Subgroups — #26121926

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688893	C₂	1	2	yes	maximal
3	C₂×#15688882	C₂	1	2	yes	maximal
4	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	M₁₆×Dic₄₇	—	2	2	no	—
10	M₁₆×Dic₄₇	—	2	2	no	—
11	C₂×#13093934	C₄	1	2	yes	complemented
12	C₂×C₈×Dic₄₇	C₄	1	2	yes	complemented
13	#15688893	C₄	1	8	yes	complemented
14	#15688882	—	2	4	no	—
15	C₄₇⋊#338	—	2	2	no	—
16	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₇	C₂²	1	2	yes	—
18	C₄₇×H(3,ℤ/4)	—	2	2	no	—
19	C₂³×Dic₄₇	—	2	1	no	—
20	C₂×#10625329	—	2	1	no	—
21	C₂×#10625329	—	2	1	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	C₂×C₄	1	4	yes	—
29	C₂×#10625329	C₂×C₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	—	2	2	no	—
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₄₇	—	2	1	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	—	2	1	no	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
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	D₄	1	2	yes	—
45	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	#117×C₄₇	—	2	2	no	—
48	#117×C₄₇	—	2	2	no	—
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	C₂⁴×C₄₇	#41	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×Dic₄₇	—	2	1	no	—
67	C₂²×Dic₄₇	—	2	1	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	—	4	4	no	—
71	M₁₆×C₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	4	2	no	—
78	C₂²⋊C₄×C₄₇	—	2	4	no	—
79	C₂²⋊C₄×C₄₇	—	4	2	no	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	4	2	no	—
82	C₄×Dic₄₇	—	2	1	no	—
83	C₄×Dic₄₇	—	2	1	no	—
84	C₄×Dic₄₇	—	2	2	no	—
85	C₄×Dic₄₇	—	2	2	no	—
86	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
88	#10625329	—	2	4	no	—
89	#10625329	—	2	4	no	—
90	#10625329	—	2	4	no	—
91	#10625329	—	2	4	no	—
92	#10625329	—	4	2	no	—
93	#10625329	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂×C₄₇⋊C₈	—	4	2	no	—
99	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	#117	1	2	yes	—
102	C₂³×C₄₇	#117	1	1	yes	characteristic
103	C₂³×C₄₇	#121	1	1	yes	characteristic
104	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₄₇	#126	1	2	yes	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	8	4	no	—
108	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#117	1	2	yes	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#123	1	2	yes	—
122	C₂×C₄×C₄₇	#123	1	2	yes	—
123	C₂×C₄×C₄₇	#124	1	2	yes	—
124	C₂×C₄×C₄₇	#125	1	2	yes	—
125	C₂×C₄×C₄₇	#126	1	2	yes	—
126	C₂×Dic₄₇	—	2	4	no	—
127	C₂×Dic₄₇	—	2	2	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₈×C₄₇	—	2	4	no	—
133	C₄₇⋊C₈	—	2	4	no	—
134	C₄₇⋊C₈	—	4	4	no	—
135	#10805	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	4	no	—
138	C₂²×C₄₇	—	4	1	no	—
139	C₂²×C₄₇	—	4	1	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	#376	1	1	yes	characteristic
142	C₂²×C₄₇	#380	1	1	yes	characteristic
143	C₂²×C₄₇	#383	1	1	yes	characteristic
144	C₂²×C₄₇	#387	1	2	yes	—
145	C₂²×C₄₇	#388	1	2	yes	—
146	C₄×C₄₇	—	8	4	no	—
147	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₄₇	#381	1	1	yes	characteristic
150	C₄×C₄₇	#382	1	1	yes	characteristic
151	C₄×C₄₇	#387	1	2	yes	—
152	C₄×C₄₇	#388	1	2	yes	—
153	Dic₄₇	—	2	4	no	—
154	Dic₄₇	—	4	2	no	—
155	C₂×C₄×M₁₆	—	47	1	no	—
156	C₂×#344	—	47	2	no	—
157	C₂×#339	—	47	2	no	—
158	C₂×#338	—	47	1	no	—
159	C₂×H(3,ℤ/4)	D₄₇	1	1	yes	characteristic
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	—	2	1	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	#1434	1	1	yes	characteristic
165	C₂×C₄₇	#1841	1	2	yes	—
166	C₂²×M₁₆	—	47	1	no	—
167	C₂²×M₁₆	—	47	2	no	—
168	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
169	C₂×#117	C₂×D₄₇	1	2	yes	—
170	C₂²×C₄²	—	47	1	no	—
171	C₄×M₁₆	—	94	2	no	—
172	C₄×M₁₆	—	94	2	no	—
173	C₂×#97	—	47	2	no	—
174	C₂×C₄×C₈	—	47	2	no	—
175	#344	—	47	8	no	—
176	#339	—	94	4	no	—
177	#338	—	94	2	no	—
178	H(3,ℤ/4)	—	2	2	no	—
179	C₄₇	#10805	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	—	94	1	no	—
181	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	47	4	no	—
185	C₂×M₁₆	—	47	4	no	—
186	C₂×M₁₆	—	94	1	no	—
187	C₂×M₁₆	—	94	2	no	—
188	C₂×M₁₆	—	94	1	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂×M₁₆	—	94	2	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂×M₁₆	D₁₈₈	1	2	yes	—
195	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
196	C₂²×C₈	—	47	2	no	—
197	C₂²×C₈	—	94	2	no	—
198	C₂²×C₈	C₄×D₄₇	1	2	yes	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	C₄×D₄₇	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	47	1	no	—
207	C₂×C₄²	—	47	1	no	—
208	C₂×C₄²	—	47	2	no	—
209	C₂×C₄²	—	47	2	no	—
210	C₂×C₄²	—	94	1	no	—
211	C₂×C₄²	—	94	1	no	—
212	C₂×C₄²	—	94	1	no	—
213	#97	—	94	4	no	—
214	C₄×C₈	—	94	4	no	—
215	C₂⁴	#10625352	1	1	yes	characteristic
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	94	1	no	—
218	C₂²×C₄	—	94	1	no	—
219	C₂²×C₄	—	94	2	no	—
220	C₂²×C₄	—	94	2	no	—
221	C₂²×C₄	—	188	4	no	—
222	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
224	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
225	C₂²×C₄	#10625350	1	2	yes	—
226	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	#10625352	1	2	yes	—
228	C₂²×C₄	#10625352	1	2	yes	—
229	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	#10625354	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	94	4	no	—
235	M₁₆	—	94	4	no	—
236	M₁₆	—	94	4	no	—
237	M₁₆	—	94	4	no	—
238	M₁₆	—	188	2	no	—
239	M₁₆	—	188	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	47	4	no	—
243	C₂×C₈	—	94	2	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	—	94	4	no	—
246	C₂×C₈	—	188	2	no	—
247	C₂×C₈	—	188	2	no	—
248	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	47	2	no	—
256	C₄²	—	47	2	no	—
257	C₄²	—	94	1	no	—
258	C₄²	—	94	1	no	—
259	C₄²	—	94	2	no	—
260	C₄²	—	94	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	#13093980	1	2	yes	—
263	C₂³	#13093982	1	1	yes	characteristic
264	C₂³	#13093995	1	1	yes	characteristic
265	C₂³	#13094004	1	1	yes	characteristic
266	C₂³	#13094017	1	2	yes	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	94	4	no	—
270	C₂×C₄	—	94	2	no	—
271	C₂×C₄	—	94	2	no	—
272	C₂×C₄	—	188	4	no	—
273	C₂×C₄	—	188	4	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
278	C₂×C₄	#13093980	1	1	yes	characteristic
279	C₂×C₄	#13093980	1	1	yes	characteristic
280	C₂×C₄	#13093981	1	1	yes	characteristic
281	C₂×C₄	#13093982	1	2	yes	—
282	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
283	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
284	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
285	C₂×C₄	#13093989	1	2	yes	—
286	C₂×C₄	#13093999	1	1	yes	characteristic
287	C₂×C₄	#13094000	1	1	yes	characteristic
288	C₂×C₄	#13094005	1	2	yes	—
289	C₂×C₄	#13094008	1	2	yes	—
290	C₂×C₄	#13094010	1	2	yes	—
291	C₂×C₄	#13094014	1	2	yes	—
292	C₂×C₄	#13094019	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	94	4	no	—
295	C₈	—	188	4	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	#15688968	1	1	yes	characteristic
302	C₂²	#15688980	1	1	yes	characteristic
303	C₂²	#15688990	1	1	yes	characteristic
304	C₂²	#15689008	1	2	yes	—
305	C₂²	#15689011	1	2	yes	—
306	C₄	—	8	4	no	—
307	C₄	—	94	4	no	—
308	C₄	—	188	2	no	—
309	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
310	C₄	#15688974	1	1	yes	characteristic
311	C₄	#15688984	1	1	yes	characteristic
312	C₄	#15688987	1	1	yes	characteristic
313	C₄	#15689004	1	2	yes	—
314	C₄	#15689015	1	2	yes	—
315	C₂	—	2	2	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	#20241612	1	1	yes	characteristic
320	C₂	#20243535	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093981
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×H(3,ℤ/4)	C₂
socle	C₂²×C₄₇	#383
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₄₇
hypocenter	C₄₇	#10805
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
11. C₂×#13093934	C₄	semidirect
12. C₂×C₈×Dic₄₇	C₄	semidirect
13. #15688893	C₄	semidirect
179. C₄₇	#10805	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₄₇	2
C₂×#15688893	2
C₂×#15688882	2
C₂×C₄₇⋊#338	2
C₂×C₄₇×H(3,ℤ/4)	2
#10805	47