Subgroups — C₂³×#88064 #16726403

Order 3328.

4990 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	C₂⁴×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×#6099	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88064	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂⁴×Dic₁₃	C₄	1	2	yes	—
7	C₂³×Dic₂₆	—	2	1	no	—
8	C₂³×Dic₂₆	—	2	7	no	—
9	C₂³×Dic₂₆	—	2	1	no	—
10	C₂³×Dic₂₆	—	2	7	no	—
11	C₂³×Dic₂₆	C₄	1	14	yes	—
12	C₂³×Dic₂₆	C₂²	1	14	yes	—
13	C₂²×C₁₃×M₁₆	—	2	7	no	—
14	C₂²×C₁₃×M₁₆	C₂²	1	14	yes	—
15	C₂³×C₈×C₁₃	—	2	1	no	—
16	C₂²×#6099	—	2	7	no	—
17	C₂²×#6099	C₂²	1	14	yes	—
18	C₂³×#1055	—	2	1	no	—
19	C₂×#88064	C₂²	1	112	yes	direct product factor
20	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂×C₄	1	7	yes	—
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	7	yes	—
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	D₄	1	7	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
27	C₂³×Dic₁₃	—	2	14	no	—
28	C₂³×Dic₁₃	—	2	2	no	—
29	C₂³×Dic₁₃	C₂×C₄	1	28	yes	—
30	C₂²×Dic₂₆	—	2	84	no	—
31	C₂²×Dic₂₆	—	2	14	no	—
32	C₂²×Dic₂₆	—	2	14	no	—
33	C₂²×Dic₂₆	—	2	14	no	—
34	C₂²×Dic₂₆	—	2	42	no	—
35	C₂²×Dic₂₆	—	2	14	no	—
36	C₂²×Dic₂₆	—	2	42	no	—
37	C₂²×Dic₂₆	C₂×C₄	1	84	yes	—
38	C₂²×Dic₂₆	C₂³	1	28	yes	—
39	C₂×C₁₃×M₁₆	—	2	42	no	—
40	C₂×C₁₃×M₁₆	C₂³	1	28	yes	—
41	C₂²×C₈×C₁₃	—	2	14	no	—
42	C₂×#6099	—	2	42	no	—
43	C₂×#6099	C₂³	1	28	yes	—
44	C₂²×#1055	—	2	14	no	—
45	#88064	C₂³	1	64	yes	direct product factor
46	C₂³×#101	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	—	2	8	no	—
48	C₂⁴×C₁₃	C₂²⋊C₄	1	7	yes	—
49	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₁₃	C₂²×C₄	1	7	yes	—
51	C₂²×C₄×C₁₃	C₂²⋊C₄	1	42	yes	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	7	yes	—
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	7	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	21	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
60	C₂²×Dic₁₃	—	2	84	no	—
61	C₂²×Dic₁₃	—	2	28	no	—
62	C₂²×Dic₁₃	C₂²×C₄	1	56	yes	—
63	C₂×Dic₂₆	—	2	168	no	—
64	C₂×Dic₂₆	—	2	84	no	—
65	C₂×Dic₂₆	—	2	84	no	—
66	C₂×Dic₂₆	—	2	28	no	—
67	C₂×Dic₂₆	—	2	28	no	—
68	C₂×Dic₂₆	—	2	28	no	—
69	C₂×Dic₂₆	—	2	28	no	—
70	C₂×Dic₂₆	C₂²×C₄	1	56	yes	—
71	C₂×Dic₂₆	C₂⁴	1	8	yes	—
72	C₁₃×M₁₆	—	2	28	no	—
73	C₁₃×M₁₆	C₂⁴	1	8	yes	—
74	C₂×C₈×C₁₃	—	2	28	no	—
75	#6099	—	2	28	no	—
76	#6099	C₂⁴	1	8	yes	—
77	C₂×#1055	—	2	28	no	—
78	C₂⁴×Q₈	—	13	1	no	—
79	C₂³×M₁₆	—	13	1	no	—
80	C₂³×M₁₆	D₁₃	1	1	yes	characteristic
81	C₂²×#101	—	13	28	no	—
82	C₂³×C₁₃	—	2	28	no	—
83	C₂³×C₁₃	—	2	28	no	—
84	C₂³×C₁₃	#101	1	8	yes	complemented
85	C₂³×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
86	C₂³×C₁₃	C₂×C₂²⋊C₄	1	7	yes	—
87	C₂³×C₁₃	C₂³×C₄	1	7	yes	—
88	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	42	yes	—
89	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
90	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
91	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
92	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
93	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
94	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
95	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
96	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
97	C₂×Dic₁₃	—	2	56	no	—
98	C₂×Dic₁₃	—	2	56	no	—
99	C₂×Dic₁₃	C₂³×C₄	1	16	yes	—
100	Dic₂₆	—	2	56	no	—
101	Dic₂₆	—	2	56	no	—
102	Dic₂₆	—	2	8	no	—
103	Dic₂₆	—	2	8	no	—
104	C₈×C₁₃	—	2	8	no	—
105	#1055	—	2	8	no	—
106	C₂³×Q₈	—	13	14	no	—
107	C₂³×Q₈	—	13	14	no	—
108	C₂³×Q₈	—	26	7	no	—
109	C₂³×Q₈	—	26	1	no	—
110	C₂³×Q₈	—	26	7	no	—
111	C₂³×Q₈	—	26	1	no	—
112	C₂⁴×C₄	—	13	2	no	—
113	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
114	C₂²×M₁₆	—	2	7	no	—
115	C₂²×M₁₆	—	13	14	no	—
116	C₂²×M₁₆	—	26	7	no	—
117	C₂²×M₁₆	C₂×D₁₃	1	14	yes	—
118	C₂³×C₈	—	2	1	no	—
119	C₂³×C₈	—	26	1	no	—
120	C₂×#101	—	13	112	no	—
121	C₂²×C₁₃	—	2	42	no	—
122	C₂²×C₁₃	—	2	14	no	—
123	C₂²×C₁₃	C₂×#101	1	28	yes	complemented
124	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
125	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
126	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
127	C₄×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
128	C₄×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
129	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
130	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
131	Dic₁₃	—	2	16	no	—
132	C₂⁵	C₄×D₁₃	1	1	yes	characteristic
133	C₂²×Q₈	—	13	84	no	—
134	C₂²×Q₈	—	13	28	no	—
135	C₂²×Q₈	—	26	84	no	—
136	C₂²×Q₈	—	26	14	no	—
137	C₂²×Q₈	—	26	14	no	—
138	C₂²×Q₈	—	26	14	no	—
139	C₂²×Q₈	—	26	42	no	—
140	C₂²×Q₈	—	26	14	no	—
141	C₂²×Q₈	—	26	42	no	—
142	C₂³×C₄	—	13	28	no	—
143	C₂³×C₄	—	26	14	no	—
144	C₂³×C₄	—	26	2	no	—
145	C₂³×C₄	C₄×D₁₃	1	7	yes	—
146	C₂³×C₄	D₅₂	1	7	yes	—
147	C₂³×C₄	D₅₂	1	1	yes	characteristic
148	C₂³×C₄	C₂²⋊D₁₃	1	7	yes	—
149	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
151	C₂×M₁₆	—	2	42	no	—
152	C₂×M₁₆	—	13	28	no	—
153	C₂×M₁₆	—	26	42	no	—
154	C₂×M₁₆	C₂²×D₁₃	1	28	yes	—
155	C₂²×C₈	—	2	14	no	—
156	C₂²×C₈	—	26	14	no	—
157	#101	—	13	64	no	—
158	C₂×C₁₃	—	2	7	no	—
159	C₂×C₁₃	—	2	1	no	—
160	C₂×C₁₃	C₂²×#101	1	14	yes	complemented
161	C₂×C₁₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
162	C₂⁴	—	2	8	no	—
163	C₂⁴	#6103	1	7	yes	—
164	C₂⁴	#6103	1	1	yes	characteristic
165	C₂⁴	C₂×C₄×D₁₃	1	7	yes	—
166	C₂×Q₈	—	13	56	no	—
167	C₂×Q₈	—	13	8	no	—
168	C₂×Q₈	—	26	168	no	—
169	C₂×Q₈	—	26	84	no	—
170	C₂×Q₈	—	26	84	no	—
171	C₂×Q₈	—	26	28	no	—
172	C₂×Q₈	—	26	28	no	—
173	C₂×Q₈	—	26	28	no	—
174	C₂×Q₈	—	26	28	no	—
175	C₂²×C₄	—	13	56	no	—
176	C₂²×C₄	—	26	84	no	—
177	C₂²×C₄	—	26	28	no	—
178	C₂²×C₄	#6103	1	42	yes	—
179	C₂²×C₄	#6103	1	7	yes	—
180	C₂²×C₄	#6103	1	7	yes	—
181	C₂²×C₄	C₂×C₄×D₁₃	1	21	yes	—
182	C₂²×C₄	C₂×D₅₂	1	7	yes	—
183	C₂²×C₄	C₂×D₅₂	1	21	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₁₃	1	7	yes	—
185	C₂²×C₄	C₂×C₂²⋊D₁₃	1	21	yes	—
186	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
187	M₁₆	—	2	28	no	—
188	M₁₆	—	13	8	no	—
189	M₁₆	—	26	28	no	—
190	M₁₆	C₂³×D₁₃	1	8	yes	—
191	C₂×C₈	—	2	28	no	—
192	C₂×C₈	—	26	28	no	—
193	C₁₃	C₂³×#101	1	1	yes	characteristic, complemented, Sylow
194	C₂³	—	2	28	no	—
195	C₂³	—	2	28	no	—
196	C₂³	#88064	1	8	yes	direct product factor
197	C₂³	C₂×#6103	1	21	yes	—
198	C₂³	C₂×#6103	1	7	yes	—
199	C₂³	C₂²×C₄×D₁₃	1	7	yes	—
200	Q₈	—	26	56	no	—
201	Q₈	—	26	56	no	—
202	Q₈	—	26	8	no	—
203	Q₈	—	26	8	no	—
204	C₂×C₄	—	13	16	no	—
205	C₂×C₄	—	26	56	no	—
206	C₂×C₄	—	26	56	no	—
207	C₂×C₄	C₂×#6103	1	42	yes	—
208	C₂×C₄	C₂×#6103	1	21	yes	—
209	C₂×C₄	C₂×#6103	1	21	yes	—
210	C₂×C₄	C₂²×C₄×D₁₃	1	7	yes	—
211	C₂×C₄	C₂²×D₅₂	1	7	yes	—
212	C₂×C₄	C₂²×D₅₂	1	7	yes	—
213	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
215	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
216	C₈	—	2	8	no	—
217	C₈	—	26	8	no	—
218	C₂²	—	2	42	no	—
219	C₂²	—	2	14	no	—
220	C₂²	C₂×#88064	1	28	yes	direct product factor
221	C₂²	C₂²×#6103	1	7	yes	—
222	C₂²	C₂²×#6103	1	7	yes	—
223	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
224	C₄	—	26	16	no	—
225	C₄	C₂²×#6103	1	7	yes	—
226	C₄	C₂²×#6103	1	7	yes	—
227	C₄	C₂³×D₅₂	1	1	yes	characteristic
228	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂	—	2	7	no	—
230	C₂	—	2	1	no	—
231	C₂	C₂²×#88064	1	14	yes	direct product factor
232	C₂	C₂³×#6103	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#6103
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×M₁₆	C₂
socle	C₂⁴×C₁₃	C₂²⋊C₄
prime radical	C₂⁵×C₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×M₁₆	D₁₃
hypocenter	C₁₃	C₂³×#101
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂²×#88064	C₂	direct (N × H)
19. C₂×#88064	C₂²	direct (N × H)
45. #88064	C₂³	direct (N × H)
84. C₂³×C₁₃	#101	semidirect
123. C₂²×C₁₃	C₂×#101	semidirect
160. C₂×C₁₃	C₂²×#101	semidirect
193. C₁₃	C₂³×#101	semidirect
196. C₂³	#88064	direct (N × H)
220. C₂²	C₂×#88064	direct (N × H)
231. C₂	C₂²×#88064	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×Dic₂₆	2
C₂³×#6099	2
C₂²×#88064	2
C₂³×C₁₃×M₁₆	2
C₂³×#101	13

Subgroups — C23×#88064 #16726403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88064 #16726403