Subgroups — C₂²×#15689159 #27079926

Order 12032.

3006 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093983	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689159	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13093957	C₂	1	2	yes	maximal, complemented
5	C₂²×#13093955	C₂	1	2	yes	maximal, complemented
6	C₂²×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₄	1	16	yes	complemented
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₄₇⋊C₈	C₂²	1	12	yes	—
11	C₂×#13093983	—	2	6	no	—
12	C₂×#13093983	C₂²	1	12	yes	—
13	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂²×C₂²⋊Dic₄₇	—	2	1	no	—
15	C₂²×C₄×Dic₄₇	—	2	1	no	—
16	#15689159	C₂²	1	64	yes	direct product factor
17	C₂×#13093957	C₂²	1	24	yes	complemented
18	C₂×#13093955	C₂²	1	24	yes	complemented
19	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₄₇	—	2	6	no	—
21	C₂×#117×C₄₇	C₂²	1	12	yes	—
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₄₇	—	2	1	no	—
24	C₂²×C₄²×C₄₇	—	2	1	no	—
25	C₂³×Dic₄₇	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	96	yes	complemented
29	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	16	yes	—
32	#13093983	—	2	24	no	—
33	#13093983	C₂³	1	16	yes	—
34	C₂×C₄⋊Dic₄₇	—	2	6	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
36	C₂×C₂²⋊Dic₄₇	—	2	12	no	—
37	C₂×C₄×Dic₄₇	—	2	12	no	—
38	#13093957	C₂³	1	32	yes	complemented
39	#13093955	C₂³	1	32	yes	complemented
40	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	—	2	2	no	—
42	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
43	C₂³×C₄×C₄₇	D₄	1	6	yes	—
44	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	Q₈	1	8	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
48	#117×C₄₇	—	2	24	no	—
49	#117×C₄₇	C₂³	1	16	yes	—
50	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
52	C₂×C₂²⋊C₄×C₄₇	—	2	12	no	—
53	C₂×C₄²×C₄₇	—	2	12	no	—
54	C₂⁴×C₄₇	—	2	8	no	—
55	C₂⁴×C₄₇	C₄⋊C₄	1	8	yes	—
56	C₂⁴×C₄₇	C₂×D₄	1	6	yes	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	12	no	—
59	C₂²×C₄×C₄₇	—	4	8	no	—
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	48	yes	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	24	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
71	C₂²×Dic₄₇	—	2	12	no	—
72	C₂²×Dic₄₇	—	4	8	no	—
73	C₄⋊C₄×C₄₇	—	2	12	no	—
74	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
75	C₂²⋊C₄×C₄₇	—	2	16	no	—
76	C₄²×C₄₇	—	2	16	no	—
77	C₂²⋊Dic₄₇	—	2	16	no	—
78	C₄⋊Dic₄₇	—	2	12	no	—
79	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
80	C₄×Dic₄₇	—	2	16	no	—
81	C₂×C₄₇⋊C₈	C₂²×C₄	1	24	yes	—
82	C₂×C₄₇⋊C₈	C₂²×C₄	1	24	yes	—
83	C₂×C₄₇⋊C₈	C₂²×C₄	1	64	yes	complemented
84	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
85	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
86	C₂³×C₄₇	—	2	24	no	—
87	C₂³×C₄₇	—	2	32	no	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	24	yes	—
89	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂³×C₄₇	#135	1	8	yes	—
91	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
92	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	4	24	no	—
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	48	yes	—
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	24	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
100	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
101	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
102	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
103	C₂×C₄×C₄₇	C₂²×Q₈	1	6	yes	—
104	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
105	C₂×Dic₄₇	—	2	8	no	—
106	C₂×Dic₄₇	—	4	24	no	—
107	C₄₇⋊C₈	C₂³×C₄	1	8	yes	—
108	C₄₇⋊C₈	C₂³×C₄	1	8	yes	—
109	C₂²×#427	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	48	no	—
111	C₂²×C₄₇	—	2	8	no	—
112	C₂²×C₄₇	#427	1	16	yes	complemented
113	C₂²×C₄₇	C₂²×C₄⋊C₄	1	8	yes	—
114	C₂²×C₄₇	C₂²×C₄⋊C₄	1	6	yes	—
115	C₂²×C₄₇	C₂×#135	1	12	yes	—
116	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
117	C₄×C₄₇	—	4	8	no	—
118	C₄×C₄₇	C₂²×C₄⋊C₄	1	6	yes	—
119	C₄×C₄₇	C₂²×C₄⋊C₄	1	8	yes	—
120	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
121	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
122	Dic₄₇	—	4	8	no	—
123	C₂⁴×C₈	—	47	1	no	—
124	C₂²×#117	—	47	1	no	—
125	C₂²×#117	D₄₇	1	1	yes	characteristic
126	C₂×#427	—	47	24	no	—
127	C₂²×#107	—	47	2	no	—
128	C₂²×#106	—	47	2	no	—
129	C₂×C₄₇	—	2	8	no	—
130	C₂×C₄₇	C₂×#427	1	12	yes	complemented
131	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄₇	C₂²×#135	1	2	yes	—
133	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
134	C₂³×C₈	—	47	16	no	—
135	C₂³×C₈	—	47	1	no	—
136	C₂³×C₈	—	47	1	no	—
137	C₂³×C₈	—	47	12	no	—
138	C₂×#117	—	2	6	no	—
139	C₂×#117	—	47	12	no	—
140	C₂×#117	—	94	6	no	—
141	C₂×#117	C₂×D₄₇	1	12	yes	—
142	C₂²×C₄⋊C₄	—	47	2	no	—
143	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
144	C₂²×C₂²⋊C₄	—	2	1	no	—
145	C₂²×C₂²⋊C₄	—	94	1	no	—
146	C₂²×C₄²	—	2	1	no	—
147	C₂²×C₄²	—	94	1	no	—
148	#427	—	47	64	no	—
149	C₂×#107	—	47	24	no	—
150	C₂×#106	—	47	24	no	—
151	C₄₇	C₂²×#427	1	1	yes	characteristic, complemented, Sylow
152	C₂⁵	C₂²⋊D₄₇	1	1	yes	characteristic
153	C₂³×C₄	—	2	2	no	—
154	C₂³×C₄	—	94	2	no	—
155	C₂³×C₄	Dic₉₄	1	8	yes	—
156	C₂³×C₄	C₄×D₄₇	1	8	yes	—
157	C₂³×C₄	C₂²⋊D₄₇	1	6	yes	—
158	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
159	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
160	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
161	C₂²×C₈	—	47	8	no	—
162	C₂²×C₈	—	47	8	no	—
163	C₂²×C₈	—	47	96	no	—
164	C₂²×C₈	—	47	6	no	—
165	C₂²×C₈	—	47	6	no	—
166	C₂²×C₈	—	47	16	no	—
167	#117	—	2	24	no	—
168	#117	—	47	16	no	—
169	#117	—	94	24	no	—
170	#117	C₂²×D₄₇	1	16	yes	—
171	C₂×C₄⋊C₄	—	2	6	no	—
172	C₂×C₄⋊C₄	—	47	12	no	—
173	C₂×C₄⋊C₄	—	94	6	no	—
174	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
175	C₂×C₂²⋊C₄	—	2	12	no	—
176	C₂×C₂²⋊C₄	—	94	12	no	—
177	C₂×C₄²	—	2	12	no	—
178	C₂×C₄²	—	94	12	no	—
179	#107	—	47	32	no	—
180	#106	—	47	32	no	—
181	C₂⁴	—	2	8	no	—
182	C₂⁴	Dic₄₇⋊C₄	1	8	yes	—
183	C₂⁴	C₂×C₂²⋊D₄₇	1	6	yes	—
184	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂²×C₄	—	2	12	no	—
186	C₂²×C₄	—	4	8	no	—
187	C₂²×C₄	—	94	12	no	—
188	C₂²×C₄	—	188	8	no	—
189	C₂²×C₄	Dic₄₇⋊C₄	1	48	yes	—
190	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
191	C₂²×C₄	C₂×Dic₉₄	1	24	yes	—
192	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
193	C₂²×C₄	C₂×C₄×D₄₇	1	24	yes	—
194	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
197	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
198	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
199	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
200	C₂×C₈	—	47	24	no	—
201	C₂×C₈	—	47	24	no	—
202	C₂×C₈	—	47	64	no	—
203	C₂×C₈	—	47	4	no	—
204	C₂×C₈	—	47	4	no	—
205	C₄⋊C₄	—	2	12	no	—
206	C₄⋊C₄	—	47	8	no	—
207	C₄⋊C₄	—	94	12	no	—
208	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
209	C₂²⋊C₄	—	2	16	no	—
210	C₂²⋊C₄	—	94	16	no	—
211	C₄²	—	2	16	no	—
212	C₄²	—	94	16	no	—
213	C₂³	—	2	24	no	—
214	C₂³	—	2	32	no	—
215	C₂³	C₂×Dic₄₇⋊C₄	1	24	yes	—
216	C₂³	C₂×Dic₄₇⋊C₄	1	4	yes	—
217	C₂³	#13094044	1	8	yes	—
218	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
219	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	4	24	no	—
222	C₂×C₄	—	94	8	no	—
223	C₂×C₄	—	188	24	no	—
224	C₂×C₄	C₂×Dic₄₇⋊C₄	1	48	yes	—
225	C₂×C₄	C₂×Dic₄₇⋊C₄	1	12	yes	—
226	C₂×C₄	C₂×Dic₄₇⋊C₄	1	24	yes	—
227	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
228	C₂×C₄	C₂²×C₄×D₄₇	1	6	yes	—
229	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
230	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
231	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
232	C₂×C₄	C₂²×Dic₉₄	1	6	yes	—
233	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
234	C₈	—	47	8	no	—
235	C₈	—	47	8	no	—
236	C₂²	—	2	48	no	—
237	C₂²	—	2	8	no	—
238	C₂²	#15689159	1	16	yes	direct product factor
239	C₂²	C₂²×Dic₄₇⋊C₄	1	8	yes	—
240	C₂²	C₂²×Dic₄₇⋊C₄	1	6	yes	—
241	C₂²	C₂×#13094044	1	12	yes	—
242	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₄	—	4	8	no	—
244	C₄	—	188	8	no	—
245	C₄	C₂²×Dic₄₇⋊C₄	1	8	yes	—
246	C₄	C₂²×Dic₄₇⋊C₄	1	6	yes	—
247	C₄	C₂³×C₄×D₄₇	1	1	yes	characteristic
248	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
249	C₂	—	2	8	no	—
250	C₂	C₂×#15689159	1	12	yes	direct product factor
251	C₂	C₂³×Dic₄₇⋊C₄	1	1	yes	characteristic
252	C₂	C₂²×#13094044	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#117×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×#117	D₄₇
hypocenter	C₄₇	C₂²×#427
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
3. C₂×#15689159	C₂	direct (N × H)
4. C₂²×#13093957	C₂	semidirect
5. C₂²×#13093955	C₂	semidirect
7. C₂³×C₄₇⋊C₈	C₄	semidirect
16. #15689159	C₂²	direct (N × H)
17. C₂×#13093957	C₂²	semidirect
18. C₂×#13093955	C₂²	semidirect
28. C₂²×C₄₇⋊C₈	C₂×C₄	semidirect
38. #13093957	C₂³	semidirect
39. #13093955	C₂³	semidirect
83. C₂×C₄₇⋊C₈	C₂²×C₄	semidirect
112. C₂²×C₄₇	#427	semidirect
130. C₂×C₄₇	C₂×#427	semidirect
151. C₄₇	C₂²×#427	semidirect
238. C₂²	#15689159	direct (N × H)
250. C₂	C₂×#15689159	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄₇⋊C₈	2
C₂²×#13093983	2
C₂×#15689159	2
C₂²×#13093957	2
C₂²×#13093955	2
C₂²×#117×C₄₇	2
C₂²×#427	47

Subgroups — C22×#15689159 #27079926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689159 #27079926