Subgroups — C₂²×#11718780 #16726536

Order 3328.

3006 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#1055	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88180	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718780	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#88054	C₂	1	4	yes	maximal, complemented
5	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×M₁₆	—	2	8	no	—
8	C₂²×C₁₃×M₁₆	C₂²	1	12	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂²×C₂²⋊Dic₁₃	C₄	1	2	yes	—
11	C₂×#88180	C₄	1	12	yes	—
12	C₂×#88180	C₂²	1	12	yes	—
13	C₂²×C₄⋊Dic₁₃	C₄	1	2	yes	—
14	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
15	C₂³×#1055	—	2	8	no	—
16	C₂³×#1055	C₂²	1	2	yes	—
17	C₂³×#1055	C₂²	1	12	yes	—
18	#11718780	C₂²	1	64	yes	direct product factor
19	C₂×#88054	C₂²	1	48	yes	complemented
20	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
23	C₂³×C₄×C₁₃	D₄	1	8	yes	—
24	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₃	C₂×C₄	1	4	yes	—
28	C₂×C₁₃×M₁₆	—	2	48	no	—
29	C₂×C₁₃×M₁₆	C₂³	1	16	yes	—
30	C₂²×C₈×C₁₃	—	2	8	no	—
31	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
32	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	24	yes	—
33	#88180	C₂×C₄	1	48	yes	—
34	#88180	C₂³	1	16	yes	—
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	24	yes	—
36	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
37	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
38	C₂²×#1055	—	2	8	no	—
39	C₂²×#1055	—	2	48	no	—
40	C₂²×#1055	C₂³	1	12	yes	—
41	C₂²×#1055	C₂³	1	16	yes	—
42	#88054	C₂³	1	64	yes	complemented
43	C₂²×#424	—	13	1	no	maximal, Sylow
44	C₂⁴×C₁₃	—	2	8	no	—
45	C₂⁴×C₁₃	C₄⋊C₄	1	8	yes	—
46	C₂⁴×C₁₃	C₂²×C₄	1	6	yes	—
47	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	48	yes	—
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
55	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
56	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
59	C₂²×Dic₁₃	—	2	16	no	—
60	C₂²×Dic₁₃	C₂²×C₄	1	24	yes	—
61	C₁₃×M₁₆	—	2	32	no	—
62	C₂×C₈×C₁₃	—	2	24	no	—
63	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
64	C₂²⋊Dic₁₃	C₂²×C₄	1	32	yes	—
65	C₄⋊Dic₁₃	C₂²×C₄	1	32	yes	—
66	C₄×Dic₁₃	C₂²×C₄	1	24	yes	—
67	C₄×Dic₁₃	C₂⁴	1	8	yes	—
68	C₂×#1055	—	2	24	no	—
69	C₂×#1055	—	2	32	no	—
70	C₂×#1055	C₂⁴	1	8	yes	—
71	C₂³×M₁₆	D₁₃	1	1	yes	characteristic
72	C₂⁴×C₈	—	13	1	no	—
73	C₂²×#117	—	13	1	no	—
74	C₂×#424	—	13	24	no	—
75	C₂²×C₄⋊C₈	—	13	4	no	—
76	C₂³×C₁₃	—	2	24	no	—
77	C₂³×C₁₃	—	2	32	no	—
78	C₂³×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
79	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
80	C₂³×C₁₃	#131	1	8	yes	—
81	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
82	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
84	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
85	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
86	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
87	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
89	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
90	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
91	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
92	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₁₃	—	2	48	no	—
94	C₂×Dic₁₃	C₂³×C₄	1	16	yes	—
95	C₈×C₁₃	—	2	8	no	—
96	#1055	—	2	8	no	—
97	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
98	C₂²×M₁₆	—	2	8	no	—
99	C₂²×M₁₆	C₂×D₁₃	1	12	yes	—
100	C₂³×C₈	—	13	12	no	—
101	C₂³×C₈	—	13	2	no	—
102	C₂³×C₈	—	26	8	no	—
103	C₂³×C₈	C₂×D₁₃	1	2	yes	—
104	C₂×#117	—	13	12	no	—
105	C₂×#117	—	13	12	no	—
106	C₂²×C₄⋊C₄	—	13	2	no	—
107	C₂²×C₂²⋊C₄	—	13	2	no	—
108	C₂²×C₄²	—	13	2	no	—
109	#424	—	13	64	no	—
110	C₂×C₄⋊C₈	—	13	48	no	—
111	C₂²×C₁₃	—	2	48	no	—
112	C₂²×C₁₃	—	2	8	no	—
113	C₂²×C₁₃	#424	1	16	yes	complemented
114	C₂²×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
115	C₂²×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
116	C₂²×C₁₃	C₂×#131	1	12	yes	—
117	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
118	C₄×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
119	C₄×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
120	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
121	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
122	Dic₁₃	—	2	16	no	—
123	C₂⁵	C₄×D₁₃	1	1	yes	characteristic
124	C₂³×C₄	—	13	4	no	—
125	C₂³×C₄	Dic₂₆	1	8	yes	—
126	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
127	C₂³×C₄	C₄×D₁₃	1	6	yes	—
128	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
129	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
130	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
131	C₂×M₁₆	—	2	48	no	—
132	C₂×M₁₆	C₂²×D₁₃	1	16	yes	—
133	C₂²×C₈	—	2	8	no	—
134	C₂²×C₈	—	13	12	no	—
135	C₂²×C₈	—	13	16	no	—
136	C₂²×C₈	—	26	8	no	—
137	C₂²×C₈	—	26	48	no	—
138	C₂²×C₈	C₂²×D₁₃	1	12	yes	—
139	#117	—	13	48	no	—
140	#117	—	13	16	no	—
141	C₂×C₄⋊C₄	—	13	24	no	—
142	C₂×C₂²⋊C₄	—	13	24	no	—
143	C₂×C₄²	—	13	12	no	—
144	C₂×C₄²	—	13	12	no	—
145	C₄⋊C₈	—	13	64	no	—
146	C₂×C₁₃	—	2	8	no	—
147	C₂×C₁₃	C₂×#424	1	12	yes	complemented
148	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₁₃	C₂²×#131	1	2	yes	—
150	C₂⁴	—	2	8	no	—
151	C₂⁴	#6101	1	8	yes	—
152	C₂⁴	C₂×C₄×D₁₃	1	6	yes	—
153	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
154	C₂²×C₄	—	13	24	no	—
155	C₂²×C₄	—	26	16	no	—
156	C₂²×C₄	#6101	1	48	yes	—
157	C₂²×C₄	#6101	1	8	yes	—
158	C₂²×C₄	C₂×Dic₂₆	1	24	yes	—
159	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
160	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
161	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
162	C₂²×C₄	C₂×C₄×D₁₃	1	12	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₁₃	1	24	yes	—
165	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
166	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
167	M₁₆	—	2	32	no	—
168	C₂×C₈	—	2	24	no	—
169	C₂×C₈	—	13	8	no	—
170	C₂×C₈	—	26	24	no	—
171	C₂×C₈	—	26	32	no	—
172	C₂×C₈	C₂³×D₁₃	1	8	yes	—
173	C₄⋊C₄	—	13	32	no	—
174	C₂²⋊C₄	—	13	32	no	—
175	C₄²	—	13	24	no	—
176	C₄²	—	13	8	no	—
177	C₁₃	C₂²×#424	1	1	yes	characteristic, complemented, Sylow
178	C₂³	—	2	24	no	—
179	C₂³	—	2	32	no	—
180	C₂³	#88161	1	8	yes	—
181	C₂³	C₂×#6101	1	24	yes	—
182	C₂³	C₂×#6101	1	4	yes	—
183	C₂³	C₂²×C₄×D₁₃	1	4	yes	—
184	C₂³	C₂²×C₄×D₁₃	1	3	yes	—
185	C₂×C₄	—	13	16	no	—
186	C₂×C₄	—	26	48	no	—
187	C₂×C₄	C₂×#6101	1	48	yes	—
188	C₂×C₄	C₂×#6101	1	24	yes	—
189	C₂×C₄	C₂×#6101	1	12	yes	—
190	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
191	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
192	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
193	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
195	C₂×C₄	C₂²×C₂²⋊D₁₃	1	8	yes	—
196	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
197	C₈	—	2	8	no	—
198	C₈	—	26	8	no	—
199	C₂²	—	2	48	no	—
200	C₂²	—	2	8	no	—
201	C₂²	#11718780	1	16	yes	direct product factor
202	C₂²	C₂×#88161	1	12	yes	—
203	C₂²	C₂²×#6101	1	8	yes	—
204	C₂²	C₂²×#6101	1	6	yes	—
205	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
206	C₄	—	26	16	no	—
207	C₄	C₂²×#6101	1	6	yes	—
208	C₄	C₂²×#6101	1	8	yes	—
209	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
210	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×#11718780	1	12	yes	direct product factor
213	C₂	C₂³×#6101	1	1	yes	characteristic
214	C₂	C₂²×#88161	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
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	C₂²×#424
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#11718780	C₂	direct (N × H)
4. C₂²×#88054	C₂	semidirect
18. #11718780	C₂²	direct (N × H)
19. C₂×#88054	C₂²	semidirect
42. #88054	C₂³	semidirect
113. C₂²×C₁₃	#424	semidirect
147. C₂×C₁₃	C₂×#424	semidirect
177. C₁₃	C₂²×#424	semidirect
201. C₂²	#11718780	direct (N × H)
212. C₂	C₂×#11718780	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×#1055	2
C₂²×#88180	2
C₂×#11718780	2
C₂²×#88054	2
C₂³×C₁₃×M₁₆	2
C₂²×#424	13

Subgroups — C22×#11718780 #16726536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718780 #16726536