Subgroups — C₂²×#11718568 #16769384

Order 3328.

2118 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88155	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718568	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88038	C₂	1	2	yes	maximal, complemented
4	C₂³×#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₁₆×D₁₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#6127	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₄	1	1	yes	characteristic
10	C₂×#88155	C₄	1	12	yes	—
11	C₂×#88155	C₂²	1	12	yes	—
12	C₂²×#6094	C₄	1	2	yes	—
13	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
14	C₂×C₁₃×M₃₂	C₂²	1	12	yes	complemented
15	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
16	C₂²×#6090	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	1	yes	characteristic
18	C₂²×#6090	C₂²	1	12	yes	complemented
19	#11718568	C₂²	1	64	yes	direct product factor
20	C₂×#88038	C₂²	1	24	yes	complemented
21	C₂×C₁₆×D₁₃	C₂²	1	24	yes	complemented
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₂²⋊D₁₃	C₈	1	2	yes	—
24	C₂×#6127	C₈	1	12	yes	—
25	C₂×#6127	C₂×C₄	1	12	yes	—
26	C₂²×D₅₂	C₈	1	1	yes	characteristic
27	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
28	C₂²×Dic₂₆	C₈	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×#6099	C₂×C₄	1	12	yes	—
34	C₂²×#1055	C₂×C₄	1	2	yes	—
35	#88155	C₂×C₄	1	48	yes	—
36	#88155	C₂³	1	16	yes	—
37	C₂×#6094	C₂×C₄	1	24	yes	—
38	C₂×C₈×D₁₃	C₂×C₄	1	12	yes	—
39	C₂×C₈×D₁₃	C₂³	1	12	yes	—
40	C₁₃×M₃₂	C₂³	1	16	yes	complemented
41	C₂×C₁₃×C₁₆	C₂³	1	12	yes	—
42	C₂×#6090	C₂³	1	6	yes	—
43	C₂×#6090	C₂³	1	6	yes	—
44	C₂×#6090	C₂³	1	16	yes	complemented
45	#88038	C₂³	1	32	yes	complemented
46	C₁₆×D₁₃	C₂³	1	32	yes	complemented
47	C₂²×#504	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂³×D₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
54	C₂×C₂²⋊D₁₃	C₂×C₈	1	24	yes	—
55	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
56	#6127	C₂×C₈	1	48	yes	—
57	#6127	C₂²×C₄	1	16	yes	—
58	C₂×D₅₂	C₂×C₈	1	12	yes	—
59	C₂×C₄×D₁₃	C₂×C₈	1	12	yes	—
60	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
61	C₂×Dic₂₆	C₂×C₈	1	12	yes	—
62	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
63	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
65	C₂×C₈×C₁₃	C₂⁴	1	3	yes	—
66	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
67	#6099	C₂²×C₄	1	16	yes	—
68	C₂×#1055	C₂²×C₄	1	12	yes	—
69	#6094	C₂²×C₄	1	32	yes	—
70	C₈×D₁₃	C₂²×C₄	1	24	yes	—
71	C₈×D₁₃	C₂⁴	1	8	yes	—
72	C₁₃×C₁₆	C₂⁴	1	8	yes	—
73	#6090	C₂⁴	1	4	yes	—
74	#6090	C₂⁴	1	4	yes	—
75	C₂²×#131	—	13	1	no	—
76	C₂×#504	—	13	24	no	—
77	C₂²×M₃₂	—	13	2	no	—
78	C₂²×M₃₂	D₁₃	1	1	yes	characteristic, complemented
79	C₂³×C₁₆	—	13	2	no	—
80	C₂³×C₁₆	—	13	1	no	—
81	C₂³×C₁₃	—	2	4	no	—
82	C₂³×C₁₃	C₂²×C₈	1	6	yes	—
83	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
84	C₂²×D₁₃	—	2	8	no	—
85	C₂²×D₁₃	C₂²×C₈	1	12	yes	—
86	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
87	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
88	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
89	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
91	C₂²⋊D₁₃	C₂²×C₈	1	32	yes	—
92	C₂×Dic₁₃	C₂²×C₈	1	12	yes	—
93	D₅₂	C₂²×C₈	1	16	yes	—
94	C₄×D₁₃	C₂²×C₈	1	24	yes	—
95	C₄×D₁₃	C₂³×C₄	1	8	yes	—
96	Dic₂₆	C₂²×C₈	1	16	yes	—
97	C₈×C₁₃	C₂³×C₄	1	3	yes	—
98	C₈×C₁₃	C₂³×C₄	1	4	yes	—
99	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
100	#1055	C₂³×C₄	1	8	yes	—
101	C₂²×#41	—	13	1	no	—
102	C₂×#131	—	13	12	no	—
103	C₂×#131	—	13	12	no	—
104	C₂²×M₁₆	—	13	1	no	—
105	C₂²×M₁₆	—	13	2	no	—
106	C₂³×C₈	—	13	2	no	—
107	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
108	#504	—	13	64	no	—
109	C₂×M₃₂	—	13	24	no	—
110	C₂×M₃₂	C₂×D₁₃	1	12	yes	complemented
111	C₂²×C₁₆	—	13	12	no	—
112	C₂²×C₁₆	—	13	1	no	—
113	C₂²×C₁₆	—	13	1	no	—
114	C₂²×C₁₆	—	13	24	no	—
115	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
116	C₂²×C₁₃	—	2	12	no	—
117	C₂²×C₁₃	#504	1	4	yes	complemented
118	C₂²×C₁₃	C₂³×C₈	1	4	yes	—
119	C₂²×C₁₃	C₂³×C₈	1	3	yes	—
120	C₂×D₁₃	—	2	24	no	—
121	C₂×D₁₃	C₂³×C₈	1	8	yes	—
122	C₄×C₁₃	C₂³×C₈	1	4	yes	—
123	C₄×C₁₃	C₂³×C₈	1	3	yes	—
124	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
125	Dic₁₃	C₂³×C₈	1	8	yes	—
126	C₂×#41	—	13	12	no	—
127	C₂×#41	—	13	12	no	—
128	C₂²×Q₈	—	13	1	no	—
129	C₂²×D₄	—	13	2	no	—
130	C₂²×D₄	—	13	1	no	—
131	C₂³×C₄	—	13	2	no	—
132	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
133	#131	—	13	48	no	—
134	#131	—	13	16	no	—
135	C₂×M₁₆	—	13	12	no	—
136	C₂×M₁₆	—	13	24	no	—
137	C₂²×C₈	—	13	2	no	—
138	C₂²×C₈	—	13	12	no	—
139	C₂²×C₈	—	13	12	no	—
140	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
141	C₂²×C₈	C₄×D₁₃	1	6	yes	—
142	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
143	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
144	M₃₂	—	13	32	no	—
145	M₃₂	C₂²×D₁₃	1	16	yes	complemented
146	C₂×C₁₆	—	13	6	no	—
147	C₂×C₁₆	—	13	6	no	—
148	C₂×C₁₆	—	13	16	no	—
149	C₂×C₁₆	—	13	32	no	—
150	C₂×C₁₆	C₂²×D₁₃	1	12	yes	—
151	C₂×C₁₃	—	2	4	no	—
152	C₂×C₁₃	C₂×#504	1	6	yes	complemented
153	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
154	D₁₃	—	2	8	no	—
155	C₂⁴	—	13	2	no	—
156	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
157	#41	—	13	48	no	—
158	#41	—	13	16	no	—
159	C₂×Q₈	—	13	12	no	—
160	C₂×D₄	—	13	24	no	—
161	C₂×D₄	—	13	12	no	—
162	C₂²×C₄	—	13	2	no	—
163	C₂²×C₄	—	13	12	no	—
164	C₂²×C₄	—	13	12	no	—
165	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
166	C₂²×C₄	C₈×D₁₃	1	6	yes	—
167	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
169	M₁₆	—	13	16	no	—
170	M₁₆	—	13	32	no	—
171	C₂×C₈	—	13	12	no	—
172	C₂×C₈	—	13	24	no	—
173	C₂×C₈	—	13	8	no	—
174	C₂×C₈	C₂×C₄×D₁₃	1	3	yes	—
175	C₂×C₈	C₂×C₄×D₁₃	1	6	yes	—
176	C₂×C₈	C₂×C₄×D₁₃	1	12	yes	—
177	C₂×C₈	C₂³×D₁₃	1	3	yes	—
178	C₂×C₈	C₂³×D₁₃	1	4	yes	—
179	C₁₆	—	13	4	no	—
180	C₁₆	—	13	4	no	—
181	C₁₆	C₂³×D₁₃	1	8	yes	—
182	C₁₃	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	2	4	no	—
184	C₂³	—	13	12	no	—
185	C₂³	—	26	8	no	—
186	C₂³	C₂×C₈×D₁₃	1	6	yes	—
187	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
188	Q₈	—	13	16	no	—
189	D₄	—	13	32	no	—
190	D₄	—	13	16	no	—
191	C₂×C₄	—	13	12	no	—
192	C₂×C₄	—	13	24	no	—
193	C₂×C₄	—	13	8	no	—
194	C₂×C₄	C₂×C₈×D₁₃	1	3	yes	—
195	C₂×C₄	C₂×C₈×D₁₃	1	6	yes	—
196	C₂×C₄	C₂×C₈×D₁₃	1	12	yes	—
197	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
198	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
199	C₈	—	13	8	no	—
200	C₈	C₂²×C₄×D₁₃	1	3	yes	—
201	C₈	C₂²×C₄×D₁₃	1	4	yes	—
202	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
203	C₂²	—	2	12	no	—
204	C₂²	—	13	8	no	—
205	C₂²	—	26	24	no	—
206	C₂²	#11718568	1	4	yes	direct product factor
207	C₂²	C₂²×C₈×D₁₃	1	4	yes	—
208	C₂²	C₂²×C₈×D₁₃	1	3	yes	—
209	C₄	—	13	8	no	—
210	C₄	C₂²×C₈×D₁₃	1	3	yes	—
211	C₄	C₂²×C₈×D₁₃	1	4	yes	—
212	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
213	C₂	—	2	4	no	—
214	C₂	—	26	8	no	—
215	C₂	C₂×#11718568	1	6	yes	direct product factor
216	C₂	C₂³×C₈×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₈	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×M₃₂	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×#6127	C₄
cosocle	C₈×C₁₃	C₂⁵
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	C₂²×#504
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#11718568	C₂	direct (N × H)
3. C₂²×#88038	C₂	semidirect
4. C₂³×#6090	C₂	semidirect
5. C₂²×C₁₆×D₁₃	C₂	semidirect
6. C₂²×C₁₃×M₃₂	C₂	semidirect
14. C₂×C₁₃×M₃₂	C₂²	semidirect
18. C₂²×#6090	C₂²	semidirect
19. #11718568	C₂²	direct (N × H)
20. C₂×#88038	C₂²	semidirect
21. C₂×C₁₆×D₁₃	C₂²	semidirect
40. C₁₃×M₃₂	C₂³	semidirect
44. C₂×#6090	C₂³	semidirect
45. #88038	C₂³	semidirect
46. C₁₆×D₁₃	C₂³	semidirect
78. C₂²×M₃₂	D₁₃	semidirect
110. C₂×M₃₂	C₂×D₁₃	semidirect
117. C₂²×C₁₃	#504	semidirect
145. M₃₂	C₂²×D₁₃	semidirect
152. C₂×C₁₃	C₂×#504	semidirect
182. C₁₃	C₂²×#504	semidirect
206. C₂²	#11718568	direct (N × H)
215. C₂	C₂×#11718568	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×#88155	2
C₂×#11718568	2
C₂²×#88038	2
C₂³×#6090	2
C₂²×C₁₆×D₁₃	2
C₂²×C₁₃×M₃₂	2
C₂²×#504	13

Subgroups — C22×#11718568 #16769384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718568 #16769384