Subgroups — #26214190

Order 12032.

1138 conjugacy classes of subgroups.

The subgroup lattice has 318 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	2	yes	maximal
2	C₂×#15689079	C₂	1	2	yes	maximal
3	#20243245	C₂	1	4	yes	maximal, complemented
4	#20243241	C₂	1	2	yes	maximal, complemented
5	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
6	#20243082	C₂	1	1	yes	maximal, characteristic
7	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
8	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	2	yes	—
10	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
11	#15689150	C₄	1	16	yes	complemented
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
14	#15689079	—	2	8	no	—
15	C₂×#13093935	C₂²	1	4	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
17	#15688878	C₂²	1	1	yes	characteristic
18	#15688878	C₂²	1	2	yes	—
19	#15688878	C₂²	1	1	yes	characteristic
20	#15688878	C₂²	1	2	yes	—
21	#15688878	C₂²	1	2	yes	—
22	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
27	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
28	C₄₇×#376	C₂²	1	2	yes	—
29	C₄³×C₄₇	C₂²	1	2	yes	—
30	C₂×#10625329	—	2	4	no	—
31	C₂×#10625329	—	2	4	no	—
32	C₂×#10625329	C₂×C₄	1	8	yes	—
33	C₂²×C₄₇⋊C₈	—	2	4	no	—
34	C₂²×C₄₇⋊C₈	—	2	4	no	—
35	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
40	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
41	#13093935	—	2	8	no	—
42	C₄×C₄₇⋊C₈	—	2	8	no	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	#117×C₄₇	C₂×C₄	1	8	yes	—
46	#117×C₄₇	D₄	1	4	yes	—
47	#117×C₄₇	Q₈	1	4	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₇	—	2	4	no	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
61	#95×C₄₇	C₂×C₄	1	4	yes	—
62	#95×C₄₇	C₂×C₄	1	4	yes	—
63	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₄₇	#41	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
79	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
80	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
81	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
82	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	8	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	4	4	no	—
88	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
89	C₄²×C₄₇	C₂×D₄	1	4	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
91	#10625329	—	2	16	no	—
92	C₂×C₄₇⋊C₈	—	2	16	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
105	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₄₇	#117	1	2	yes	—
107	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	4	8	no	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₇	#117	1	2	yes	—
124	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
125	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
126	C₂×C₄×C₄₇	#131	1	4	yes	—
127	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
131	C₄₇⋊C₈	—	2	4	no	—
132	C₄₇⋊C₈	—	2	4	no	—
133	C₄₇⋊C₈	—	2	8	no	—
134	#20430	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	4	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
140	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
141	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#131	1	2	yes	—
144	C₂²×C₄₇	#568	1	2	yes	—
145	C₄×C₄₇	—	2	4	no	—
146	C₄×C₄₇	—	2	4	no	—
147	C₄×C₄₇	—	2	8	no	—
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	—	4	2	no	—
150	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
151	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
152	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
153	C₂×#424	—	47	2	no	—
154	C₂×#405	—	47	2	no	—
155	C₄×#117	D₄₇	1	1	yes	characteristic
156	#1768	—	47	4	no	—
157	#1767	—	47	2	no	—
158	C₄×C₄⋊C₈	—	47	2	no	—
159	#1728	—	47	1	no	—
160	C₂×#336	—	47	1	no	—
161	C₂×C₄₇	—	2	4	no	—
162	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄₇	C₄×#131	1	1	yes	characteristic
164	C₂×C₄₇	#2875	1	1	yes	characteristic
165	C₂×C₄₇	#2906	1	2	yes	—
166	C₂×C₄₇	#2924	1	2	yes	—
167	C₂²×M₁₆	—	47	2	no	—
168	C₂³×C₈	—	47	2	no	—
169	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
170	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	#424	—	47	16	no	—
173	C₂×C₄⋊C₈	—	47	4	no	—
174	C₂×C₄⋊C₈	—	47	4	no	—
175	#405	—	94	8	no	—
176	C₂×#97	—	47	4	no	—
177	C₂×C₄×C₈	—	47	4	no	—
178	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
179	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
180	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
181	#376	C₂×D₄₇	1	2	yes	—
182	C₄³	C₂×D₄₇	1	2	yes	—
183	#336	—	47	1	no	—
184	#336	—	47	1	no	—
185	#336	—	47	2	no	—
186	#336	—	47	2	no	—
187	#336	—	47	2	no	—
188	C₄₇	#20430	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
190	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
191	C₂×M₁₆	—	47	8	no	—
192	C₂×M₁₆	—	94	4	no	—
193	C₂×M₁₆	—	94	4	no	—
194	C₂²×C₈	—	47	8	no	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	47	2	no	—
197	C₂²×C₈	—	47	4	no	—
198	C₂²×C₈	—	94	4	no	—
199	C₂²×C₈	—	94	4	no	—
200	#117	Dic₉₄	1	4	yes	—
201	#117	C₄×D₄₇	1	8	yes	—
202	#117	D₁₈₈	1	4	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
206	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
210	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
211	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
215	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
216	C₄⋊C₈	—	47	16	no	—
217	C₄⋊C₈	—	47	16	no	—
218	#97	—	94	8	no	—
219	C₄×C₈	—	94	8	no	—
220	#95	C₂×Dic₄₇	1	4	yes	—
221	#95	C₂×Dic₄₇	1	4	yes	—
222	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
228	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
229	C₂²×C₄	#10625350	1	4	yes	—
230	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
231	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
232	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
234	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
235	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
237	M₁₆	—	94	16	no	—
238	C₂×C₈	—	47	4	no	—
239	C₂×C₈	—	47	4	no	—
240	C₂×C₈	—	47	8	no	—
241	C₂×C₈	—	94	16	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	94	4	no	—
248	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	—
249	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	—
250	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
251	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
252	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	8	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	4	4	no	—
258	C₄²	C₂×Dic₉₄	1	4	yes	—
259	C₄²	C₂×C₄×D₄₇	1	8	yes	—
260	C₄²	C₂×D₁₈₈	1	4	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
264	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
265	C₂³	#13093983	1	2	yes	—
266	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
278	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
279	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
282	C₂×C₄	#13093983	1	2	yes	—
283	C₂×C₄	C₄×D₁₈₈	1	8	yes	—
284	C₂×C₄	C₄×Dic₉₄	1	8	yes	—
285	C₂×C₄	#13094031	1	4	yes	—
286	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
287	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
288	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
289	C₂×C₄	C₂×#10625350	1	2	yes	—
290	C₈	—	94	4	no	—
291	C₈	—	94	4	no	—
292	C₈	—	94	8	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	C₄×C₄⋊Dic₄₇	1	4	yes	—
298	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
299	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
300	C₂²	C₂×#13093983	1	1	yes	characteristic
301	C₂²	C₂×#13094031	1	2	yes	—
302	C₂²	#15689890	1	2	yes	—
303	C₄	—	2	8	no	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	C₄×C₄⋊Dic₄₇	1	4	yes	—
309	C₄	C₂×C₄×D₁₈₈	1	2	yes	—
310	C₄	C₂×C₄×Dic₉₄	1	2	yes	—
311	C₂	—	2	4	no	—
312	C₂	C₂×C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
313	C₂	C₄×#13094031	1	1	yes	characteristic
314	C₂	#20251816	1	1	yes	characteristic
315	C₂	#20251956	1	2	yes	—
316	C₂	#20252046	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20430
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20243245	C₂	semidirect
4. #20243241	C₂	semidirect
5. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
11. #15689150	C₄	semidirect
39. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
40. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
188. C₄₇	#20430	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×#15689079	2
#20243245	2
#20243241	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243082	2
C₂×#15688878	2
C₄×#117×C₄₇	2
#20430	47