Subgroups — #26214719

Order 12032.

1234 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689201	C₂	1	2	yes	maximal
2	C₂×C₄×#10625329	C₂	1	2	yes	maximal, complemented
3	C₄⋊C₄×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	#20243241	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243172	C₂	1	4	yes	maximal, complemented
6	#20243150	C₂	1	2	yes	maximal, complemented
7	C₄×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	4	yes	—
10	#15689201	C₄	1	16	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
13	C₄×#10625329	—	2	8	no	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	1	yes	characteristic
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	#15688878	C₂²	1	2	yes	—
21	#15688878	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
23	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
28	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
29	C₄³×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#10625329	—	2	8	no	—
31	C₂×#10625329	—	2	8	no	—
32	C₂×#10625329	C₂×C₄	1	16	yes	complemented
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
37	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
38	#13093935	—	2	4	no	—
39	#13093935	—	2	2	no	—
40	#13093935	—	2	4	no	—
41	#13093935	D₄	1	4	yes	complemented
42	C₄×C₄₇⋊C₈	—	2	8	no	—
43	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
44	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
47	C₄×D₄×C₄₇	—	2	8	no	—
48	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	—
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	4	yes	—
52	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₇	—	2	4	no	—
54	C₂×C₄²×C₄₇	—	2	4	no	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	#95×C₄₇	C₂×C₄	1	2	yes	—
63	#95×C₄₇	C₂×C₄	1	4	yes	—
64	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂×D₄×C₄₇	—	2	4	no	—
66	C₂×D₄×C₄₇	C₄²	1	4	yes	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	8	no	—
71	C₂²×C₄×C₄₇	—	2	8	no	—
72	C₂²×C₄×C₄₇	—	4	4	no	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
82	C₄⋊C₄×C₄₇	M₁₆	1	8	yes	complemented
83	C₂²⋊C₄×C₄₇	—	2	8	no	—
84	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
85	C₄²×C₄₇	—	2	8	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	—	4	2	no	—
91	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
92	C₄²×C₄₇	C₂×D₄	1	2	yes	—
93	C₄²×C₄₇	#41	1	2	yes	—
94	#10625329	—	2	32	no	—
95	C₂×C₄₇⋊C₈	—	2	4	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	8	yes	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
102	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
103	C₂×C₄₇⋊C₈	#41	1	4	yes	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	—	2	4	no	—
108	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
109	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	D₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	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	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	8	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	4	16	no	—
124	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
126	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
128	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
129	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
130	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₄₇⋊C₈	—	2	8	no	—
135	C₄₇⋊C₈	—	2	4	no	—
136	C₄₇⋊C₈	C₄×D₄	1	8	yes	complemented
137	#20494	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	4	4	no	—
142	C₂²×C₄₇	—	4	4	no	—
143	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
146	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
147	C₂²×C₄₇	#568	1	2	yes	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	—	2	8	no	—
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	4	2	no	—
153	C₄×C₄₇	C₄×M₁₆	1	4	yes	complemented
154	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
155	C₄×C₄₇	C₄×#41	1	2	yes	—
156	C₂×#436	—	47	2	no	—
157	C₂×C₄×M₁₆	—	47	2	no	—
158	C₄²×D₄	D₄₇	1	1	yes	characteristic
159	C₈×C₄⋊C₄	—	47	2	no	—
160	#1767	—	47	1	no	—
161	#1753	—	47	4	no	—
162	#1749	—	47	2	no	—
163	C₄×#97	—	47	1	no	—
164	C₂×C₄₇	—	4	4	no	—
165	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
166	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
167	C₂×C₄₇	#2875	1	1	yes	characteristic
168	C₂×C₄₇	D₄×M₁₆	1	2	yes	—
169	C₂×C₄₇	#2971	1	2	yes	—
170	C₂²×M₁₆	—	47	4	no	—
171	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
172	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
173	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
174	#436	—	47	16	no	—
175	C₂×C₄⋊C₈	—	47	2	no	—
176	C₂×C₂²⋊C₈	—	47	4	no	—
177	C₄×M₁₆	—	94	8	no	—
178	C₂×#97	—	47	1	no	—
179	C₂×#97	—	47	2	no	—
180	C₂×#97	—	47	1	no	—
181	C₂×#97	—	47	2	no	—
182	C₂×C₄×C₈	—	47	2	no	—
183	C₂×C₄×C₈	—	47	4	no	—
184	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
185	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
186	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
187	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
188	C₄³	C₂×D₄₇	1	1	yes	characteristic
189	#336	—	47	4	no	—
190	#336	—	47	2	no	—
191	C₄₇	#20494	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
193	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
194	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
195	C₂×M₁₆	—	47	16	no	—
196	C₂×M₁₆	—	94	8	no	—
197	C₂×M₁₆	—	94	8	no	—
198	C₂²×C₈	—	47	2	no	—
199	C₂²×C₈	—	47	4	no	—
200	C₂²×C₈	—	47	2	no	—
201	C₄×D₄	—	2	8	no	—
202	C₄×D₄	C₄×D₄₇	1	8	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
206	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
210	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
211	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₄⋊C₈	—	47	8	no	—
217	C₂²⋊C₈	—	47	16	no	—
218	#97	—	47	4	no	—
219	#97	—	94	4	no	—
220	#97	—	94	4	no	—
221	#97	—	94	2	no	—
222	C₄×C₈	—	47	8	no	—
223	C₄×C₈	—	94	8	no	—
224	#95	C₂×Dic₄₇	1	2	yes	—
225	#95	C₂×Dic₄₇	1	4	yes	—
226	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	C₄×Dic₄₇	1	4	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
236	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
239	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
240	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
241	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
243	M₁₆	—	94	32	no	—
244	C₂×C₈	—	47	4	no	—
245	C₂×C₈	—	47	4	no	—
246	C₂×C₈	—	47	4	no	—
247	C₂×C₈	—	47	4	no	—
248	C₂×C₈	—	47	8	no	—
249	C₂×C₈	—	94	4	no	—
250	C₂×C₈	—	94	2	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	—	94	2	no	—
253	C₄⋊C₄	#10625329	1	8	yes	complemented
254	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	—
255	C₂²⋊C₄	—	2	8	no	—
256	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
257	C₄²	—	2	8	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	C₂×C₄×D₄₇	1	4	yes	—
264	C₄²	D₄×D₄₇	1	2	yes	—
265	C₄²	#10625352	1	2	yes	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	C₂×C₄×Dic₄₇	1	4	yes	—
271	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
272	D₄	—	2	8	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	16	no	—
286	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
288	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
289	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
290	C₂×C₄	C₂×#10625329	1	8	yes	—
291	C₂×C₄	C₂×#10625329	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
293	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
294	C₂×C₄	#41×D₄₇	1	2	yes	—
295	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
296	C₈	—	47	8	no	—
297	C₈	—	94	8	no	—
298	C₈	—	94	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	4	4	no	—
304	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
305	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
306	C₂²	#41×Dic₄₇	1	1	yes	characteristic
307	C₂²	C₂²×#10625329	1	2	yes	—
308	C₂²	#15689890	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	2	8	no	—
312	C₄	—	2	4	no	—
313	C₄	—	4	2	no	—
314	C₄	C₄×#10625329	1	4	yes	complemented
315	C₄	C₄×D₄×D₄₇	1	2	yes	—
316	C₄	C₄×#10625352	1	2	yes	—
317	C₂	—	4	4	no	—
318	C₂	C₄×D₄×Dic₄₇	1	1	yes	characteristic
319	C₂	C₂×C₄×#10625329	1	1	yes	characteristic
320	C₂	#20251816	1	1	yes	characteristic
321	C₂	D₄×#10625329	1	2	yes	—
322	C₂	#20252408	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄²×D₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×D₄	D₄₇
hypocenter	C₄₇	#20494
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#10625329	C₂	semidirect
3. C₄⋊C₄×C₄₇⋊C₈	C₂	semidirect
4. #20243241	C₂	semidirect
5. #20243172	C₂	semidirect
6. #20243150	C₂	semidirect
7. C₄×#13093935	C₂	semidirect
10. #15689201	C₄	semidirect
32. C₂×#10625329	C₂×C₄	semidirect
36. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
37. C₂²⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
41. #13093935	D₄	semidirect
43. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
82. C₄⋊C₄×C₄₇	M₁₆	semidirect
136. C₄₇⋊C₈	C₄×D₄	semidirect
153. C₄×C₄₇	C₄×M₁₆	semidirect
191. C₄₇	#20494	semidirect
253. C₄⋊C₄	#10625329	semidirect
314. C₄	C₄×#10625329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689201	2
C₂×C₄×#10625329	2
C₄⋊C₄×C₄₇⋊C₈	2
#20243241	2
#20243172	2
#20243150	2
C₄×#13093935	2
C₄²×D₄×C₄₇	2
#20494	47