Subgroups — #26122198

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093959	C₂	1	2	yes	maximal
2	C₂×#15689081	C₂	1	1	yes	maximal, characteristic
3	#20243126	C₂	1	2	yes	maximal
4	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	1	yes	characteristic
9	C₂×#13093959	—	2	4	no	—
10	C₂×#13093959	—	2	4	no	—
11	C₂×#13093959	C₄	1	8	yes	complemented
12	#15689081	—	2	4	no	—
13	C₂×#13093934	C₄	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
15	#15688891	—	2	4	no	—
16	C₄₇⋊#338	—	2	2	no	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#338	—	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	—	2	2	no	—
25	C₂×#10625329	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
31	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
32	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
33	#13093983	—	2	2	no	—
34	#13093983	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
36	C₂×C₂²⋊Dic₄₇	Q₈	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	#13093959	—	2	16	no	—
39	#13093934	—	2	4	no	—
40	C₈×Dic₄₇	—	2	4	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	—	2	2	no	—
49	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
51	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
52	C₂²×C₈×C₄₇	—	2	2	no	—
53	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
54	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	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	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	2	4	no	—
70	M₁₆×C₄₇	—	4	2	no	—
71	M₁₆×C₄₇	—	4	2	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
78	C₂²⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄×Dic₄₇	—	2	2	no	—
81	C₄×Dic₄₇	—	2	2	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	4	2	no	—
85	#10625329	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
90	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
91	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
92	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
93	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	8	yes	—
94	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
95	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
100	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
102	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	#121	1	2	yes	—
115	C₂×C₄×C₄₇	#121	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	#124	1	2	yes	—
118	C₂×C₄×C₄₇	#125	1	2	yes	—
119	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₈×C₄₇	—	2	4	no	—
124	C₈×C₄₇	—	4	4	no	—
125	C₄₇⋊C₈	—	4	4	no	—
126	C₄₇⋊C₈	#95	1	4	yes	—
127	C₄₇⋊C₈	#95	1	4	yes	—
128	#10867	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	—	4	1	no	—
133	C₂²×C₄₇	—	4	1	no	—
134	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₇	#380	1	1	yes	characteristic
136	C₂²×C₄₇	#384	1	1	yes	characteristic
137	C₂²×C₄₇	#389	1	2	yes	—
138	C₂²×C₄₇	#391	1	2	yes	—
139	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
140	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₄₇	#380	1	1	yes	characteristic
142	C₄×C₄₇	#382	1	1	yes	characteristic
143	C₄×C₄₇	#389	1	2	yes	—
144	C₄×C₄₇	#390	1	2	yes	—
145	Dic₄₇	—	4	4	no	—
146	C₂²×#108	—	47	2	no	—
147	C₂×#405	—	47	1	no	—
148	#1742	—	47	2	no	—
149	C₂×#338	—	47	1	no	—
150	C₂×#338	D₄₇	1	1	yes	characteristic
151	C₂×C₄₇	—	2	1	no	—
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
156	C₂×C₄₇	#1813	1	2	yes	—
157	C₂²×M₁₆	—	47	2	no	—
158	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
159	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
160	C₂³×C₈	—	47	1	no	—
161	C₂×#117	—	47	1	no	—
162	C₂×#108	—	47	8	no	—
163	C₂×#108	—	94	4	no	—
164	C₂×#108	—	94	4	no	—
165	#405	—	94	4	no	—
166	C₂×#97	—	47	2	no	—
167	C₂×C₄×C₈	—	47	2	no	—
168	#343	—	94	4	no	—
169	#338	—	2	2	no	—
170	#338	—	94	2	no	—
171	C₄₇	#10867	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	47	4	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	94	2	no	—
184	C₂×M₁₆	—	94	2	no	—
185	C₂×M₁₆	—	94	2	no	—
186	C₂×M₁₆	Dic₉₄	1	2	yes	—
187	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
188	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	47	1	no	—
191	C₂²×C₈	—	47	1	no	—
192	C₂²×C₈	—	47	4	no	—
193	C₂²×C₈	—	47	4	no	—
194	C₂²×C₈	—	47	2	no	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	C₄×D₄₇	1	2	yes	—
198	#117	—	94	2	no	—
199	#117	—	94	2	no	—
200	C₂×C₄⋊C₄	—	47	2	no	—
201	C₂×C₂²⋊C₄	—	47	2	no	—
202	C₂×C₄²	—	47	2	no	—
203	#108	—	94	16	no	—
204	#97	—	94	4	no	—
205	C₄×C₈	—	94	4	no	—
206	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	—	47	4	no	—
208	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
211	C₂²×C₄	#10625350	1	2	yes	—
212	C₂²×C₄	D₄×D₄₇	1	2	yes	—
213	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
214	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	#10625352	1	2	yes	—
216	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	94	4	no	—
225	M₁₆	—	94	4	no	—
226	M₁₆	—	188	2	no	—
227	M₁₆	—	188	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	47	2	no	—
234	C₂×C₈	—	47	2	no	—
235	C₂×C₈	—	47	2	no	—
236	C₂×C₈	—	47	2	no	—
237	C₂×C₈	—	47	8	no	—
238	C₂×C₈	—	47	2	no	—
239	C₂×C₈	—	47	2	no	—
240	C₂×C₈	—	47	8	no	—
241	C₂×C₈	—	94	4	no	—
242	C₂×C₈	—	188	2	no	—
243	C₂×C₈	—	188	2	no	—
244	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
245	C₄⋊C₄	—	94	4	no	—
246	C₂²⋊C₄	—	94	4	no	—
247	C₄²	—	94	2	no	—
248	C₄²	—	94	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
251	C₂³	#13093989	1	2	yes	—
252	C₂³	#13094002	1	2	yes	—
253	C₂³	#13094022	1	1	yes	characteristic
254	C₂³	#13094023	1	1	yes	characteristic
255	C₂×C₄	—	94	4	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
258	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
259	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
260	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
261	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
262	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
263	C₂×C₄	#13093980	1	1	yes	characteristic
264	C₂×C₄	#13093981	1	1	yes	characteristic
265	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
266	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
267	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
268	C₂×C₄	#13093996	1	2	yes	—
269	C₂×C₄	#13093997	1	2	yes	—
270	C₂×C₄	#13094001	1	2	yes	—
271	C₂×C₄	#13094010	1	2	yes	—
272	C₂×C₄	#13094014	1	2	yes	—
273	C₂×C₄	#13094020	1	1	yes	characteristic
274	C₂×C₄	#13094023	1	1	yes	characteristic
275	C₈	—	2	4	no	—
276	C₈	—	4	4	no	—
277	C₈	—	47	4	no	—
278	C₈	—	47	4	no	—
279	C₈	—	188	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	#15688974	1	1	yes	characteristic
286	C₂²	#15688982	1	1	yes	characteristic
287	C₂²	#15688992	1	1	yes	characteristic
288	C₂²	#15689018	1	2	yes	—
289	C₂²	#15689027	1	2	yes	—
290	C₄	—	188	4	no	—
291	C₄	#95×D₄₇	1	1	yes	characteristic
292	C₄	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
293	C₄	#15688981	1	1	yes	characteristic
294	C₄	#15688987	1	1	yes	characteristic
295	C₄	#15689020	1	2	yes	—
296	C₄	#15689023	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	Dic₄₇⋊#95	1	1	yes	characteristic
302	C₂	#20243437	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093981
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#338	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#338	D₄₇
hypocenter	C₄₇	#10867
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
11. C₂×#13093959	C₄	semidirect
171. C₄₇	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093959	2
C₂×#15689081	2
#20243126	2
C₂×C₄₇⋊#338	2
C₂×C₄₇×#338	2
#10867	47