Subgroups — #26121945

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688883	C₂	1	2	yes	maximal
3	C₂×#15688883	C₂	1	2	yes	maximal
4	#20243082	C₂	1	1	yes	maximal, characteristic
5	C₄×#95×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	complemented
10	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	complemented
11	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
12	#15688969	—	2	1	no	—
13	C₄²×Dic₄₇	—	2	1	no	—
14	#15688883	—	2	4	no	—
15	#15688883	C₄	1	8	yes	complemented
16	#15688878	—	2	1	no	—
17	#15688878	—	2	1	no	—
18	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#95×C₄₇	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	—	2	1	no	—
30	#13093983	—	2	1	no	—
31	#13093983	C₂×C₄	1	4	yes	—
32	#13093983	D₄	1	2	yes	—
33	#13093983	Q₈	1	2	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
38	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
39	C₂×C₄×Dic₄₇	—	2	1	no	—
40	C₂×C₄×Dic₄₇	—	2	1	no	—
41	C₂×C₄×Dic₄₇	—	2	2	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	C₄₇⋊#95	—	2	2	no	—
44	C₄₇⋊#95	—	2	2	no	—
45	C₂³×C₄×C₄₇	—	2	2	no	—
46	C₂³×C₄×C₄₇	D₄	1	2	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	—	2	2	no	—
52	C₂×C₄²×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
59	#95×C₄₇	—	2	8	no	—
60	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	—	4	4	no	—
67	C₂²×C₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	#41	1	2	yes	—
79	C₂²×C₄×C₄₇	#41	1	2	yes	—
80	C₂²×C₄×C₄₇	#41	1	2	yes	—
81	C₂²×Dic₄₇	—	2	1	no	—
82	C₂²×Dic₄₇	—	2	1	no	—
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	4	no	—
89	C₄²×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	4	2	no	—
91	C₄²×C₄₇	—	4	2	no	—
92	C₂²⋊Dic₄₇	—	2	4	no	—
93	C₂²⋊Dic₄₇	—	2	2	no	—
94	C₂²⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	1	no	—
96	C₄⋊Dic₄₇	—	2	1	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	C₄²	1	2	yes	—
99	C₄⋊Dic₄₇	C₄²	1	2	yes	—
100	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
101	C₄×Dic₄₇	—	2	4	no	—
102	C₄×Dic₄₇	—	2	2	no	—
103	C₄×Dic₄₇	—	2	2	no	—
104	C₄×Dic₄₇	—	4	4	no	—
105	C₄×Dic₄₇	—	4	4	no	—
106	#10625329	—	2	4	no	—
107	#10625329	—	2	4	no	—
108	#10625329	—	4	2	no	—
109	#10625329	—	4	2	no	—
110	C₂×C₄₇⋊C₈	—	2	4	no	—
111	C₂×C₄₇⋊C₈	—	4	2	no	—
112	C₂×C₄₇⋊C₈	—	4	2	no	—
113	C₂³×C₄₇	—	2	2	no	—
114	C₂³×C₄₇	—	2	2	no	—
115	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂³×C₄₇	C₄×D₄	1	2	yes	—
117	C₂³×C₄₇	#121	1	1	yes	characteristic
118	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂³×C₄₇	#123	1	2	yes	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	8	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	4	8	no	—
127	C₂×C₄×C₄₇	—	4	8	no	—
128	C₂×C₄×C₄₇	—	4	8	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	#104	1	4	yes	—
132	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#117	1	2	yes	—
136	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#117	1	2	yes	—
140	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#123	1	2	yes	—
146	C₂×C₄×C₄₇	#124	1	2	yes	—
147	C₂×C₄×C₄₇	#125	1	2	yes	—
148	C₂×C₄×C₄₇	#126	1	2	yes	—
149	C₂×C₄×C₄₇	#126	1	2	yes	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	2	4	no	—
153	C₂×Dic₄₇	—	4	2	no	—
154	C₂×Dic₄₇	—	4	2	no	—
155	C₂×Dic₄₇	—	4	2	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	C₄₇⋊C₈	—	4	4	no	—
158	#10808	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	2	2	no	—
161	C₂²×C₄₇	—	2	4	no	—
162	C₂²×C₄₇	—	2	2	no	—
163	C₂²×C₄₇	—	2	2	no	—
164	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄₇	#380	1	1	yes	characteristic
166	C₂²×C₄₇	#381	1	1	yes	characteristic
167	C₂²×C₄₇	#387	1	2	yes	—
168	C₂²×C₄₇	#388	1	2	yes	—
169	C₂²×C₄₇	C₂×#104	1	2	yes	—
170	C₂²×C₄₇	#421	1	2	yes	—
171	C₄×C₄₇	—	2	4	no	—
172	C₄×C₄₇	—	4	8	no	—
173	C₄×C₄₇	—	4	2	no	—
174	C₄×C₄₇	#376	1	1	yes	characteristic
175	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
176	C₄×C₄₇	#382	1	1	yes	characteristic
177	C₄×C₄₇	#383	1	1	yes	characteristic
178	C₄×C₄₇	#387	1	2	yes	—
179	C₄×C₄₇	#388	1	2	yes	—
180	Dic₄₇	—	4	4	no	—
181	Dic₄₇	—	4	2	no	—
182	Dic₄₇	—	4	2	no	—
183	C₄×#117	—	47	1	no	—
184	C₂×#339	—	47	2	no	—
185	C₂×#339	—	47	2	no	—
186	#1728	—	47	1	no	—
187	C₄×#95	D₄₇	1	1	yes	characteristic
188	C₂×C₄₇	—	2	2	no	—
189	C₂×C₄₇	—	2	2	no	—
190	C₂×C₄₇	#1434	1	1	yes	characteristic
191	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
192	C₂×C₄₇	#1759	1	1	yes	characteristic
193	C₂×C₄₇	#1806	1	2	yes	—
194	C₂×C₄₇	#1826	1	2	yes	—
195	C₂²×M₁₆	—	47	2	no	—
196	C₂×#117	—	47	1	no	—
197	C₂×#117	—	47	1	no	—
198	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
199	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
200	C₄×C₄⋊C₄	—	47	2	no	—
201	C₄×C₄⋊C₄	—	47	2	no	—
202	C₄×C₂²⋊C₄	—	94	2	no	—
203	#376	—	94	1	no	—
204	C₂×#95	—	2	2	no	—
205	C₄³	—	94	1	no	—
206	#339	—	47	8	no	—
207	#339	—	94	4	no	—
208	#336	—	94	1	no	—
209	#336	—	94	1	no	—
210	C₄₇	#10808	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₄	—	2	2	no	—
212	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
213	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×M₁₆	—	47	4	no	—
215	C₂×M₁₆	—	94	2	no	—
216	C₂×M₁₆	—	94	2	no	—
217	C₂×M₁₆	—	94	2	no	—
218	C₂×M₁₆	—	94	2	no	—
219	C₂×M₁₆	—	94	2	no	—
220	C₂²×C₈	—	94	2	no	—
221	#117	—	47	4	no	—
222	#117	—	47	2	no	—
223	#117	—	47	2	no	—
224	#117	—	94	2	no	—
225	#117	—	94	1	no	—
226	#117	—	94	1	no	—
227	C₂×C₄⋊C₄	—	47	1	no	—
228	C₂×C₄⋊C₄	—	47	1	no	—
229	C₂×C₄⋊C₄	—	47	2	no	—
230	C₂×C₂²⋊C₄	—	94	1	no	—
231	C₂×C₂²⋊C₄	—	94	1	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	—	2	2	no	—
236	C₂×C₄²	—	2	2	no	—
237	C₂×C₄²	—	2	2	no	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	—	94	1	no	—
240	C₂×C₄²	—	94	1	no	—
241	C₂×C₄²	—	94	2	no	—
242	C₂×C₄²	—	94	2	no	—
243	C₂×C₄²	C₄×D₄₇	1	4	yes	—
244	C₂×C₄²	C₄×D₄₇	1	2	yes	—
245	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
246	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
247	#95	—	2	8	no	—
248	#95	—	94	2	no	—
249	#95	—	94	2	no	—
250	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
251	C₂²×C₄	—	2	8	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	4	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	94	1	no	—
260	C₂²×C₄	—	94	1	no	—
261	C₂²×C₄	—	94	2	no	—
262	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
263	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
264	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
266	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
267	C₂²×C₄	#10625350	1	2	yes	—
268	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
269	C₂²×C₄	#10625352	1	1	yes	characteristic
270	C₂²×C₄	#10625352	1	2	yes	—
271	C₂²×C₄	#10625352	1	2	yes	—
272	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
273	C₂²×C₄	#10625354	1	1	yes	characteristic
274	M₁₆	—	94	4	no	—
275	M₁₆	—	94	4	no	—
276	M₁₆	—	188	2	no	—
277	M₁₆	—	188	2	no	—
278	C₂×C₈	—	94	4	no	—
279	C₂×C₈	—	188	2	no	—
280	C₂×C₈	—	188	2	no	—
281	C₄⋊C₄	—	47	2	no	—
282	C₄⋊C₄	—	47	2	no	—
283	C₄⋊C₄	—	47	4	no	—
284	C₄⋊C₄	—	94	1	no	—
285	C₄⋊C₄	—	94	1	no	—
286	C₄⋊C₄	—	94	2	no	—
287	C₂²⋊C₄	—	94	4	no	—
288	C₂²⋊C₄	—	94	2	no	—
289	C₂²⋊C₄	—	94	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	4	no	—
295	C₄²	—	2	4	no	—
296	C₄²	—	4	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	—	94	4	no	—
299	C₄²	—	94	2	no	—
300	C₄²	—	94	2	no	—
301	C₄²	—	188	4	no	—
302	C₄²	—	188	4	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
306	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
307	C₂³	#13093995	1	1	yes	characteristic
308	C₂³	#13094000	1	1	yes	characteristic
309	C₂³	#13094005	1	2	yes	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	8	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	8	no	—
317	C₂×C₄	—	4	8	no	—
318	C₂×C₄	—	4	8	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	4	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	#13093952	1	4	yes	—
329	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
330	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
331	C₂×C₄	#13093980	1	1	yes	characteristic
332	C₂×C₄	#13093980	1	2	yes	—
333	C₂×C₄	#13093980	1	1	yes	characteristic
334	C₂×C₄	#13093981	1	1	yes	characteristic
335	C₂×C₄	#13093982	1	1	yes	characteristic
336	C₂×C₄	#13093982	1	2	yes	—
337	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
338	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
339	C₂×C₄	#13093989	1	2	yes	—
340	C₂×C₄	#13093999	1	1	yes	characteristic
341	C₂×C₄	#13094004	1	1	yes	characteristic
342	C₂×C₄	#13094008	1	2	yes	—
343	C₂×C₄	#13094010	1	2	yes	—
344	C₂×C₄	#13094014	1	2	yes	—
345	C₂×C₄	#13094017	1	2	yes	—
346	C₂×C₄	#13094019	1	2	yes	—
347	C₈	—	188	4	no	—
348	C₂²	—	2	4	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
354	C₂²	#15688980	1	1	yes	characteristic
355	C₂²	#15688984	1	1	yes	characteristic
356	C₂²	#15689004	1	2	yes	—
357	C₂²	#15689011	1	2	yes	—
358	C₂²	C₂×#13093952	1	2	yes	—
359	C₂²	#15689141	1	2	yes	—
360	C₄	—	2	4	no	—
361	C₄	—	4	8	no	—
362	C₄	—	4	2	no	—
363	C₄	—	188	4	no	—
364	C₄	—	188	2	no	—
365	C₄	—	188	2	no	—
366	C₄	#15688968	1	1	yes	characteristic
367	C₄	#15688974	1	1	yes	characteristic
368	C₄	#15688987	1	1	yes	characteristic
369	C₄	#15688990	1	1	yes	characteristic
370	C₄	#15689008	1	2	yes	—
371	C₄	#15689015	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	2	2	no	—
374	C₂	#20241612	1	1	yes	characteristic
375	C₂	C₄×#13093952	1	1	yes	characteristic
376	C₂	#20243199	1	1	yes	characteristic
377	C₂	#20243409	1	2	yes	—
378	C₂	#20243488	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄×#95×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₄×#95	D₄₇
hypocenter	C₄₇	#10808
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
9. C₄×C₄⋊Dic₄₇	C₄	semidirect
10. C₄×C₄⋊Dic₄₇	C₄	semidirect
15. #15688883	C₄	semidirect
210. C₄₇	#10808	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093983	2
C₂×#15688883	2
#20243082	2
C₄×#95×C₄₇	2
#10808	47