Subgroups — #26121692

Order 12032.

1202 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689097	C₂	1	2	yes	maximal
2	#20243105	C₂	1	2	yes	maximal, complemented
3	C₂×#15688880	C₂	1	2	yes	maximal, complemented
4	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₂²	1	4	yes	—
6	#15689097	—	2	4	no	—
7	#15689097	C₄	1	8	yes	complemented
8	C₂×#13093939	—	2	4	no	—
9	#15688886	—	2	4	no	—
10	#15688880	—	2	4	no	—
11	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#131×C₄₇	—	2	4	no	—
13	C₂×#131×C₄₇	—	2	4	no	—
14	C₂×#131×C₄₇	—	2	4	no	—
15	C₂²×M₁₆×C₄₇	C₄	1	2	yes	complemented
16	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₄₇	C₄	1	2	yes	complemented
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	#13093983	—	2	4	no	—
20	#13093983	—	2	4	no	—
21	#13093983	—	2	4	no	—
22	#13093983	C₂×C₄	1	8	yes	—
23	C₂×C₄⋊Dic₄₇	—	2	4	no	—
24	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
25	C₂×C₄×Dic₄₇	—	2	4	no	—
26	#13093939	—	2	16	no	—
27	C₂×#41×C₄₇	—	2	2	no	—
28	C₂×#41×C₄₇	—	2	2	no	—
29	C₂×#41×C₄₇	—	2	2	no	—
30	C₂×#41×C₄₇	—	2	2	no	—
31	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×#41×C₄₇	D₄	1	4	yes	—
33	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
34	C₂²×D₄×C₄₇	D₄	1	2	yes	—
35	C₂²×D₄×C₄₇	Q₈	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	#131×C₄₇	—	2	32	no	—
39	C₂×M₁₆×C₄₇	—	2	4	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	4	no	—
43	C₂×M₁₆×C₄₇	—	2	4	no	—
44	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
45	C₂²×C₈×C₄₇	—	2	4	no	—
46	C₂²×C₈×C₄₇	—	2	2	no	—
47	C₂²×C₈×C₄₇	—	2	2	no	—
48	C₂²×C₈×C₄₇	—	2	4	no	—
49	C₂²×C₈×C₄₇	—	2	4	no	—
50	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	D₄	1	4	yes	—
54	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
56	#41×C₄₇	—	2	8	no	—
57	#41×C₄₇	—	2	8	no	—
58	#41×C₄₇	—	2	8	no	—
59	#41×C₄₇	—	2	8	no	—
60	C₂×Q₈×C₄₇	—	2	2	no	—
61	C₂×Q₈×C₄₇	—	2	2	no	—
62	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	—	2	2	no	—
66	C₂×D₄×C₄₇	—	2	2	no	—
67	C₂×D₄×C₄₇	C₂²⋊C₄	1	8	yes	—
68	C₂×D₄×C₄₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	#41	1	2	yes	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×Dic₄₇	—	4	4	no	—
85	M₁₆×C₄₇	—	2	16	no	—
86	M₁₆×C₄₇	—	2	8	no	—
87	C₂×C₈×C₄₇	—	2	16	no	—
88	C₂×C₈×C₄₇	—	2	1	no	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	—	2	8	no	—
95	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
97	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
98	C₂²⋊Dic₄₇	—	2	8	no	—
99	C₂²⋊Dic₄₇	—	4	4	no	—
100	C₄⋊Dic₄₇	—	2	8	no	—
101	C₄⋊Dic₄₇	—	4	4	no	—
102	C₄×Dic₄₇	—	2	8	no	—
103	C₄×Dic₄₇	—	4	4	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	4	2	no	—
107	C₂³×C₄₇	—	4	4	no	—
108	C₂³×C₄₇	#95	1	4	yes	—
109	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₄₇	#121	1	2	yes	—
112	C₂³×C₄₇	#121	1	2	yes	—
113	Q₈×C₄₇	—	2	8	no	—
114	D₄×C₄₇	—	2	16	no	—
115	D₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	#95	1	4	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
131	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
132	C₂×C₄×C₄₇	#120	1	4	yes	—
133	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₄₇	#123	1	4	yes	—
136	C₂×Dic₄₇	—	4	8	no	—
137	C₂×Dic₄₇	—	8	8	no	—
138	C₈×C₄₇	—	2	2	no	—
139	C₈×C₄₇	—	2	4	no	—
140	C₈×C₄₇	—	2	2	no	—
141	#10759	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	4	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	—	4	2	no	—
147	C₂²×C₄₇	—	4	4	no	—
148	C₂²×C₄₇	—	4	4	no	—
149	C₂²×C₄₇	—	4	4	no	—
150	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
151	C₂²×C₄₇	#380	1	1	yes	characteristic
152	C₂²×C₄₇	#380	1	1	yes	characteristic
153	C₂²×C₄₇	#386	1	4	yes	—
154	C₄×C₄₇	—	2	4	no	—
155	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₄₇	#379	1	1	yes	characteristic
158	C₄×C₄₇	#381	1	1	yes	characteristic
159	C₄×C₄₇	#385	1	4	yes	—
160	Dic₄₇	—	8	8	no	—
161	C₂²×#131	D₄₇	1	1	yes	characteristic
162	C₂×#410	—	47	2	no	—
163	#1735	—	47	2	no	—
164	C₂×H(3,ℤ/4)	—	47	2	no	—
165	C₂×C₄₇	—	2	4	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	—	4	4	no	—
169	C₂×C₄₇	C₂²⋊#95	1	1	yes	characteristic
170	C₂×C₄₇	#1812	1	2	yes	—
171	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
172	C₂×#131	—	2	4	no	—
173	C₂×#131	—	2	4	no	—
174	C₂×#131	—	2	4	no	—
175	C₂²×M₁₆	Dic₄₇	1	2	yes	complemented
176	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
177	C₂³×C₈	Dic₄₇	1	2	yes	complemented
178	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
179	C₂×#117	—	47	4	no	—
180	#410	—	47	8	no	—
181	#410	—	94	4	no	—
182	C₂×#99	—	94	4	no	—
183	#340	—	94	4	no	—
184	H(3,ℤ/4)	—	94	4	no	—
185	C₄₇	#10759	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	—	2	2	no	—
187	C₂×#41	—	2	2	no	—
188	C₂×#41	—	2	2	no	—
189	C₂×#41	—	2	2	no	—
190	C₂×#41	C₄×D₄₇	1	4	yes	—
191	C₂×#41	C₂²⋊D₄₇	1	4	yes	—
192	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
193	C₂²×D₄	Dic₉₄	1	1	yes	characteristic
194	C₂²×D₄	D₁₈₈	1	2	yes	—
195	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	#131	—	2	32	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	4	no	—
202	C₂×M₁₆	—	2	4	no	—
203	C₂×M₁₆	C₂²⋊D₄₇	1	4	yes	—
204	C₂²×C₈	—	2	4	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	4	no	—
208	C₂²×C₈	—	2	4	no	—
209	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
210	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
211	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
212	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
213	#117	—	47	8	no	—
214	#117	—	94	4	no	—
215	#117	—	94	4	no	—
216	#117	—	94	4	no	—
217	C₂×C₄⋊C₄	—	94	4	no	—
218	C₂×C₂²⋊C₄	—	94	4	no	—
219	C₂×C₄²	—	94	4	no	—
220	#99	—	94	16	no	—
221	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
222	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
223	#41	—	2	8	no	—
224	#41	—	2	8	no	—
225	#41	—	2	8	no	—
226	#41	—	2	8	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	Dic₄₇⋊C₄	1	4	yes	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	Dic₄₇⋊C₄	1	4	yes	—
235	C₂×D₄	(C₂×D₄₇)⋊C₄	1	8	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	188	4	no	—
241	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
242	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
243	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
244	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
246	C₂²×C₄	#10625350	1	2	yes	—
247	C₂²×C₄	#10625350	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
251	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
252	M₁₆	—	2	16	no	—
253	M₁₆	—	2	8	no	—
254	C₂×C₈	—	2	16	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	8	no	—
262	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
263	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
264	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
265	C₄⋊C₄	—	94	8	no	—
266	C₄⋊C₄	—	188	4	no	—
267	C₂²⋊C₄	—	94	8	no	—
268	C₂²⋊C₄	—	188	4	no	—
269	C₄²	—	94	8	no	—
270	C₄²	—	188	4	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	4	no	—
275	C₂³	C₄₇⋊#95	1	4	yes	—
276	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
277	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
278	C₂³	#13093998	1	2	yes	—
279	C₂³	#13093998	1	2	yes	—
280	Q₈	—	2	8	no	—
281	D₄	—	2	16	no	—
282	D₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	376	8	no	—
290	C₂×C₄	—	188	8	no	—
291	C₂×C₄	C₄₇⋊#95	1	4	yes	—
292	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
297	C₂×C₄	#13093983	1	1	yes	characteristic
298	C₂×C₄	#13093983	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
300	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
301	C₂×C₄	#13093993	1	4	yes	—
302	C₂×C₄	#13094003	1	2	yes	—
303	C₂×C₄	#13094003	1	2	yes	—
304	C₂×C₄	#13094007	1	4	yes	—
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	—	2	2	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
317	C₂²	#15688983	1	1	yes	characteristic
318	C₂²	#15688983	1	1	yes	characteristic
319	C₂²	#15689002	1	4	yes	—
320	C₄	—	2	4	no	—
321	C₄	—	376	8	no	—
322	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
323	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
324	C₄	#15688979	1	1	yes	characteristic
325	C₄	#15688986	1	1	yes	characteristic
326	C₄	#15688996	1	4	yes	—
327	C₂	—	2	4	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	4	4	no	—
331	C₂	#20241611	1	1	yes	characteristic
332	C₂	#20243434	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093983
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#10759
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #20243105	C₂	semidirect
3. C₂×#15688880	C₂	semidirect
7. #15689097	C₄	semidirect
15. C₂²×M₁₆×C₄₇	C₄	semidirect
17. C₂³×C₈×C₄₇	C₄	semidirect
175. C₂²×M₁₆	Dic₄₇	semidirect
177. C₂³×C₈	Dic₄₇	semidirect
185. C₄₇	#10759	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689097	2
#20243105	2
C₂×#15688880	2
C₂²×#131×C₄₇	2
#10759	47