Subgroups — #26216652

Order 12032.

608 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251817	C₂	1	1	yes	maximal, characteristic
2	#20243537	C₂	1	1	yes	maximal, characteristic, complemented
3	#20243509	C₂	1	2	yes	maximal, complemented
4	#20243491	C₂	1	2	yes	maximal
5	#20243437	C₂	1	1	yes	maximal, characteristic
6	#20243433	C₂	1	1	yes	maximal, characteristic
7	#20243271	C₂	1	2	yes	maximal, complemented
8	#20243187	C₂	1	2	yes	maximal
9	#20243144	C₂	1	1	yes	maximal, characteristic, complemented
10	#20243142	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1841	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094032	C₂²	1	1	yes	characteristic
13	C₂×#13094032	C₂²	1	1	yes	characteristic
14	C₄×#10625350	C₂²	1	1	yes	characteristic
15	C₂×#13093959	C₂²	1	2	yes	—
16	#15689159	C₂²	1	2	yes	—
17	#15689105	C₂²	1	2	yes	—
18	#15689096	C₂²	1	2	yes	—
19	#15689081	C₂²	1	2	yes	—
20	#15689081	C₂²	1	1	yes	characteristic
21	#15689081	C₂²	1	1	yes	characteristic
22	#15689079	C₂²	1	2	yes	—
23	#15689077	C₄	1	2	yes	complemented
24	#15689077	C₄	1	2	yes	complemented
25	#15689075	C₄	1	2	yes	complemented
26	C₂×#13093935	C₂²	1	1	yes	characteristic
27	#97×D₄₇	C₄	1	2	yes	complemented
28	#15688894	C₂²	1	2	yes	—
29	#15688883	C₂²	1	2	yes	—
30	C₄₇⋊#338	C₂²	1	1	yes	characteristic
31	#15688880	C₂²	1	2	yes	—
32	#15688880	C₂²	1	1	yes	characteristic
33	#15688879	C₂²	1	2	yes	—
34	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#344	C₂²	1	2	yes	—
36	C₄₇×#339	C₂²	1	2	yes	—
37	C₄₇×#338	C₂²	1	1	yes	characteristic
38	C₄₇×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
39	C₂×#10625350	C₂³	1	1	yes	characteristic
40	#13094032	—	2	2	no	—
41	#13094032	—	2	2	no	—
42	#13094032	—	2	2	no	—
43	#13094032	—	2	2	no	—
44	C₂×#10625329	C₂³	1	2	yes	—
45	C₂×#10625324	C₂×C₄	1	2	yes	—
46	C₂×#10625324	C₂×C₄	1	2	yes	—
47	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
51	C₄×Dic₉₄	—	2	1	no	—
52	C₄×D₁₈₈	—	2	1	no	—
53	C₄×C₂²⋊D₄₇	—	2	2	no	—
54	#13093983	C₂³	1	2	yes	—
55	#13093983	C₂³	1	1	yes	characteristic
56	#13093981	C₂×C₄	1	2	yes	—
57	C₄²×D₄₇	C₂×C₄	1	2	yes	—
58	#13093959	—	2	2	no	—
59	#13093959	—	2	2	no	—
60	#13093957	—	2	2	no	—
61	#13093955	—	2	2	no	—
62	#13093942	—	2	2	no	—
63	#13093940	—	2	2	no	—
64	#13093938	—	2	4	no	—
65	#13093935	—	2	2	no	—
66	#13093935	C₂×C₄	1	1	yes	characteristic
67	#13093935	C₂×C₄	1	1	yes	characteristic
68	#13093935	C₂×C₄	1	1	yes	characteristic
69	#13093935	C₂×C₄	1	1	yes	characteristic
70	#13093934	—	2	2	no	—
71	#13093934	C₂×C₄	1	2	yes	—
72	#13093934	C₂×C₄	1	2	yes	—
73	C₄×C₄₇⋊C₈	—	2	2	no	—
74	C₈×Dic₄₇	—	2	2	no	—
75	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
76	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
77	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
78	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
80	#117×C₄₇	C₂³	1	2	yes	—
81	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
82	#97×C₄₇	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
84	C₂²×C₄×C₄₇	—	2	1	no	—
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₂²⋊D₄₇	#41	1	2	yes	—
87	#10625350	—	2	2	no	—
88	#10625350	—	2	2	no	—
89	C₂×D₁₈₈	#41	1	1	yes	characteristic
90	C₂×C₄×D₄₇	—	2	2	no	—
91	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₉₄	#41	1	1	yes	characteristic
93	M₁₆×C₄₇	—	2	2	no	—
94	M₁₆×C₄₇	—	2	2	no	—
95	M₁₆×C₄₇	—	2	2	no	—
96	M₁₆×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
100	C₄⋊C₄×C₄₇	—	2	2	no	—
101	C₂²⋊C₄×C₄₇	—	2	2	no	—
102	C₄²×C₄₇	—	2	1	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²⋊Dic₄₇	—	2	1	no	—
107	C₂²⋊Dic₄₇	—	2	2	no	—
108	(C₂×D₄₇)⋊C₄	—	2	2	no	—
109	C₄⋊Dic₄₇	—	2	1	no	—
110	C₄⋊Dic₄₇	—	2	2	no	—
111	Dic₄₇⋊C₄	—	2	2	no	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	2	2	no	—
114	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
115	#10625329	—	2	2	no	—
116	#10625329	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄₇⋊C₈	#41	1	2	yes	—
123	C₂×C₄₇⋊C₈	#41	1	2	yes	—
124	C₂×C₄₇⋊C₈	#41	1	2	yes	—
125	C₂×C₄₇⋊C₈	#41	1	2	yes	—
126	#10625324	—	2	4	no	—
127	#10625324	—	2	2	no	—
128	#10625324	—	2	2	no	—
129	C₈×D₄₇	—	2	2	no	—
130	C₈×D₄₇	—	2	2	no	—
131	C₈×D₄₇	—	2	4	no	—
132	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂²×D₄₇	#117	1	2	yes	—
134	C₂×C₄×C₄₇	—	2	1	no	—
135	C₂×C₄×C₄₇	—	2	1	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	8	1	no	—
138	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂²⋊D₄₇	—	2	4	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	#117	1	2	yes	—
146	D₁₈₈	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	2	2	no	—
149	C₄×D₄₇	—	8	2	no	—
150	Dic₉₄	—	2	2	no	—
151	C₈×C₄₇	—	2	2	no	—
152	C₈×C₄₇	—	2	2	no	—
153	C₈×C₄₇	—	4	2	no	—
154	C₄₇⋊C₈	—	4	2	no	—
155	C₄₇⋊C₈	#117	1	1	yes	characteristic
156	C₄₇⋊C₈	#117	1	1	yes	characteristic
157	C₄₇⋊C₈	#117	1	2	yes	—
158	C₄₇⋊C₈	#117	1	2	yes	—
159	C₄₇⋊C₈	#117	1	1	yes	characteristic
160	C₄₇⋊C₈	#117	1	1	yes	characteristic
161	#20788	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	4	1	no	—
163	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
164	C₂²×C₄₇	#529	1	2	yes	—
165	C₂×D₄₇	—	2	2	no	—
166	C₂×D₄₇	—	4	2	no	—
167	C₄×C₄₇	—	4	1	no	—
168	C₄×C₄₇	—	4	1	no	—
169	C₄×C₄₇	—	8	2	no	—
170	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
171	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
172	C₄×C₄₇	#529	1	2	yes	—
173	Dic₄₇	—	2	2	no	—
174	Dic₄₇	—	4	2	no	—
175	Dic₄₇	—	8	2	no	—
176	#2875	—	47	1	no	—
177	#1841	—	47	1	no	—
178	#1841	D₄₇	1	1	yes	characteristic, complemented
179	#1833	—	47	2	no	—
180	#1827	—	47	2	no	—
181	#1813	—	47	1	no	—
182	#1812	—	47	1	no	—
183	#1773	—	47	2	no	—
184	#1757	—	47	2	no	—
185	#1747	—	47	1	no	—
186	#1747	—	47	1	no	—
187	C₂×C₄₇	—	2	1	no	—
188	C₂×C₄₇	—	2	2	no	—
189	C₂×C₄₇	#2283	1	1	yes	characteristic
190	D₄₇	—	4	2	no	—
191	C₂×#131	—	47	1	no	—
192	C₂×#131	—	47	1	no	—
193	C₄×#41	—	47	1	no	—
194	C₂×#108	—	47	2	no	—
195	#427	—	47	2	no	—
196	#413	—	47	2	no	—
197	#410	—	47	2	no	—
198	#405	—	47	1	no	—
199	#405	—	47	1	no	—
200	#405	—	47	2	no	—
201	#405	—	47	2	no	—
202	#405	—	47	2	no	—
203	#405	—	47	2	no	—
204	C₄×M₁₆	—	47	2	no	—
205	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
206	C₂×#97	—	47	1	no	—
207	C₂×#97	—	47	2	no	—
208	#344	—	47	2	no	—
209	#344	C₂×D₄₇	1	2	yes	—
210	#339	—	47	2	no	—
211	#339	C₂×D₄₇	1	2	yes	—
212	#338	—	47	1	no	—
213	#338	C₂×D₄₇	1	1	yes	characteristic
214	H(3,ℤ/4)	—	47	2	no	—
215	H(3,ℤ/4)	—	47	1	no	—
216	H(3,ℤ/4)	—	47	2	no	—
217	H(3,ℤ/4)	C₂×D₄₇	1	1	yes	characteristic
218	C₄₇	#20788	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	47	1	no	—
220	#131	—	94	2	no	—
221	#131	—	94	2	no	—
222	#131	—	94	2	no	—
223	#131	—	94	2	no	—
224	C₂×M₁₆	—	47	2	no	—
225	C₂×M₁₆	—	47	2	no	—
226	C₂×M₁₆	—	47	2	no	—
227	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
228	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
229	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
230	C₂²×C₈	—	47	1	no	—
231	C₂²×C₈	—	47	1	no	—
232	C₂²×C₈	—	47	2	no	—
233	C₂²×C₈	—	47	2	no	—
234	C₄×Q₈	—	94	1	no	—
235	C₄×D₄	—	94	1	no	—
236	C₄×D₄	—	94	2	no	—
237	#117	—	47	1	no	—
238	#117	—	47	2	no	—
239	#117	—	47	2	no	—
240	#117	C₂²×D₄₇	1	2	yes	—
241	C₂×C₄²	—	47	2	no	—
242	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
243	#108	—	94	2	no	—
244	#108	—	94	2	no	—
245	#107	—	94	2	no	—
246	#106	—	94	2	no	—
247	#101	—	94	2	no	—
248	#100	—	94	2	no	—
249	#99	—	94	4	no	—
250	#97	—	47	2	no	—
251	#97	—	47	2	no	—
252	#97	—	47	1	no	—
253	#97	—	47	1	no	—
254	#97	—	47	1	no	—
255	#97	—	47	1	no	—
256	#97	—	94	2	no	—
257	#97	—	94	2	no	—
258	#97	C₄×D₄₇	1	2	yes	—
259	C₄×C₈	—	47	2	no	—
260	C₄×C₈	—	47	2	no	—
261	C₄×C₈	—	94	2	no	—
262	C₄×C₈	—	94	2	no	—
263	C₄×C₈	C₄×D₄₇	1	2	yes	—
264	#41	—	94	2	no	—
265	#41	—	94	2	no	—
266	C₂×Q₈	—	47	1	no	—
267	C₂×D₄	—	47	2	no	—
268	C₂×D₄	—	47	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	47	2	no	—
271	C₂²×C₄	—	94	2	no	—
272	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	94	4	no	—
278	M₁₆	—	94	2	no	—
279	M₁₆	—	94	2	no	—
280	M₁₆	—	94	2	no	—
281	M₁₆	—	94	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	47	1	no	—
284	C₂×C₈	—	47	1	no	—
285	C₂×C₈	—	47	1	no	—
286	C₂×C₈	—	47	1	no	—
287	C₂×C₈	—	47	2	no	—
288	C₂×C₈	—	47	2	no	—
289	C₂×C₈	—	47	2	no	—
290	C₂×C₈	—	47	2	no	—
291	C₂×C₈	—	94	2	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	—	94	4	no	—
294	C₂×C₈	—	94	2	no	—
295	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
296	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	94	2	no	—
299	C₄⋊C₄	—	94	1	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	94	2	no	—
303	C₂²⋊C₄	—	94	1	no	—
304	C₂²⋊C₄	—	94	2	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	47	2	no	—
308	C₄²	—	94	2	no	—
309	C₄²	—	94	2	no	—
310	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
311	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
312	C₂³	—	47	2	no	—
313	C₂³	#41×D₄₇	1	1	yes	characteristic
314	Q₈	—	94	2	no	—
315	D₄	—	94	4	no	—
316	D₄	—	94	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	1	no	—
321	C₂×C₄	—	47	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	94	2	no	—
325	C₂×C₄	—	376	2	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
328	C₂×C₄	#41×D₄₇	1	2	yes	—
329	C₂×C₄	#41×D₄₇	1	2	yes	—
330	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	47	1	no	—
335	C₈	—	47	1	no	—
336	C₈	—	47	2	no	—
337	C₈	—	47	2	no	—
338	C₈	—	47	1	no	—
339	C₈	—	47	1	no	—
340	C₈	—	188	2	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	94	2	no	—
343	C₂²	—	188	2	no	—
344	C₂²	#117×D₄₇	1	1	yes	characteristic
345	C₂²	#15689656	1	2	yes	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	94	2	no	—
350	C₄	—	376	2	no	—
351	C₄	—	188	2	no	—
352	C₄	#117×D₄₇	1	1	yes	characteristic
353	C₄	#117×D₄₇	1	1	yes	characteristic
354	C₄	#15689656	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	188	2	no	—
358	C₂	#20245785	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1841	C₂
socle	C₂×C₄₇	#2283
prime radical	C₂×#10625350	C₂³
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1841	D₄₇
hypocenter	C₄₇	#20788
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1841

Complements

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

N	H ≅ G/N	type
2. #20243537	C₂	semidirect
3. #20243509	C₂	semidirect
7. #20243271	C₂	semidirect
9. #20243144	C₂	semidirect
10. #20243142	C₂	semidirect
11. C₄₇×#1841	C₂	semidirect
23. #15689077	C₄	semidirect
24. #15689077	C₄	semidirect
25. #15689075	C₄	semidirect
27. #97×D₄₇	C₄	semidirect
178. #1841	D₄₇	semidirect
218. C₄₇	#20788	semidirect

Maximal subgroups

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

subgroup	index
#20251817	2
#20243537	2
#20243509	2
#20243491	2
#20243437	2
#20243433	2
#20243271	2
#20243187	2
#20243144	2
#20243142	2
C₄₇×#1841	2
#20788	47