes	characteristic
185	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
186	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
187	Dic₄₇	—	2	1	no	—
188	Dic₄₇	—	2	1	no	—
189	Dic₄₇	—	2	1	no	—
190	Dic₄₇	—	4	2	no	—
191	Dic₄₇	—	4	2	no	—
192	#3168	—	47	1	no	—
193	#2969	—	47	2	no	—
194	#1591	—	47	1	no	—
195	#1589	D₄₇	1	1	yes	characteristic, complemented
196	#1583	—	47	2	no	—
197	#1582	—	47	2	no	—
198	C₈×Q₁₆	—	47	1	no	—
199	C₈×D₈	—	47	1	no	—
200	#1564	—	47	1	no	—
201	#1482	—	47	2	no	—
202	C₈×M₁₆	—	47	1	no	—
203	C₂×C₄₇	—	8	2	no	—
204	C₂×C₄₇	#2935	1	1	yes	characteristic
205	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
206	C₂×C₄₇	#2957	1	1	yes	characteristic
207	D₄₇	—	4	1	no	—
208	#529	—	47	2	no	—
209	#499	—	47	1	no	—
210	#495	—	47	1	no	—
211	#488	—	47	2	no	—
212	#467	—	47	1	no	—
213	#466	—	47	1	no	—
214	#465	—	47	2	no	—
215	#446	—	47	2	no	—
216	C₈×Q₈	—	47	2	no	—
217	C₄×Q₁₆	—	47	1	no	—
218	C₄×SD₁₆	—	47	2	no	—
219	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
220	#436	—	94	2	no	—
221	#436	C₂×D₄₇	1	2	yes	—
222	#435	—	94	2	no	—
223	C₈×D₄	—	47	2	no	—
224	#433	—	94	2	no	—
225	#431	—	47	1	no	—
226	#431	—	47	2	no	—
227	#405	—	94	2	no	—
228	C₄×M₁₆	—	47	1	no	—
229	C₂×C₄×C₈	—	47	1	no	—
230	#335	—	47	1	no	—
231	#335	C₂×D₄₇	1	1	yes	characteristic
232	#334	—	47	2	no	—
233	Q₈⋊C₈	—	47	2	no	—
234	Q₈⋊C₈	—	47	2	no	—
235	D₄⋊C₈	—	47	2	no	—
236	D₄⋊C₈	C₂×D₄₇	1	2	yes	—
237	#322	—	47	1	no	—
238	#322	—	47	1	no	—
239	C₈²	—	47	1	no	—
240	C₈²	C₂×D₄₇	1	1	yes	characteristic
241	C₄₇	#18069	1	1	yes	characteristic, complemented, Sylow
242	C₂×Q₁₆	—	47	1	no	—
243	C₂×SD₁₆	—	47	2	no	—
244	C₂×D₈	C₄×D₄₇	1	1	yes	characteristic
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	—	47	1	no	—
247	C₂×M₁₆	—	47	1	no	—
248	C₂²×C₈	—	47	1	no	—
249	C₂²×C₈	—	47	1	no	—
250	C₂²×C₈	—	94	2	no	—
251	#126	—	94	2	no	—
252	#125	—	47	2	no	—
253	#124	—	47	2	no	—
254	#123	—	94	2	no	—
255	C₂²⋊Q₈	—	47	2	no	—
256	#121	—	47	2	no	—
257	C₄×Q₈	—	47	2	no	—
258	C₄×D₄	—	94	2	no	—
259	C₄×D₄	C₂²×D₄₇	1	2	yes	—
260	#117	—	94	2	no	—
261	C₂×C₄²	—	47	1	no	—
262	#107	—	47	1	no	—
263	#107	C₄×D₄₇	1	1	yes	characteristic
264	#106	—	47	2	no	—
265	C₄⋊C₈	—	47	2	no	—
266	C₄⋊C₈	—	94	1	no	—
267	C₄⋊C₈	—	94	2	no	—
268	C₄⋊C₈	—	94	2	no	—
269	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
270	#103	—	47	2	no	—
271	#103	—	47	2	no	—
272	#102	—	47	2	no	—
273	#102	C₄×D₄₇	1	2	yes	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₂²⋊C₈	—	94	1	no	—
276	C₂²⋊C₈	—	94	2	no	—
277	C₂²⋊C₈	—	94	2	no	—
278	#97	—	47	1	no	—
279	#97	—	47	1	no	—
280	#97	—	94	2	no	—
281	C₄×C₈	—	47	1	no	—
282	C₄×C₈	—	47	1	no	—
283	C₄×C₈	—	47	1	no	—
284	C₄×C₈	—	47	1	no	—
285	C₄×C₈	—	47	1	no	—
286	C₄×C₈	—	94	2	no	—
287	C₄×C₈	—	94	2	no	—
288	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
291	C₂×Q₈	—	47	2	no	—
292	C₂×D₄	—	94	2	no	—
293	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	47	1	no	—
296	C₂²×C₄	—	47	1	no	—
297	C₂²×C₄	—	47	1	no	—
298	Q₁₆	—	94	2	no	—
299	SD₁₆	—	188	2	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	4	4	no	—
302	M₁₆	—	94	2	no	—
303	M₁₆	—	94	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	47	1	no	—
307	C₂×C₈	—	47	1	no	—
308	C₂×C₈	—	47	1	no	—
309	C₂×C₈	—	47	1	no	—
310	C₂×C₈	—	94	2	no	—
311	C₂×C₈	—	94	2	no	—
312	C₂×C₈	—	94	1	no	—
313	C₂×C₈	—	188	1	no	—
314	C₂×C₈	—	188	4	no	—
315	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
316	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
317	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
318	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
319	C₄⋊C₄	—	47	2	no	—
320	C₄⋊C₄	—	94	2	no	—
321	C₄⋊C₄	—	94	2	no	—
322	C₄⋊C₄	—	94	2	no	—
323	C₄⋊C₄	—	94	2	no	—
324	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	94	2	no	—
327	C₂²⋊C₄	—	94	2	no	—
328	C₂²⋊C₄	—	94	2	no	—
329	C₂²⋊C₄	—	94	2	no	—
330	C₄²	—	47	1	no	—
331	C₄²	—	47	1	no	—
332	C₄²	—	47	1	no	—
333	C₄²	—	94	2	no	—
334	C₄²	C₂³×D₄₇	1	1	yes	characteristic
335	C₂³	—	2	2	no	—
336	C₂³	—	47	1	no	—
337	Q₈	—	94	2	no	—
338	Q₈	—	94	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	376	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	47	1	no	—
346	C₂×C₄	—	47	1	no	—
347	C₂×C₄	—	47	1	no	—
348	C₂×C₄	—	94	1	no	—
349	C₂×C₄	—	94	1	no	—
350	C₂×C₄	—	94	1	no	—
351	C₂×C₄	—	94	1	no	—
352	C₂×C₄	—	94	2	no	—
353	C₂×C₄	—	94	2	no	—
354	C₂×C₄	—	188	1	no	—
355	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
357	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	1	no	—
361	C₈	—	4	2	no	—
362	C₈	—	47	2	no	—
363	C₈	—	94	1	no	—
364	C₈	—	94	1	no	—
365	C₈	—	94	1	no	—
366	C₈	—	188	1	no	—
367	C₈	—	188	2	no	—
368	C₈	#13094032	1	2	yes	—
369	C₈	#13094043	1	2	yes	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	94	1	no	—
374	C₂²	—	94	1	no	—
375	C₂²	—	94	1	no	—
376	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	94	1	no	—
380	C₄	—	94	1	no	—
381	C₄	—	94	1	no	—
382	C₄	—	188	2	no	—
383	C₄	—	188	2	no	—
384	C₄	C₂×#13094032	1	1	yes	characteristic
385	C₄	#15689888	1	1	yes	characteristic
386	C₄	#135×D₄₇	1	1	yes	characteristic
387	C₄	C₂×#13094043	1	1	yes	characteristic
388	C₂	—	8	2	no	—
389	C₂	—	188	1	no	—
390	C₂	#20252133	1	1	yes	characteristic
391	C₂	C₄×#13094043	1	1	yes	characteristic
392	C₂	#20252300	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×#1589	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689356	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1589	D₄₇
hypocenter	C₄₇	#18069
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1589

Complements

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

N	H ≅ G/N	type
2. #20252395	C₂	semidirect
3. #20242284	C₂	semidirect
4. #20242229	C₂	semidirect
5. #20242221	C₂	semidirect
6. C₈×Dic₁₈₈	C₂	semidirect
7. D₈×C₄₇⋊C₈	C₂	semidirect
8. #20242135	C₂	semidirect
9. #20241793	C₂	semidirect
10. C₈×#10625324	C₂	semidirect
11. C₄₇×#1589	C₂	semidirect
195. #1589	D₄₇	semidirect
241. C₄₇	#18069	semidirect

Maximal subgroups

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

subgroup	index
#20253956	2
#20252395	2
#20242284	2
#20242229	2
#20242221	2
C₈×Dic₁₈₈	2
D₈×C₄₇⋊C₈	2
#20242135	2
#20241793	2
C₈×#10625324	2
C₄₇×#1589	2
#18069	47