Subgroups — #26230171

Order 12032.

826 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253833	C₂	1	1	yes	maximal, characteristic
2	#20252107	C₂	1	1	yes	maximal, characteristic, complemented
3	#20244217	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244215	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243718	C₂	1	4	yes	maximal, complemented
6	#20243715	C₂	1	4	yes	maximal, complemented
7	#20243528	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243136	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	#15689518	C₄	1	2	yes	complemented
14	#15689515	C₄	1	2	yes	complemented
15	#15689513	C₄	1	2	yes	complemented
16	#15689510	C₄	1	2	yes	complemented
17	#15689196	—	2	4	no	—
18	#15689147	C₂²	1	1	yes	characteristic
19	C₂×#13093951	C₂²	1	4	yes	—
20	C₂×#13093950	C₂²	1	4	yes	—
21	#15689089	C₂²	1	2	yes	—
22	C₂×#13093935	C₂²	1	1	yes	characteristic
23	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
24	#15688887	C₂²	1	4	yes	—
25	#15688877	C₂²	1	2	yes	—
26	#15688877	C₂²	1	2	yes	—
27	C₂×#131×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#532	C₂²	1	1	yes	characteristic
29	C₄₇×#435	—	2	4	no	—
30	C₄₇×#423	C₂²	1	1	yes	characteristic
31	C₄₇×#408	C₂²	1	2	yes	—
32	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#10625350	C₂³	1	2	yes	—
34	#13094044	—	4	4	no	—
35	#13094044	—	4	4	no	—
36	C₂×#10625337	C₂×C₄	1	4	yes	—
37	C₂×#10625335	C₂×C₄	1	4	yes	—
38	C₂×#10625336	C₂×C₄	1	4	yes	—
39	C₂×#10625334	C₂×C₄	1	4	yes	—
40	#13094030	—	4	4	no	—
41	C₂×#10625329	C₂³	1	2	yes	—
42	C₂×#10625324	—	2	2	no	—
43	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
44	C₂×C₈×D₄₇	—	2	2	no	—
45	#13094024	C₂×C₄	1	1	yes	characteristic
46	#13094021	C₂×C₄	1	1	yes	characteristic
47	#13094012	C₂×C₄	1	2	yes	—
48	#13093998	—	2	2	no	—
49	C₄×D₁₈₈	—	2	2	no	—
50	C₂×C₄×Dic₄₇	—	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	#13093935	C₂×C₄	1	2	yes	—
57	#13093935	C₂×C₄	1	2	yes	—
58	C₂×#41×C₄₇	C₂³	1	2	yes	—
59	#131×C₄₇	—	4	4	no	—
60	C₂×M₁₆×C₄₇	—	2	2	no	—
61	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₄₇	—	2	2	no	—
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
66	#127×C₄₇	C₂×C₄	1	1	yes	characteristic
67	#124×C₄₇	C₂×C₄	1	2	yes	—
68	#121×C₄₇	—	2	2	no	—
69	C₄×D₄×C₄₇	—	2	2	no	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₄₇	—	2	2	no	—
72	C₂²⋊C₈×C₄₇	—	2	4	no	—
73	#97×C₄₇	D₄	1	4	yes	—
74	#41×C₄₇	—	2	4	no	—
75	#41×C₄₇	—	2	4	no	—
76	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×D₄×C₄₇	—	2	2	no	—
78	C₂×D₄×C₄₇	—	2	2	no	—
79	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	#41	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₂²⋊D₄₇	—	2	2	no	—
84	C₂²×Dic₄₇	—	2	2	no	—
85	#10625350	—	2	4	no	—
86	#10625350	—	2	4	no	—
87	C₂×D₁₈₈	—	2	2	no	—
88	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
89	C₂×C₄×D₄₇	—	2	2	no	—
90	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
91	M₁₆×C₄₇	—	2	4	no	—
92	M₁₆×C₄₇	—	4	4	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	4	4	no	—
97	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₂²⋊C₄×C₄₇	—	2	4	no	—
101	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₄₇	C₂×D₄	1	2	yes	—
104	#10625337	—	4	4	no	—
105	#10625336	—	4	4	no	—
106	#10625335	—	4	4	no	—
107	#10625334	—	4	4	no	—
108	(C₂×D₄₇)⋊C₄	—	2	4	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄×Dic₄₇	—	2	2	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	4	4	no	—
113	#10625329	—	2	4	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
118	#10625324	—	4	4	no	—
119	C₈×D₄₇	—	4	4	no	—
120	C₂³×C₄₇	—	2	2	no	—
121	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂²×D₄₇	—	2	2	no	—
123	Q₈×C₄₇	—	2	4	no	—
124	D₄×C₄₇	—	2	4	no	—
125	D₄×C₄₇	—	4	4	no	—
126	D₄×C₄₇	—	8	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
136	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
137	C₂²⋊D₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	D₁₈₈	—	2	4	no	—
143	D₁₈₈	—	8	2	no	—
144	C₄×D₄₇	—	4	2	no	—
145	C₄×D₄₇	—	4	2	no	—
146	Dic₉₄	—	2	4	no	—
147	C₈×C₄₇	—	2	2	no	—
148	C₈×C₄₇	—	2	2	no	—
149	C₈×C₄₇	—	4	2	no	—
150	C₄₇⋊C₈	—	2	4	no	—
151	C₄₇⋊C₈	—	4	2	no	—
152	#22283	—	47	1	no	maximal, Sylow
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	—	4	2	no	—
155	C₂²×C₄₇	—	4	4	no	—
156	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₄₇	#540	1	2	yes	—
158	C₂×D₄₇	—	4	2	no	—
159	C₂×D₄₇	—	4	4	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	4	2	no	—
163	C₄×C₄₇	#518	1	2	yes	—
164	C₄×C₄₇	#535	1	1	yes	characteristic
165	C₄×C₄₇	#535	1	1	yes	characteristic
166	Dic₄₇	—	4	4	no	—
167	Dic₄₇	—	4	2	no	—
168	#3150	—	47	1	no	—
169	#2932	—	47	1	no	—
170	#2932	D₄₇	1	1	yes	characteristic, complemented
171	#1962	—	47	1	no	—
172	#1962	—	47	1	no	—
173	#1876	—	47	4	no	—
174	#1876	—	47	4	no	—
175	#1838	—	47	1	no	—
176	#1745	—	47	1	no	—
177	C₂×C₄₇	—	2	2	no	—
178	C₂×C₄₇	—	8	2	no	—
179	C₂×C₄₇	#2328	1	1	yes	characteristic
180	C₂×C₄₇	#2956	1	2	yes	—
181	D₄₇	—	8	2	no	—
182	C₂×#135	—	47	4	no	—
183	C₂×#131	—	47	2	no	—
184	C₂×#131	C₂×D₄₇	1	2	yes	—
185	#532	—	47	1	no	—
186	#532	C₂×D₄₇	1	1	yes	characteristic
187	#497	—	47	2	no	—
188	#497	—	47	2	no	—
189	#496	—	47	2	no	—
190	#496	—	47	2	no	—
191	#435	—	2	4	no	—
192	#435	—	94	4	no	—
193	#423	—	47	1	no	—
194	#423	C₂×D₄₇	1	1	yes	characteristic
195	C₂×#104	—	47	4	no	—
196	C₂×#104	—	47	4	no	—
197	#408	—	47	2	no	—
198	#408	C₂×D₄₇	1	2	yes	—
199	C₂×#97	—	47	1	no	—
200	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
201	C₄³	—	47	1	no	—
202	#341	—	47	4	no	—
203	#336	—	47	2	no	—
204	#336	—	47	2	no	—
205	C₄₇	#22283	1	1	yes	characteristic, complemented, Sylow
206	C₂×#41	—	47	2	no	—
207	C₂×#41	C₂²×D₄₇	1	2	yes	—
208	#135	—	188	4	no	—
209	#135	—	188	4	no	—
210	C₂×Q₁₆	—	47	4	no	—
211	C₂×SD₁₆	—	47	4	no	—
212	C₂×SD₁₆	—	47	4	no	—
213	C₂×D₈	—	47	4	no	—
214	#131	—	4	4	no	—
215	#131	—	188	4	no	—
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	47	2	no	—
218	C₂×M₁₆	—	94	2	no	—
219	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	—	47	2	no	—
222	C₂²×C₈	—	94	2	no	—
223	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
224	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
225	C₄⋊Q₈	—	47	1	no	—
226	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
227	#127	—	47	1	no	—
228	#127	C₄×D₄₇	1	1	yes	characteristic
229	#124	—	47	2	no	—
230	#124	C₄×D₄₇	1	2	yes	—
231	#121	—	2	2	no	—
232	#121	—	94	2	no	—
233	C₄×D₄	—	2	2	no	—
234	C₄×D₄	—	94	2	no	—
235	C₂×C₄²	—	47	2	no	—
236	C₂×C₄²	—	94	2	no	—
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	94	2	no	—
240	#104	—	188	4	no	—
241	#104	—	188	4	no	—
242	C₂²⋊C₈	—	2	4	no	—
243	C₂²⋊C₈	—	94	4	no	—
244	#97	—	47	2	no	—
245	#97	—	47	2	no	—
246	#97	C₂²⋊D₄₇	1	4	yes	—
247	#41	—	2	4	no	—
248	#41	—	2	4	no	—
249	#41	—	94	4	no	—
250	#41	—	94	4	no	—
251	C₂×Q₈	—	47	2	no	—
252	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	47	2	no	—
256	C₂×D₄	—	94	2	no	—
257	C₂×D₄	—	94	2	no	—
258	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	94	2	no	—
261	C₂²×C₄	—	94	2	no	—
262	C₂²×C₄	#10625350	1	2	yes	—
263	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
264	Q₁₆	—	188	4	no	—
265	SD₁₆	—	188	4	no	—
266	SD₁₆	—	188	4	no	—
267	D₈	—	188	4	no	—
268	M₁₆	—	2	4	no	—
269	M₁₆	—	4	4	no	—
270	M₁₆	—	94	4	no	—
271	M₁₆	—	188	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	4	no	—
276	C₂×C₈	—	47	4	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	—	94	2	no	—
279	C₂×C₈	—	94	2	no	—
280	C₂×C₈	—	188	4	no	—
281	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
282	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	94	2	no	—
285	C₂²⋊C₄	—	2	4	no	—
286	C₂²⋊C₄	—	94	4	no	—
287	C₄²	—	94	2	no	—
288	C₄²	—	94	2	no	—
289	C₄²	—	188	4	no	—
290	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
291	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
292	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
293	C₂³	—	2	2	no	—
294	C₂³	—	94	2	no	—
295	C₂³	C₂×#10625350	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	Q₈	—	94	4	no	—
298	D₄	—	2	4	no	—
299	D₄	—	4	4	no	—
300	D₄	—	8	2	no	—
301	D₄	—	94	4	no	—
302	D₄	—	376	2	no	—
303	D₄	—	188	4	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	94	4	no	—
309	C₂×C₄	—	94	2	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
315	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
316	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
318	C₂×C₄	#13094069	1	2	yes	—
319	C₂×C₄	#13094070	1	2	yes	—
320	C₈	—	2	2	no	—
321	C₈	—	2	2	no	—
322	C₈	—	4	2	no	—
323	C₈	—	94	4	no	—
324	C₈	—	188	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	188	2	no	—
329	C₂²	—	188	4	no	—
330	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
331	C₂²	#15689724	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	188	4	no	—
336	C₄	—	188	2	no	—
337	C₄	#15689595	1	2	yes	—
338	C₄	#15689687	1	1	yes	characteristic
339	C₄	#15689690	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	8	2	no	—
342	C₂	—	376	2	no	—
343	C₂	#20246110	1	1	yes	characteristic
344	C₂	#20252295	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	#20253833	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2932	D₄₇
hypocenter	C₄₇	#22283
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. #20244217	C₂	semidirect
4. #20244215	C₂	semidirect
5. #20243718	C₂	semidirect
6. #20243715	C₂	semidirect
7. #20243528	C₂	semidirect
8. #20243136	C₂	semidirect
9. C₄₇×#2932	C₂	semidirect
13. #15689518	C₄	semidirect
14. #15689515	C₄	semidirect
15. #15689513	C₄	semidirect
16. #15689510	C₄	semidirect
170. #2932	D₄₇	semidirect
205. C₄₇	#22283	semidirect

Maximal subgroups

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

subgroup	index
#20253833	2
#20252107	2
#20244217	2
#20244215	2
#20243718	2
#20243715	2
#20243528	2
#20243136	2
C₄₇×#2932	2
#22283	47