Subgroups — C₄²×#6106 #15861391

Order 3328.

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×#6106	C₂	1	3	yes	maximal
2	C₄²×D₅₂	C₂	1	1	yes	maximal, characteristic
3	C₄×#88048	C₂	1	3	yes	maximal, complemented
4	C₄×#88075	C₂	1	3	yes	maximal, complemented
5	C₄×#88049	C₂	1	3	yes	maximal
6	C₄²×#1055	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
9	C₂²×#6106	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₅₂	C₂²	1	3	yes	—
11	C₂×C₄²×D₁₃	—	2	1	no	—
12	C₄×C₁₃×C₄⋊C₄	—	2	3	no	—
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
14	C₄³×C₁₃	—	2	1	no	—
15	C₄³×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88075	C₂²	1	3	yes	—
17	C₄×#6106	C₄	1	24	yes	direct product factor
18	C₂×#88049	C₂²	1	3	yes	—
19	C₂×#88048	C₂²	1	3	yes	—
20	C₄×#6103	—	2	3	no	—
21	C₄×C₄⋊Dic₁₃	C₂²	1	3	yes	—
22	C₂×C₄×#1055	C₂²	1	3	yes	—
23	C₂×C₄×#1055	C₂²	1	3	yes	—
24	#11718121	C₂²	1	6	yes	—
25	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×D₅₂	C₂³	1	1	yes	characteristic
27	C₂²×C₄×D₁₃	—	2	3	no	—
28	C₄×Q₈×C₁₃	—	2	6	no	—
29	C₄×Q₈×C₁₃	C₂×C₄	1	12	yes	complemented
30	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
32	C₂×C₄²×C₁₃	—	2	3	no	—
33	C₂×C₄²×C₁₃	—	2	3	no	—
34	C₂×C₄²×C₁₃	D₄	1	3	yes	—
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
37	C₂×#6106	C₂×C₄	1	12	yes	—
38	C₂×#6103	—	2	3	no	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	3	yes	—
40	C₂×C₄×Dic₁₃	—	2	3	no	—
41	C₂²×#1055	C₂³	1	3	yes	—
42	C₂²×#1055	C₂³	1	1	yes	characteristic
43	C₄×D₅₂	—	2	6	no	—
44	C₄×D₅₂	C₂×C₄	1	12	yes	—
45	C₄²×D₁₃	—	4	4	no	—
46	C₁₃×#95	—	2	3	no	—
47	#88075	C₂×C₄	1	12	yes	complemented
48	#88071	—	2	3	no	—
49	#88049	C₂×C₄	1	12	yes	—
50	#88048	C₂×C₄	1	12	yes	complemented
51	C₄×#1055	—	2	6	no	—
52	C₄×#1055	C₂×C₄	1	12	yes	complemented
53	C₄²×SD₁₆	—	13	1	no	maximal, Sylow
54	C₂³×D₁₃	—	2	1	no	—
55	C₂×Q₈×C₁₃	—	2	3	no	—
56	C₂×Q₈×C₁₃	C₂²×C₄	1	6	yes	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	—	2	3	no	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₁₃	—	2	3	no	—
63	C₂×D₅₂	—	2	3	no	—
64	C₂×D₅₂	C₂²×C₄	1	6	yes	—
65	C₂×C₄×D₁₃	—	2	12	no	—
66	C₂×C₄×D₁₃	—	4	6	no	—
67	C₂×C₄×D₁₃	—	4	6	no	—
68	C₁₃×C₄⋊C₄	—	2	12	no	—
69	C₁₃×C₄⋊C₄	—	2	3	no	—
70	C₁₃×C₄⋊C₄	C₂²×C₄	1	6	yes	—
71	C₄²×C₁₃	—	2	12	no	—
72	C₄²×C₁₃	—	2	6	no	—
73	C₄²×C₁₃	—	4	6	no	—
74	C₄²×C₁₃	SD₁₆	1	4	yes	complemented
75	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
76	C₄²×C₁₃	#41	1	6	yes	—
77	#6106	C₄²	1	16	yes	direct product factor
78	#6103	—	2	12	no	—
79	C₄⋊Dic₁₃	—	2	3	no	—
80	C₄⋊Dic₁₃	C₂²×C₄	1	6	yes	—
81	C₄×Dic₁₃	—	4	6	no	—
82	C₂×#1055	—	2	6	no	—
83	C₂×#1055	C₂²×C₄	1	6	yes	—
84	C₂×#1055	C₂²×C₄	1	6	yes	—
85	C₂×C₄×SD₁₆	—	13	3	no	—
86	C₄²×Q₈	D₁₃	1	1	yes	characteristic, complemented
87	C₄²×D₄	—	13	1	no	—
88	C₄×#106	—	13	3	no	—
89	C₄×#103	—	13	3	no	—
90	C₄×#102	—	13	3	no	—
91	C₄²×C₈	—	13	1	no	—
92	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₂²×D₁₃	—	2	6	no	—
94	C₂²×D₁₃	—	4	1	no	—
95	C₂²×D₁₃	—	4	3	no	—
96	Q₈×C₁₃	—	2	6	no	—
97	Q₈×C₁₃	C₂×C₄²	1	4	yes	complemented
98	C₂×C₄×C₁₃	—	2	6	no	—
99	C₂×C₄×C₁₃	—	2	6	no	—
100	C₂×C₄×C₁₃	—	2	6	no	—
101	C₂×C₄×C₁₃	—	4	6	no	—
102	C₂×C₄×C₁₃	C₄×D₄	1	6	yes	—
103	C₂×C₄×C₁₃	C₄×D₄	1	6	yes	—
104	C₂×C₄×C₁₃	C₂×SD₁₆	1	6	yes	—
105	C₂×C₄×C₁₃	#135	1	6	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
107	C₂×C₄×C₁₃	C₂×#41	1	3	yes	—
108	C₂×Dic₁₃	—	2	6	no	—
109	C₂×Dic₁₃	—	4	3	no	—
110	C₂×Dic₁₃	—	4	3	no	—
111	D₅₂	—	2	6	no	—
112	D₅₂	C₂×C₄²	1	4	yes	—
113	C₄×D₁₃	—	4	24	no	—
114	#1055	—	2	6	no	—
115	#1055	C₂×C₄²	1	4	yes	complemented
116	C₂²×SD₁₆	—	13	1	no	—
117	C₂×C₄×Q₈	C₂×D₁₃	1	3	yes	—
118	C₂×C₄×D₄	—	13	3	no	—
119	C₂²×C₄²	—	26	1	no	—
120	C₄×SD₁₆	—	13	24	no	—
121	C₂×#106	—	13	3	no	—
122	C₂×#103	—	13	3	no	—
123	C₂×#102	—	13	3	no	—
124	C₂×C₄×C₈	—	13	3	no	—
125	C₂×C₄×C₈	—	13	3	no	—
126	C₄×C₄⋊C₄	—	2	3	no	—
127	C₄×C₄⋊C₄	—	13	3	no	—
128	C₄×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
129	C₄×C₂²⋊C₄	—	26	3	no	—
130	C₄³	—	2	1	no	—
131	C₄³	C₂×D₁₃	1	1	yes	characteristic
132	#340	—	13	6	no	—
133	C₂²×C₁₃	C₂×C₄×D₄	1	3	yes	—
134	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#135	1	3	yes	—
136	C₂×D₁₃	—	2	4	no	—
137	C₂×D₁₃	—	4	6	no	—
138	C₂×D₁₃	—	4	6	no	—
139	C₄×C₁₃	—	2	6	no	—
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	—	4	6	no	—
142	C₄×C₁₃	C₄×SD₁₆	1	12	yes	complemented
143	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
144	C₄×C₁₃	C₄×#41	1	3	yes	—
145	Dic₁₃	—	4	6	no	—
146	C₂²×Q₈	C₂²×D₁₃	1	1	yes	characteristic
147	C₂²×D₄	—	13	1	no	—
148	C₂³×C₄	—	26	3	no	—
149	C₂×SD₁₆	—	13	12	no	—
150	C₂²×C₈	—	13	1	no	—
151	C₂²×C₈	—	13	3	no	—
152	C₄×Q₈	—	2	6	no	—
153	C₄×Q₈	C₄×D₁₃	1	12	yes	complemented
154	C₄×D₄	—	13	12	no	—
155	C₄×D₄	—	26	6	no	—
156	C₂×C₄⋊C₄	—	2	3	no	—
157	C₂×C₄⋊C₄	—	13	3	no	—
158	C₂×C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
159	C₂×C₂²⋊C₄	—	26	3	no	—
160	C₂×C₄²	—	2	3	no	—
161	C₂×C₄²	—	2	3	no	—
162	C₂×C₄²	—	26	3	no	—
163	C₂×C₄²	—	52	4	no	—
164	C₂×C₄²	C₂²⋊D₁₃	1	3	yes	—
165	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
167	#106	—	13	12	no	—
168	#103	—	13	12	no	—
169	#102	—	13	12	no	—
170	C₄×C₈	—	13	12	no	—
171	C₄×C₈	—	26	6	no	—
172	#95	—	2	3	no	—
173	#95	—	26	3	no	—
174	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
175	C₂×C₁₃	C₂×C₄×SD₁₆	1	3	yes	—
176	C₂×C₁₃	C₄×#135	1	3	yes	—
177	D₁₃	—	4	4	no	—
178	C₂⁴	—	26	1	no	—
179	C₂×Q₈	—	2	3	no	—
180	C₂×Q₈	C₂×C₄×D₁₃	1	6	yes	—
181	C₂×D₄	—	13	6	no	—
182	C₂×D₄	—	26	3	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	3	no	—
185	C₂²×C₄	—	26	3	no	—
186	C₂²×C₄	—	26	12	no	—
187	C₂²×C₄	—	52	6	no	—
188	C₂²×C₄	—	52	6	no	—
189	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
190	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
191	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
192	SD₁₆	—	13	16	no	—
193	C₂×C₈	—	13	6	no	—
194	C₂×C₈	—	13	6	no	—
195	C₂×C₈	—	26	6	no	—
196	C₄⋊C₄	—	2	12	no	—
197	C₄⋊C₄	—	2	3	no	—
198	C₄⋊C₄	—	13	6	no	—
199	C₄⋊C₄	—	26	3	no	—
200	C₄⋊C₄	C₂×C₄×D₁₃	1	6	yes	—
201	C₂²⋊C₄	—	26	12	no	—
202	C₄²	—	2	12	no	—
203	C₄²	—	2	6	no	—
204	C₄²	—	4	6	no	—
205	C₄²	—	52	6	no	—
206	C₄²	#6106	1	4	yes	direct product factor
207	C₄²	C₂×C₄×D₁₃	1	6	yes	—
208	C₄²	#6127	1	6	yes	—
209	C₁₃	C₄²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	26	6	no	—
211	C₂³	—	52	1	no	—
212	C₂³	—	52	3	no	—
213	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	C₄²×D₁₃	1	4	yes	complemented
216	D₄	—	13	4	no	—
217	D₄	—	26	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	—	26	6	no	—
223	C₂×C₄	—	52	3	no	—
224	C₂×C₄	—	52	3	no	—
225	C₂×C₄	—	52	24	no	—
226	C₂×C₄	C₄×C₂²⋊D₁₃	1	6	yes	—
227	C₂×C₄	C₄×C₂²⋊D₁₃	1	6	yes	—
228	C₂×C₄	C₂×#6106	1	6	yes	—
229	C₂×C₄	#88204	1	6	yes	—
230	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
231	C₂×C₄	C₂×#6127	1	3	yes	—
232	C₈	—	13	4	no	—
233	C₈	—	26	6	no	—
234	C₂²	—	26	4	no	—
235	C₂²	—	52	6	no	—
236	C₂²	—	52	6	no	—
237	C₂²	C₂×C₄×C₂²⋊D₁₃	1	3	yes	—
238	C₂²	C₂²×#6106	1	1	yes	characteristic
239	C₂²	C₂×#88204	1	3	yes	—
240	C₄	—	2	6	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	6	no	—
243	C₄	—	52	6	no	—
244	C₄	C₄×#6106	1	12	yes	direct product factor
245	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
246	C₄	C₄×#6127	1	3	yes	—
247	C₂	—	52	4	no	—
248	C₂	C₄²×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂	C₂×C₄×#6106	1	3	yes	—
250	C₂	C₄×#88204	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄²×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×D₅₂	C₂³
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×Q₈	D₁₃
hypocenter	C₁₃	C₄²×SD₁₆
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
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×#88048	C₂	semidirect
4. C₄×#88075	C₂	semidirect
6. C₄²×#1055	C₂	semidirect
7. C₄²×Q₈×C₁₃	C₂	semidirect
17. C₄×#6106	C₄	direct (N × H)
29. C₄×Q₈×C₁₃	C₂×C₄	semidirect
47. #88075	C₂×C₄	semidirect
50. #88048	C₂×C₄	semidirect
52. C₄×#1055	C₂×C₄	semidirect
74. C₄²×C₁₃	SD₁₆	semidirect
77. #6106	C₄²	direct (N × H)
86. C₄²×Q₈	D₁₃	semidirect
97. Q₈×C₁₃	C₂×C₄²	semidirect
115. #1055	C₂×C₄²	semidirect
142. C₄×C₁₃	C₄×SD₁₆	semidirect
153. C₄×Q₈	C₄×D₁₃	semidirect
206. C₄²	#6106	direct (N × H)
209. C₁₃	C₄²×SD₁₆	semidirect
215. Q₈	C₄²×D₁₃	semidirect
244. C₄	C₄×#6106	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×#6106	2
C₄²×D₅₂	2
C₄×#88048	2
C₄×#88075	2
C₄×#88049	2
C₄²×#1055	2
C₄²×Q₈×C₁₃	2
C₄²×SD₁₆	13

Subgroups — C42×#6106 #15861391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#6106 #15861391