Subgroups — #26122022

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243127	C₂	1	1	yes	maximal, characteristic
2	#20243121	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688883	C₂	1	2	yes	maximal
4	C₂×#15688879	C₂	1	2	yes	maximal
5	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15688892	—	2	2	no	—
9	#15688890	—	2	2	no	—
10	#15688883	C₄	1	8	yes	complemented
11	#15688879	—	2	4	no	—
12	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
16	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
17	C₄₇×#376	C₄	1	2	yes	complemented
18	C₄³×C₄₇	C₄	1	2	yes	complemented
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	C₂×C₄	1	4	yes	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	#13093983	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂³×C₄×C₄₇	—	2	1	no	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	#117×C₄₇	—	2	1	no	—
36	#117×C₄₇	—	2	1	no	—
37	#117×C₄₇	—	2	2	no	—
38	#117×C₄₇	—	2	2	no	—
39	#117×C₄₇	D₄	1	2	yes	—
40	#117×C₄₇	Q₈	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	—	2	1	no	—
46	C₂×C₄²×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	—	2	4	no	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
53	#95×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	1	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	—	2	2	no	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₂²⋊C₄×C₄₇	—	2	2	no	—
76	C₂²⋊C₄×C₄₇	—	2	2	no	—
77	C₂²⋊C₄×C₄₇	—	2	4	no	—
78	C₄²×C₄₇	—	2	1	no	—
79	C₄²×C₄₇	—	2	1	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	4	8	no	—
85	C₄²×C₄₇	C₄²	1	2	yes	—
86	C₄²×C₄₇	C₄²	1	2	yes	—
87	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
88	C₂²⋊Dic₄₇	—	2	4	no	—
89	C₂²⋊Dic₄₇	—	4	2	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	C₄×Dic₄₇	—	2	4	no	—
93	C₄×Dic₄₇	—	4	2	no	—
94	#10625329	—	2	4	no	—
95	#10625329	—	2	4	no	—
96	#10625329	—	4	2	no	—
97	#10625329	—	4	2	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	4	2	no	—
101	C₂³×C₄₇	—	4	2	no	—
102	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₄₇	C₄×D₄	1	2	yes	—
104	C₂³×C₄₇	#121	1	1	yes	characteristic
105	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₄₇	#123	1	2	yes	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	4	8	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#117	1	2	yes	—
119	C₂×C₄×C₄₇	#117	1	2	yes	—
120	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#123	1	2	yes	—
130	C₂×C₄×C₄₇	#124	1	2	yes	—
131	C₂×C₄×C₄₇	#125	1	2	yes	—
132	C₂×C₄×C₄₇	#126	1	2	yes	—
133	C₂×C₄×C₄₇	#126	1	2	yes	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	8	4	no	—
136	C₄₇⋊C₈	—	4	4	no	—
137	#10824	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	—	4	1	no	—
142	C₂²×C₄₇	—	4	1	no	—
143	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄₇	#380	1	1	yes	characteristic
145	C₂²×C₄₇	#381	1	1	yes	characteristic
146	C₂²×C₄₇	#387	1	2	yes	—
147	C₂²×C₄₇	#388	1	2	yes	—
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	—	4	4	no	—
150	C₄×C₄₇	—	4	2	no	—
151	C₄×C₄₇	—	4	4	no	—
152	C₄×C₄₇	#376	1	1	yes	characteristic
153	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₄₇	#382	1	1	yes	characteristic
155	C₄×C₄₇	#383	1	1	yes	characteristic
156	C₄×C₄₇	#387	1	2	yes	—
157	C₄×C₄₇	#388	1	2	yes	—
158	Dic₄₇	—	8	4	no	—
159	C₄×#117	D₄₇	1	1	yes	characteristic
160	#1742	—	47	1	no	—
161	#1740	—	47	1	no	—
162	C₂×#339	—	47	2	no	—
163	C₂×H(3,ℤ/4)	—	47	2	no	—
164	C₂×C₄₇	—	2	2	no	—
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	#1434	1	1	yes	characteristic
169	C₂×C₄₇	#1827	1	2	yes	—
170	C₂²×M₁₆	—	47	2	no	—
171	C₂×#117	—	47	2	no	—
172	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
173	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
174	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
175	C₄×C₄⋊C₄	—	2	2	no	—
176	C₄×C₂²⋊C₄	—	2	2	no	—
177	#376	Dic₄₇	1	2	yes	complemented
178	C₄³	Dic₄₇	1	2	yes	complemented
179	#343	—	94	2	no	—
180	#342	—	94	2	no	—
181	#339	—	47	8	no	—
182	H(3,ℤ/4)	—	94	4	no	—
183	C₄₇	#10824	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	2	1	no	—
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×M₁₆	—	47	4	no	—
187	C₂×M₁₆	—	94	2	no	—
188	C₂×M₁₆	—	94	2	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	1	no	—
196	#117	—	2	1	no	—
197	#117	—	47	4	no	—
198	#117	—	94	2	no	—
199	#117	—	94	2	no	—
200	#117	—	94	2	no	—
201	#117	Dic₉₄	1	2	yes	—
202	#117	D₁₈₈	1	2	yes	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	94	2	no	—
205	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₂²⋊C₄	—	94	2	no	—
208	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	—	94	2	no	—
213	C₂×C₄²	C₄×D₄₇	1	4	yes	—
214	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
215	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
216	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
218	#95	—	2	4	no	—
219	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	188	2	no	—
226	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
227	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
229	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂²×C₄	#10625350	1	1	yes	characteristic
231	C₂²×C₄	#10625350	1	1	yes	characteristic
232	C₂²×C₄	#10625350	1	2	yes	—
233	C₂²×C₄	#10625350	1	2	yes	—
234	C₂²×C₄	#10625350	1	2	yes	—
235	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
237	M₁₆	—	94	4	no	—
238	M₁₆	—	94	4	no	—
239	M₁₆	—	188	2	no	—
240	M₁₆	—	188	2	no	—
241	C₂×C₈	—	94	4	no	—
242	C₂×C₈	—	188	2	no	—
243	C₂×C₈	—	188	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	94	4	no	—
248	C₄⋊C₄	—	188	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	94	4	no	—
253	C₂²⋊C₄	—	188	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	4	8	no	—
261	C₄²	—	94	4	no	—
262	C₄²	—	188	2	no	—
263	C₄²	C₄×Dic₄₇	1	2	yes	—
264	C₄²	C₄×Dic₄₇	1	2	yes	—
265	C₄²	C₄⋊Dic₄₇	1	4	yes	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
268	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
269	C₂³	#13093998	1	1	yes	characteristic
270	C₂³	#13094003	1	1	yes	characteristic
271	C₂³	#13094007	1	2	yes	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	376	4	no	—
282	C₂×C₄	—	188	4	no	—
283	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
285	C₂×C₄	#13093981	1	2	yes	—
286	C₂×C₄	#13093981	1	2	yes	—
287	C₂×C₄	#13093983	1	1	yes	characteristic
288	C₂×C₄	#13093983	1	1	yes	characteristic
289	C₂×C₄	#13093983	1	1	yes	characteristic
290	C₂×C₄	#13093983	1	1	yes	characteristic
291	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
292	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
293	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
294	C₂×C₄	#13093998	1	1	yes	characteristic
295	C₂×C₄	#13094003	1	1	yes	characteristic
296	C₂×C₄	#13094007	1	2	yes	—
297	C₂×C₄	#13094012	1	2	yes	—
298	C₂×C₄	#13094016	1	2	yes	—
299	C₂×C₄	#13094018	1	2	yes	—
300	C₂×C₄	#13094018	1	2	yes	—
301	C₈	—	188	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
308	C₂²	#15688983	1	1	yes	characteristic
309	C₂²	#15688986	1	1	yes	characteristic
310	C₂²	#15689007	1	2	yes	—
311	C₂²	#15689014	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	4	4	no	—
316	C₄	—	376	4	no	—
317	C₄	#15688969	1	1	yes	characteristic
318	C₄	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
319	C₄	#15688989	1	1	yes	characteristic
320	C₄	#15688991	1	1	yes	characteristic
321	C₄	#15689007	1	2	yes	—
322	C₄	#15689014	1	2	yes	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	2	2	no	—
327	C₂	#20241615	1	1	yes	characteristic
328	C₂	#20243492	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093983
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄×#117×C₄₇	C₂
socle	C₂²×C₄₇	#381
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#10824
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
10. #15688883	C₄	semidirect
17. C₄₇×#376	C₄	semidirect
18. C₄³×C₄₇	C₄	semidirect
177. #376	Dic₄₇	semidirect
178. C₄³	Dic₄₇	semidirect
183. C₄₇	#10824	semidirect

Maximal subgroups

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

subgroup	index
#20243127	2
#20243121	2
C₂×#15688883	2
C₂×#15688879	2
C₄×#117×C₄₇	2
#10824	47