Subgroups — #26230165

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253814	C₂	1	1	yes	maximal, characteristic
2	#20252107	C₂	1	1	yes	maximal, characteristic, complemented
3	#20244221	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244219	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243718	C₂	1	4	yes	maximal, complemented
6	#20243715	C₂	1	4	yes	maximal, complemented
7	#20243525	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2932	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094044	C₂²	1	4	yes	—
11	C₂×#13094030	C₂²	1	2	yes	—
12	#15689673	C₂²	1	1	yes	characteristic
13	#15689505	C₄	1	4	yes	complemented
14	#15689502	C₄	1	1	yes	characteristic, complemented
15	#15689499	C₄	1	1	yes	characteristic, complemented
16	#15689497	C₄	1	1	yes	characteristic, complemented
17	#15689495	C₄	1	1	yes	characteristic, complemented
18	#15689196	—	2	4	no	—
19	#15689147	C₂²	1	1	yes	characteristic
20	C₂×#13093951	C₂²	1	4	yes	—
21	C₂×#13093950	C₂²	1	4	yes	—
22	#15689089	C₂²	1	2	yes	—
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	#15688969	C₂²	1	1	yes	characteristic
25	#15688887	C₂²	1	4	yes	—
26	#15688877	C₂²	1	2	yes	—
27	#15688877	C₂²	1	2	yes	—
28	C₂×#131×C₄₇	C₂²	1	2	yes	—
29	C₄₇×#532	C₂²	1	1	yes	characteristic
30	C₄₇×#435	—	2	4	no	—
31	C₄₇×#423	C₂²	1	1	yes	characteristic
32	C₄₇×#408	C₂²	1	2	yes	—
33	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#10625350	C₂³	1	2	yes	—
35	#13094044	—	4	4	no	—
36	#13094044	—	4	4	no	—
37	C₂×#10625337	C₂×C₄	1	4	yes	—
38	C₂×#10625335	C₂×C₄	1	4	yes	—
39	C₂×#10625336	C₂×C₄	1	4	yes	—
40	C₂×#10625334	C₂×C₄	1	4	yes	—
41	#13094030	—	4	4	no	—
42	C₂×#10625329	C₂³	1	2	yes	—
43	C₂×#10625324	—	2	2	no	—
44	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
45	C₂×C₈×D₄₇	—	2	2	no	—
46	#13094024	C₂×C₄	1	1	yes	characteristic
47	#13094021	C₂×C₄	1	1	yes	characteristic
48	#13094012	C₂×C₄	1	2	yes	—
49	#13093998	—	2	2	no	—
50	C₄×D₁₈₈	—	2	2	no	—
51	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
52	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
53	#13093951	—	4	4	no	—
54	#13093950	—	4	4	no	—
55	(C₂×D₄₇)⋊C₈	—	2	4	no	—
56	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
59	C₄₇⋊#95	—	2	2	no	—
60	C₂×#41×C₄₇	C₂³	1	2	yes	—
61	#131×C₄₇	—	4	4	no	—
62	C₂×M₁₆×C₄₇	—	2	2	no	—
63	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	—	2	2	no	—
65	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
68	#127×C₄₇	C₂×C₄	1	1	yes	characteristic
69	#124×C₄₇	C₂×C₄	1	2	yes	—
70	#121×C₄₇	—	2	2	no	—
71	C₄×D₄×C₄₇	—	2	2	no	—
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₄₇	—	2	2	no	—
74	C₂²⋊C₈×C₄₇	—	2	4	no	—
75	#97×C₄₇	D₄	1	4	yes	—
76	#41×C₄₇	—	2	4	no	—
77	#41×C₄₇	—	2	4	no	—
78	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₄₇	—	2	2	no	—
80	C₂×D₄×C₄₇	—	2	2	no	—
81	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₂²⋊D₄₇	—	2	2	no	—
86	C₂²×Dic₄₇	—	2	2	no	—
87	#10625350	—	2	4	no	—
88	#10625350	—	2	4	no	—
89	C₂×D₁₈₈	—	2	2	no	—
90	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₄₇	—	2	2	no	—
92	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
93	M₁₆×C₄₇	—	2	4	no	—
94	M₁₆×C₄₇	—	4	4	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	4	4	no	—
99	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₂²⋊C₄×C₄₇	—	2	4	no	—
103	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂×D₄	1	2	yes	—
106	#10625337	—	4	4	no	—
107	#10625336	—	4	4	no	—
108	#10625335	—	4	4	no	—
109	#10625334	—	4	4	no	—
110	(C₂×D₄₇)⋊C₄	—	2	4	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	2	2	no	—
114	#10625329	—	2	4	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
120	#10625324	—	4	4	no	—
121	C₈×D₄₇	—	4	4	no	—
122	C₂³×C₄₇	—	2	2	no	—
123	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂²×D₄₇	—	2	2	no	—
125	Q₈×C₄₇	—	2	4	no	—
126	D₄×C₄₇	—	2	4	no	—
127	D₄×C₄₇	—	4	4	no	—
128	D₄×C₄₇	—	8	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
138	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
139	C₂²⋊D₄₇	—	4	4	no	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	4	4	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	D₁₈₈	—	2	4	no	—
144	D₁₈₈	—	8	2	no	—
145	C₄×D₄₇	—	4	2	no	—
146	C₄×D₄₇	—	4	2	no	—
147	Dic₉₄	—	2	4	no	—
148	C₈×C₄₇	—	2	2	no	—
149	C₈×C₄₇	—	2	2	no	—
150	C₈×C₄₇	—	4	2	no	—
151	C₄₇⋊C₈	—	2	2	no	—
152	C₄₇⋊C₈	—	2	2	no	—
153	C₄₇⋊C₈	—	4	2	no	—
154	#22282	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	4	2	no	—
157	C₂²×C₄₇	—	4	4	no	—
158	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₄₇	#540	1	2	yes	—
160	C₂×D₄₇	—	4	2	no	—
161	C₂×D₄₇	—	4	4	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	4	2	no	—
165	C₄×C₄₇	#518	1	2	yes	—
166	C₄×C₄₇	#535	1	1	yes	characteristic
167	C₄×C₄₇	#535	1	1	yes	characteristic
168	Dic₄₇	—	4	2	no	—
169	Dic₄₇	—	8	2	no	—
170	#3147	—	47	1	no	—
171	#2932	—	47	1	no	—
172	#2932	D₄₇	1	1	yes	characteristic, complemented
173	#1963	—	47	1	no	—
174	#1963	—	47	1	no	—
175	#1876	—	47	4	no	—
176	#1876	—	47	4	no	—
177	#1837	—	47	1	no	—
178	#1746	—	47	1	no	—
179	C₂×C₄₇	—	2	2	no	—
180	C₂×C₄₇	—	8	2	no	—
181	C₂×C₄₇	#2328	1	1	yes	characteristic
182	C₂×C₄₇	#2957	1	2	yes	—
183	D₄₇	—	8	2	no	—
184	C₂×#135	—	47	4	no	—
185	C₂×#131	—	47	2	no	—
186	C₂×#131	C₂×D₄₇	1	2	yes	—
187	#532	—	47	1	no	—
188	#532	C₂×D₄₇	1	1	yes	characteristic
189	#495	—	47	4	no	—
190	#494	—	47	1	no	—
191	#493	—	47	1	no	—
192	#492	—	47	1	no	—
193	#492	—	47	1	no	—
194	#435	—	2	4	no	—
195	#435	—	94	4	no	—
196	#423	—	47	1	no	—
197	#423	C₂×D₄₇	1	1	yes	characteristic
198	C₂×#104	—	47	4	no	—
199	C₂×#104	—	47	4	no	—
200	#408	—	47	2	no	—
201	#408	C₂×D₄₇	1	2	yes	—
202	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
203	C₂×C₄×C₈	—	47	1	no	—
204	#376	—	47	1	no	—
205	#341	—	47	4	no	—
206	#336	—	47	2	no	—
207	#336	—	47	2	no	—
208	C₄₇	#22282	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	47	2	no	—
210	C₂×#41	C₂²×D₄₇	1	2	yes	—
211	#135	—	188	4	no	—
212	#135	—	188	4	no	—
213	C₂×Q₁₆	—	47	4	no	—
214	C₂×SD₁₆	—	47	4	no	—
215	C₂×SD₁₆	—	47	4	no	—
216	C₂×D₈	—	47	4	no	—
217	#131	—	4	4	no	—
218	#131	—	188	4	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	47	2	no	—
221	C₂×M₁₆	—	94	2	no	—
222	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
223	C₂²×C₈	—	2	2	no	—
224	C₂²×C₈	—	47	2	no	—
225	C₂²×C₈	—	94	2	no	—
226	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
228	C₄⋊Q₈	—	47	1	no	—
229	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
230	#127	—	47	1	no	—
231	#127	C₄×D₄₇	1	1	yes	characteristic
232	#124	—	47	2	no	—
233	#124	C₄×D₄₇	1	2	yes	—
234	#121	—	2	2	no	—
235	#121	—	94	2	no	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	94	2	no	—
238	C₂×C₄²	—	47	2	no	—
239	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	94	2	no	—
242	#104	—	188	4	no	—
243	#104	—	188	4	no	—
244	C₂²⋊C₈	—	2	4	no	—
245	C₂²⋊C₈	—	94	4	no	—
246	#97	C₂²⋊D₄₇	1	4	yes	—
247	C₄×C₈	—	47	1	no	—
248	C₄×C₈	—	47	1	no	—
249	C₄×C₈	—	47	2	no	—
250	#95	—	94	2	no	—
251	#41	—	2	4	no	—
252	#41	—	2	4	no	—
253	#41	—	94	4	no	—
254	#41	—	94	4	no	—
255	C₂×Q₈	—	47	2	no	—
256	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	47	2	no	—
260	C₂×D₄	—	94	2	no	—
261	C₂×D₄	—	94	2	no	—
262	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	94	2	no	—
265	C₂²×C₄	—	94	2	no	—
266	C₂²×C₄	#10625350	1	2	yes	—
267	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
268	Q₁₆	—	188	4	no	—
269	SD₁₆	—	188	4	no	—
270	SD₁₆	—	188	4	no	—
271	D₈	—	188	4	no	—
272	M₁₆	—	2	4	no	—
273	M₁₆	—	4	4	no	—
274	M₁₆	—	94	4	no	—
275	M₁₆	—	188	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	4	4	no	—
280	C₂×C₈	—	47	2	no	—
281	C₂×C₈	—	47	2	no	—
282	C₂×C₈	—	94	2	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	—	188	4	no	—
286	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
287	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	94	2	no	—
290	C₂²⋊C₄	—	2	4	no	—
291	C₂²⋊C₄	—	94	4	no	—
292	C₄²	—	94	2	no	—
293	C₄²	—	94	2	no	—
294	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
295	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
296	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
297	C₂³	—	2	2	no	—
298	C₂³	—	94	2	no	—
299	C₂³	C₂×#10625350	1	1	yes	characteristic
300	Q₈	—	2	4	no	—
301	Q₈	—	94	4	no	—
302	D₄	—	2	4	no	—
303	D₄	—	4	4	no	—
304	D₄	—	8	2	no	—
305	D₄	—	94	4	no	—
306	D₄	—	376	2	no	—
307	D₄	—	188	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	94	2	no	—
313	C₂×C₄	—	188	4	no	—
314	C₂×C₄	—	188	2	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
318	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
319	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
321	C₂×C₄	#13094069	1	2	yes	—
322	C₂×C₄	#13094070	1	2	yes	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	4	2	no	—
326	C₈	—	94	2	no	—
327	C₈	—	94	2	no	—
328	C₈	—	188	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	4	no	—
332	C₂²	—	188	2	no	—
333	C₂²	—	188	4	no	—
334	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
335	C₂²	#15689724	1	2	yes	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	376	2	no	—
340	C₄	—	188	2	no	—
341	C₄	#15689595	1	2	yes	—
342	C₄	#15689687	1	1	yes	characteristic
343	C₄	#15689690	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	8	2	no	—
346	C₂	—	376	2	no	—
347	C₂	#20246110	1	1	yes	characteristic
348	C₂	#20252303	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2932	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#20253814	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2932	D₄₇
hypocenter	C₄₇	#22282
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #2932

Complements

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

N	H ≅ G/N	type
2. #20252107	C₂	semidirect
3. #20244221	C₂	semidirect
4. #20244219	C₂	semidirect
5. #20243718	C₂	semidirect
6. #20243715	C₂	semidirect
7. #20243525	C₂	semidirect
8. #20243141	C₂	semidirect
9. C₄₇×#2932	C₂	semidirect
13. #15689505	C₄	semidirect
14. #15689502	C₄	semidirect
15. #15689499	C₄	semidirect
16. #15689497	C₄	semidirect
17. #15689495	C₄	semidirect
172. #2932	D₄₇	semidirect
208. C₄₇	#22282	semidirect

Maximal subgroups

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

subgroup	index
#20253814	2
#20252107	2
#20244221	2
#20244219	2
#20243718	2
#20243715	2
#20243525	2
#20243141	2
C₄₇×#2932	2
#22282	47