Subgroups — #26109521

Order 12032.

430 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252099	C₂	1	1	yes	maximal, characteristic
2	#20243559	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688873	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242066	C₂	1	1	yes	maximal, characteristic
5	#20242063	C₂	1	1	yes	maximal, characteristic
6	#20241262	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094032	—	2	1	no	—
9	#15689673	C₂²	1	1	yes	characteristic
10	#15689200	—	2	1	no	—
11	#15689200	C₄	1	2	yes	—
12	#15689188	—	2	1	no	—
13	#15689188	C₄	1	2	yes	—
14	C₂×#13093959	—	2	1	no	—
15	C₂×#13093959	—	2	1	no	—
16	#15689147	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
18	#15689091	—	2	1	no	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#15688873	—	2	1	no	—
21	#15688873	—	2	1	no	—
22	#15688870	—	2	1	no	—
23	#15688868	—	2	1	no	—
24	#15688866	—	2	1	no	—
25	#15688866	—	2	1	no	—
26	C₄₇×#423	C₂²	1	1	yes	characteristic
27	C₄₇×#423	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#10625350	—	2	1	no	—
30	#13094032	—	4	2	no	—
31	#13094032	—	4	2	no	—
32	C₂×#10625329	—	2	1	no	—
33	C₂×#10625324	—	2	2	no	—
34	C₂²×C₄₇⋊C₈	—	2	1	no	—
35	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
36	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
37	C₂×C₈×D₄₇	—	2	2	no	—
38	#13094024	D₄	1	1	yes	characteristic
39	#13094021	D₄	1	1	yes	characteristic
40	#13094012	—	2	1	no	—
41	#13093998	—	2	1	no	—
42	#13093998	Q₈	1	2	yes	—
43	C₄×D₁₈₈	—	2	1	no	—
44	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
45	#13093959	—	4	2	no	—
46	#13093959	—	4	2	no	—
47	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
50	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
51	(C₂×D₄₇)⋊C₈	—	2	2	no	—
52	(C₂×D₄₇)⋊C₈	—	2	2	no	—
53	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
56	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₄₇	—	2	1	no	—
58	C₂×M₁₆×C₄₇	—	2	1	no	—
59	C₂²×C₈×C₄₇	—	2	1	no	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₇	—	2	1	no	—
62	C₄⋊C₈×C₄₇	—	2	1	no	—
63	C₄⋊C₈×C₄₇	—	2	1	no	—
64	C₄⋊C₈×C₄₇	—	2	1	no	—
65	C₄⋊C₈×C₄₇	—	2	1	no	—
66	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₇	—	2	2	no	—
71	#10625350	—	4	1	no	—
72	#10625350	—	4	1	no	—
73	#10625350	—	4	1	no	—
74	#10625350	—	4	1	no	—
75	C₂×D₁₈₈	—	2	1	no	—
76	C₂×D₁₈₈	—	2	1	no	—
77	C₂×D₁₈₈	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₄₇	—	2	2	no	—
79	C₂×Dic₉₄	—	2	1	no	—
80	M₁₆×C₄₇	—	4	1	no	—
81	M₁₆×C₄₇	—	4	1	no	—
82	M₁₆×C₄₇	—	4	1	no	—
83	M₁₆×C₄₇	—	4	1	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	4	1	no	—
86	C₂×C₈×C₄₇	—	4	1	no	—
87	C₂×C₈×C₄₇	—	8	1	no	—
88	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	#41	1	1	yes	characteristic
92	(C₂×D₄₇)⋊C₄	—	2	2	no	—
93	(C₂×D₄₇)⋊C₄	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	1	no	—
95	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
96	#10625329	—	4	1	no	—
97	#10625329	—	4	1	no	—
98	C₂×C₄₇⋊C₈	—	2	1	no	—
99	C₂×C₄₇⋊C₈	—	2	1	no	—
100	C₂×C₄₇⋊C₈	—	4	1	no	—
101	C₂×C₄₇⋊C₈	—	4	1	no	—
102	C₂×C₄₇⋊C₈	—	8	1	no	—
103	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
108	C₂×C₄₇⋊C₈	SD₁₆	1	2	yes	—
109	#10625324	—	4	4	no	—
110	C₈×D₄₇	—	4	4	no	—
111	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂²×D₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
126	C₂²⋊D₄₇	—	4	2	no	—
127	C₂²⋊D₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	D₁₈₈	—	4	2	no	—
130	D₁₈₈	—	4	1	no	—
131	D₁₈₈	—	4	1	no	—
132	D₁₈₈	—	4	1	no	—
133	D₁₈₈	—	4	1	no	—
134	C₄×D₄₇	—	4	2	no	—
135	C₄×D₄₇	—	4	2	no	—
136	Dic₉₄	—	4	1	no	—
137	Dic₉₄	—	4	1	no	—
138	C₈×C₄₇	—	4	2	no	—
139	C₈×C₄₇	—	8	1	no	—
140	C₈×C₄₇	—	8	1	no	—
141	C₄₇⋊C₈	—	2	1	no	—
142	C₄₇⋊C₈	—	2	1	no	—
143	C₄₇⋊C₈	—	8	1	no	—
144	C₄₇⋊C₈	—	8	1	no	—
145	C₄₇⋊C₈	#102	1	2	yes	—
146	C₄₇⋊C₈	#102	1	2	yes	—
147	#8682	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	#380	1	1	yes	characteristic
151	C₂²×C₄₇	#474	1	1	yes	characteristic
152	C₂²×C₄₇	#476	1	1	yes	characteristic
153	C₂×D₄₇	—	4	2	no	—
154	C₂×D₄₇	—	4	2	no	—
155	C₂×D₄₇	—	4	2	no	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
160	C₄×C₄₇	#417	1	1	yes	characteristic
161	C₄×C₄₇	#459	1	1	yes	characteristic
162	C₄×C₄₇	#461	1	1	yes	characteristic
163	C₄×C₄₇	#471	1	1	yes	characteristic
164	C₄×C₄₇	#471	1	1	yes	characteristic
165	Dic₄₇	—	4	2	no	—
166	#2931	—	47	1	no	—
167	#1848	—	47	1	no	—
168	C₂×#334	—	47	1	no	—
169	#1547	—	47	1	no	—
170	#1547	—	47	1	no	—
171	#1292	—	47	1	no	—
172	#1292	D₄₇	1	1	yes	characteristic, complemented
173	C₂×C₄₇	—	2	1	no	—
174	C₂×C₄₇	—	2	1	no	—
175	C₂×C₄₇	#1802	1	1	yes	characteristic
176	C₂×C₄₇	#1813	1	1	yes	characteristic
177	C₂×C₄₇	#1815	1	1	yes	characteristic
178	D₄₇	—	8	2	no	—
179	C₂×#131	—	94	1	no	—
180	#532	—	47	1	no	—
181	#436	—	47	2	no	—
182	#436	—	94	1	no	—
183	C₈×D₄	—	47	2	no	—
184	C₈×D₄	—	94	1	no	—
185	C₂×#108	—	94	1	no	—
186	C₂×#108	—	94	1	no	—
187	#423	—	47	1	no	—
188	#423	C₂×D₄₇	1	1	yes	characteristic
189	#423	C₂×D₄₇	1	1	yes	characteristic
190	C₂×C₄⋊C₈	—	47	1	no	—
191	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
192	#408	—	94	1	no	—
193	C₂×C₄×C₈	—	47	1	no	—
194	#334	—	94	1	no	—
195	#334	—	94	1	no	—
196	#332	—	94	1	no	—
197	#332	—	94	1	no	—
198	#331	—	94	1	no	—
199	#331	—	94	1	no	—
200	C₄₇	#8682	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	94	1	no	—
202	#131	—	188	2	no	—
203	#131	—	188	2	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	94	2	no	—
207	C₂×M₁₆	—	94	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	47	1	no	—
210	C₂²×C₈	—	47	1	no	—
211	C₂²×C₈	—	94	2	no	—
212	C₂²×C₈	—	94	1	no	—
213	C₄⋊Q₈	—	47	1	no	—
214	#127	—	47	1	no	—
215	#124	—	94	1	no	—
216	#121	—	47	2	no	—
217	#121	—	94	1	no	—
218	C₄×D₄	—	47	2	no	—
219	C₄×D₄	—	94	1	no	—
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	#108	—	188	2	no	—
222	#108	—	188	2	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	94	1	no	—
229	C₄⋊C₈	—	94	1	no	—
230	C₄⋊C₈	—	94	1	no	—
231	C₄⋊C₈	—	94	1	no	—
232	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
233	C₂²⋊C₈	—	94	2	no	—
234	C₂²⋊C₈	—	94	2	no	—
235	C₄×C₈	—	47	1	no	—
236	C₄×C₈	—	47	1	no	—
237	C₄×C₈	—	47	1	no	—
238	C₄×C₈	—	47	1	no	—
239	#41	—	188	1	no	—
240	#41	—	188	1	no	—
241	#41	—	188	1	no	—
242	#41	—	188	1	no	—
243	C₂×Q₈	—	94	1	no	—
244	C₂×D₄	—	47	2	no	—
245	C₂×D₄	—	94	2	no	—
246	C₂×D₄	—	94	1	no	—
247	C₂×D₄	—	94	1	no	—
248	C₂²×C₄	—	94	2	no	—
249	C₂²×C₄	#10625350	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
251	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	188	4	no	—
257	M₁₆	—	188	1	no	—
258	M₁₆	—	188	1	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	8	1	no	—
263	C₂×C₈	—	47	1	no	—
264	C₂×C₈	—	47	1	no	—
265	C₂×C₈	—	47	2	no	—
266	C₂×C₈	—	47	2	no	—
267	C₂×C₈	—	47	1	no	—
268	C₂×C₈	—	47	1	no	—
269	C₂×C₈	—	94	1	no	—
270	C₂×C₈	—	94	1	no	—
271	C₂×C₈	—	376	1	no	—
272	C₂×C₈	—	188	4	no	—
273	C₂×C₈	—	188	1	no	—
274	C₂×C₈	—	188	1	no	—
275	C₄⋊C₄	—	47	2	no	—
276	C₄⋊C₄	—	94	1	no	—
277	C₂²⋊C₄	—	94	2	no	—
278	C₂²⋊C₄	—	94	2	no	—
279	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
280	C₄²	#10625350	1	1	yes	characteristic
281	C₄²	D₄×D₄₇	1	1	yes	characteristic
282	C₄²	D₄×D₄₇	1	1	yes	characteristic
283	C₂³	—	94	2	no	—
284	C₂³	#13094002	1	1	yes	characteristic
285	Q₈	—	188	1	no	—
286	Q₈	—	188	1	no	—
287	D₄	—	188	2	no	—
288	D₄	—	188	2	no	—
289	D₄	—	188	2	no	—
290	D₄	—	188	1	no	—
291	D₄	—	188	1	no	—
292	D₄	—	188	1	no	—
293	D₄	—	188	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	188	2	no	—
299	C₂×C₄	—	188	2	no	—
300	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
302	C₂×C₄	#13093981	1	1	yes	characteristic
303	C₂×C₄	#13093997	1	1	yes	characteristic
304	C₂×C₄	#13093997	1	1	yes	characteristic
305	C₂×C₄	#13094002	1	1	yes	characteristic
306	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
307	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
308	C₂×C₄	#13094051	1	1	yes	characteristic
309	C₂×C₄	#13094056	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	8	1	no	—
312	C₈	—	8	1	no	—
313	C₈	—	47	2	no	—
314	C₈	—	47	2	no	—
315	C₈	—	94	1	no	—
316	C₈	—	94	1	no	—
317	C₈	—	376	1	no	—
318	C₈	—	376	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	188	2	no	—
322	C₂²	—	188	2	no	—
323	C₂²	—	188	2	no	—
324	C₂²	#15688982	1	1	yes	characteristic
325	C₂²	#15689402	1	1	yes	characteristic
326	C₂²	#15689412	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	188	2	no	—
331	C₄	#102×D₄₇	1	1	yes	characteristic
332	C₄	#15689123	1	1	yes	characteristic
333	C₄	#15689317	1	1	yes	characteristic
334	C₄	#15689327	1	1	yes	characteristic
335	C₄	#15689385	1	1	yes	characteristic
336	C₄	#15689389	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	376	2	no	—
340	C₂	#20243387	1	1	yes	characteristic
341	C₂	#20243437	1	1	yes	characteristic
342	C₂	#20243446	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688982
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1292	C₂
socle	C₂²×C₄₇	#380
prime radical	#15689673	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1292	D₄₇
hypocenter	C₄₇	#8682
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
2. #20243559	C₂	semidirect
3. C₂×#15688873	C₂	semidirect
6. #20241262	C₂	semidirect
7. C₄₇×#1292	C₂	semidirect
172. #1292	D₄₇	semidirect
200. C₄₇	#8682	semidirect

Maximal subgroups

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

subgroup	index
#20252099	2
#20243559	2
C₂×#15688873	2
#20242066	2
#20242063	2
#20241262	2
C₄₇×#1292	2
#8682	47