Subgroups — #26124184

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251952	C₂	1	1	yes	maximal, characteristic
2	#20246008	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688894	C₂	1	2	yes	maximal
4	C₂×#15688891	C₂	1	2	yes	maximal
5	C₄₇×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
9	#15689150	—	2	4	no	—
10	#15689147	—	2	2	no	—
11	#15689021	—	2	2	no	—
12	#15688995	C₄	1	4	yes	—
13	#15688992	—	2	1	no	—
14	#15688988	—	2	1	no	—
15	#15688894	—	2	4	no	—
16	#15688891	—	2	4	no	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#343	—	2	1	no	—
20	C₄₇×#343	—	2	1	no	—
21	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
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	—	2	2	no	—
27	C₂×#10625329	D₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	#13093983	—	2	1	no	—
30	#13093983	—	2	1	no	—
31	#13093983	D₄	1	2	yes	—
32	#13093983	Q₈	1	2	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	1	no	—
34	C₂×C₄⋊Dic₄₇	—	2	1	no	—
35	C₂×C₄⋊Dic₄₇	—	2	1	no	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×Dic₄₇⋊C₄	—	2	4	no	—
39	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₄₇	—	2	1	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
42	C₄₇⋊#95	—	2	4	no	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	—	2	2	no	—
49	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	#117×C₄₇	—	2	2	no	—
52	#117×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
55	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₄₇	D₄	1	2	yes	—
57	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×Dic₄₇	—	2	2	no	—
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	8	2	no	—
74	M₁₆×C₄₇	—	2	4	no	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	—	4	2	no	—
77	M₁₆×C₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₂×C₈×C₄₇	—	4	2	no	—
81	C₄⋊C₄×C₄₇	—	2	2	no	—
82	C₄⋊C₄×C₄₇	—	2	2	no	—
83	C₂²⋊C₄×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₂²⋊Dic₄₇	—	2	2	no	—
87	C₂²⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	—	4	1	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	C₄⋊Dic₄₇	—	4	1	no	—
93	Dic₄₇⋊C₄	—	8	4	no	—
94	C₄×Dic₄₇	—	2	2	no	—
95	C₄×Dic₄₇	—	2	2	no	—
96	#10625329	—	2	4	no	—
97	#10625329	—	2	4	no	—
98	#10625329	—	4	2	no	—
99	#10625329	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	—	4	2	no	—
102	C₂×C₄₇⋊C₈	—	4	2	no	—
103	C₂³×C₄₇	—	4	1	no	—
104	C₂³×C₄₇	—	4	1	no	—
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₄₇	#117	1	1	yes	characteristic
108	C₂³×C₄₇	C₄×D₄	1	2	yes	—
109	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#121	1	2	yes	—
117	C₂×C₄×C₄₇	#121	1	2	yes	—
118	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#123	1	2	yes	—
126	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#125	1	2	yes	—
128	C₂×C₄×C₄₇	#125	1	2	yes	—
129	C₂×C₄×C₄₇	#126	1	2	yes	—
130	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	8	4	no	—
138	C₈×C₄₇	—	4	4	no	—
139	C₄₇⋊C₈	—	4	4	no	—
140	#11206	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	4	1	no	—
146	C₂²×C₄₇	—	4	1	no	—
147	C₂²×C₄₇	—	4	2	no	—
148	C₂²×C₄₇	#380	1	1	yes	characteristic
149	C₂²×C₄₇	#386	1	1	yes	characteristic
150	C₂²×C₄₇	#387	1	2	yes	—
151	C₂²×C₄₇	#389	1	2	yes	—
152	C₂²×C₄₇	#391	1	1	yes	characteristic
153	C₄×C₄₇	—	4	2	no	—
154	C₄×C₄₇	—	4	2	no	—
155	C₄×C₄₇	#394	1	1	yes	characteristic
156	C₄×C₄₇	#395	1	1	yes	characteristic
157	C₄×C₄₇	#396	1	2	yes	—
158	C₄×C₄₇	#398	1	1	yes	characteristic
159	C₄×C₄₇	#398	1	2	yes	—
160	C₄×C₄₇	#400	1	1	yes	characteristic
161	Dic₄₇	—	4	4	no	—
162	Dic₄₇	—	8	2	no	—
163	#2905	—	47	1	no	—
164	#2317	—	47	1	no	—
165	C₂×#344	—	47	2	no	—
166	#1742	D₄₇	1	1	yes	characteristic
167	C₂×#343	—	47	2	no	—
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	—	2	2	no	—
170	C₂×C₄₇	—	2	2	no	—
171	C₂×C₄₇	—	2	1	no	—
172	C₂×C₄₇	#1441	1	1	yes	characteristic
173	C₂×C₄₇	#1914	1	1	yes	characteristic
174	C₂×C₄₇	#1915	1	1	yes	characteristic
175	C₂²×M₁₆	—	47	2	no	—
176	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
177	C₂×#117	—	47	1	no	—
178	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
179	C₂²×C₄⋊C₄	—	47	1	no	—
180	#424	—	94	4	no	—
181	#423	—	94	2	no	—
182	#389	—	94	2	no	—
183	#385	—	47	4	no	—
184	#384	—	94	1	no	—
185	#382	—	94	1	no	—
186	#344	—	94	4	no	—
187	#343	—	2	1	no	—
188	#343	—	2	1	no	—
189	#343	—	94	4	no	—
190	C₄₇	#11206	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	47	2	no	—
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	47	4	no	—
199	C₂×M₁₆	—	94	2	no	—
200	C₂×M₁₆	—	94	2	no	—
201	C₂×M₁₆	—	94	2	no	—
202	C₂×M₁₆	—	94	2	no	—
203	C₂×M₁₆	—	94	2	no	—
204	C₂×M₁₆	Dic₉₄	1	4	yes	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	94	2	no	—
207	#117	—	2	2	no	—
208	#117	—	2	2	no	—
209	#117	—	47	2	no	—
210	#117	—	47	2	no	—
211	#117	—	94	1	no	—
212	#117	—	94	1	no	—
213	C₂×C₄⋊C₄	—	94	1	no	—
214	C₂×C₄⋊C₄	—	94	4	no	—
215	C₂×C₄⋊C₄	—	94	1	no	—
216	C₂×C₄⋊C₄	—	94	2	no	—
217	C₂×C₄⋊C₄	—	94	2	no	—
218	C₂×C₄⋊C₄	—	94	1	no	—
219	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	—	47	2	no	—
222	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
223	C₂×C₄²	—	94	1	no	—
224	C₂×C₄²	D₁₈₈	1	2	yes	—
225	C₄⋊C₈	—	94	8	no	—
226	#95	—	94	4	no	—
227	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	94	4	no	—
230	C₂²×C₄	—	94	2	no	—
231	C₂²×C₄	—	376	2	no	—
232	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
233	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
234	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
235	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
236	C₂²×C₄	#10625350	1	2	yes	—
237	C₂²×C₄	D₄×D₄₇	1	2	yes	—
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	#10625352	1	1	yes	characteristic
240	C₂²×C₄	#10625352	1	2	yes	—
241	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	#10625354	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
244	M₁₆	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	94	4	no	—
249	M₁₆	—	94	4	no	—
250	M₁₆	—	188	2	no	—
251	M₁₆	—	188	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	94	4	no	—
256	C₂×C₈	—	188	2	no	—
257	C₂×C₈	—	188	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	94	2	no	—
261	C₄⋊C₄	—	94	2	no	—
262	C₄⋊C₄	—	376	4	no	—
263	C₄⋊C₄	—	188	1	no	—
264	C₄⋊C₄	—	188	2	no	—
265	C₄⋊C₄	—	188	1	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	47	4	no	—
268	C₂²⋊C₄	—	94	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	94	2	no	—
272	C₄²	—	94	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
276	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
277	C₂³	#13093982	1	1	yes	characteristic
278	C₂³	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
279	C₂³	C₄×Dic₉₄	1	2	yes	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	376	4	no	—
286	C₂×C₄	—	188	4	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	#13093994	1	1	yes	characteristic
291	C₂×C₄	#13093995	1	1	yes	characteristic
292	C₂×C₄	#13093996	1	2	yes	—
293	C₂×C₄	#13093997	1	2	yes	—
294	C₂×C₄	#13093999	1	1	yes	characteristic
295	C₂×C₄	#13094000	1	1	yes	characteristic
296	C₂×C₄	#13094001	1	2	yes	—
297	C₂×C₄	#13094001	1	2	yes	—
298	C₂×C₄	#13094003	1	2	yes	—
299	C₂×C₄	#13094004	1	1	yes	characteristic
300	C₂×C₄	#13094008	1	1	yes	characteristic
301	C₂×C₄	#13094009	1	2	yes	—
302	C₂×C₄	#13094012	1	1	yes	characteristic
303	C₂×C₄	#13094014	1	2	yes	—
304	C₂×C₄	#13094015	1	2	yes	—
305	C₂×C₄	#13094017	1	2	yes	—
306	C₂×C₄	#13094023	1	2	yes	—
307	C₂×C₄	#13094024	1	1	yes	characteristic
308	C₈	—	4	4	no	—
309	C₈	—	188	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	#15688980	1	1	yes	characteristic
318	C₂²	#15689000	1	1	yes	characteristic
319	C₂²	#15689010	1	2	yes	—
320	C₂²	#15689020	1	2	yes	—
321	C₂²	#15689029	1	1	yes	characteristic
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	376	2	no	—
325	C₄	—	188	4	no	—
326	C₄	#15689041	1	1	yes	characteristic
327	C₄	#15689044	1	1	yes	characteristic
328	C₄	#15689046	1	2	yes	—
329	C₄	#15689055	1	1	yes	characteristic
330	C₄	#15689056	1	2	yes	—
331	C₄	#15689063	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	—	2	1	no	—
336	C₂	#20241642	1	1	yes	characteristic
337	C₂	#20243933	1	1	yes	characteristic
338	C₂	#20243938	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689000
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1742	C₂
socle	C₂²×C₄₇	#386
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1742	D₄₇
hypocenter	C₄₇	#11206
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
190. C₄₇	#11206	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
#20246008	2
C₂×#15688894	2
C₂×#15688891	2
C₄₇×#1742	2
#11206	47