Subgroups — C₂²×#11718725 #16732510

Order 3328.

1966 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88124	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718725	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88047	C₂	1	2	yes	maximal
4	C₂²×#88048	C₂	1	2	yes	maximal
5	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#125	—	2	6	no	—
8	C₂×C₁₃×#125	C₂²	1	12	yes	—
9	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₂²×#6101	—	2	2	no	—
14	C₂³×#1055	C₂²	1	1	yes	characteristic
15	C₂³×#1055	C₂²	1	1	yes	characteristic
16	C₂×#88124	—	2	6	no	—
17	C₂×#88124	C₂²	1	12	yes	—
18	#11718725	C₂²	1	64	yes	direct product factor
19	C₂×#88048	C₂²	1	24	yes	—
20	C₂×#88047	C₂²	1	24	yes	—
21	C₂×C₄×#1055	C₂²	1	12	yes	—
22	C₂³×C₄×C₁₃	—	2	2	no	—
23	C₂³×C₄×C₁₃	D₄	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	—	2	2	no	—
26	C₁₃×#125	—	2	24	no	—
27	C₁₃×#125	C₂³	1	16	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	24	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
31	C₂×C₄²×C₁₃	D₄	1	6	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
33	C₂×C₄⋊Dic₁₃	—	2	6	no	—
34	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
35	C₂×#6101	—	2	24	no	—
36	C₂²×#1055	Q₈	1	8	yes	—
37	C₂²×#1055	Q₈	1	8	yes	—
38	C₂²×#1055	C₂³	1	6	yes	—
39	C₂²×#1055	C₂³	1	6	yes	—
40	#88124	—	2	24	no	—
41	#88124	C₂³	1	16	yes	—
42	#88048	C₂³	1	32	yes	—
43	#88047	C₂³	1	32	yes	—
44	C₄×#1055	C₂³	1	16	yes	—
45	C₂²×#499	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	8	no	—
48	C₂²×C₄×C₁₃	—	2	12	no	—
49	C₂²×C₄×C₁₃	—	4	8	no	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
52	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
55	C₂²×Dic₁₃	—	2	12	no	—
56	C₂²×Dic₁₃	—	4	8	no	—
57	C₁₃×C₄⋊C₄	—	2	32	no	—
58	C₁₃×C₄⋊C₄	—	2	12	no	—
59	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
60	C₄²×C₁₃	C₂×D₄	1	12	yes	—
61	C₄²×C₁₃	C₂⁴	1	4	yes	—
62	C₄⋊Dic₁₃	—	2	12	no	—
63	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
64	#6101	—	2	32	no	—
65	C₂×#1055	C₂×Q₈	1	24	yes	—
66	C₂×#1055	C₂×Q₈	1	24	yes	—
67	C₂×#1055	C₂⁴	1	4	yes	—
68	C₂×#1055	C₂⁴	1	4	yes	—
69	C₂²×#125	—	13	1	no	—
70	C₂²×#125	D₁₃	1	1	yes	characteristic
71	C₂×#499	—	13	24	no	—
72	C₂²×#107	—	13	2	no	—
73	C₂²×#106	—	13	2	no	—
74	C₂²×C₄×C₈	—	13	1	no	—
75	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
76	C₂³×C₁₃	#135	1	8	yes	—
77	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
78	C₂×C₄×C₁₃	—	2	24	no	—
79	C₂×C₄×C₁₃	—	2	8	no	—
80	C₂×C₄×C₁₃	—	4	24	no	—
81	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
82	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
83	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
84	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
85	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
86	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₁₃	—	2	8	no	—
88	C₂×Dic₁₃	—	4	24	no	—
89	#1055	C₂²×Q₈	1	8	yes	—
90	#1055	C₂²×Q₈	1	8	yes	—
91	C₂³×C₈	—	13	1	no	—
92	C₂³×C₈	—	13	1	no	—
93	C₂×#125	—	2	6	no	—
94	C₂×#125	—	13	12	no	—
95	C₂×#125	—	26	6	no	—
96	C₂×#125	C₂×D₁₃	1	12	yes	—
97	C₂²×C₄⋊C₄	—	2	2	no	—
98	C₂²×C₄⋊C₄	—	13	2	no	—
99	C₂²×C₄⋊C₄	—	26	2	no	—
100	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
101	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
102	#499	—	13	64	no	—
103	C₂×#107	—	13	24	no	—
104	C₂×#106	—	13	24	no	—
105	C₂×C₄×C₈	—	13	12	no	—
106	C₂²×C₁₃	#499	1	16	yes	complemented
107	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
108	C₂²×C₁₃	C₂×#135	1	12	yes	—
109	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
110	C₄×C₁₃	—	2	8	no	—
111	C₄×C₁₃	—	4	8	no	—
112	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
113	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
114	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
115	Dic₁₃	—	4	8	no	—
116	C₂³×C₄	—	2	2	no	—
117	C₂³×C₄	—	26	2	no	—
118	C₂³×C₄	C₂²⋊D₁₃	1	2	yes	—
119	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
120	C₂²×C₈	—	13	8	no	—
121	C₂²×C₈	—	13	8	no	—
122	C₂²×C₈	—	13	6	no	—
123	C₂²×C₈	—	13	6	no	—
124	#125	—	2	24	no	—
125	#125	—	13	16	no	—
126	#125	—	26	24	no	—
127	#125	C₂²×D₁₃	1	16	yes	—
128	C₂×C₄⋊C₄	—	2	24	no	—
129	C₂×C₄⋊C₄	—	2	6	no	—
130	C₂×C₄⋊C₄	—	13	12	no	—
131	C₂×C₄⋊C₄	—	26	6	no	—
132	C₂×C₄⋊C₄	—	26	24	no	—
133	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
134	C₂×C₄²	C₂²⋊D₁₃	1	6	yes	—
135	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
136	#107	—	13	32	no	—
137	#106	—	13	32	no	—
138	C₄×C₈	—	13	16	no	—
139	C₂×C₁₃	C₂×#499	1	12	yes	complemented
140	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₁₃	C₂²×#135	1	2	yes	—
142	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	—	2	12	no	—
145	C₂²×C₄	—	4	8	no	—
146	C₂²×C₄	—	26	12	no	—
147	C₂²×C₄	—	52	8	no	—
148	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
149	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₁₃	1	12	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
152	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
153	C₂×C₈	—	13	24	no	—
154	C₂×C₈	—	13	24	no	—
155	C₂×C₈	—	13	4	no	—
156	C₂×C₈	—	13	4	no	—
157	C₄⋊C₄	—	2	32	no	—
158	C₄⋊C₄	—	2	12	no	—
159	C₄⋊C₄	—	13	8	no	—
160	C₄⋊C₄	—	26	12	no	—
161	C₄⋊C₄	—	26	32	no	—
162	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
163	C₄²	C₂×C₂²⋊D₁₃	1	12	yes	—
164	C₄²	C₂³×D₁₃	1	4	yes	—
165	C₁₃	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
166	C₂³	#88198	1	4	yes	—
167	C₂³	#88204	1	8	yes	—
168	C₂³	C₂²×C₂²⋊D₁₃	1	3	yes	—
169	C₂×C₄	—	2	24	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	4	24	no	—
172	C₂×C₄	—	26	8	no	—
173	C₂×C₄	—	52	24	no	—
174	C₂×C₄	#88198	1	12	yes	—
175	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
176	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
177	C₂×C₄	C₂²×C₂²⋊D₁₃	1	8	yes	—
178	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
179	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
180	C₈	—	13	8	no	—
181	C₈	—	13	8	no	—
182	C₂²	#11718725	1	16	yes	direct product factor
183	C₂²	C₂×#88198	1	6	yes	—
184	C₂²	C₂×#88204	1	12	yes	—
185	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	—	4	8	no	—
188	C₄	—	52	8	no	—
189	C₄	C₂×#88198	1	6	yes	—
190	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
191	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
192	C₂	C₂×#11718725	1	12	yes	direct product factor
193	C₂	C₂²×#88198	1	1	yes	characteristic
194	C₂	C₂²×#88204	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#125	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×#125	D₁₃
hypocenter	C₁₃	C₂²×#499
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
2. C₂×#11718725	C₂	direct (N × H)
18. #11718725	C₂²	direct (N × H)
106. C₂²×C₁₃	#499	semidirect
139. C₂×C₁₃	C₂×#499	semidirect
165. C₁₃	C₂²×#499	semidirect
182. C₂²	#11718725	direct (N × H)
192. C₂	C₂×#11718725	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×#88124	2
C₂×#11718725	2
C₂²×#88047	2
C₂²×#88048	2
C₂²×C₄×#1055	2
C₂²×C₁₃×#125	2
C₂²×#499	13

Subgroups — C22×#11718725 #16732510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718725 #16732510