Subgroups — #27085823

Order 12032.

3386 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689928	C₂	1	4	yes	maximal, complemented
2	#20257166	C₂	1	2	yes	maximal, complemented
3	#20253874	C₂	1	12	yes	maximal, complemented
4	#20253840	C₂	1	6	yes	maximal, complemented
5	#20253816	C₂	1	3	yes	maximal, complemented
6	C₂×#15689500	C₂	1	3	yes	maximal, complemented
7	C₄×#131×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094070	C₂²	1	4	yes	—
9	#15689928	—	2	32	no	—
10	C₂×#13094052	C₂²	1	24	yes	complemented
11	C₂×#13094046	C₂²	1	12	yes	complemented
12	C₂²×D₃₇₆	C₂²	1	12	yes	complemented
13	#15689795	—	2	4	no	—
14	D₄×D₁₈₈	—	2	12	no	—
15	#15689692	C₂²	1	12	yes	complemented
16	#15689673	C₂²	1	6	yes	complemented
17	C₂×#13094021	C₂²	1	6	yes	complemented
18	#15689512	C₂²	1	24	yes	complemented
19	#15689507	C₂²	1	12	yes	complemented
20	#15689503	C₂²	1	1	yes	characteristic, complemented
21	#15689500	C₂²	1	3	yes	complemented
22	#15689500	C₂²	1	6	yes	complemented
23	#15689500	C₂²	1	12	yes	complemented
24	#15689496	C₂²	1	6	yes	complemented
25	C₂×#131×C₄₇	C₂²	1	2	yes	—
26	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#405	C₂²	1	6	yes	complemented
28	C₄×M₁₆×C₄₇	C₂²	1	3	yes	complemented
29	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	complemented
30	#13094070	—	2	32	no	—
31	C₂×#10625354	—	2	4	no	—
32	C₂×#10625350	C₂³	1	12	yes	—
33	C₂²×D₁₈₈	—	2	12	no	—
34	C₂²×D₁₈₈	C₂³	1	12	yes	—
35	C₂×D₄×D₄₇	—	2	12	no	—
36	#13094052	—	2	96	no	—
37	#13094046	—	2	48	no	—
38	C₂×Dic₁₈₈	C₂³	1	4	yes	—
39	C₂×#10625325	C₂³	1	24	yes	—
40	C₂×D₃₇₆	—	2	48	no	—
41	C₂×D₃₇₆	C₂³	1	12	yes	—
42	C₂×D₃₇₆	C₂³	1	24	yes	—
43	#13094024	C₂³	1	2	yes	—
44	#13094021	—	2	12	no	—
45	#13094021	C₂³	1	6	yes	—
46	#13094021	C₂³	1	12	yes	—
47	#13094012	C₂³	1	12	yes	—
48	#13093999	—	2	24	no	—
49	#13093998	—	2	12	no	—
50	#13093994	—	2	24	no	—
51	C₄×D₁₈₈	—	2	12	no	—
52	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
53	#131×C₄₇	D₄	1	16	yes	complemented
54	C₂×M₁₆×C₄₇	C₂³	1	6	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
56	C₄×Q₈×C₄₇	D₄	1	2	yes	—
57	C₄×D₄×C₄₇	D₄	1	6	yes	—
58	#117×C₄₇	C₂³	1	3	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
60	#97×C₄₇	C₂³	1	6	yes	—
61	C₄×C₈×C₄₇	C₂³	1	3	yes	—
62	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₇	C₂³	1	6	yes	—
64	C₂³×D₄₇	—	2	12	no	—
65	#41×C₄₇	C₂×D₄	1	8	yes	—
66	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₄₇	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
70	C₂×C₂²⋊D₄₇	—	2	12	no	—
71	#10625354	—	4	16	no	—
72	D₄×D₄₇	—	4	48	no	—
73	#10625350	—	2	48	no	—
74	C₂×D₁₈₈	—	2	48	no	—
75	C₂×D₁₈₈	—	2	12	no	—
76	C₂×D₁₈₈	—	2	12	no	—
77	C₂×D₁₈₈	—	2	24	no	—
78	C₂×D₁₈₈	—	8	24	no	—
79	C₂×D₁₈₈	C₂⁴	1	12	yes	—
80	C₂×C₄×D₄₇	—	2	12	no	—
81	C₂×Dic₉₄	C₂⁴	1	4	yes	—
82	M₁₆×C₄₇	C₂×D₄	1	24	yes	complemented
83	C₂×C₈×C₄₇	C₂×D₄	1	24	yes	complemented
84	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
85	C₂×C₈×C₄₇	C₂⁴	1	6	yes	—
86	C₄⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
87	C₂²⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
88	C₄²×C₄₇	C₂×D₄	1	6	yes	—
89	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₄²×C₄₇	C₂⁴	1	3	yes	—
91	(C₂×D₄₇)⋊C₄	—	2	24	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	Dic₁₈₈	—	2	8	no	—
94	D₃₇₆	—	2	24	no	—
95	D₃₇₆	—	2	48	no	—
96	#10625325	—	2	48	no	—
97	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
98	C₂²×D₄₇	—	2	12	no	—
99	C₂²×D₄₇	—	4	24	no	—
100	C₂²×D₄₇	—	8	24	no	—
101	Q₈×C₄₇	#127	1	4	yes	—
102	D₄×C₄₇	#127	1	12	yes	—
103	C₂×C₄×C₄₇	—	2	12	no	—
104	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
105	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
107	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
109	C₂²⋊D₄₇	—	4	24	no	—
110	C₂×Dic₄₇	—	2	4	no	—
111	D₁₈₈	—	2	24	no	—
112	D₁₈₈	—	4	24	no	—
113	D₁₈₈	—	4	24	no	—
114	D₁₈₈	—	8	24	no	—
115	C₄×D₄₇	—	4	24	no	—
116	Dic₉₄	—	2	8	no	—
117	C₈×C₄₇	C₂²×D₄	1	12	yes	—
118	C₈×C₄₇	C₂²×D₄	1	4	yes	—
119	#60647	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	6	no	—
121	C₂²×C₄₇	C₂×#127	1	6	yes	—
122	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
123	C₂×D₄₇	—	4	12	no	—
124	C₂×D₄₇	—	4	24	no	—
125	C₂×D₄₇	—	16	24	no	—
126	C₄×C₄₇	—	2	6	no	—
127	C₄×C₄₇	C₂×#127	1	6	yes	—
128	C₄×C₄₇	#576	1	4	yes	complemented
129	C₄×C₄₇	C₂³×D₄	1	2	yes	—
130	Dic₄₇	—	4	4	no	—
131	C₂×#576	—	47	4	no	—
132	#3498	—	47	2	no	—
133	#3155	—	47	12	no	—
134	#3151	—	47	6	no	—
135	#3147	—	47	3	no	—
136	C₂×#493	—	47	3	no	—
137	C₄×#131	D₄₇	1	1	yes	characteristic, complemented
138	C₂×C₄₇	—	2	6	no	—
139	C₂×C₄₇	C₂²×#127	1	1	yes	characteristic
140	C₂×C₄₇	C₂×#576	1	2	yes	—
141	D₄₇	—	8	12	no	—
142	C₂×H(5,𝔽₂)	—	47	4	no	—
143	#576	—	94	32	no	—
144	C₂×AGL(1,ℤ/8)	—	47	24	no	—
145	C₂×#135	—	47	12	no	—
146	C₂²×D₈	—	47	12	no	—
147	C₂×#131	C₂×D₄₇	1	2	yes	—
148	#550	—	94	4	no	—
149	D₄²	—	94	12	no	—
150	#535	—	47	12	no	—
151	#532	—	47	6	no	—
152	C₂×#127	—	47	6	no	—
153	C₄×#41	C₂×D₄₇	1	1	yes	characteristic
154	#496	—	47	24	no	—
155	#495	—	47	12	no	—
156	#494	—	47	1	no	—
157	#493	—	47	3	no	—
158	#493	—	47	6	no	—
159	#493	—	47	12	no	—
160	#492	—	47	6	no	—
161	#405	C₂×D₄₇	1	6	yes	complemented
162	C₄×M₁₆	C₂×D₄₇	1	3	yes	complemented
163	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	complemented
164	C₄₇	#60647	1	1	yes	characteristic, complemented, Sylow
165	H(5,𝔽₂)	—	94	32	no	—
166	C₂×#41	—	47	12	no	—
167	C₂×#41	—	94	4	no	—
168	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
169	C₂²×D₄	—	47	12	no	—
170	C₂²×D₄	—	94	12	no	—
171	C₂²×D₄	—	94	12	no	—
172	AGL(1,ℤ/8)	—	94	96	no	—
173	#135	—	94	48	no	—
174	C₂×Q₁₆	—	47	4	no	—
175	C₂×SD₁₆	—	47	24	no	—
176	C₂×D₈	—	47	12	no	—
177	C₂×D₈	—	47	24	no	—
178	C₂×D₈	—	94	48	no	—
179	#131	D₁₈₈	1	16	yes	complemented
180	C₂×M₁₆	C₂²×D₄₇	1	6	yes	—
181	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
182	C₄⋊Q₈	—	47	2	no	—
183	#127	—	47	6	no	—
184	#127	—	47	12	no	—
185	#127	—	94	12	no	—
186	#124	—	47	12	no	—
187	#121	—	94	12	no	—
188	#121	—	94	24	no	—
189	#120	—	94	24	no	—
190	C₄×Q₈	D₁₈₈	1	2	yes	—
191	C₄×D₄	—	94	12	no	—
192	C₄×D₄	D₁₈₈	1	6	yes	—
193	#117	C₂²×D₄₇	1	3	yes	—
194	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
195	#97	C₂²×D₄₇	1	6	yes	—
196	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₄₇	1	3	yes	—
198	C₄×C₈	C₂²×D₄₇	1	6	yes	—
199	C₂⁴	—	94	12	no	—
200	#41	—	94	48	no	—
201	#41	—	188	16	no	—
202	#41	C₂×D₁₈₈	1	8	yes	—
203	C₂×Q₈	—	47	4	no	—
204	C₂×Q₈	C₂×D₁₈₈	1	1	yes	characteristic
205	C₂×D₄	—	47	12	no	—
206	C₂×D₄	—	94	12	no	—
207	C₂×D₄	—	94	48	no	—
208	C₂×D₄	—	94	24	no	—
209	C₂×D₄	—	94	12	no	—
210	C₂×D₄	—	94	12	no	—
211	C₂×D₄	—	376	24	no	—
212	C₂×D₄	—	188	48	no	—
213	C₂×D₄	C₂×D₁₈₈	1	3	yes	—
214	C₂²×C₄	—	94	12	no	—
215	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
216	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
217	Q₁₆	—	94	8	no	—
218	SD₁₆	—	94	48	no	—
219	D₈	—	94	24	no	—
220	D₈	—	94	48	no	—
221	M₁₆	C₂×D₁₈₈	1	24	yes	complemented
222	C₂×C₈	C₂×D₁₈₈	1	24	yes	complemented
223	C₂×C₈	C₂³×D₄₇	1	2	yes	—
224	C₂×C₈	C₂³×D₄₇	1	6	yes	—
225	C₄⋊C₄	—	94	4	no	—
226	C₄⋊C₄	C₂×D₁₈₈	1	6	yes	—
227	C₂²⋊C₄	—	94	24	no	—
228	C₂²⋊C₄	C₂×D₁₈₈	1	6	yes	—
229	C₄²	C₂×D₁₈₈	1	6	yes	—
230	C₄²	C₂³×D₄₇	1	1	yes	characteristic
231	C₄²	C₂³×D₄₇	1	3	yes	—
232	C₂³	—	94	12	no	—
233	C₂³	—	376	24	no	—
234	C₂³	—	188	24	no	—
235	C₂³	C₂²×D₁₈₈	1	3	yes	—
236	Q₈	—	94	8	no	—
237	Q₈	#13094021	1	4	yes	—
238	D₄	—	94	24	no	—
239	D₄	—	376	24	no	—
240	D₄	—	188	24	no	—
241	D₄	—	188	24	no	—
242	D₄	—	188	24	no	—
243	D₄	#13094021	1	12	yes	—
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	94	4	no	—
246	C₂×C₄	—	188	24	no	—
247	C₂×C₄	C₂²×D₁₈₈	1	6	yes	—
248	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
249	C₂×C₄	C₂²×D₁₈₈	1	12	yes	—
250	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
251	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
252	C₈	C₂²×D₁₈₈	1	4	yes	—
253	C₈	C₂²×D₁₈₈	1	12	yes	—
254	C₂²	—	2	6	no	—
255	C₂²	—	188	24	no	—
256	C₂²	—	188	12	no	—
257	C₂²	—	752	24	no	—
258	C₂²	C₂×#13094021	1	6	yes	—
259	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
260	C₄	—	2	6	no	—
261	C₄	—	188	4	no	—
262	C₄	C₂×#13094021	1	6	yes	—
263	C₄	#15689928	1	4	yes	complemented
264	C₄	C₂³×D₁₈₈	1	2	yes	—
265	C₂	—	2	6	no	—
266	C₂	—	376	12	no	—
267	C₂	C₂²×#13094021	1	1	yes	characteristic
268	C₂	C₂×#15689928	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄×#131×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	C₄×#131	D₄₇
hypocenter	C₄₇	#60647
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#15689928	C₂	semidirect
2. #20257166	C₂	semidirect
3. #20253874	C₂	semidirect
4. #20253840	C₂	semidirect
5. #20253816	C₂	semidirect
6. C₂×#15689500	C₂	semidirect
7. C₄×#131×C₄₇	C₂	semidirect
10. C₂×#13094052	C₂²	semidirect
11. C₂×#13094046	C₂²	semidirect
12. C₂²×D₃₇₆	C₂²	semidirect
15. #15689692	C₂²	semidirect
16. #15689673	C₂²	semidirect
17. C₂×#13094021	C₂²	semidirect
18. #15689512	C₂²	semidirect
19. #15689507	C₂²	semidirect
20. #15689503	C₂²	semidirect
21. #15689500	C₂²	semidirect
22. #15689500	C₂²	semidirect
23. #15689500	C₂²	semidirect
24. #15689496	C₂²	semidirect
27. C₄₇×#405	C₂²	semidirect
28. C₄×M₁₆×C₄₇	C₂²	semidirect
29. C₂×C₄×C₈×C₄₇	C₂²	semidirect
53. #131×C₄₇	D₄	semidirect
82. M₁₆×C₄₇	C₂×D₄	semidirect
83. C₂×C₈×C₄₇	C₂×D₄	semidirect
128. C₄×C₄₇	#576	semidirect
137. C₄×#131	D₄₇	semidirect
161. #405	C₂×D₄₇	semidirect
162. C₄×M₁₆	C₂×D₄₇	semidirect
163. C₂×C₄×C₈	C₂×D₄₇	semidirect
164. C₄₇	#60647	semidirect
179. #131	D₁₈₈	semidirect
221. M₁₆	C₂×D₁₈₈	semidirect
222. C₂×C₈	C₂×D₁₈₈	semidirect
263. C₄	#15689928	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689928	2
#20257166	2
#20253874	2
#20253840	2
#20253816	2
C₂×#15689500	2
C₄×#131×C₄₇	2
#60647	47