Subgroups — #26110246

Order 12032.

558 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252412	C₂	1	1	yes	maximal, characteristic
2	#20243681	C₂	1	1	yes	maximal, characteristic
3	#20243568	C₂	1	2	yes	maximal, complemented
4	#20241236	C₂	1	2	yes	maximal, complemented
5	C₄×#99×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	D₄×C₄₇⋊C₈	—	2	2	no	—
8	D₄×C₄₇⋊C₈	—	2	2	no	—
9	#15689176	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	#15688878	—	2	2	no	—
16	#15688878	—	2	1	no	—
17	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#99×C₄₇	C₄	1	1	yes	characteristic
19	C₂×#99×C₄₇	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#342	C₄	1	2	yes	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	1	no	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	C₂²×C₄₇⋊C₈	—	2	1	no	—
27	C₂²×C₄₇⋊C₈	—	4	2	no	—
28	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
29	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
30	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
33	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
34	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
36	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
37	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
38	C₄×C₄₇⋊C₈	—	2	2	no	—
39	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₄×D₄×C₄₇	—	2	2	no	—
43	C₄×D₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	Q₈	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₄₇	Q₈	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	#99×C₄₇	—	2	2	no	—
52	#99×C₄₇	C₈	1	4	yes	complemented
53	#95×C₄₇	—	2	2	no	—
54	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×D₄×C₄₇	—	2	2	no	—
57	C₂×D₄×C₄₇	—	2	2	no	—
58	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₄₇	M₁₆	1	2	yes	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	4	1	no	—
65	C₂²×C₄×C₄₇	—	4	1	no	—
66	C₂²×C₄×C₄₇	—	4	1	no	—
67	C₂²×C₄×C₄₇	—	4	1	no	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	#41	1	2	yes	—
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₂²⋊C₄×C₄₇	—	2	2	no	—
77	C₂²⋊C₄×C₄₇	—	2	2	no	—
78	C₂²⋊C₄×C₄₇	—	2	2	no	—
79	C₂²⋊C₄×C₄₇	—	2	2	no	—
80	C₂²⋊C₄×C₄₇	—	2	2	no	—
81	C₂²⋊C₄×C₄₇	C₂×C₈	1	4	yes	complemented
82	C₄²×C₄₇	—	2	1	no	—
83	C₄²×C₄₇	—	2	1	no	—
84	C₄²×C₄₇	—	4	2	no	—
85	C₄²×C₄₇	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	2	2	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	1	no	—
92	C₂×C₄₇⋊C₈	—	4	1	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	1	no	—
96	C₂×C₄₇⋊C₈	—	4	1	no	—
97	C₂×C₄₇⋊C₈	—	4	1	no	—
98	C₂×C₄₇⋊C₈	—	8	1	no	—
99	C₂×C₄₇⋊C₈	—	8	1	no	—
100	C₂³×C₄₇	—	2	1	no	—
101	C₂³×C₄₇	—	2	1	no	—
102	C₂³×C₄₇	—	4	1	no	—
103	C₂³×C₄₇	—	4	1	no	—
104	C₂³×C₄₇	—	4	1	no	—
105	C₂³×C₄₇	—	4	1	no	—
106	C₂³×C₄₇	C₄⋊C₈	1	2	yes	complemented
107	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₄₇	C₄×D₄	1	2	yes	—
109	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
110	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂³×C₄₇	#131	1	2	yes	—
112	D₄×C₄₇	—	2	4	no	—
113	D₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	4	1	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	1	no	—
128	C₂×C₄×C₄₇	—	4	1	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	—	4	1	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	8	1	no	—
135	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	complemented
136	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
139	C₄₇⋊C₈	—	4	2	no	—
140	C₄₇⋊C₈	—	4	2	no	—
141	C₄₇⋊C₈	—	4	2	no	—
142	C₄₇⋊C₈	—	8	1	no	—
143	#8851	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	4	1	no	—
150	C₂²×C₄₇	—	4	1	no	—
151	C₂²×C₄₇	—	4	2	no	—
152	C₂²×C₄₇	—	4	2	no	—
153	C₂²×C₄₇	—	4	1	no	—
154	C₂²×C₄₇	—	4	1	no	—
155	C₂²×C₄₇	—	8	1	no	—
156	C₂²×C₄₇	#385	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
158	C₂²×C₄₇	#424	1	1	yes	characteristic
159	C₂²×C₄₇	C₈×D₄	1	2	yes	—
160	C₂²×C₄₇	#435	1	2	yes	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	4	4	no	—
165	C₄×C₄₇	—	4	1	no	—
166	C₄×C₄₇	—	4	1	no	—
167	C₄×C₄₇	—	8	2	no	—
168	C₄×C₄₇	B(4,𝔽₂)	1	1	yes	characteristic
169	C₄×C₄₇	#458	1	1	yes	characteristic
170	#2972	—	47	1	no	—
171	#1869	—	47	1	no	—
172	#1850	—	47	2	no	—
173	C₄×#99	D₄₇	1	1	yes	characteristic
174	#1282	—	47	2	no	—
175	C₂×C₄₇	—	2	2	no	—
176	C₂×C₄₇	—	2	2	no	—
177	C₂×C₄₇	—	2	1	no	—
178	C₂×C₄₇	—	2	1	no	—
179	C₂×C₄₇	—	4	2	no	—
180	C₂×C₄₇	#1850	1	1	yes	characteristic
181	C₂×C₄₇	#1867	1	1	yes	characteristic
182	C₂×C₄₇	#1868	1	1	yes	characteristic
183	C₂³×C₈	—	47	1	no	—
184	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
185	C₈×D₄	—	94	2	no	—
186	C₈×D₄	—	94	2	no	—
187	#431	—	94	1	no	—
188	C₂×C₄⋊C₈	—	94	2	no	—
189	C₂×#99	Dic₄₇	1	1	yes	characteristic
190	C₂×#99	Dic₄₇	1	1	yes	characteristic
191	C₂×C₂²⋊C₈	—	47	2	no	—
192	C₂×C₂²⋊C₈	—	47	1	no	—
193	C₂×C₂²⋊C₈	—	94	1	no	—
194	C₂×C₂²⋊C₈	—	94	1	no	—
195	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
196	#342	Dic₄₇	1	2	yes	—
197	#336	—	94	2	no	—
198	#336	—	94	1	no	—
199	C₄₇	#8851	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
201	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
202	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
203	C₂²×C₈	—	47	2	no	—
204	C₂²×C₈	—	47	2	no	—
205	C₂²×C₈	—	94	1	no	—
206	C₂²×C₈	—	94	1	no	—
207	C₂²×C₈	—	94	1	no	—
208	C₂²×C₈	—	94	2	no	—
209	C₂²×C₈	—	94	1	no	—
210	C₂²×C₈	—	188	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₂×C₄⋊C₄	Dic₉₄	1	1	yes	characteristic
214	C₂×C₂²⋊C₄	Dic₉₄	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
216	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
220	C₄⋊C₈	—	94	1	no	—
221	C₄⋊C₈	—	94	1	no	—
222	C₄⋊C₈	—	188	4	no	—
223	#99	—	2	2	no	—
224	#99	C₄₇⋊C₈	1	4	yes	complemented
225	C₂²⋊C₈	—	94	2	no	—
226	C₂²⋊C₈	—	94	2	no	—
227	C₂²⋊C₈	—	94	2	no	—
228	C₂²⋊C₈	—	94	2	no	—
229	C₂²⋊C₈	—	188	2	no	—
230	C₄×C₈	—	94	2	no	—
231	#95	—	2	2	no	—
232	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
233	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
237	C₂×D₄	#10625329	1	2	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₄₇	1	2	yes	—
252	C₂²×C₄	#10625352	1	2	yes	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	—	94	4	no	—
255	C₂×C₈	—	94	2	no	—
256	C₂×C₈	—	94	2	no	—
257	C₂×C₈	—	376	1	no	—
258	C₂×C₈	—	376	1	no	—
259	C₂×C₈	—	188	1	no	—
260	C₂×C₈	—	188	1	no	—
261	C₂×C₈	—	188	1	no	—
262	C₂×C₈	—	188	2	no	—
263	C₂×C₈	—	188	1	no	—
264	C₂×C₈	—	188	1	no	—
265	C₂×C₈	—	188	2	no	—
266	C₂×C₈	—	188	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	complemented
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	4	1	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	C₄⋊(C₄₇⋊C₈)	1	2	yes	complemented
285	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
286	C₂³	D₄×Dic₄₇	1	2	yes	—
287	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
288	C₂³	C₂×#10625329	1	1	yes	characteristic
289	C₂³	#13094031	1	2	yes	—
290	D₄	—	2	4	no	—
291	D₄	—	2	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	1	no	—
313	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	complemented
314	C₂×C₄	#13093994	1	1	yes	characteristic
315	C₂×C₄	#13094001	1	2	yes	—
316	C₂×C₄	#13094008	1	1	yes	characteristic
317	C₈	—	376	1	no	—
318	C₈	—	188	2	no	—
319	C₈	—	188	2	no	—
320	C₈	—	188	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	#15688998	1	1	yes	characteristic
334	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
335	C₂²	#15689150	1	1	yes	characteristic
336	C₂²	D₄×C₄₇⋊C₈	1	2	yes	—
337	C₂²	#15689195	1	2	yes	—
338	C₄	—	2	2	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	4	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	8	2	no	—
345	C₄	#15689311	1	1	yes	characteristic
346	C₄	#15689314	1	1	yes	characteristic
347	C₂	—	2	2	no	—
348	C₂	—	2	2	no	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	4	2	no	—
352	C₂	#20243568	1	1	yes	characteristic
353	C₂	#20243671	1	1	yes	characteristic
354	C₂	#20243676	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093994
commutator	C₂³×C₄₇	C₂²×C₈
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄×#99×C₄₇	C₂
socle	C₂²×C₄₇	#385
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₄×#99	D₄₇
hypocenter	C₄₇	#8851
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₄×#99

Complements

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

N	H ≅ G/N	type
3. #20243568	C₂	semidirect
4. #20241236	C₂	semidirect
52. #99×C₄₇	C₈	semidirect
81. C₂²⋊C₄×C₄₇	C₂×C₈	semidirect
106. C₂³×C₄₇	C₄⋊C₈	semidirect
135. C₂×C₄×C₄₇	C₄⋊C₈	semidirect
199. C₄₇	#8851	semidirect
224. #99	C₄₇⋊C₈	semidirect
273. C₂²⋊C₄	C₂×C₄₇⋊C₈	semidirect
284. C₂³	C₄⋊(C₄₇⋊C₈)	semidirect
313. C₂×C₄	C₄⋊(C₄₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#20252412	2
#20243681	2
#20243568	2
#20241236	2
C₄×#99×C₄₇	2
#8851	47