Subgroups — #26115515

Order 12032.

586 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689497	C₂	1	1	yes	maximal, characteristic
2	#20244315	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688873	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688869	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688866	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625336	—	2	1	no	—
9	C₂²×#10625336	—	2	1	no	—
10	C₂×#13094021	C₂²	1	1	yes	characteristic
11	#15689497	—	2	2	no	—
12	#15689497	C₄	1	4	yes	complemented
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×#13093944	—	2	2	no	—
16	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	#15688878	—	2	1	no	—
19	#15688873	—	2	2	no	—
20	#15688869	—	2	2	no	—
21	#15688866	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#340	—	2	2	no	—
26	C₂²×D₁₈₈	—	2	1	no	—
27	C₂²×D₁₈₈	—	2	1	no	—
28	C₂²×D₁₈₈	—	2	1	no	—
29	C₂×#10625336	—	2	4	no	—
30	C₂×#10625336	—	2	4	no	—
31	C₂×#10625336	—	4	2	no	—
32	C₂×#10625336	—	4	2	no	—
33	C₂×#10625336	—	4	2	no	—
34	C₂×#10625336	—	4	2	no	—
35	C₂²×C₄₇⋊C₈	—	2	1	no	—
36	C₂²×C₄₇⋊C₈	—	2	1	no	—
37	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
38	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
39	#13094021	—	2	1	no	—
40	#13094021	—	2	1	no	—
41	#13094021	—	2	1	no	—
42	#13094021	C₂×C₄	1	2	yes	—
43	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
47	#13093944	—	4	4	no	—
48	C₄×C₄₇⋊C₈	—	4	2	no	—
49	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
50	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
51	C₂²×Q₈×C₄₇	—	2	1	no	—
52	C₂²×C₈×C₄₇	—	2	1	no	—
53	C₄⋊Q₈×C₄₇	—	2	1	no	—
54	C₄⋊Q₈×C₄₇	—	2	1	no	—
55	C₄⋊Q₈×C₄₇	—	2	1	no	—
56	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
60	C₂×C₄²×C₄₇	—	2	1	no	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₄₇	—	2	1	no	—
63	C₄⋊C₈×C₄₇	—	2	1	no	—
64	#95×C₄₇	—	2	1	no	—
65	C₂³×D₄₇	—	4	1	no	—
66	C₂×Q₈×C₄₇	—	2	2	no	—
67	C₂×Q₈×C₄₇	—	4	1	no	—
68	C₂×Q₈×C₄₇	—	4	1	no	—
69	C₂×Q₈×C₄₇	—	4	1	no	—
70	C₂×Q₈×C₄₇	—	4	1	no	—
71	C₂×Q₈×C₄₇	—	4	1	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	4	1	no	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
77	C₂×D₁₈₈	—	2	2	no	—
78	C₂×D₁₈₈	—	2	2	no	—
79	C₂×D₁₈₈	—	2	2	no	—
80	C₂×D₁₈₈	—	4	1	no	—
81	C₂×D₁₈₈	—	4	2	no	—
82	C₂×D₁₈₈	—	4	1	no	—
83	C₂×D₁₈₈	—	4	2	no	—
84	C₂×D₁₈₈	—	4	1	no	—
85	C₂×D₁₈₈	—	4	1	no	—
86	C₂×D₁₈₈	—	4	1	no	—
87	C₂×D₁₈₈	—	4	1	no	—
88	C₂×D₁₈₈	—	4	1	no	—
89	C₂×D₁₈₈	—	8	2	no	—
90	C₂×C₈×C₄₇	—	4	1	no	—
91	C₂×C₈×C₄₇	—	4	2	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	—	2	2	no	—
95	C₄⋊C₄×C₄₇	—	4	1	no	—
96	C₄⋊C₄×C₄₇	—	4	1	no	—
97	C₄⋊C₄×C₄₇	SD₁₆	1	4	yes	complemented
98	C₄²×C₄₇	—	4	1	no	—
99	C₄²×C₄₇	—	4	1	no	—
100	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₇	#41	1	1	yes	characteristic
104	#10625336	—	4	8	no	—
105	#10625336	—	4	8	no	—
106	C₂×C₄₇⋊C₈	—	2	1	no	—
107	C₂×C₄₇⋊C₈	—	2	1	no	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	4	1	no	—
110	C₂×C₄₇⋊C₈	—	4	1	no	—
111	C₂×C₄₇⋊C₈	—	4	1	no	—
112	C₂×C₄₇⋊C₈	—	4	2	no	—
113	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
114	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
115	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
116	C₂×C₄₇⋊C₈	SD₁₆	1	2	yes	—
117	C₂³×C₄₇	#121	1	1	yes	characteristic
118	C₂²×D₄₇	—	4	2	no	—
119	C₂²×D₄₇	—	8	1	no	—
120	C₂²×D₄₇	—	8	1	no	—
121	C₂²×D₄₇	—	8	2	no	—
122	C₂²×D₄₇	—	8	2	no	—
123	Q₈×C₄₇	—	4	2	no	—
124	Q₈×C₄₇	—	4	2	no	—
125	Q₈×C₄₇	—	8	1	no	—
126	Q₈×C₄₇	—	8	1	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	4	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	8	1	no	—
133	C₂×C₄×C₄₇	—	8	1	no	—
134	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
143	C₂×C₄×C₄₇	#135	1	2	yes	—
144	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
146	D₁₈₈	—	4	2	no	—
147	D₁₈₈	—	4	2	no	—
148	D₁₈₈	—	4	4	no	—
149	D₁₈₈	—	8	2	no	—
150	D₁₈₈	—	8	2	no	—
151	D₁₈₈	—	8	1	no	—
152	D₁₈₈	—	8	1	no	—
153	D₁₈₈	—	8	1	no	—
154	D₁₈₈	—	8	1	no	—
155	C₈×C₄₇	—	8	1	no	—
156	C₄₇⋊C₈	—	2	2	no	—
157	C₄₇⋊C₈	—	4	2	no	—
158	C₄₇⋊C₈	—	8	1	no	—
159	C₄₇⋊C₈	#102	1	4	yes	complemented
160	#9732	—	47	1	no	maximal, Sylow
161	C₂²×C₄₇	#390	1	1	yes	characteristic
162	C₂²×C₄₇	#459	1	1	yes	characteristic
163	C₂²×C₄₇	#460	1	1	yes	characteristic
164	C₂²×C₄₇	#465	1	2	yes	—
165	C₂²×C₄₇	#471	1	2	yes	—
166	C₂×D₄₇	—	8	2	no	—
167	C₂×D₄₇	—	8	2	no	—
168	C₂×D₄₇	—	8	2	no	—
169	C₂×D₄₇	—	16	2	no	—
170	C₂×D₄₇	—	16	2	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	—	4	4	no	—
174	C₄×C₄₇	—	8	2	no	—
175	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
176	C₄×C₄₇	#418	1	1	yes	characteristic
177	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
178	C₄×C₄₇	#459	1	1	yes	characteristic
179	C₄×C₄₇	#460	1	1	yes	characteristic
180	C₄×C₄₇	#492	1	1	yes	characteristic
181	C₄×C₄₇	#495	1	1	yes	characteristic
182	C₂×#492	—	47	1	no	—
183	#1988	D₄₇	1	1	yes	characteristic, complemented
184	#1982	—	47	1	no	—
185	C₈×C₄⋊C₄	—	47	1	no	—
186	C₂×#334	—	47	1	no	—
187	C₂×#332	—	47	1	no	—
188	C₂×#331	—	47	1	no	—
189	C₂×C₄₇	#1665	1	2	yes	—
190	C₂×C₄₇	#1666	1	2	yes	—
191	C₂×C₄₇	#1968	1	1	yes	characteristic
192	C₂×C₄₇	#1977	1	1	yes	characteristic
193	C₂×C₄₇	#1982	1	1	yes	characteristic
194	D₄₇	—	16	2	no	—
195	C₂²×SD₁₆	—	94	1	no	—
196	C₂²×SD₁₆	—	94	1	no	—
197	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
198	C₂×#127	—	47	1	no	—
199	#492	—	47	4	no	—
200	#492	—	94	2	no	—
201	C₂×C₄⋊C₈	—	47	1	no	—
202	C₂×C₄⋊C₈	—	47	1	no	—
203	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
204	C₂×#102	—	94	2	no	—
205	C₂×C₄×C₈	—	47	1	no	—
206	C₂×C₄×C₈	—	94	1	no	—
207	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
208	#340	—	2	2	no	—
209	#336	—	94	1	no	—
210	#334	—	94	2	no	—
211	#332	—	94	2	no	—
212	#331	—	94	2	no	—
213	C₄₇	#9732	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	—	2	1	no	—
215	C₂²×D₄	—	94	1	no	—
216	C₂²×D₄	—	94	1	no	—
217	C₂²×D₄	—	94	1	no	—
218	C₂×SD₁₆	—	94	4	no	—
219	C₂×SD₁₆	—	94	4	no	—
220	C₂×SD₁₆	—	188	2	no	—
221	C₂×SD₁₆	—	188	2	no	—
222	C₂×SD₁₆	—	188	2	no	—
223	C₂×SD₁₆	—	188	2	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	47	1	no	—
226	C₂²×C₈	—	47	1	no	—
227	C₂²×C₈	—	94	1	no	—
228	C₂²×C₈	—	94	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	C₄×D₄₇	1	2	yes	—
233	#127	—	47	2	no	—
234	#127	—	94	1	no	—
235	#127	—	94	1	no	—
236	#127	—	94	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	47	2	no	—
245	C₄⋊C₈	—	94	1	no	—
246	C₄⋊C₈	—	94	1	no	—
247	C₄⋊C₈	—	94	1	no	—
248	#102	—	188	4	no	—
249	C₄×C₈	—	47	2	no	—
250	C₄×C₈	—	47	2	no	—
251	C₄×C₈	—	188	2	no	—
252	#95	—	2	1	no	—
253	C₂⁴	—	188	1	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×D₄	—	94	2	no	—
261	C₂×D₄	—	94	2	no	—
262	C₂×D₄	—	94	2	no	—
263	C₂×D₄	—	376	2	no	—
264	C₂×D₄	—	188	1	no	—
265	C₂×D₄	—	188	1	no	—
266	C₂×D₄	—	188	1	no	—
267	C₂×D₄	—	188	1	no	—
268	C₂×D₄	—	188	1	no	—
269	C₂×D₄	—	188	2	no	—
270	C₂×D₄	—	188	2	no	—
271	C₂×D₄	—	188	1	no	—
272	C₂×D₄	—	188	1	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
276	C₂²×C₄	#10625352	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
278	SD₁₆	—	188	8	no	—
279	SD₁₆	—	188	8	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	47	2	no	—
283	C₂×C₈	—	47	2	no	—
284	C₂×C₈	—	47	2	no	—
285	C₂×C₈	—	47	2	no	—
286	C₂×C₈	—	94	1	no	—
287	C₂×C₈	—	94	1	no	—
288	C₂×C₈	—	94	2	no	—
289	C₂×C₈	—	188	1	no	—
290	C₂×C₈	—	188	2	no	—
291	C₂×C₈	—	188	1	no	—
292	C₂×C₈	—	188	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	#10625336	1	4	yes	complemented
299	C₄²	—	4	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
302	C₄²	#10625350	1	1	yes	characteristic
303	C₄²	D₄×D₄₇	1	1	yes	characteristic
304	C₄²	D₄×D₄₇	1	1	yes	characteristic
305	C₂³	—	376	1	no	—
306	C₂³	—	376	1	no	—
307	C₂³	—	376	2	no	—
308	C₂³	—	376	2	no	—
309	C₂³	—	188	2	no	—
310	C₂³	#13093996	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	1	no	—
314	Q₈	—	8	1	no	—
315	D₄	—	376	2	no	—
316	D₄	—	376	2	no	—
317	D₄	—	376	1	no	—
318	D₄	—	376	1	no	—
319	D₄	—	376	1	no	—
320	D₄	—	376	1	no	—
321	D₄	—	188	2	no	—
322	D₄	—	188	2	no	—
323	D₄	—	188	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	8	1	no	—
331	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
332	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
333	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
334	C₂×C₄	#13093997	1	1	yes	characteristic
335	C₂×C₄	#13094010	1	1	yes	characteristic
336	C₂×C₄	#13094020	1	1	yes	characteristic
337	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
338	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
339	C₂×C₄	C₂×#10625336	1	2	yes	—
340	C₂×C₄	#13094044	1	2	yes	—
341	C₂×C₄	#13094048	1	1	yes	characteristic
342	C₂×C₄	#13094051	1	1	yes	characteristic
343	C₈	—	8	1	no	—
344	C₈	—	47	4	no	—
345	C₈	—	94	2	no	—
346	C₈	—	376	1	no	—
347	C₈	—	188	2	no	—
348	C₂²	—	376	2	no	—
349	C₂²	—	376	2	no	—
350	C₂²	—	376	2	no	—
351	C₂²	—	752	2	no	—
352	C₂²	—	752	2	no	—
353	C₂²	#15689023	1	1	yes	characteristic
354	C₂²	#15689316	1	1	yes	characteristic
355	C₂²	#15689321	1	1	yes	characteristic
356	C₂²	#15689354	1	2	yes	—
357	C₂²	#15689388	1	2	yes	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	4	no	—
361	C₄	—	8	2	no	—
362	C₄	#102×D₄₇	1	1	yes	characteristic
363	C₄	#15689126	1	1	yes	characteristic
364	C₄	C₄×#10625336	1	2	yes	—
365	C₄	#15689317	1	1	yes	characteristic
366	C₄	#15689322	1	1	yes	characteristic
367	C₄	#15689497	1	1	yes	characteristic
368	C₄	#15689505	1	1	yes	characteristic
369	C₂	—	752	2	no	—
370	C₂	#20242731	1	2	yes	—
371	C₂	#20242738	1	2	yes	—
372	C₂	#20244245	1	1	yes	characteristic
373	C₂	#20244290	1	1	yes	characteristic
374	C₂	#20244313	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093996
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1988	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂×#13094021	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1988	D₄₇
hypocenter	C₄₇	#9732
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1988

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×C₄₇⋊C₈	C₂	semidirect
4. C₂×#15688873	C₂	semidirect
5. C₂×#15688869	C₂	semidirect
7. C₄₇×#1988	C₂	semidirect
12. #15689497	C₄	semidirect
97. C₄⋊C₄×C₄₇	SD₁₆	semidirect
159. C₄₇⋊C₈	#102	semidirect
183. #1988	D₄₇	semidirect
213. C₄₇	#9732	semidirect
298. C₄⋊C₄	#10625336	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689497	2
#20244315	2
C₄⋊C₄×C₄₇⋊C₈	2
C₂×#15688873	2
C₂×#15688869	2
C₂×#15688866	2
C₄₇×#1988	2
#9732	47