Subgroups — C₂²×#11718516 #16732507

Order 3328.

2398 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88142	C₂	1	2	yes	maximal
2	C₂×#11718516	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88057	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88048	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₆	—	2	2	no	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂²×#6101	—	2	2	no	—
12	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂³×#1055	C₂²	1	1	yes	characteristic
14	C₂×#88142	—	2	12	no	—
15	C₂×#88142	C₂²	1	24	yes	—
16	C₂×C₁₃×#106	C₂²	1	12	yes	—
17	C₂×#88057	C₂²	1	12	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	12	yes	—
19	C₂×#88048	C₂²	1	24	yes	—
20	#11718516	C₂²	1	64	yes	direct product factor
21	C₂³×C₄×C₁₃	—	2	2	no	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₃	—	2	2	no	—
24	C₂³×Dic₁₃	D₄	1	2	yes	—
25	C₂²×Dic₂₆	—	2	24	no	—
26	C₂²×Dic₂₆	—	4	16	no	—
27	C₂²×C₈×C₁₃	Q₈	1	8	yes	complemented
28	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
29	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
31	C₂×C₄⋊Dic₁₃	—	2	6	no	—
32	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
33	C₂×#6101	—	2	24	no	—
34	C₂×C₄×Dic₁₃	D₄	1	6	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
36	C₂²×#1055	Q₈	1	8	yes	complemented
37	C₂²×#1055	C₂³	1	6	yes	—
38	#88142	—	2	48	no	—
39	#88142	C₂³	1	32	yes	—
40	C₁₃×#106	C₂³	1	16	yes	—
41	#88057	C₂³	1	16	yes	—
42	C₈×Dic₁₃	C₂³	1	16	yes	—
43	#88048	C₂³	1	32	yes	—
44	C₂²×#498	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	12	no	—
47	C₂²×C₄×C₁₃	—	4	8	no	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
52	C₂²×Dic₁₃	—	2	12	no	—
53	C₂²×Dic₁₃	—	4	8	no	—
54	C₂²×Dic₁₃	SD₁₆	1	16	yes	—
55	C₂²×Dic₁₃	C₂×D₄	1	12	yes	—
56	C₂×Dic₂₆	—	2	32	no	—
57	C₂×Dic₂₆	—	4	96	no	—
58	C₂×C₈×C₁₃	C₂×Q₈	1	24	yes	complemented
59	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
60	C₁₃×C₄⋊C₄	—	2	12	no	—
61	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
62	C₄⋊Dic₁₃	—	2	12	no	—
63	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
64	#6101	—	2	32	no	—
65	C₄×Dic₁₃	C₂×D₄	1	12	yes	—
66	C₄×Dic₁₃	C₂⁴	1	4	yes	—
67	C₂×#1055	C₂×Q₈	1	24	yes	complemented
68	C₂×#1055	C₂⁴	1	4	yes	—
69	C₂²×C₄⋊Q₈	—	13	2	no	—
70	C₂×#498	—	13	24	no	—
71	C₂²×#106	—	13	1	no	—
72	C₂²×#106	—	13	2	no	—
73	C₂²×#106	D₁₃	1	1	yes	characteristic
74	C₂²×C₄×C₈	—	13	1	no	—
75	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
76	C₂³×C₁₃	C₂×SD₁₆	1	8	yes	—
77	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
78	C₂×C₄×C₁₃	—	2	8	no	—
79	C₂×C₄×C₁₃	—	4	24	no	—
80	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
81	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
82	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
83	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
84	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
85	C₂×Dic₁₃	—	2	8	no	—
86	C₂×Dic₁₃	—	4	24	no	—
87	C₂×Dic₁₃	C₂×SD₁₆	1	48	yes	—
88	C₂×Dic₁₃	C₂²×D₄	1	8	yes	—
89	Dic₂₆	—	4	64	no	—
90	C₈×C₁₃	C₂²×Q₈	1	8	yes	complemented
91	#1055	C₂²×Q₈	1	8	yes	complemented
92	C₂³×Q₈	—	26	2	no	—
93	C₂³×C₈	—	13	1	no	—
94	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
95	C₂×C₄⋊Q₈	—	13	24	no	—
96	C₂×C₄⋊Q₈	—	26	12	no	—
97	C₂²×C₄⋊C₄	—	13	2	no	—
98	C₂²×C₄⋊C₄	—	26	2	no	—
99	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
100	C₂²×C₄²	—	13	1	no	—
101	#498	—	13	64	no	—
102	C₂×#106	—	13	12	no	—
103	C₂×#106	—	13	24	no	—
104	C₂×#106	C₂×D₁₃	1	12	yes	—
105	C₂×C₄×C₈	—	13	12	no	—
106	C₂²×C₁₃	#498	1	16	yes	complemented
107	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
108	C₂²×C₁₃	C₂²×SD₁₆	1	12	yes	—
109	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
110	C₄×C₁₃	—	4	8	no	—
111	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
112	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
113	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
114	Dic₁₃	—	4	8	no	—
115	Dic₁₃	C₂²×SD₁₆	1	16	yes	—
116	C₂²×Q₈	—	26	24	no	—
117	C₂²×Q₈	—	52	16	no	—
118	C₂³×C₄	—	2	2	no	—
119	C₂³×C₄	—	13	2	no	—
120	C₂³×C₄	—	26	2	no	—
121	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
122	C₂²×C₈	—	13	8	no	—
123	C₂²×C₈	—	13	6	no	—
124	C₂²×C₈	Dic₂₆	1	8	yes	complemented
125	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
126	C₄⋊Q₈	—	13	32	no	—
127	C₄⋊Q₈	—	26	48	no	—
128	C₂×C₄⋊C₄	—	2	6	no	—
129	C₂×C₄⋊C₄	—	13	12	no	—
130	C₂×C₄⋊C₄	—	26	24	no	—
131	C₂×C₄⋊C₄	—	26	6	no	—
132	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
133	C₂×C₄²	—	13	6	no	—
134	C₂×C₄²	—	13	6	no	—
135	#106	—	13	16	no	—
136	#106	—	13	32	no	—
137	#106	C₂²×D₁₃	1	16	yes	—
138	C₄×C₈	—	13	16	no	—
139	C₂×C₁₃	C₂×#498	1	12	yes	complemented
140	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₁₃	C₂³×SD₁₆	1	2	yes	—
142	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
143	C₂×Q₈	—	26	32	no	—
144	C₂×Q₈	—	52	96	no	—
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	—	4	8	no	—
147	C₂²×C₄	—	13	16	no	—
148	C₂²×C₄	—	13	12	no	—
149	C₂²×C₄	—	26	12	no	—
150	C₂²×C₄	—	52	8	no	—
151	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
152	C₂²×C₄	D₄×D₁₃	1	3	yes	—
153	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
154	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
155	C₂×C₈	—	13	24	no	—
156	C₂×C₈	—	13	4	no	—
157	C₂×C₈	C₂×Dic₂₆	1	24	yes	complemented
158	C₂×C₈	C₂³×D₁₃	1	4	yes	—
159	C₄⋊C₄	—	2	12	no	—
160	C₄⋊C₄	—	13	8	no	—
161	C₄⋊C₄	—	26	32	no	—
162	C₄⋊C₄	—	26	12	no	—
163	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
164	C₄²	—	13	12	no	—
165	C₄²	—	13	4	no	—
166	C₁₃	C₂²×#498	1	1	yes	characteristic, complemented, Sylow
167	C₂³	#88142	1	4	yes	—
168	C₂³	SD₁₆×D₁₃	1	8	yes	—
169	C₂³	C₂×D₄×D₁₃	1	3	yes	—
170	Q₈	—	52	64	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	4	24	no	—
173	C₂×C₄	—	13	48	no	—
174	C₂×C₄	—	13	8	no	—
175	C₂×C₄	—	26	8	no	—
176	C₂×C₄	—	52	24	no	—
177	C₂×C₄	#88142	1	12	yes	—
178	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
179	C₂×C₄	C₂×D₄×D₁₃	1	3	yes	—
180	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
181	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
182	C₈	—	13	8	no	—
183	C₈	C₂²×Dic₂₆	1	8	yes	complemented
184	C₂²	#11718516	1	16	yes	direct product factor
185	C₂²	C₂×#88142	1	6	yes	—
186	C₂²	C₂×SD₁₆×D₁₃	1	12	yes	—
187	C₂²	C₂²×D₄×D₁₃	1	1	yes	characteristic
188	C₄	—	4	8	no	—
189	C₄	—	13	16	no	—
190	C₄	—	52	8	no	—
191	C₄	C₂×#88142	1	6	yes	—
192	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
193	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
194	C₂	C₂×#11718516	1	12	yes	direct product factor
195	C₂	C₂²×#88142	1	1	yes	characteristic
196	C₂	C₂²×SD₁₆×D₁₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#106	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×#106	D₁₃
hypocenter	C₁₃	C₂²×#498
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
2. C₂×#11718516	C₂	direct (N × H)
20. #11718516	C₂²	direct (N × H)
27. C₂²×C₈×C₁₃	Q₈	semidirect
36. C₂²×#1055	Q₈	semidirect
58. C₂×C₈×C₁₃	C₂×Q₈	semidirect
67. C₂×#1055	C₂×Q₈	semidirect
90. C₈×C₁₃	C₂²×Q₈	semidirect
91. #1055	C₂²×Q₈	semidirect
106. C₂²×C₁₃	#498	semidirect
124. C₂²×C₈	Dic₂₆	semidirect
139. C₂×C₁₃	C₂×#498	semidirect
157. C₂×C₈	C₂×Dic₂₆	semidirect
166. C₁₃	C₂²×#498	semidirect
183. C₈	C₂²×Dic₂₆	semidirect
184. C₂²	#11718516	direct (N × H)
194. C₂	C₂×#11718516	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88142	2
C₂×#11718516	2
C₂²×#88057	2
C₂²×#88048	2
C₂²×C₈×Dic₁₃	2
C₂²×C₁₃×#106	2
C₂²×#498	13

Subgroups — C22×#11718516 #16732507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718516 #16732507