Subgroups — C₄×#15688927 #26131487

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688927	C₂	1	2	yes	maximal
2	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093958	C₂	1	1	yes	maximal, characteristic
4	#20241479	C₂	1	2	yes	maximal
5	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₄₇	—	2	1	no	—
9	#15688927	C₄	1	8	yes	direct product factor
10	C₄×C₄₇⋊C₁₆	—	2	2	no	—
11	#15688886	—	2	1	no	—
12	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
14	C₄×Q₁₆×C₄₇	—	2	2	no	—
15	C₄×Q₁₆×C₄₇	C₄	1	4	yes	complemented
16	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#103×C₄₇	—	2	1	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄×Dic₄₇	—	4	1	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
24	#13093958	—	2	2	no	—
25	#13093958	C₂×C₄	1	4	yes	—
26	C₄₇⋊#95	—	4	1	no	—
27	C₂²×Q₈×C₄₇	—	2	1	no	—
28	C₂×Q₁₆×C₄₇	—	2	1	no	—
29	C₂×Q₁₆×C₄₇	—	2	2	no	—
30	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₄₇	—	2	4	no	—
35	C₄×Q₈×C₄₇	—	4	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
38	C₂×C₄²×C₄₇	—	4	1	no	—
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	#107×C₄₇	—	2	1	no	—
41	#107×C₄₇	C₂×C₄	1	2	yes	—
42	#103×C₄₇	—	2	4	no	—
43	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₂×Q₈×C₄₇	—	2	2	no	—
46	C₂×Q₈×C₄₇	—	2	4	no	—
47	C₂×Q₈×C₄₇	—	4	1	no	—
48	C₂×Q₈×C₄₇	—	4	2	no	—
49	C₂²×C₄×C₄₇	—	4	1	no	—
50	C₂²×C₄×C₄₇	—	4	1	no	—
51	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	4	2	no	—
55	Q₁₆×C₄₇	—	2	2	no	—
56	Q₁₆×C₄₇	—	2	4	no	—
57	Q₁₆×C₄₇	C₄²	1	4	yes	complemented
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	#41	1	2	yes	—
64	C₄⋊C₄×C₄₇	—	2	2	no	—
65	C₄⋊C₄×C₄₇	—	4	4	no	—
66	C₄⋊C₄×C₄₇	—	4	1	no	—
67	C₄²×C₄₇	—	4	4	no	—
68	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₄₇	SD₁₆	1	2	yes	—
70	C₄⋊Dic₄₇	—	2	4	no	—
71	C₄⋊Dic₄₇	—	4	2	no	—
72	C₄×Dic₄₇	—	8	2	no	—
73	C₄₇⋊C₁₆	—	2	4	no	—
74	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₄₇	—	2	4	no	—
76	Q₈×C₄₇	—	4	2	no	—
77	Q₈×C₄₇	—	4	4	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	—	8	2	no	—
83	C₂×C₄×C₄₇	#102	1	2	yes	—
84	C₂×C₄×C₄₇	#102	1	2	yes	—
85	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
86	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#135	1	2	yes	—
91	C₂×Dic₄₇	—	4	4	no	—
92	C₂×Dic₄₇	—	8	2	no	—
93	C₂×Dic₄₇	—	8	2	no	—
94	C₈×C₄₇	—	2	2	no	—
95	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
96	C₈×C₄₇	#117	1	1	yes	characteristic
97	C₈×C₄₇	C₄×D₄	1	2	yes	—
98	C₄×#358	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
100	C₂²×C₄₇	C₄×D₈	1	2	yes	—
101	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
103	C₂²×C₄₇	#508	1	2	yes	—
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	4	4	no	—
106	C₄×C₄₇	—	8	2	no	—
107	C₄×C₄₇	#358	1	4	yes	complemented
108	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₄₇	#416	1	1	yes	characteristic
110	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
111	Dic₄₇	—	8	4	no	—
112	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
113	C₂×#358	—	47	2	no	—
114	C₂×C₄×C₁₆	—	47	1	no	—
115	C₄×#107	—	47	1	no	—
116	#1380	—	47	2	no	—
117	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
118	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
119	C₂×C₄₇	#2138	1	1	yes	characteristic
120	C₂×C₄₇	C₄×SD₃₂	1	2	yes	—
121	C₂×C₄₇	C₄×Q₃₂	1	2	yes	—
122	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
123	C₂×C₄×Q₈	—	2	1	no	—
124	C₂²×C₁₆	—	47	2	no	—
125	C₄×Q₁₆	—	2	2	no	—
126	C₄×Q₁₆	Dic₄₇	1	4	yes	complemented
127	C₂×#107	—	47	2	no	—
128	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
129	C₂×#103	—	2	1	no	—
130	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	C₄×C₄⋊C₄	—	94	1	no	—
132	#358	—	47	8	no	—
133	C₄×C₁₆	—	94	2	no	—
134	#340	—	94	1	no	—
135	C₄₇	C₄×#358	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	2	1	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	1	no	—
139	C₂×Q₁₆	C₄×D₄₇	1	4	yes	—
140	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
141	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
143	C₄×Q₈	—	2	4	no	—
144	C₄×Q₈	—	4	2	no	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	4	1	no	—
147	C₂×C₄⋊C₄	—	94	2	no	—
148	C₂×C₄²	—	4	1	no	—
149	C₂×C₄²	—	188	1	no	—
150	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
151	C₂×C₁₆	—	47	4	no	—
152	C₂×C₁₆	—	94	2	no	—
153	C₂×C₁₆	—	94	2	no	—
154	#107	—	2	1	no	—
155	#107	—	47	4	no	—
156	#107	—	94	2	no	—
157	#107	C₂×Dic₄₇	1	2	yes	—
158	#103	—	2	4	no	—
159	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
160	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
161	#95	—	188	1	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	188	2	no	—
169	C₂²×C₄	#10625334	1	1	yes	characteristic
170	C₂²×C₄	#10625334	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
172	Q₁₆	—	2	2	no	—
173	Q₁₆	—	2	4	no	—
174	Q₁₆	C₄×Dic₄₇	1	4	yes	complemented
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
178	C₂×C₈	#10625350	1	2	yes	—
179	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	94	4	no	—
185	C₄⋊C₄	—	188	2	no	—
186	C₄²	—	4	4	no	—
187	C₄²	—	376	2	no	—
188	C₄²	#10625335	1	2	yes	—
189	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
190	C₁₆	—	94	4	no	—
191	C₂³	C₂×#10625334	1	1	yes	characteristic
192	Q₈	—	2	4	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	4	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	376	2	no	—
201	C₂×C₄	—	376	2	no	—
202	C₂×C₄	—	188	4	no	—
203	C₂×C₄	#13093945	1	2	yes	—
204	C₂×C₄	#13093945	1	2	yes	—
205	C₂×C₄	#13093967	1	2	yes	—
206	C₂×C₄	#13093970	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
209	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
210	C₂×C₄	#13094044	1	2	yes	—
211	C₈	—	2	2	no	—
212	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
213	C₈	#13093983	1	1	yes	characteristic
214	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
215	C₂²	C₂×#13093945	1	1	yes	characteristic
216	C₂²	C₄×#10625334	1	2	yes	—
217	C₂²	C₂×#13093967	1	1	yes	characteristic
218	C₂²	C₂×#13093970	1	1	yes	characteristic
219	C₂²	#15689553	1	2	yes	—
220	C₄	—	2	2	no	—
221	C₄	—	4	4	no	—
222	C₄	—	8	2	no	—
223	C₄	—	376	4	no	—
224	C₄	#15688927	1	4	yes	direct product factor
225	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
226	C₄	#15689119	1	1	yes	characteristic
227	C₄	C₄×#10625335	1	2	yes	—
228	C₂	C₄×#13093945	1	1	yes	characteristic
229	C₂	C₂×#15688927	1	1	yes	characteristic
230	C₂	#20245095	1	1	yes	characteristic
231	C₂	C₄×#13093967	1	2	yes	—
232	C₂	C₄×#13093970	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#10625334
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	C₄×#358
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
9. #15688927	C₄	direct (N × H)
15. C₄×Q₁₆×C₄₇	C₄	semidirect
57. Q₁₆×C₄₇	C₄²	semidirect
107. C₄×C₄₇	#358	semidirect
126. C₄×Q₁₆	Dic₄₇	semidirect
135. C₄₇	C₄×#358	semidirect
174. Q₁₆	C₄×Dic₄₇	semidirect
224. C₄	#15688927	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×#15688927	2
C₂×C₄×C₄₇⋊C₁₆	2
C₄×#13093958	2
#20241479	2
C₂×C₄×Q₁₆×C₄₇	2
C₄×#358	47

Subgroups — C4×#15688927 #26131487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688927 #26131487