Subgroups — #26247177

Order 12032.

608 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252327	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252290	C₂	1	1	yes	maximal, characteristic
3	#20244962	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244844	C₂	1	2	yes	maximal, complemented
5	#20244694	C₂	1	1	yes	maximal, characteristic
6	#20244676	C₂	1	1	yes	maximal, characteristic
7	#20244563	C₂	1	2	yes	maximal, complemented
8	#20243231	C₂	1	2	yes	maximal, complemented
9	#20243187	C₂	1	2	yes	maximal, complemented
10	#20243144	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2103	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689923	C₂²	1	1	yes	characteristic
13	C₂×#13094032	C₂²	1	2	yes	—
14	#15689676	C₂²	1	1	yes	characteristic
15	#15689252	—	2	2	no	—
16	#15689250	—	2	2	no	—
17	#15689244	—	2	2	no	—
18	#15689239	—	2	2	no	—
19	#15689149	C₂²	1	1	yes	characteristic
20	#15689146	C₂²	1	1	yes	characteristic
21	#15689139	C₂²	1	2	yes	—
22	C₂×#13093950	C₂²	1	2	yes	—
23	#15689120	C₂²	1	2	yes	—
24	#15689105	C₂²	1	2	yes	—
25	#15689096	C₂²	1	2	yes	—
26	#15689081	C₂²	1	2	yes	—
27	#15689081	C₂²	1	1	yes	characteristic
28	#15689079	C₂²	1	1	yes	characteristic
29	C₂×#13093934	C₂²	1	1	yes	characteristic
30	#15688894	C₂²	1	2	yes	—
31	#15688884	C₂²	1	2	yes	—
32	#15688880	C₂²	1	2	yes	—
33	#15688880	C₂²	1	1	yes	characteristic
34	C₄₇×#430	C₂²	1	1	yes	characteristic
35	C₄₇×#424	C₂²	1	1	yes	characteristic
36	C₄₇×#344	C₂²	1	2	yes	—
37	C₄₇×#341	C₂²	1	2	yes	—
38	C₄₇×#338	C₂²	1	1	yes	characteristic
39	C₂×#10625350	C₂³	1	2	yes	—
40	#13094057	—	2	2	no	—
41	#13094052	—	2	2	no	—
42	#13094046	—	2	2	no	—
43	#13094032	—	2	2	no	—
44	#13094032	—	2	2	no	—
45	#13094032	—	2	2	no	—
46	#13094032	—	2	2	no	—
47	C₂×#10625329	C₂³	1	1	yes	characteristic
48	C₂×#10625324	—	2	2	no	—
49	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₈×D₄₇	—	2	2	no	—
51	#13094023	—	2	1	no	—
52	#13094015	—	2	1	no	—
53	#13094001	—	2	2	no	—
54	C₄×Dic₉₄	—	2	2	no	—
55	#13093983	C₂³	1	2	yes	—
56	#13093983	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
58	Dic₄₇⋊C₈	—	2	2	no	—
59	Dic₄₇⋊C₈	—	2	2	no	—
60	#13093950	—	2	4	no	—
61	#13093950	—	2	2	no	—
62	#13093950	—	2	2	no	—
63	#13093949	—	2	2	no	—
64	#13093946	—	2	2	no	—
65	#13093942	—	2	2	no	—
66	#13093940	—	2	2	no	—
67	#13093938	—	2	4	no	—
68	#13093935	—	2	1	no	—
69	#13093934	—	2	1	no	—
70	#13093934	—	2	1	no	—
71	#13093934	—	2	2	no	—
72	#13093934	D₄	1	2	yes	—
73	C₄×C₄₇⋊C₈	—	2	1	no	—
74	C₈×Dic₄₇	—	2	2	no	—
75	C₈×Dic₄₇	D₄	1	2	yes	—
76	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
77	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
78	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	#117×C₄₇	C₂³	1	1	yes	characteristic
81	#108×C₄₇	—	2	2	no	—
82	C₄⋊C₈×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₂²⋊D₄₇	—	2	2	no	—
85	C₂²×Dic₄₇	—	2	1	no	—
86	#10625350	—	2	2	no	—
87	#10625350	—	2	2	no	—
88	#10625350	—	2	2	no	—
89	#10625350	—	2	2	no	—
90	C₂×D₁₈₈	#41	1	2	yes	—
91	C₂×C₄×D₄₇	—	2	2	no	—
92	C₂×Dic₉₄	—	2	2	no	—
93	C₂×Dic₉₄	#41	1	2	yes	—
94	M₁₆×C₄₇	—	2	2	no	—
95	M₁₆×C₄₇	—	2	2	no	—
96	M₁₆×C₄₇	—	2	2	no	—
97	M₁₆×C₄₇	—	2	2	no	—
98	M₁₆×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	1	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₄₇	—	2	1	no	—
105	C₂²⋊C₄×C₄₇	#41	1	2	yes	—
106	C₄²×C₄₇	—	2	1	no	—
107	C₂²⋊Dic₄₇	—	2	1	no	—
108	C₂²⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	—	2	1	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	Dic₄₇⋊C₄	—	2	2	no	—
113	Dic₄₇⋊C₄	—	2	2	no	—
114	C₄×Dic₄₇	—	2	1	no	—
115	C₄×Dic₄₇	—	2	2	no	—
116	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
117	C₄×Dic₄₇	#41	1	1	yes	characteristic
118	C₄×Dic₄₇	#41	1	1	yes	characteristic
119	#10625329	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	1	no	—
121	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄₇⋊C₈	#41	1	2	yes	—
124	C₂×C₄₇⋊C₈	#41	1	2	yes	—
125	Dic₁₈₈	—	2	2	no	—
126	D₃₇₆	—	2	2	no	—
127	#10625325	—	2	4	no	—
128	#10625324	—	4	2	no	—
129	#10625324	—	4	2	no	—
130	C₈×D₄₇	—	4	2	no	—
131	C₈×D₄₇	—	4	2	no	—
132	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂²×D₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
137	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂²⋊D₄₇	—	4	2	no	—
140	C₂²⋊D₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	2	1	no	—
142	C₂×Dic₄₇	—	2	1	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	8	1	no	—
147	D₁₈₈	—	2	2	no	—
148	D₁₈₈	—	2	2	no	—
149	C₄×D₄₇	—	4	2	no	—
150	C₄×D₄₇	—	4	2	no	—
151	Dic₉₄	—	2	2	no	—
152	Dic₉₄	—	2	2	no	—
153	Dic₉₄	—	8	2	no	—
154	C₈×C₄₇	—	2	2	no	—
155	C₈×C₄₇	—	4	1	no	—
156	C₈×C₄₇	—	4	2	no	—
157	C₈×C₄₇	—	4	2	no	—
158	C₄₇⋊C₈	—	4	1	no	—
159	C₄₇⋊C₈	#124	1	1	yes	characteristic
160	C₄₇⋊C₈	#124	1	2	yes	—
161	C₄₇⋊C₈	#124	1	1	yes	characteristic
162	#23834	—	47	1	no	maximal, Sylow
163	C₂²×C₄₇	—	4	1	no	—
164	C₂²×C₄₇	#525	1	1	yes	characteristic
165	C₂²×C₄₇	#529	1	2	yes	—
166	C₂×D₄₇	—	4	2	no	—
167	C₂×D₄₇	—	8	2	no	—
168	C₄×C₄₇	—	4	2	no	—
169	C₄×C₄₇	#525	1	1	yes	characteristic
170	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
171	C₄×C₄₇	#529	1	2	yes	—
172	Dic₄₇	—	4	1	no	—
173	Dic₄₇	—	4	2	no	—
174	Dic₄₇	—	4	1	no	—
175	Dic₄₇	—	4	2	no	—
176	Dic₄₇	—	8	2	no	—
177	#2961	—	47	1	no	—
178	#2955	—	47	1	no	—
179	#2103	D₄₇	1	1	yes	characteristic, complemented
180	#2102	—	47	1	no	—
181	#2080	—	47	2	no	—
182	#2053	—	47	1	no	—
183	#2050	—	47	1	no	—
184	#2031	—	47	2	no	—
185	#1765	—	47	2	no	—
186	#1757	—	47	2	no	—
187	#1747	—	47	1	no	—
188	C₂×C₄₇	—	2	1	no	—
189	C₂×C₄₇	—	2	2	no	—
190	C₂×C₄₇	#2398	1	1	yes	characteristic
191	D₄₇	—	8	2	no	—
192	#575	—	47	1	no	—
193	C₂×#131	—	47	2	no	—
194	#533	—	47	1	no	—
195	#446	—	94	2	no	—
196	#446	—	94	2	no	—
197	#444	—	94	2	no	—
198	#443	—	94	2	no	—
199	#430	C₂×D₄₇	1	1	yes	characteristic
200	#424	—	47	1	no	—
201	#424	C₂×D₄₇	1	1	yes	characteristic
202	#423	—	47	1	no	—
203	#421	—	47	2	no	—
204	C₂×#104	—	47	2	no	—
205	#416	—	47	2	no	—
206	#413	—	47	2	no	—
207	#410	—	47	2	no	—
208	#405	—	47	1	no	—
209	#405	—	47	2	no	—
210	#405	—	47	1	no	—
211	C₂×#97	—	47	1	no	—
212	#344	—	47	2	no	—
213	#344	C₂×D₄₇	1	2	yes	—
214	#341	C₂×D₄₇	1	2	yes	—
215	#339	—	47	2	no	—
216	#338	C₂×D₄₇	1	1	yes	characteristic
217	H(3,ℤ/4)	—	47	1	no	—
218	H(3,ℤ/4)	—	47	2	no	—
219	C₄₇	#23834	1	1	yes	characteristic, complemented, Sylow
220	C₂×#41	—	47	2	no	—
221	#137	—	94	2	no	—
222	AGL(1,ℤ/8)	—	94	2	no	—
223	#135	—	94	2	no	—
224	#131	—	94	2	no	—
225	#131	—	94	2	no	—
226	#131	—	94	2	no	—
227	#131	—	94	2	no	—
228	C₂×M₁₆	—	47	1	no	—
229	C₂×M₁₆	—	94	2	no	—
230	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
231	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
232	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
233	C₂²×C₈	—	47	1	no	—
234	C₂²×C₈	—	94	2	no	—
235	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₄⋊Q₈	—	94	1	no	—
237	#125	—	94	1	no	—
238	C₂²⋊Q₈	—	94	2	no	—
239	C₄×Q₈	—	94	2	no	—
240	#117	—	47	1	no	—
241	#117	—	47	2	no	—
242	#117	C₂²×D₄₇	1	1	yes	characteristic
243	C₂×C₄²	—	47	1	no	—
244	#108	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	94	2	no	—
247	C₄⋊C₈	—	94	2	no	—
248	#104	—	94	2	no	—
249	#104	—	94	2	no	—
250	#104	—	94	4	no	—
251	#103	—	94	2	no	—
252	#102	—	94	2	no	—
253	#101	—	94	2	no	—
254	#100	—	94	2	no	—
255	#99	—	94	4	no	—
256	#97	—	47	2	no	—
257	#97	—	94	1	no	—
258	#97	—	94	1	no	—
259	#97	—	94	1	no	—
260	#97	—	94	2	no	—
261	C₄×C₈	—	47	2	no	—
262	C₄×C₈	—	94	1	no	—
263	C₄×C₈	—	94	2	no	—
264	#41	—	94	2	no	—
265	#41	—	94	2	no	—
266	#41	—	94	2	no	—
267	#41	—	94	2	no	—
268	C₂×Q₈	—	47	2	no	—
269	C₂×Q₈	—	94	2	no	—
270	C₂×D₄	—	47	2	no	—
271	C₂×D₄	—	94	2	no	—
272	C₂²×C₄	—	94	1	no	—
273	C₂²×C₄	—	94	2	no	—
274	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
275	Q₁₆	—	94	2	no	—
276	SD₁₆	—	94	4	no	—
277	D₈	—	94	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	2	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	94	2	no	—
284	M₁₆	—	188	2	no	—
285	M₁₆	—	188	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	47	1	no	—
291	C₂×C₈	—	47	1	no	—
292	C₂×C₈	—	47	2	no	—
293	C₂×C₈	—	47	2	no	—
294	C₂×C₈	—	94	1	no	—
295	C₂×C₈	—	188	2	no	—
296	C₂×C₈	—	188	2	no	—
297	C₂×C₈	D₄×D₄₇	1	2	yes	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	94	2	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₄⋊C₄	—	94	2	no	—
302	C₄⋊C₄	—	94	1	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₂²⋊C₄	—	94	1	no	—
305	C₂²⋊C₄	—	94	2	no	—
306	C₂²⋊C₄	#10625354	1	2	yes	—
307	C₄²	—	2	1	no	—
308	C₄²	—	47	2	no	—
309	C₄²	—	47	1	no	—
310	C₄²	—	47	1	no	—
311	C₄²	—	94	1	no	—
312	C₄²	—	94	2	no	—
313	C₂³	—	94	2	no	—
314	C₂³	C₂×#10625354	1	1	yes	characteristic
315	Q₈	—	94	2	no	—
316	Q₈	—	94	2	no	—
317	Q₈	—	376	2	no	—
318	D₄	—	94	2	no	—
319	D₄	—	94	2	no	—
320	D₄	—	188	2	no	—
321	D₄	—	188	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	94	1	no	—
324	C₂×C₄	—	94	1	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	376	1	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
332	C₂×C₄	#41×D₄₇	1	2	yes	—
333	C₂×C₄	#41×D₄₇	1	2	yes	—
334	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	2	no	—
339	C₈	—	47	1	no	—
340	C₈	—	47	2	no	—
341	C₈	—	47	1	no	—
342	C₈	—	188	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	376	2	no	—
345	C₂²	—	188	2	no	—
346	C₂²	#15689636	1	1	yes	characteristic
347	C₂²	#15689658	1	2	yes	—
348	C₄	—	4	2	no	—
349	C₄	—	376	2	no	—
350	C₄	—	188	1	no	—
351	C₄	—	188	1	no	—
352	C₄	—	188	2	no	—
353	C₄	—	188	2	no	—
354	C₄	#15689631	1	1	yes	characteristic
355	C₄	#124×D₄₇	1	1	yes	characteristic
356	C₄	#15689656	1	2	yes	—
357	C₂	—	2	1	no	—
358	C₂	—	2	2	no	—
359	C₂	—	376	2	no	—
360	C₂	#20246677	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#124×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2103	C₂
socle	C₂×C₄₇	#2398
prime radical	#15689923	C₂²
hypercenter	#2103	D₄₇
hypocenter	C₄₇	#23834
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2103

Complements

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

N	H ≅ G/N	type
1. #20252327	C₂	semidirect
3. #20244962	C₂	semidirect
4. #20244844	C₂	semidirect
7. #20244563	C₂	semidirect
8. #20243231	C₂	semidirect
9. #20243187	C₂	semidirect
10. #20243144	C₂	semidirect
11. C₄₇×#2103	C₂	semidirect
179. #2103	D₄₇	semidirect
219. C₄₇	#23834	semidirect

Maximal subgroups

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

subgroup	index
#20252327	2
#20252290	2
#20244962	2
#20244844	2
#20244694	2
#20244676	2
#20244563	2
#20243231	2
#20243187	2
#20243144	2
C₄₇×#2103	2
#23834	47