Subgroups — C₄×#15689497 #26224701

Order 12032.

1254 conjugacy classes of subgroups.

The subgroup lattice has 352 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	C₂×C₄×#10625336	C₂	1	4	yes	maximal, complemented
3	C₄×#13094021	C₂	1	1	yes	maximal, characteristic
4	#20244245	C₂	1	4	yes	maximal, complemented
5	#20244193	C₂	1	1	yes	maximal, characteristic
6	#20244191	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243544	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625336	C₂²	1	4	yes	—
11	C₂×#13094021	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₈₈	—	2	2	no	—
13	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
14	#15689497	C₄	1	8	yes	direct product factor
15	C₄×#10625336	—	2	16	no	—
16	C₂×#13093955	C₂²	1	4	yes	—
17	C₂×#13093948	C₂²	1	4	yes	—
18	C₂×#13093944	C₂²	1	4	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
23	#15689025	—	2	2	no	—
24	#15689025	C₂²	1	2	yes	—
25	#15688993	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#391	C₂²	1	2	yes	—
29	C₄₇×#389	—	2	2	no	—
30	C₄₇×#384	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₄³×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²×D₁₈₈	—	2	2	no	—
34	C₂²×D₁₈₈	C₂³	1	2	yes	—
35	C₂²×C₄×D₄₇	—	2	2	no	—
36	C₂×#10625336	—	2	8	no	—
37	C₂×#10625336	—	2	8	no	—
38	C₂×#10625336	C₂×C₄	1	16	yes	complemented
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
41	#13094021	—	2	2	no	—
42	#13094021	C₂×C₄	1	4	yes	—
43	C₄×D₁₈₈	—	2	8	no	—
44	C₄×D₁₈₈	—	4	8	no	—
45	C₂×C₄⋊Dic₄₇	—	2	2	no	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
48	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
49	#13093955	—	2	8	no	—
50	#13093948	—	2	8	no	—
51	#13093944	—	2	8	no	—
52	C₄×C₄₇⋊C₈	—	2	2	no	—
53	C₄×C₄₇⋊C₈	—	2	4	no	—
54	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
55	C₄×C₄₇⋊C₈	D₄	1	8	yes	complemented
56	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₄₇	—	2	2	no	—
58	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	complemented
59	C₄×Q₈×C₄₇	—	2	8	no	—
60	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₇	—	2	2	no	—
65	C₂×C₄²×C₄₇	D₄	1	2	yes	—
66	C₂×C₄²×C₄₇	D₄	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	#95×C₄₇	—	2	4	no	—
71	C₂³×D₄₇	—	2	2	no	—
72	C₂×Q₈×C₄₇	—	2	2	no	—
73	C₂×Q₈×C₄₇	—	2	4	no	—
74	C₂×Q₈×C₄₇	—	2	4	no	—
75	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₄₇	—	2	2	no	—
77	C₂²×C₄×C₄₇	—	2	2	no	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₂×D₁₈₈	—	2	4	no	—
85	C₂×D₁₈₈	—	2	4	no	—
86	C₂×D₁₈₈	—	2	8	no	—
87	C₂×D₁₈₈	—	2	2	no	—
88	C₂×D₁₈₈	—	4	4	no	—
89	C₂×D₁₈₈	—	4	4	no	—
90	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
91	C₂×C₄×D₄₇	—	4	4	no	—
92	C₂×C₄×D₄₇	—	4	8	no	—
93	C₄⋊C₄×C₄₇	—	2	8	no	—
94	C₄⋊C₄×C₄₇	—	2	4	no	—
95	C₄⋊C₄×C₄₇	—	4	4	no	—
96	C₄⋊C₄×C₄₇	—	4	4	no	—
97	C₄²×C₄₇	—	2	4	no	—
98	C₄²×C₄₇	—	2	2	no	—
99	C₄²×C₄₇	—	4	4	no	—
100	C₄²×C₄₇	SD₁₆	1	8	yes	complemented
101	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
102	C₄²×C₄₇	C₂×D₄	1	4	yes	—
103	C₄²×C₄₇	#41	1	2	yes	—
104	#10625336	—	2	32	no	—
105	(C₂×D₄₇)⋊C₄	—	4	4	no	—
106	(C₂×D₄₇)⋊C₄	—	4	8	no	—
107	C₄⋊Dic₄₇	—	2	4	no	—
108	C₄⋊Dic₄₇	—	4	4	no	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	—	2	2	no	—
111	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
112	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
113	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
114	C₂×C₄₇⋊C₈	#41	1	4	yes	—
115	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂²×D₄₇	—	2	4	no	—
117	C₂²×D₄₇	—	4	2	no	—
118	C₂²×D₄₇	—	4	2	no	—
119	C₂²×D₄₇	—	4	4	no	—
120	C₂²×D₄₇	—	4	4	no	—
121	Q₈×C₄₇	—	2	8	no	—
122	Q₈×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
132	C₂×C₄×C₄₇	#127	1	2	yes	—
133	C₂×C₄×C₄₇	#127	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
135	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
136	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
137	C₂×C₄×C₄₇	#135	1	4	yes	—
138	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	4	no	—
144	D₁₈₈	—	2	8	no	—
145	D₁₈₈	—	4	4	no	—
146	D₁₈₈	—	4	16	no	—
147	C₄×D₄₇	—	8	8	no	—
148	C₄₇⋊C₈	—	2	4	no	—
149	C₄₇⋊C₈	C₄×D₄	1	8	yes	complemented
150	C₄×#492	—	47	1	no	maximal, Sylow
151	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
153	C₂²×C₄₇	#532	1	1	yes	characteristic
154	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
155	C₂²×C₄₇	C₂×#135	1	2	yes	—
156	C₂×D₄₇	—	4	4	no	—
157	C₂×D₄₇	—	4	8	no	—
158	C₂×D₄₇	—	8	2	no	—
159	C₂×D₄₇	—	8	2	no	—
160	C₂×D₄₇	—	8	4	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	—	4	4	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	C₄×SD₁₆	1	8	yes	complemented
166	C₄×C₄₇	#492	1	4	yes	complemented
167	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
168	C₄×C₄₇	#532	1	2	yes	—
169	Dic₄₇	—	8	2	no	—
170	C₂×#492	—	47	1	no	—
171	C₂×C₄×SD₁₆	—	47	4	no	—
172	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
173	C₄×#127	—	47	1	no	—
174	#1968	—	47	4	no	—
175	#1957	—	47	1	no	—
176	#1956	—	47	1	no	—
177	#1844	—	47	1	no	—
178	C₄²×C₈	—	47	1	no	—
179	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
180	C₂×C₄₇	C₂×C₄×SD₁₆	1	2	yes	—
181	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
182	C₂×C₄₇	#3147	1	1	yes	characteristic
183	C₂×C₄₇	#3167	1	2	yes	—
184	D₄₇	—	8	4	no	—
185	C₂²×SD₁₆	—	47	4	no	—
186	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
187	C₂×#127	—	47	1	no	—
188	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
189	C₂×C₄×D₄	—	47	2	no	—
190	C₂×C₄×D₄	—	94	2	no	—
191	#492	—	47	8	no	—
192	C₄×SD₁₆	—	94	16	no	—
193	C₂×#106	—	47	4	no	—
194	C₂×#103	—	47	4	no	—
195	C₂×#102	—	47	4	no	—
196	C₂×C₄×C₈	—	47	1	no	—
197	C₂×C₄×C₈	—	47	2	no	—
198	C₂×C₄×C₈	—	47	1	no	—
199	C₂×C₄×C₈	—	47	2	no	—
200	#391	C₂×D₄₇	1	2	yes	—
201	#390	—	47	2	no	—
202	#390	—	94	2	no	—
203	#389	—	2	2	no	—
204	#384	—	47	1	no	—
205	#384	C₂×D₄₇	1	1	yes	characteristic
206	C₄×C₄⋊C₄	—	2	2	no	—
207	C₄³	C₂×D₄₇	1	1	yes	characteristic
208	C₄₇	C₄×#492	1	1	yes	characteristic, complemented, Sylow
209	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
210	C₂²×D₄	—	47	2	no	—
211	C₂²×D₄	—	94	2	no	—
212	C₂³×C₄	—	94	2	no	—
213	C₂×SD₁₆	—	47	16	no	—
214	C₂×SD₁₆	—	94	8	no	—
215	C₂×SD₁₆	—	94	8	no	—
216	C₂²×C₈	—	47	2	no	—
217	C₂²×C₈	—	47	2	no	—
218	C₄⋊Q₈	—	2	2	no	—
219	C₄⋊Q₈	C₄×D₄₇	1	4	yes	complemented
220	#127	—	47	4	no	—
221	#127	—	94	2	no	—
222	C₄×Q₈	—	2	8	no	—
223	C₄×D₄	—	94	8	no	—
224	C₄×D₄	—	188	8	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	47	2	no	—
229	C₂×C₄⋊C₄	—	94	2	no	—
230	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
231	C₂×C₂²⋊C₄	—	94	2	no	—
232	C₂×C₂²⋊C₄	—	94	4	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
235	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
236	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
239	#106	—	94	8	no	—
240	#103	—	94	8	no	—
241	#102	—	94	8	no	—
242	C₄×C₈	—	47	4	no	—
243	C₄×C₈	—	47	8	no	—
244	C₄×C₈	—	94	4	no	—
245	C₄×C₈	—	94	2	no	—
246	#95	—	2	4	no	—
247	C₂⁴	—	94	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	C₂×C₄×D₄₇	1	4	yes	—
252	C₂×D₄	—	47	4	no	—
253	C₂×D₄	—	94	8	no	—
254	C₂×D₄	—	94	4	no	—
255	C₂×D₄	—	94	4	no	—
256	C₂×D₄	—	94	2	no	—
257	C₂×D₄	—	188	4	no	—
258	C₂×D₄	—	188	4	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	94	2	no	—
262	C₂²×C₄	—	188	4	no	—
263	C₂²×C₄	—	188	8	no	—
264	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
269	SD₁₆	—	94	32	no	—
270	C₂×C₈	—	47	4	no	—
271	C₂×C₈	—	47	4	no	—
272	C₂×C₈	—	47	4	no	—
273	C₂×C₈	—	47	4	no	—
274	C₂×C₈	—	94	2	no	—
275	C₂×C₈	—	94	2	no	—
276	C₄⋊C₄	—	2	8	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₄⋊C₄	—	94	4	no	—
281	C₄⋊C₄	—	188	4	no	—
282	C₂²⋊C₄	—	188	4	no	—
283	C₂²⋊C₄	—	188	8	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	4	no	—
287	C₄²	#10625336	1	8	yes	complemented
288	C₄²	C₂×C₄×D₄₇	1	2	yes	—
289	C₄²	#10625350	1	2	yes	—
290	C₄²	D₄×D₄₇	1	4	yes	—
291	C₂³	—	94	4	no	—
292	C₂³	—	188	4	no	—
293	C₂³	—	188	4	no	—
294	C₂³	—	188	2	no	—
295	C₂³	—	188	2	no	—
296	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
297	Q₈	—	2	8	no	—
298	Q₈	—	4	4	no	—
299	D₄	—	94	8	no	—
300	D₄	—	188	4	no	—
301	D₄	—	188	16	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	376	8	no	—
311	C₂×C₄	—	188	4	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
314	C₂×C₄	#13094020	1	2	yes	—
315	C₂×C₄	#13094020	1	2	yes	—
316	C₂×C₄	C₂×#10625336	1	4	yes	—
317	C₂×C₄	C₂×#10625336	1	4	yes	—
318	C₂×C₄	C₂×#10625336	1	4	yes	—
319	C₂×C₄	#13094044	1	4	yes	—
320	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
321	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
322	C₂×C₄	#41×D₄₇	1	2	yes	—
323	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
324	C₈	—	47	8	no	—
325	C₈	—	94	4	no	—
326	C₂²	—	376	2	no	—
327	C₂²	—	376	2	no	—
328	C₂²	—	376	4	no	—
329	C₂²	—	188	4	no	—
330	C₂²	—	188	8	no	—
331	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
332	C₂²	C₂×#13094020	1	1	yes	characteristic
333	C₂²	#15689669	1	1	yes	characteristic
334	C₂²	C₂²×#10625336	1	2	yes	—
335	C₂²	C₂×#13094044	1	2	yes	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	4	4	no	—
339	C₄	—	8	2	no	—
340	C₄	—	376	2	no	—
341	C₄	C₄×#10625336	1	8	yes	complemented
342	C₄	#15689497	1	4	yes	direct product factor
343	C₄	C₄×D₄×D₄₇	1	2	yes	—
344	C₄	#15689670	1	2	yes	—
345	C₂	—	376	4	no	—
346	C₂	C₄×#13094020	1	1	yes	characteristic
347	C₂	C₂×C₄×#10625336	1	2	yes	—
348	C₂	C₂×#15689497	1	1	yes	characteristic
349	C₂	#20253814	1	1	yes	characteristic
350	C₂	#20253948	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#13094021	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₄×#492
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#10625336	C₂	semidirect
4. #20244245	C₂	semidirect
6. #20244191	C₂	semidirect
7. #20243544	C₂	semidirect
8. C₄²×C₄₇⋊C₈	C₂	semidirect
9. C₄×C₄⋊Q₈×C₄₇	C₂	semidirect
14. #15689497	C₄	direct (N × H)
38. C₂×#10625336	C₂×C₄	semidirect
54. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
55. C₄×C₄₇⋊C₈	D₄	semidirect
58. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
100. C₄²×C₄₇	SD₁₆	semidirect
149. C₄₇⋊C₈	C₄×D₄	semidirect
165. C₄×C₄₇	C₄×SD₁₆	semidirect
166. C₄×C₄₇	#492	semidirect
172. C₄×C₄⋊Q₈	D₄₇	semidirect
208. C₄₇	C₄×#492	semidirect
219. C₄⋊Q₈	C₄×D₄₇	semidirect
287. C₄²	#10625336	semidirect
341. C₄	C₄×#10625336	semidirect
342. C₄	#15689497	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15689497	2
C₂×C₄×#10625336	2
C₄×#13094021	2
#20244245	2
#20244193	2
#20244191	2
#20243544	2
C₄²×C₄₇⋊C₈	2
C₄×C₄⋊Q₈×C₄₇	2
C₄×#492	47

Subgroups — C4×#15689497 #26224701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689497 #26224701