Subgroups — #26122051

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245745	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688895	C₂	1	2	yes	maximal
3	C₂×#15688894	C₂	1	2	yes	maximal
4	C₂×#15688892	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#343	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×Dic₄₇	—	2	2	no	—
10	C₄⋊C₄×Dic₄₇	—	2	2	no	—
11	C₄×C₂²⋊Dic₄₇	C₄	1	4	yes	—
12	#15688895	—	2	4	no	—
13	#15688894	C₄	1	8	yes	complemented
14	#15688892	—	2	2	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#343	—	2	2	no	—
18	C₂×#10625329	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	#13093983	—	2	2	no	—
26	#13093983	—	2	1	no	—
27	#13093983	—	2	1	no	—
28	#13093983	C₂×C₄	1	4	yes	—
29	#13093983	D₄	1	2	yes	—
30	#13093983	Q₈	1	2	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	—	2	1	no	—
36	C₂×C₄×Dic₄₇	—	2	1	no	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	C₄₇⋊#95	—	2	4	no	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	8	2	no	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	M₁₆×C₄₇	—	2	4	no	—
70	M₁₆×C₄₇	—	2	4	no	—
71	M₁₆×C₄₇	—	4	2	no	—
72	M₁₆×C₄₇	—	4	2	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₂²⋊Dic₄₇	—	2	2	no	—
79	C₂²⋊Dic₄₇	—	2	2	no	—
80	C₂²⋊Dic₄₇	C₄²	1	4	yes	—
81	C₂²⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	2	no	—
84	C₄⋊Dic₄₇	—	2	2	no	—
85	C₄×Dic₄₇	—	2	4	no	—
86	C₄×Dic₄₇	—	2	2	no	—
87	C₄×Dic₄₇	—	2	2	no	—
88	C₄×Dic₄₇	—	8	4	no	—
89	#10625329	—	2	4	no	—
90	#10625329	—	2	4	no	—
91	#10625329	—	4	2	no	—
92	#10625329	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂³×C₄₇	—	4	1	no	—
97	C₂³×C₄₇	—	4	1	no	—
98	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₇	#117	1	1	yes	characteristic
102	C₂³×C₄₇	#117	1	1	yes	characteristic
103	C₂³×C₄₇	#117	1	1	yes	characteristic
104	C₂³×C₄₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	8	2	no	—
108	C₂×C₄×C₄₇	—	8	2	no	—
109	C₂×C₄×C₄₇	#117	1	2	yes	—
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#123	1	2	yes	—
120	C₂×C₄×C₄₇	#123	1	2	yes	—
121	C₂×C₄×C₄₇	#124	1	2	yes	—
122	C₂×C₄×C₄₇	#125	1	2	yes	—
123	C₂×C₄×C₄₇	#126	1	2	yes	—
124	C₂×C₄×C₄₇	#126	1	2	yes	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	2	4	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₈×C₄₇	—	4	4	no	—
132	C₄₇⋊C₈	—	4	4	no	—
133	#10830	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	4	1	no	—
141	C₂²×C₄₇	—	4	1	no	—
142	C₂²×C₄₇	—	4	1	no	—
143	C₂²×C₄₇	—	4	1	no	—
144	C₂²×C₄₇	#376	1	1	yes	characteristic
145	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₇	#380	1	1	yes	characteristic
148	C₂²×C₄₇	#381	1	1	yes	characteristic
149	C₂²×C₄₇	#382	1	1	yes	characteristic
150	C₂²×C₄₇	#383	1	1	yes	characteristic
151	C₄×C₄₇	—	4	4	no	—
152	C₄×C₄₇	#387	1	2	yes	—
153	C₄×C₄₇	#387	1	2	yes	—
154	C₄×C₄₇	#388	1	2	yes	—
155	C₄×C₄₇	#388	1	2	yes	—
156	Dic₄₇	—	4	4	no	—
157	Dic₄₇	—	4	4	no	—
158	#2274	—	47	1	no	—
159	C₂×#344	—	47	2	no	—
160	C₂×#344	—	47	2	no	—
161	C₂×#343	—	47	1	no	—
162	C₂×#343	D₄₇	1	1	yes	characteristic
163	C₂×C₄₇	—	2	1	no	—
164	C₂×C₄₇	—	2	1	no	—
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	#1434	1	1	yes	characteristic
170	C₂×C₄₇	#1829	1	1	yes	characteristic
171	C₂×C₄₇	#1830	1	1	yes	characteristic
172	C₂²×M₁₆	—	47	2	no	—
173	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
174	C₂×#117	—	47	1	no	—
175	C₂×#117	—	47	1	no	—
176	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
177	C₄×C₄⋊C₄	—	94	2	no	—
178	C₄×C₄⋊C₄	—	94	2	no	—
179	C₄×C₂²⋊C₄	—	47	4	no	—
180	#344	—	47	8	no	—
181	#344	—	94	4	no	—
182	#343	—	2	2	no	—
183	#343	—	94	2	no	—
184	C₄₇	#10830	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	C₄×D₄₇	1	2	yes	—
186	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	47	4	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂×M₁₆	—	94	2	no	—
195	C₂×M₁₆	—	94	2	no	—
196	C₂×M₁₆	—	94	2	no	—
197	C₂×M₁₆	—	94	2	no	—
198	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	94	2	no	—
201	#117	—	47	4	no	—
202	#117	—	47	2	no	—
203	#117	—	47	2	no	—
204	#117	—	94	2	no	—
205	#117	—	94	1	no	—
206	#117	—	94	1	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	94	1	no	—
210	C₂×C₄⋊C₄	—	94	1	no	—
211	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
212	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
213	C₂×C₂²⋊C₄	—	47	2	no	—
214	C₂×C₂²⋊C₄	—	47	2	no	—
215	C₂×C₄²	—	94	1	no	—
216	C₂×C₄²	—	94	1	no	—
217	C₂×C₄²	—	94	4	no	—
218	#95	—	94	4	no	—
219	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	8	2	no	—
223	C₂²×C₄	—	94	2	no	—
224	C₂²×C₄	—	94	2	no	—
225	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
226	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
227	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
228	C₂²×C₄	#10625350	1	2	yes	—
229	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	#10625352	1	1	yes	characteristic
231	C₂²×C₄	#10625352	1	2	yes	—
232	C₂²×C₄	#10625352	1	2	yes	—
233	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	#10625354	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	94	4	no	—
241	M₁₆	—	94	4	no	—
242	M₁₆	—	188	2	no	—
243	M₁₆	—	188	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	94	4	no	—
248	C₂×C₈	—	188	2	no	—
249	C₂×C₈	—	188	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	94	4	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄⋊C₄	—	94	2	no	—
255	C₂²⋊C₄	—	47	4	no	—
256	C₂²⋊C₄	—	47	4	no	—
257	C₂²⋊C₄	—	94	2	no	—
258	C₂²⋊C₄	—	94	2	no	—
259	C₄²	—	94	4	no	—
260	C₄²	—	94	2	no	—
261	C₄²	—	94	2	no	—
262	C₄²	—	376	4	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	C₄²×D₄₇	1	1	yes	characteristic
266	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
267	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
268	C₂³	#13093980	1	1	yes	characteristic
269	C₂³	#13093980	1	1	yes	characteristic
270	C₂³	#13093981	1	1	yes	characteristic
271	C₂³	#13093982	1	1	yes	characteristic
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	—	94	4	no	—
277	C₂×C₄	—	94	4	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	#13093980	1	2	yes	—
283	C₂×C₄	#13093982	1	2	yes	—
284	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
285	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
286	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
287	C₂×C₄	#13093989	1	2	yes	—
288	C₂×C₄	#13093995	1	1	yes	characteristic
289	C₂×C₄	#13093999	1	1	yes	characteristic
290	C₂×C₄	#13094000	1	1	yes	characteristic
291	C₂×C₄	#13094004	1	1	yes	characteristic
292	C₂×C₄	#13094005	1	2	yes	—
293	C₂×C₄	#13094008	1	2	yes	—
294	C₂×C₄	#13094010	1	2	yes	—
295	C₂×C₄	#13094014	1	2	yes	—
296	C₂×C₄	#13094017	1	2	yes	—
297	C₂×C₄	#13094019	1	2	yes	—
298	C₈	—	4	4	no	—
299	C₈	—	188	4	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	#15688968	1	1	yes	characteristic
311	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
312	C₂²	#15688974	1	1	yes	characteristic
313	C₂²	#15688980	1	1	yes	characteristic
314	C₂²	#15688984	1	1	yes	characteristic
315	C₂²	#15688987	1	1	yes	characteristic
316	C₂²	#15688990	1	1	yes	characteristic
317	C₄	—	4	4	no	—
318	C₄	—	188	4	no	—
319	C₄	—	188	4	no	—
320	C₄	#15689004	1	2	yes	—
321	C₄	#15689008	1	2	yes	—
322	C₄	#15689011	1	2	yes	—
323	C₄	#15689015	1	2	yes	—
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	#20241612	1	1	yes	characteristic
331	C₂	#20243497	1	1	yes	characteristic
332	C₂	#20243500	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688984
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#343	C₂
socle	C₂²×C₄₇	#381
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#343	D₄₇
hypocenter	C₄₇	#10830
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#343

Complements

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

N	H ≅ G/N	type
13. #15688894	C₄	semidirect
184. C₄₇	#10830	semidirect

Maximal subgroups

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

subgroup	index
#20245745	2
C₂×#15688895	2
C₂×#15688894	2
C₂×#15688892	2
C₂×C₄₇×#343	2
#10830	47