Subgroups — C₄²×#10625336 #26214750

Order 12032.

1230 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625336	C₂	1	3	yes	maximal
2	C₄²×D₁₈₈	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093955	C₂	1	3	yes	maximal, complemented
4	C₄×#13093948	C₂	1	3	yes	maximal, complemented
5	C₄×#13093944	C₂	1	3	yes	maximal
6	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625336	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₈₈	C₂²	1	3	yes	—
10	C₂×C₄²×D₄₇	—	2	1	no	—
11	C₄×#10625336	C₄	1	24	yes	direct product factor
12	C₂×#13093955	C₂²	1	3	yes	—
13	C₂×#13093948	C₂²	1	3	yes	—
14	C₂×#13093944	C₂²	1	3	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	3	yes	—
18	C₄×(C₂×D₄₇)⋊C₄	—	2	3	no	—
19	#15688885	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
21	C₄×C₄⋊C₄×C₄₇	—	2	3	no	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
23	C₄³×C₄₇	—	2	1	no	—
24	C₄³×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₈₈	C₂³	1	1	yes	characteristic
26	C₂²×C₄×D₄₇	—	2	3	no	—
27	C₂×#10625336	C₂×C₄	1	12	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
30	C₄×D₁₈₈	—	2	6	no	—
31	C₄×D₁₈₈	C₂×C₄	1	12	yes	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	3	yes	—
33	C₂×(C₂×D₄₇)⋊C₄	—	2	3	no	—
34	C₂×C₄×Dic₄₇	—	2	3	no	—
35	C₄²×D₄₇	—	4	4	no	—
36	#13093955	C₂×C₄	1	12	yes	complemented
37	#13093948	C₂×C₄	1	12	yes	complemented
38	#13093944	C₂×C₄	1	12	yes	—
39	C₄×C₄₇⋊C₈	—	2	6	no	—
40	C₄×C₄₇⋊C₈	C₂×C₄	1	12	yes	complemented
41	C₄₇⋊#95	—	2	3	no	—
42	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₄₇	—	2	6	no	—
44	C₄×Q₈×C₄₇	C₂×C₄	1	12	yes	complemented
45	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
47	C₂×C₄²×C₄₇	—	2	3	no	—
48	C₂×C₄²×C₄₇	—	2	3	no	—
49	C₂×C₄²×C₄₇	D₄	1	3	yes	—
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
52	#95×C₄₇	—	2	3	no	—
53	C₂³×D₄₇	—	2	1	no	—
54	C₂×Q₈×C₄₇	—	2	3	no	—
55	C₂×Q₈×C₄₇	C₂²×C₄	1	6	yes	—
56	C₂²×C₄×C₄₇	—	2	1	no	—
57	C₂²×C₄×C₄₇	—	2	3	no	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	3	no	—
62	C₂×D₁₈₈	—	2	3	no	—
63	C₂×D₁₈₈	C₂²×C₄	1	6	yes	—
64	C₂×C₄×D₄₇	—	2	12	no	—
65	C₂×C₄×D₄₇	—	4	6	no	—
66	C₂×C₄×D₄₇	—	4	6	no	—
67	C₄⋊C₄×C₄₇	—	2	12	no	—
68	C₄⋊C₄×C₄₇	—	2	3	no	—
69	C₄⋊C₄×C₄₇	C₂²×C₄	1	6	yes	—
70	C₄²×C₄₇	—	2	12	no	—
71	C₄²×C₄₇	—	2	6	no	—
72	C₄²×C₄₇	—	4	6	no	—
73	C₄²×C₄₇	SD₁₆	1	4	yes	complemented
74	C₄²×C₄₇	C₂²×C₄	1	6	yes	—
75	C₄²×C₄₇	#41	1	6	yes	—
76	#10625336	C₄²	1	16	yes	direct product factor
77	(C₂×D₄₇)⋊C₄	—	2	12	no	—
78	C₄⋊Dic₄₇	—	2	3	no	—
79	C₄⋊Dic₄₇	C₂²×C₄	1	6	yes	—
80	C₄×Dic₄₇	—	4	6	no	—
81	C₂×C₄₇⋊C₈	—	2	6	no	—
82	C₂×C₄₇⋊C₈	C₂²×C₄	1	6	yes	—
83	C₂×C₄₇⋊C₈	C₂²×C₄	1	6	yes	—
84	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	C₂²×D₄₇	—	2	6	no	—
86	C₂²×D₄₇	—	4	1	no	—
87	C₂²×D₄₇	—	4	3	no	—
88	Q₈×C₄₇	—	2	6	no	—
89	Q₈×C₄₇	C₂×C₄²	1	4	yes	complemented
90	C₂×C₄×C₄₇	—	2	6	no	—
91	C₂×C₄×C₄₇	—	2	6	no	—
92	C₂×C₄×C₄₇	—	2	6	no	—
93	C₂×C₄×C₄₇	—	4	6	no	—
94	C₂×C₄×C₄₇	C₄×D₄	1	6	yes	—
95	C₂×C₄×C₄₇	C₄×D₄	1	6	yes	—
96	C₂×C₄×C₄₇	C₂×SD₁₆	1	6	yes	—
97	C₂×C₄×C₄₇	#135	1	6	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	3	yes	—
99	C₂×C₄×C₄₇	C₂×#41	1	3	yes	—
100	C₂×Dic₄₇	—	2	6	no	—
101	C₂×Dic₄₇	—	4	3	no	—
102	C₂×Dic₄₇	—	4	3	no	—
103	D₁₈₈	—	2	6	no	—
104	D₁₈₈	C₂×C₄²	1	4	yes	—
105	C₄×D₄₇	—	4	24	no	—
106	C₄₇⋊C₈	—	2	6	no	—
107	C₄₇⋊C₈	C₂×C₄²	1	4	yes	complemented
108	C₄²×SD₁₆	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	C₂×C₄×D₄	1	3	yes	—
110	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#135	1	3	yes	—
112	C₂×D₄₇	—	2	4	no	—
113	C₂×D₄₇	—	4	6	no	—
114	C₂×D₄₇	—	4	6	no	—
115	C₄×C₄₇	—	2	6	no	—
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	—	4	6	no	—
118	C₄×C₄₇	C₄×SD₁₆	1	12	yes	complemented
119	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
120	C₄×C₄₇	C₄×#41	1	3	yes	—
121	Dic₄₇	—	4	6	no	—
122	C₂×C₄×SD₁₆	—	47	3	no	—
123	C₄²×Q₈	D₄₇	1	1	yes	characteristic, complemented
124	C₄²×D₄	—	47	1	no	—
125	C₄×#106	—	47	3	no	—
126	C₄×#103	—	47	3	no	—
127	C₄×#102	—	47	3	no	—
128	C₄²×C₈	—	47	1	no	—
129	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
130	C₂×C₄₇	C₂×C₄×SD₁₆	1	3	yes	—
131	C₂×C₄₇	C₄×#135	1	3	yes	—
132	D₄₇	—	4	4	no	—
133	C₂²×SD₁₆	—	47	1	no	—
134	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
135	C₂×C₄×D₄	—	47	3	no	—
136	C₂²×C₄²	—	94	1	no	—
137	C₄×SD₁₆	—	47	24	no	—
138	C₂×#106	—	47	3	no	—
139	C₂×#103	—	47	3	no	—
140	C₂×#102	—	47	3	no	—
141	C₂×C₄×C₈	—	47	3	no	—
142	C₂×C₄×C₈	—	47	3	no	—
143	C₄×C₄⋊C₄	—	2	3	no	—
144	C₄×C₄⋊C₄	—	47	3	no	—
145	C₄×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
146	C₄×C₂²⋊C₄	—	94	3	no	—
147	C₄³	—	2	1	no	—
148	C₄³	C₂×D₄₇	1	1	yes	characteristic
149	#340	—	47	6	no	—
150	C₄₇	C₄²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
152	C₂²×D₄	—	47	1	no	—
153	C₂³×C₄	—	94	3	no	—
154	C₂×SD₁₆	—	47	12	no	—
155	C₂²×C₈	—	47	1	no	—
156	C₂²×C₈	—	47	3	no	—
157	C₄×Q₈	—	2	6	no	—
158	C₄×Q₈	C₄×D₄₇	1	12	yes	complemented
159	C₄×D₄	—	47	12	no	—
160	C₄×D₄	—	94	6	no	—
161	C₂×C₄⋊C₄	—	2	3	no	—
162	C₂×C₄⋊C₄	—	47	3	no	—
163	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
164	C₂×C₂²⋊C₄	—	94	3	no	—
165	C₂×C₄²	—	2	3	no	—
166	C₂×C₄²	—	2	3	no	—
167	C₂×C₄²	—	94	3	no	—
168	C₂×C₄²	—	188	4	no	—
169	C₂×C₄²	C₂²⋊D₄₇	1	3	yes	—
170	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
172	#106	—	47	12	no	—
173	#103	—	47	12	no	—
174	#102	—	47	12	no	—
175	C₄×C₈	—	47	12	no	—
176	C₄×C₈	—	94	6	no	—
177	#95	—	2	3	no	—
178	#95	—	94	3	no	—
179	C₂⁴	—	94	1	no	—
180	C₂×Q₈	—	2	3	no	—
181	C₂×Q₈	C₂×C₄×D₄₇	1	6	yes	—
182	C₂×D₄	—	47	6	no	—
183	C₂×D₄	—	94	3	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	3	no	—
186	C₂²×C₄	—	94	3	no	—
187	C₂²×C₄	—	94	12	no	—
188	C₂²×C₄	—	188	6	no	—
189	C₂²×C₄	—	188	6	no	—
190	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
192	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
193	SD₁₆	—	47	16	no	—
194	C₂×C₈	—	47	6	no	—
195	C₂×C₈	—	47	6	no	—
196	C₂×C₈	—	94	6	no	—
197	C₄⋊C₄	—	2	12	no	—
198	C₄⋊C₄	—	2	3	no	—
199	C₄⋊C₄	—	47	6	no	—
200	C₄⋊C₄	—	94	3	no	—
201	C₄⋊C₄	C₂×C₄×D₄₇	1	6	yes	—
202	C₂²⋊C₄	—	94	12	no	—
203	C₄²	—	2	12	no	—
204	C₄²	—	2	6	no	—
205	C₄²	—	4	6	no	—
206	C₄²	—	188	6	no	—
207	C₄²	#10625336	1	4	yes	direct product factor
208	C₄²	C₂×C₄×D₄₇	1	6	yes	—
209	C₄²	#10625350	1	6	yes	—
210	C₂³	—	94	6	no	—
211	C₂³	—	188	1	no	—
212	C₂³	—	188	3	no	—
213	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	C₄²×D₄₇	1	4	yes	complemented
216	D₄	—	47	4	no	—
217	D₄	—	94	6	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	6	no	—
221	C₂×C₄	—	4	6	no	—
222	C₂×C₄	—	94	6	no	—
223	C₂×C₄	—	188	3	no	—
224	C₂×C₄	—	188	3	no	—
225	C₂×C₄	—	188	24	no	—
226	C₂×C₄	C₄×C₂²⋊D₄₇	1	6	yes	—
227	C₂×C₄	C₄×C₂²⋊D₄₇	1	6	yes	—
228	C₂×C₄	C₂×#10625336	1	6	yes	—
229	C₂×C₄	#13094044	1	6	yes	—
230	C₂×C₄	C₂²×C₄×D₄₇	1	3	yes	—
231	C₂×C₄	C₂×#10625350	1	3	yes	—
232	C₈	—	47	4	no	—
233	C₈	—	94	6	no	—
234	C₂²	—	94	4	no	—
235	C₂²	—	188	6	no	—
236	C₂²	—	188	6	no	—
237	C₂²	C₂×C₄×C₂²⋊D₄₇	1	3	yes	—
238	C₂²	C₂²×#10625336	1	1	yes	characteristic
239	C₂²	C₂×#13094044	1	3	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	2	6	no	—
242	C₄	—	4	6	no	—
243	C₄	—	188	6	no	—
244	C₄	C₄×#10625336	1	12	yes	direct product factor
245	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
246	C₄	C₄×#10625350	1	3	yes	—
247	C₂	—	188	4	no	—
248	C₂	C₄²×C₂²⋊D₄₇	1	1	yes	characteristic
249	C₂	C₂×C₄×#10625336	1	3	yes	—
250	C₂	C₄×#13094044	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄²×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₁₈₈	C₂³
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×Q₈	D₄₇
hypocenter	C₄₇	C₄²×SD₁₆
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
3. C₄×#13093955	C₂	semidirect
4. C₄×#13093948	C₂	semidirect
6. C₄²×C₄₇⋊C₈	C₂	semidirect
7. C₄²×Q₈×C₄₇	C₂	semidirect
11. C₄×#10625336	C₄	direct (N × H)
36. #13093955	C₂×C₄	semidirect
37. #13093948	C₂×C₄	semidirect
40. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
44. C₄×Q₈×C₄₇	C₂×C₄	semidirect
73. C₄²×C₄₇	SD₁₆	semidirect
76. #10625336	C₄²	direct (N × H)
89. Q₈×C₄₇	C₂×C₄²	semidirect
107. C₄₇⋊C₈	C₂×C₄²	semidirect
118. C₄×C₄₇	C₄×SD₁₆	semidirect
123. C₄²×Q₈	D₄₇	semidirect
150. C₄₇	C₄²×SD₁₆	semidirect
158. C₄×Q₈	C₄×D₄₇	semidirect
207. C₄²	#10625336	direct (N × H)
215. Q₈	C₄²×D₄₇	semidirect
244. C₄	C₄×#10625336	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625336	2
C₄²×D₁₈₈	2
C₄×#13093955	2
C₄×#13093948	2
C₄×#13093944	2
C₄²×C₄₇⋊C₈	2
C₄²×Q₈×C₄₇	2
C₄²×SD₁₆	47

Subgroups — C42×#10625336 #26214750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#10625336 #26214750