Subgroups — C₂²×#11718232 #16806746

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	C₂×#11718232	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#88043	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×#1055	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂×#88106	C₄	1	12	yes	complemented
8	C₂×#88106	C₂²	1	12	yes	—
9	C₂²×#6092	C₄	1	4	yes	complemented
10	C₂×#88043	C₄	1	6	yes	—
11	C₂×#88043	C₂²	1	6	yes	—
12	#11718232	C₂²	1	16	yes	direct product factor
13	C₂³×C₄×C₁₃	—	2	1	no	—
14	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	—	2	3	no	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
17	C₂²×#1055	—	2	3	no	—
18	C₂²×#1055	—	2	1	no	—
19	C₂²×#1055	C₂×C₄	1	6	yes	—
20	C₂²×#1055	C₂×C₄	1	3	yes	—
21	C₂²×#1055	C₂×C₄	1	2	yes	—
22	C₂²×#1055	C₂×C₄	1	6	yes	—
23	C₂²×#1055	C₂³	1	3	yes	—
24	#88106	C₂×C₄	1	24	yes	complemented
25	#88106	C₂³	1	8	yes	—
26	C₂×#6092	C₂×C₄	1	24	yes	complemented
27	#88043	C₂×C₄	1	12	yes	—
28	#88043	C₂³	1	4	yes	—
29	C₂²×#347	—	13	1	no	maximal, Sylow
30	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	—	2	6	no	—
32	C₂²×C₄×C₁₃	—	4	4	no	—
33	C₂²×C₄×C₁₃	C₄²	1	3	yes	—
34	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	M₁₆	1	3	yes	—
36	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	M₁₆	1	3	yes	—
38	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
39	C₄²×C₁₃	—	2	6	no	—
40	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
41	C₂×#1055	—	2	6	no	—
42	C₂×#1055	—	2	6	no	—
43	C₂×#1055	C₄²	1	6	yes	—
44	C₂×#1055	C₄²	1	6	yes	—
45	C₂×#1055	C₂²×C₄	1	4	yes	—
46	C₂×#1055	C₂²×C₄	1	3	yes	—
47	C₂×#1055	C₂²×C₄	1	6	yes	—
48	C₂×#1055	C₂²×C₄	1	6	yes	—
49	C₂×#1055	C₂⁴	1	1	yes	characteristic
50	#6092	C₂²×C₄	1	16	yes	complemented
51	C₂²×M₃₂	—	13	2	no	—
52	C₂²×#97	—	13	1	no	—
53	C₂×#347	—	13	12	no	—
54	C₂³×C₁₃	—	2	4	no	—
55	C₂³×C₁₃	#97	1	3	yes	—
56	C₂³×C₁₃	#97	1	1	yes	characteristic
57	C₂³×C₁₃	C₂×M₁₆	1	3	yes	—
58	C₂×C₄×C₁₃	—	2	4	no	—
59	C₂×C₄×C₁₃	—	4	12	no	—
60	C₂×C₄×C₁₃	#97	1	6	yes	—
61	C₂×C₄×C₁₃	#97	1	3	yes	—
62	C₂×C₄×C₁₃	#97	1	3	yes	—
63	C₂×C₄×C₁₃	C₂×C₄²	1	3	yes	—
64	C₂×C₄×C₁₃	C₂×C₄²	1	3	yes	—
65	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
66	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
67	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
68	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
69	#1055	—	2	4	no	—
70	#1055	C₂×C₄²	1	6	yes	—
71	#1055	C₂³×C₄	1	2	yes	—
72	C₂³×C₈	—	13	1	no	—
73	C₂³×C₈	—	13	1	no	—
74	C₂²×C₄²	#261	1	1	yes	characteristic
75	C₂×M₃₂	—	13	12	no	—
76	C₂×M₃₂	—	13	12	no	—
77	C₂²×C₁₆	—	13	4	no	—
78	C₂×#97	—	13	6	no	—
79	C₂×#97	—	13	6	no	—
80	#347	—	13	16	no	—
81	C₂²×C₁₃	—	2	6	no	—
82	C₂²×C₁₃	—	2	6	no	—
83	C₂²×C₁₃	#347	1	4	yes	complemented
84	C₂²×C₁₃	C₂×#97	1	3	yes	—
85	C₂²×C₁₃	C₂×#97	1	3	yes	—
86	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
87	C₄×C₁₃	—	4	4	no	—
88	C₄×C₁₃	C₂×#97	1	3	yes	—
89	C₄×C₁₃	C₂×#97	1	3	yes	—
90	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
91	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
92	C₂³×C₄	—	2	1	no	—
93	C₂³×C₄	C₂×#261	1	1	yes	characteristic
94	C₂²×C₈	—	13	6	no	—
95	C₂²×C₈	—	13	2	no	—
96	C₂²×C₈	—	13	3	no	—
97	C₂²×C₈	—	13	3	no	—
98	C₂²×C₈	—	13	6	no	—
99	C₂²×C₈	—	26	1	no	—
100	C₂²×C₈	—	26	3	no	—
101	C₂×C₄²	—	2	3	no	—
102	C₂×C₄²	C₂×#261	1	6	yes	—
103	M₃₂	—	13	24	no	—
104	M₃₂	—	13	8	no	—
105	C₂×C₁₆	—	13	24	no	—
106	#97	—	13	4	no	—
107	#97	—	13	12	no	—
108	C₂×C₁₃	—	2	1	no	—
109	C₂×C₁₃	—	2	3	no	—
110	C₂×C₁₃	C₂×#347	1	6	yes	complemented
111	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
112	C₂⁴	#6122	1	1	yes	characteristic
113	C₂²×C₄	—	2	6	no	—
114	C₂²×C₄	—	4	4	no	—
115	C₂²×C₄	#6118	1	1	yes	characteristic
116	C₂²×C₄	#6118	1	3	yes	—
117	C₂²×C₄	C₄×#261	1	3	yes	—
118	C₂²×C₄	C₄×#261	1	1	yes	characteristic
119	C₂²×C₄	#6122	1	3	yes	—
120	C₂²×C₄	C₂²×#261	1	3	yes	—
121	C₂×C₈	—	13	6	no	—
122	C₂×C₈	—	13	6	no	—
123	C₂×C₈	—	13	6	no	—
124	C₂×C₈	—	13	6	no	—
125	C₂×C₈	—	13	1	no	—
126	C₂×C₈	—	13	3	no	—
127	C₂×C₈	—	13	4	no	—
128	C₂×C₈	—	26	6	no	—
129	C₂×C₈	—	26	6	no	—
130	C₄²	—	2	6	no	—
131	C₄²	C₂²×#261	1	4	yes	—
132	C₁₆	—	13	16	no	—
133	C₁₃	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
134	C₂³	—	2	4	no	—
135	C₂³	#88110	1	3	yes	—
136	C₂³	#88110	1	1	yes	characteristic
137	C₂³	C₂×#6122	1	3	yes	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	12	no	—
140	C₂×C₄	#88110	1	6	yes	—
141	C₂×C₄	#88110	1	3	yes	—
142	C₂×C₄	#88110	1	3	yes	—
143	C₂×C₄	C₂×#6118	1	3	yes	—
144	C₂×C₄	C₂×#6118	1	3	yes	—
145	C₂×C₄	C₂×C₄×#261	1	3	yes	—
146	C₂×C₄	C₂×C₄×#261	1	3	yes	—
147	C₂×C₄	C₂×#6122	1	3	yes	—
148	C₂×C₄	C₂³×#261	1	1	yes	characteristic
149	C₈	—	13	2	no	—
150	C₈	—	13	6	no	—
151	C₈	—	26	4	no	—
152	C₂²	—	2	6	no	—
153	C₂²	—	2	6	no	—
154	C₂²	#11718232	1	4	yes	direct product factor
155	C₂²	C₂×#88110	1	3	yes	—
156	C₂²	C₂×#88110	1	3	yes	—
157	C₂²	C₂²×#6122	1	1	yes	characteristic
158	C₄	—	4	4	no	—
159	C₄	C₂×#88110	1	3	yes	—
160	C₄	C₂×#88110	1	3	yes	—
161	C₄	C₂²×#6118	1	1	yes	characteristic
162	C₄	C₂²×C₄×#261	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	3	no	—
165	C₂	C₂×#11718232	1	6	yes	direct product factor
166	C₂	C₂²×#88110	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	#97
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₂²×#347
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#11718232	C₂	direct (N × H)
7. C₂×#88106	C₄	semidirect
9. C₂²×#6092	C₄	semidirect
12. #11718232	C₂²	direct (N × H)
24. #88106	C₂×C₄	semidirect
26. C₂×#6092	C₂×C₄	semidirect
50. #6092	C₂²×C₄	semidirect
83. C₂²×C₁₃	#347	semidirect
110. C₂×C₁₃	C₂×#347	semidirect
133. C₁₃	C₂²×#347	semidirect
154. C₂²	#11718232	direct (N × H)
165. C₂	C₂×#11718232	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×#88106	2
C₂×#11718232	2
C₂²×#88043	2
C₂²×#347	13

Subgroups — C22×#11718232 #16806746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718232 #16806746