Subgroups — #26184528

Order 12032.

876 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252402	C₂	1	2	yes	maximal
2	#20242026	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242015	C₂	1	2	yes	maximal, complemented
4	#20242003	C₂	1	4	yes	maximal, complemented
5	#20241990	C₂	1	2	yes	maximal, complemented
6	#20241983	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688865	C₂	1	2	yes	maximal, complemented
8	C₄₇×#3504	C₂	1	1	yes	maximal, characteristic
9	#15689254	—	2	2	no	—
10	#15689195	—	2	4	no	—
11	D₄×C₄₇⋊C₈	—	2	2	no	—
12	#15689180	C₂²	1	2	yes	—
13	#15689180	C₂²	1	4	yes	—
14	C₄×#10625329	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	4	yes	—
16	#15688865	C₂²	1	1	yes	characteristic
17	#15688865	C₂²	1	4	yes	—
18	#15688865	C₂²	1	2	yes	—
19	#15688865	C₂²	1	2	yes	—
20	#15688865	C₂²	1	1	yes	characteristic
21	#15688865	C₂²	1	4	yes	—
22	#15688865	C₂²	1	2	yes	—
23	C₄₇×#557	—	2	2	no	—
24	C₄₇×#554	—	2	2	no	—
25	C₄₇×#554	—	2	4	no	—
26	C₄₇×#544	C₄	1	4	yes	—
27	C₄₇×#544	C₄	1	2	yes	—
28	C₄₇×#533	C₂²	1	1	yes	characteristic
29	C₄₇×#533	C₂²	1	2	yes	—
30	C₂×#125×C₄₇	C₂²	1	2	yes	—
31	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#520	C₄	1	2	yes	—
33	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#10625329	—	2	2	no	—
35	C₂²×C₄₇⋊C₈	—	2	4	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
38	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
39	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
40	#13093935	—	2	4	no	—
41	#13093935	C₂³	1	2	yes	—
42	#13093935	C₂³	1	4	yes	—
43	#13093935	C₂³	1	2	yes	—
44	C₄×C₄₇⋊C₈	—	2	2	no	—
45	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₄₇	—	2	4	no	—
47	C₄⋊Q₈×C₄₇	—	2	2	no	—
48	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
49	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
50	#125×C₄₇	—	2	4	no	—
51	#125×C₄₇	—	2	2	no	—
52	#125×C₄₇	—	2	2	no	—
53	#125×C₄₇	—	2	4	no	—
54	#125×C₄₇	C₂×C₄	1	4	yes	—
55	#125×C₄₇	C₂×C₄	1	2	yes	—
56	#125×C₄₇	C₂³	1	1	yes	characteristic
57	#125×C₄₇	C₂³	1	2	yes	—
58	#125×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²⋊Q₈×C₄₇	—	2	4	no	—
60	C₂²⋊Q₈×C₄₇	—	2	2	no	—
61	C₂²⋊Q₈×C₄₇	—	2	4	no	—
62	C₂²⋊Q₈×C₄₇	—	2	4	no	—
63	C₂²⋊Q₈×C₄₇	—	2	4	no	—
64	C₄×Q₈×C₄₇	—	2	2	no	—
65	C₄×Q₈×C₄₇	—	2	2	no	—
66	C₄×Q₈×C₄₇	—	2	4	no	—
67	C₄×Q₈×C₄₇	D₄	1	2	yes	—
68	C₄×D₄×C₄₇	—	2	4	no	—
69	C₄×D₄×C₄₇	—	2	2	no	—
70	C₄×D₄×C₄₇	D₄	1	2	yes	—
71	C₄×D₄×C₄₇	D₄	1	4	yes	—
72	#117×C₄₇	C₂×C₄	1	2	yes	—
73	#117×C₄₇	C₂×C₄	1	4	yes	—
74	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
75	#117×C₄₇	C₂×C₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
79	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
81	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
82	#41×C₄₇	—	2	4	no	—
83	C₂×Q₈×C₄₇	—	2	2	no	—
84	C₂×Q₈×C₄₇	—	2	4	no	—
85	C₂×Q₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×D₄×C₄₇	C₂²⋊C₄	1	2	yes	—
87	C₂×D₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	—	2	2	no	—
89	C₂²×C₄×C₄₇	—	2	4	no	—
90	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₄₇	—	2	4	no	—
95	C₄⋊C₄×C₄₇	—	2	4	no	—
96	C₄⋊C₄×C₄₇	—	2	4	no	—
97	C₄⋊C₄×C₄₇	—	2	4	no	—
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	2	4	no	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄⋊C₄×C₄₇	—	2	4	no	—
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
105	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
106	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
107	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
108	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
109	C₂²⋊C₄×C₄₇	—	2	2	no	—
110	C₂²⋊C₄×C₄₇	—	2	4	no	—
111	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
112	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
113	C₄²×C₄₇	—	2	4	no	—
114	C₄²×C₄₇	—	2	2	no	—
115	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
116	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₄₇	C₂×D₄	1	2	yes	—
118	C₄²×C₄₇	C₂×D₄	1	4	yes	—
119	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
120	#10625329	—	4	4	no	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	2	4	no	—
124	C₂×C₄₇⋊C₈	—	4	8	no	—
125	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
127	Q₈×C₄₇	—	2	2	no	—
128	Q₈×C₄₇	—	8	4	no	—
129	Q₈×C₄₇	—	8	2	no	—
130	D₄×C₄₇	—	2	4	no	—
131	D₄×C₄₇	#101	1	4	yes	—
132	C₂×C₄×C₄₇	—	2	4	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	4	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	4	4	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	8	4	no	—
140	C₂×C₄×C₄₇	—	8	2	no	—
141	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
143	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
144	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
147	C₄₇⋊C₈	—	2	4	no	—
148	C₄₇⋊C₈	—	4	4	no	—
149	C₄₇⋊C₈	—	4	2	no	—
150	#17689	—	47	1	no	maximal, Sylow
151	C₂²×C₄₇	—	2	4	no	—
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	C₂×#101	1	4	yes	—
155	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	4	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	4	2	no	—
162	C₄×C₄₇	—	4	4	no	—
163	C₄×C₄₇	—	4	2	no	—
164	C₄×C₄₇	C₂×#101	1	2	yes	—
165	#3504	D₄₇	1	1	yes	characteristic
166	#2970	—	47	2	no	—
167	#1537	—	47	1	no	—
168	#1534	—	47	2	no	—
169	#1531	—	47	4	no	—
170	#1528	—	47	2	no	—
171	#1526	—	47	1	no	—
172	C₂×#330	—	47	2	no	—
173	C₂×C₄₇	—	2	4	no	—
174	C₂×C₄₇	—	4	1	no	—
175	C₂×C₄₇	C₂²×#101	1	1	yes	characteristic
176	C₂×C₄₇	#2901	1	2	yes	—
177	#557	—	2	2	no	—
178	#554	—	2	2	no	—
179	#554	—	2	4	no	—
180	#544	Dic₄₇	1	4	yes	—
181	#544	Dic₄₇	1	2	yes	—
182	#533	C₂×D₄₇	1	1	yes	characteristic
183	#533	C₂×D₄₇	1	2	yes	—
184	C₂×#125	C₂×D₄₇	1	2	yes	—
185	C₂×#125	C₂×D₄₇	1	1	yes	characteristic
186	#520	Dic₄₇	1	2	yes	—
187	C₄×#41	C₂×D₄₇	1	1	yes	characteristic
188	#446	—	94	2	no	—
189	#435	—	94	4	no	—
190	C₈×D₄	—	94	2	no	—
191	#432	—	47	2	no	—
192	#432	—	47	4	no	—
193	C₄×M₁₆	—	47	2	no	—
194	C₂×#97	—	47	4	no	—
195	#330	—	47	2	no	—
196	#330	—	47	2	no	—
197	#330	—	47	1	no	—
198	#330	—	47	4	no	—
199	#330	—	47	2	no	—
200	#330	—	47	1	no	—
201	#330	—	47	4	no	—
202	C₄₇	#17689	1	1	yes	characteristic, complemented, Sylow
203	C₂×#41	C₂×Dic₄₇	1	1	yes	characteristic
204	C₂×M₁₆	—	94	2	no	—
205	C₂²×C₈	—	94	4	no	—
206	C₄⋊Q₈	—	2	4	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
209	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
210	#125	—	2	4	no	—
211	#125	—	2	2	no	—
212	#125	—	2	2	no	—
213	#125	—	2	4	no	—
214	#125	C₂×Dic₄₇	1	4	yes	—
215	#125	C₂×Dic₄₇	1	2	yes	—
216	#125	C₂²×D₄₇	1	1	yes	characteristic
217	#125	C₂²×D₄₇	1	2	yes	—
218	#125	C₂²×D₄₇	1	1	yes	characteristic
219	C₂²⋊Q₈	—	2	4	no	—
220	C₂²⋊Q₈	—	2	2	no	—
221	C₂²⋊Q₈	—	2	4	no	—
222	C₂²⋊Q₈	—	2	4	no	—
223	C₂²⋊Q₈	—	2	4	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	—	2	4	no	—
227	C₄×Q₈	C₂²⋊D₄₇	1	2	yes	—
228	C₄×D₄	—	2	4	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	C₂²⋊D₄₇	1	2	yes	—
231	C₄×D₄	C₂²⋊D₄₇	1	4	yes	—
232	#117	C₂×Dic₄₇	1	2	yes	—
233	#117	C₂×Dic₄₇	1	4	yes	—
234	#117	C₂×Dic₄₇	1	1	yes	characteristic
235	#117	C₂×Dic₄₇	1	2	yes	—
236	C₂×C₄⋊C₄	—	2	4	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
239	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
240	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
241	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
242	C₄⋊C₈	—	94	2	no	—
243	C₄⋊C₈	—	94	4	no	—
244	C₂²⋊C₈	—	94	4	no	—
245	C₂²⋊C₈	—	94	2	no	—
246	#97	—	47	2	no	—
247	#97	—	47	4	no	—
248	#97	—	47	2	no	—
249	#97	—	94	4	no	—
250	C₄×C₈	—	94	2	no	—
251	#41	—	2	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
255	C₂×D₄	C₂²⋊Dic₄₇	1	2	yes	—
256	C₂×D₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
262	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
263	M₁₆	—	188	4	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	—	94	2	no	—
266	C₂×C₈	—	94	4	no	—
267	C₂×C₈	—	188	8	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
279	C₄⋊C₄	C₂²⋊Dic₄₇	1	2	yes	—
280	C₄⋊C₄	C₂²×Dic₄₇	1	2	yes	—
281	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
282	C₄⋊C₄	C₂²×Dic₄₇	1	2	yes	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₂²⋊C₄	—	2	4	no	—
285	C₂²⋊C₄	C₂²⋊Dic₄₇	1	2	yes	—
286	C₂²⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	2	no	—
289	C₄²	C₂²×Dic₄₇	1	2	yes	—
290	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
291	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
292	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
293	C₄²	C₂³×D₄₇	1	1	yes	characteristic
294	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
295	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
296	Q₈	—	2	2	no	—
297	Q₈	—	8	4	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	2	4	no	—
300	D₄	#13093943	1	4	yes	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
311	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
312	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
313	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
316	C₈	—	94	4	no	—
317	C₈	—	188	4	no	—
318	C₈	—	188	2	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	1	no	—
322	C₂²	C₂×#13093943	1	4	yes	—
323	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	4	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	2	no	—
332	C₄	C₂×#13093943	1	2	yes	—
333	C₂	—	2	4	no	—
334	C₂	—	4	1	no	—
335	C₂	C₂²×#13093943	1	1	yes	characteristic
336	C₂	#20251935	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3504	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#3504	D₄₇
hypocenter	C₄₇	#17689
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #3504

Complements

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

N	H ≅ G/N	type
2. #20242026	C₂	semidirect
3. #20242015	C₂	semidirect
4. #20242003	C₂	semidirect
5. #20241990	C₂	semidirect
6. #20241983	C₂	semidirect
7. C₂×#15688865	C₂	semidirect
202. C₄₇	#17689	semidirect

Maximal subgroups

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

subgroup	index
#20252402	2
#20242026	2
#20242015	2
#20242003	2
#20241990	2
#20241983	2
C₂×#15688865	2
C₄₇×#3504	2
#17689	47