Subgroups — #26122192

Order 12032.

746 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689159	C₂	1	2	yes	maximal
2	C₂×#15689079	C₂	1	1	yes	maximal, characteristic
3	#20243121	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688879	C₂	1	2	yes	maximal
5	C₄₇×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	1	yes	characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689159	—	2	4	no	—
10	#15689159	C₄	1	8	yes	complemented
11	C₂×#13093957	—	2	2	no	—
12	C₂×#13093955	—	2	2	no	—
13	#15689079	—	2	2	no	—
14	#15689079	—	2	2	no	—
15	C₂×#13093935	C₄	1	2	yes	complemented
16	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	complemented
17	#15688890	—	2	2	no	—
18	#15688879	—	2	4	no	—
19	C₂×#117×C₄₇	C₂²	1	2	yes	—
20	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#342	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	D₄	1	2	yes	—
25	C₂×#10625329	Q₈	1	2	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
31	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
32	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
33	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
34	#13093983	—	2	2	no	—
35	#13093983	—	2	2	no	—
36	#13093983	—	2	2	no	—
37	#13093983	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	—	2	2	no	—
41	#13093957	—	2	8	no	—
42	#13093955	—	2	8	no	—
43	#13093935	—	2	4	no	—
44	C₄×C₄₇⋊C₈	—	2	2	no	—
45	C₄×C₄₇⋊C₈	—	2	2	no	—
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	#117×C₄₇	—	2	2	no	—
48	#117×C₄₇	—	2	2	no	—
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	—	2	2	no	—
52	#117×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
57	C₂×C₄²×C₄₇	—	2	2	no	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
59	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×Dic₄₇	—	4	2	no	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	4	2	no	—
76	C₂²⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	—	2	4	no	—
78	C₂²⋊C₄×C₄₇	—	4	2	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄²×C₄₇	—	4	2	no	—
83	C₂²⋊Dic₄₇	—	2	4	no	—
84	C₂²⋊Dic₄₇	—	4	2	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	4	2	no	—
87	C₄×Dic₄₇	—	2	4	no	—
88	C₄×Dic₄₇	—	4	2	no	—
89	#10625329	—	2	4	no	—
90	#10625329	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
94	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
95	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
98	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
99	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
100	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
101	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
102	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
103	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
104	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
105	C₂³×C₄₇	—	4	2	no	—
106	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₄₇	C₄×D₄	1	2	yes	—
108	C₂³×C₄₇	#121	1	2	yes	—
109	C₂³×C₄₇	#127	1	1	yes	characteristic
110	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	8	4	no	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₄₇	#121	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	#124	1	2	yes	—
130	C₂×C₄×C₄₇	#125	1	2	yes	—
131	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	8	4	no	—
135	C₄₇⋊C₈	—	2	4	no	—
136	C₄₇⋊C₈	#95	1	2	yes	—
137	C₄₇⋊C₈	#95	1	2	yes	—
138	C₄₇⋊C₈	#95	1	4	yes	—
139	#10866	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	4	2	no	—
143	C₂²×C₄₇	—	4	1	no	—
144	C₂²×C₄₇	—	4	1	no	—
145	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	#380	1	1	yes	characteristic
147	C₂²×C₄₇	#384	1	1	yes	characteristic
148	C₂²×C₄₇	#389	1	2	yes	—
149	C₂²×C₄₇	#390	1	2	yes	—
150	C₄×C₄₇	—	4	4	no	—
151	C₄×C₄₇	—	8	4	no	—
152	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
153	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₄₇	#380	1	1	yes	characteristic
155	C₄×C₄₇	#382	1	1	yes	characteristic
156	C₄×C₄₇	#389	1	2	yes	—
157	C₄×C₄₇	#391	1	2	yes	—
158	Dic₄₇	—	8	4	no	—
159	C₂×#427	—	47	2	no	—
160	C₂×#405	—	47	1	no	—
161	#1740	—	47	1	no	—
162	#1740	D₄₇	1	1	yes	characteristic
163	C₂×H(3,ℤ/4)	—	47	2	no	—
164	C₂×C₄₇	—	2	2	no	—
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
169	C₂×C₄₇	#1812	1	2	yes	—
170	C₂²×M₁₆	—	47	1	no	—
171	C₂³×C₈	—	47	1	no	—
172	C₂×#117	—	47	2	no	—
173	C₂×#117	C₂×D₄₇	1	2	yes	—
174	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
175	#427	—	47	8	no	—
176	#427	—	94	4	no	—
177	C₂×#107	—	94	2	no	—
178	C₂×#106	—	94	2	no	—
179	#405	—	94	2	no	—
180	#405	—	94	2	no	—
181	C₂×#97	—	47	2	no	—
182	C₂×C₄×C₈	—	47	2	no	—
183	#342	—	2	2	no	—
184	#342	—	94	2	no	—
185	H(3,ℤ/4)	—	94	4	no	—
186	C₄₇	#10866	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
188	C₂×M₁₆	—	47	2	no	—
189	C₂×M₁₆	—	47	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂²×C₈	—	47	1	no	—
193	C₂²×C₈	—	47	1	no	—
194	C₂²×C₈	—	47	4	no	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	47	2	no	—
197	C₂²×C₈	—	47	2	no	—
198	C₂²×C₈	—	47	2	no	—
199	C₂²×C₈	—	47	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	2	2	no	—
205	#117	—	47	4	no	—
206	#117	—	94	2	no	—
207	#117	—	94	2	no	—
208	#117	—	94	2	no	—
209	#117	C₄×D₄₇	1	4	yes	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	94	2	no	—
212	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	—	94	2	no	—
215	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	—	94	2	no	—
218	C₂×C₄²	C₄×D₄₇	1	2	yes	—
219	#107	—	94	8	no	—
220	#106	—	94	8	no	—
221	#97	—	94	4	no	—
222	C₄×C₈	—	94	2	no	—
223	C₄×C₈	—	94	2	no	—
224	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	188	2	no	—
227	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
228	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
231	C₂²×C₄	#10625350	1	2	yes	—
232	C₂²×C₄	D₄×D₄₇	1	2	yes	—
233	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
235	C₂²×C₄	#10625352	1	2	yes	—
236	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
238	M₁₆	—	94	4	no	—
239	M₁₆	—	94	4	no	—
240	C₂×C₈	—	47	2	no	—
241	C₂×C₈	—	47	2	no	—
242	C₂×C₈	—	47	1	no	—
243	C₂×C₈	—	47	1	no	—
244	C₂×C₈	—	47	2	no	—
245	C₂×C₈	—	47	4	no	—
246	C₂×C₈	—	47	4	no	—
247	C₂×C₈	—	47	2	no	—
248	C₂×C₈	—	47	2	no	—
249	C₂×C₈	—	47	4	no	—
250	C₂×C₈	—	47	4	no	—
251	C₂×C₈	—	47	4	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	—	94	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	94	4	no	—
258	C₄⋊C₄	—	188	2	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	94	4	no	—
263	C₂²⋊C₄	—	188	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	94	4	no	—
269	C₄²	—	188	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
272	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
273	C₂³	#13093996	1	2	yes	—
274	C₂³	#13094020	1	1	yes	characteristic
275	C₂³	#13094022	1	1	yes	characteristic
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	—	376	4	no	—
281	C₂×C₄	—	188	4	no	—
282	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
285	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
286	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
287	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
288	C₂×C₄	#13093980	1	1	yes	characteristic
289	C₂×C₄	#13093981	1	1	yes	characteristic
290	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
291	C₂×C₄	#13093989	1	2	yes	—
292	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
293	C₂×C₄	#13093997	1	2	yes	—
294	C₂×C₄	#13094001	1	2	yes	—
295	C₂×C₄	#13094002	1	2	yes	—
296	C₂×C₄	#13094010	1	2	yes	—
297	C₂×C₄	#13094014	1	2	yes	—
298	C₂×C₄	#13094023	1	1	yes	characteristic
299	C₂×C₄	#13094023	1	1	yes	characteristic
300	C₈	—	47	2	no	—
301	C₈	—	47	2	no	—
302	C₈	—	47	4	no	—
303	C₈	—	94	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
310	C₂²	#15688981	1	1	yes	characteristic
311	C₂²	#15688992	1	1	yes	characteristic
312	C₂²	#15689018	1	2	yes	—
313	C₂²	#15689023	1	2	yes	—
314	C₄	—	4	4	no	—
315	C₄	—	8	4	no	—
316	C₄	—	376	4	no	—
317	C₄	#95×D₄₇	1	1	yes	characteristic
318	C₄	#15688974	1	1	yes	characteristic
319	C₄	#15688982	1	1	yes	characteristic
320	C₄	#15688987	1	1	yes	characteristic
321	C₄	#15689020	1	2	yes	—
322	C₄	#15689027	1	2	yes	—
323	C₂	—	2	2	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	2	no	—
327	C₂	Dic₄₇⋊#95	1	1	yes	characteristic
328	C₂	#20243432	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093980
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1740	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1740	D₄₇
hypocenter	C₄₇	#10866
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
10. #15689159	C₄	semidirect
15. C₂×#13093935	C₄	semidirect
16. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
186. C₄₇	#10866	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689159	2
C₂×#15689079	2
#20243121	2
C₂×#15688879	2
C₄₇×#1740	2
#10866	47