Subgroups — #26214175

Order 12032.

1118 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689150	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
4	#117×Dic₄₇	C₂	1	1	yes	maximal, characteristic
5	#20243247	C₂	1	4	yes	maximal, complemented
6	#20243243	C₂	1	4	yes	maximal, complemented
7	#20243086	C₂	1	1	yes	maximal, characteristic
8	#20243084	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#424	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	#15689150	C₄	1	8	yes	complemented
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
16	M₁₆×Dic₄₇	—	2	4	no	—
17	C₂×#13093934	—	2	4	no	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×#13093934	C₂²	1	2	yes	—
20	C₂×#13093934	C₂²	1	2	yes	—
21	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₄₇	C₄	1	4	yes	—
23	C₂²⋊C₄×Dic₄₇	C₄	1	4	yes	—
24	#15688969	C₂²	1	2	yes	—
25	C₄²×Dic₄₇	C₂²	1	2	yes	—
26	#15688877	C₂²	1	4	yes	—
27	#15688877	C₂²	1	4	yes	—
28	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#424	C₄	1	8	yes	complemented
32	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
33	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×#10625329	—	2	2	no	—
35	C₂×#10625329	—	2	2	no	—
36	C₂×#10625329	C₂×C₄	1	4	yes	—
37	C₂²×C₄₇⋊C₈	—	2	2	no	—
38	C₂²×C₄₇⋊C₈	—	2	2	no	—
39	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
42	#13093983	—	2	2	no	—
43	#13093983	—	2	2	no	—
44	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
49	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
54	#13093934	—	2	4	no	—
55	#13093934	—	2	4	no	—
56	#13093934	—	2	4	no	—
57	C₈×Dic₄₇	—	2	4	no	—
58	C₄₇⋊#95	C₂×C₄	1	8	yes	—
59	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₄₇	—	2	2	no	—
61	C₂×M₁₆×C₄₇	—	2	2	no	—
62	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
63	C₂²×C₈×C₄₇	—	2	2	no	—
64	C₂²×C₈×C₄₇	—	2	2	no	—
65	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
66	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
68	#117×C₄₇	C₂×C₄	1	4	yes	—
69	#117×C₄₇	C₂×C₄	1	4	yes	—
70	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
73	C₄⋊C₈×C₄₇	C₂×C₄	1	16	yes	complemented
74	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₄₇	#41	1	2	yes	—
82	C₂²×C₄×C₄₇	#41	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₇	—	4	4	no	—
85	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
86	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
87	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
88	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
89	M₁₆×C₄₇	—	2	8	no	—
90	C₂×C₈×C₄₇	—	2	8	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
97	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
98	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
100	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
101	C₂²⋊Dic₄₇	—	2	4	no	—
102	C₄⋊Dic₄₇	—	2	4	no	—
103	C₄×Dic₄₇	—	2	2	no	—
104	C₄×Dic₄₇	—	2	2	no	—
105	C₄×Dic₄₇	—	4	8	no	—
106	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
107	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
108	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
109	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
110	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
111	#10625329	—	2	8	no	—
112	C₂×C₄₇⋊C₈	—	2	8	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
118	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
119	C₂³×C₄₇	—	2	4	no	—
120	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂³×C₄₇	#117	1	2	yes	—
123	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
126	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	#117	1	2	yes	—
129	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
130	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
131	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
132	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	2	4	no	—
141	C₂×Dic₄₇	—	2	4	no	—
142	C₂×Dic₄₇	—	2	8	no	—
143	C₂×Dic₄₇	—	4	8	no	—
144	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
145	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
146	C₈×C₄₇	—	2	4	no	—
147	C₈×C₄₇	—	2	4	no	—
148	C₄₇⋊C₈	—	2	4	no	—
149	C₄₇⋊C₈	—	2	4	no	—
150	#20427	—	47	1	no	maximal, Sylow
151	C₂²×C₄₇	—	2	4	no	—
152	C₂²×C₄₇	—	2	4	no	—
153	C₂²×C₄₇	—	2	4	no	—
154	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
155	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
156	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
158	C₂²×C₄₇	#568	1	4	yes	—
159	C₄×C₄₇	—	2	8	no	—
160	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
161	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
162	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
165	Dic₄₇	—	2	4	no	—
166	Dic₄₇	—	2	4	no	—
167	Dic₄₇	—	4	4	no	—
168	C₂×#424	—	47	1	no	—
169	C₂×#424	D₄₇	1	1	yes	characteristic
170	C₂×C₄×M₁₆	—	47	1	no	—
171	C₂²×#97	—	47	1	no	—
172	C₄×#117	—	47	1	no	—
173	#1768	—	47	4	no	—
174	#1767	—	47	4	no	—
175	#1729	—	47	1	no	—
176	#1729	—	47	1	no	—
177	C₂×C₄₇	—	2	4	no	—
178	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄₇	#2875	1	2	yes	—
180	C₂×C₄₇	#2905	1	2	yes	—
181	C₂×C₄₇	#2923	1	2	yes	—
182	C₂²×M₁₆	—	47	1	no	—
183	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
184	C₂³×C₈	—	47	1	no	—
185	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
186	C₂×#117	—	47	1	no	—
187	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
188	C₂²×C₄²	—	47	1	no	—
189	#424	—	47	8	no	—
190	#424	Dic₄₇	1	8	yes	complemented
191	C₂×C₄⋊C₈	—	47	4	no	—
192	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
193	C₄×M₁₆	—	94	4	no	—
194	C₂×#97	—	47	2	no	—
195	C₂×#97	—	47	2	no	—
196	C₂×#97	—	47	2	no	—
197	C₂×#97	—	94	4	no	—
198	C₂×C₄×C₈	—	47	2	no	—
199	C₄×C₄⋊C₄	—	47	4	no	—
200	C₄×C₂²⋊C₄	—	47	4	no	—
201	#376	—	47	2	no	—
202	C₄³	—	47	2	no	—
203	#336	—	47	4	no	—
204	#336	—	47	4	no	—
205	C₄₇	#20427	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₄	—	47	2	no	—
207	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	47	4	no	—
211	C₂×M₁₆	—	94	2	no	—
212	C₂×M₁₆	—	94	2	no	—
213	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	47	4	no	—
217	C₂²×C₈	—	47	2	no	—
218	C₂²×C₈	—	47	2	no	—
219	C₂²×C₈	—	94	2	no	—
220	C₂²×C₈	—	94	2	no	—
221	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
222	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
223	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
224	#117	—	94	2	no	—
225	#117	—	94	2	no	—
226	#117	C₄×D₄₇	1	4	yes	—
227	#117	C₂×Dic₄₇	1	4	yes	—
228	C₂×C₄⋊C₄	—	47	2	no	—
229	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
230	C₂×C₂²⋊C₄	—	47	2	no	—
231	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
232	C₂×C₄²	—	47	2	no	—
233	C₂×C₄²	—	47	4	no	—
234	C₂×C₄²	—	47	4	no	—
235	C₂×C₄²	—	47	1	no	—
236	C₂×C₄²	—	47	1	no	—
237	C₂×C₄²	—	47	2	no	—
238	C₂×C₄²	—	47	8	no	—
239	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
240	C₄⋊C₈	—	47	16	no	—
241	C₄⋊C₈	C₂×Dic₄₇	1	16	yes	complemented
242	#97	—	94	4	no	—
243	#97	—	94	4	no	—
244	#97	—	94	4	no	—
245	C₄×C₈	—	94	4	no	—
246	#95	—	47	8	no	—
247	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	—	47	2	no	—
249	C₂²×C₄	—	47	2	no	—
250	C₂²×C₄	—	47	8	no	—
251	C₂²×C₄	—	47	4	no	—
252	C₂²×C₄	—	188	4	no	—
253	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
254	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
256	C₂²×C₄	D₄×D₄₇	1	2	yes	—
257	C₂²×C₄	#10625352	1	2	yes	—
258	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
259	C₂²×C₄	#10625354	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
261	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
262	M₁₆	—	2	8	no	—
263	M₁₆	—	94	8	no	—
264	C₂×C₈	—	2	8	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	47	4	no	—
270	C₂×C₈	—	47	4	no	—
271	C₂×C₈	—	94	8	no	—
272	C₂×C₈	—	94	2	no	—
273	C₂×C₈	—	94	2	no	—
274	C₂×C₈	—	94	2	no	—
275	C₂×C₈	—	94	2	no	—
276	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
277	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	—
280	C₂²⋊C₄	—	94	4	no	—
281	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
282	C₄²	—	47	8	no	—
283	C₄²	—	47	2	no	—
284	C₄²	—	47	2	no	—
285	C₄²	—	47	2	no	—
286	C₄²	—	47	2	no	—
287	C₄²	—	94	2	no	—
288	C₄²	—	94	2	no	—
289	C₄²	—	188	8	no	—
290	C₄²	C₂×C₄×D₄₇	1	4	yes	—
291	C₄²	C₂²×Dic₄₇	1	4	yes	—
292	C₂³	—	2	4	no	—
293	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
294	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
295	C₂³	#13093982	1	2	yes	—
296	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	47	4	no	—
299	C₂×C₄	—	47	4	no	—
300	C₂×C₄	—	94	4	no	—
301	C₂×C₄	—	94	4	no	—
302	C₂×C₄	—	94	4	no	—
303	C₂×C₄	—	94	8	no	—
304	C₂×C₄	—	188	8	no	—
305	C₂×C₄	C₂×C₄×Dic₄₇	1	8	yes	—
306	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
307	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
308	C₂×C₄	#13093982	1	2	yes	—
309	C₂×C₄	#13093986	1	4	yes	—
310	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
311	C₂×C₄	#13093989	1	4	yes	—
312	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
313	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
315	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
318	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	94	4	no	—
322	C₈	—	94	4	no	—
323	C₂²	—	2	4	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	2	4	no	—
326	C₂²	C₄⋊C₄×Dic₄₇	1	4	yes	—
327	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
328	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
329	C₂²	C₂×#13093982	1	1	yes	characteristic
330	C₂²	#15689888	1	4	yes	—
331	C₄	—	2	8	no	—
332	C₄	—	94	4	no	—
333	C₄	—	94	4	no	—
334	C₄	—	188	4	no	—
335	C₄	C₄⋊C₄×Dic₄₇	1	4	yes	—
336	C₄	C₂×#13093986	1	1	yes	characteristic
337	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
338	C₄	C₂×#13093989	1	1	yes	characteristic
339	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
340	C₂	—	2	4	no	—
341	C₂	C₂×C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
342	C₂	#20251814	1	2	yes	—
343	C₂	#20251949	1	2	yes	—
344	C₂	#20252038	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#424	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#424	D₄₇
hypocenter	C₄₇	#20427
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
5. #20243247	C₂	semidirect
6. #20243243	C₂	semidirect
14. #15689150	C₄	semidirect
31. C₄₇×#424	C₄	semidirect
53. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
73. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
190. #424	Dic₄₇	semidirect
205. C₄₇	#20427	semidirect
241. C₄⋊C₈	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689150	2
C₂×M₁₆×Dic₄₇	2
C₂²×#13093934	2
#117×Dic₄₇	2
#20243247	2
#20243243	2
#20243086	2
#20243084	2
C₂×C₄₇×#424	2
#20427	47