Subgroups — C₂²×#11718055 #16015915

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	2	yes	maximal
2	C₂×#11718055	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₄	1	6	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	6	yes	—
9	C₂×#88052	—	2	6	no	—
10	C₂×#88052	C₂²	1	12	yes	—
11	C₂²×#6090	—	2	2	no	—
12	#11718055	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	C₂²×C₈×C₁₃	C₈	1	6	yes	—
16	C₂²×C₈×C₁₃	C₈	1	2	yes	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
19	C₂²×C₈×C₁₃	D₄	1	3	yes	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	Q₈	1	3	yes	—
23	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
24	C₂×C₄²×C₁₃	—	2	3	no	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
26	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
27	C₄×C₈×C₁₃	C₂³	1	4	yes	—
28	#88052	—	2	12	no	—
29	#88052	C₂³	1	8	yes	—
30	C₂×#6090	—	2	12	no	—
31	C₂²×#364	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	—	2	8	no	—
34	C₂²×C₄×C₁₃	—	2	6	no	—
35	C₂²×C₄×C₁₃	C₄⋊C₄	1	3	yes	—
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
38	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	M₁₆	1	3	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
41	C₂×C₈×C₁₃	C₄⋊C₄	1	6	yes	—
42	C₂×C₈×C₁₃	C₄⋊C₄	1	3	yes	—
43	C₂×C₈×C₁₃	C₄⋊C₄	1	3	yes	—
44	C₂×C₈×C₁₃	C₂×C₈	1	12	yes	—
45	C₂×C₈×C₁₃	C₂×C₈	1	12	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
48	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
49	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
50	C₂×C₈×C₁₃	C₂×Q₈	1	3	yes	—
51	C₂×C₈×C₁₃	C₂×Q₈	1	3	yes	—
52	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₄²×C₁₃	—	2	6	no	—
54	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
55	#6090	—	2	8	no	—
56	C₂²×M₃₂	—	13	2	no	—
57	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
58	C₂×#364	—	13	12	no	—
59	C₂³×C₁₃	—	2	4	no	—
60	C₂³×C₁₃	C₄⋊C₈	1	3	yes	—
61	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂³×C₁₃	C₂×M₁₆	1	3	yes	—
63	C₂×C₄×C₁₃	—	2	24	no	—
64	C₂×C₄×C₁₃	—	2	4	no	—
65	C₂×C₄×C₁₃	C₄⋊C₈	1	6	yes	—
66	C₂×C₄×C₁₃	C₄⋊C₈	1	3	yes	—
67	C₂×C₄×C₁₃	C₄⋊C₈	1	3	yes	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
72	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
73	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
74	C₈×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₈×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₈×C₁₃	C₂²×C₈	1	8	yes	—
77	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
78	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
80	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
81	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
82	C₂×M₃₂	—	13	12	no	—
83	C₂×M₃₂	—	26	6	no	—
84	C₂²×C₁₆	—	26	2	no	—
85	C₂×C₄×C₈	Dic₁₃	1	6	yes	—
86	C₂×C₄×C₈	C₂×D₁₃	1	6	yes	—
87	#364	—	13	16	no	—
88	C₂²×C₁₃	—	2	6	no	—
89	C₂²×C₁₃	—	2	6	no	—
90	C₂²×C₁₃	#364	1	4	yes	complemented
91	C₂²×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
92	C₂²×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
93	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
94	C₄×C₁₃	—	2	8	no	—
95	C₄×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
96	C₄×C₁₃	C₂×C₄⋊C₈	1	3	yes	—
97	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
99	C₂³×C₄	—	2	1	no	—
100	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
101	C₂²×C₈	#1055	1	6	yes	—
102	C₂²×C₈	#1055	1	2	yes	—
103	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
104	C₂²×C₈	Dic₂₆	1	3	yes	—
105	C₂²×C₈	D₅₂	1	3	yes	—
106	C₂²×C₈	D₅₂	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₁₃	1	6	yes	—
108	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
109	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
110	C₂×C₄²	—	2	3	no	—
111	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
112	M₃₂	—	13	8	no	—
113	M₃₂	—	26	12	no	—
114	C₂×C₁₆	—	26	12	no	—
115	C₄×C₈	C₂×Dic₁₃	1	12	yes	—
116	C₄×C₈	C₂²×D₁₃	1	4	yes	—
117	C₂×C₁₃	—	2	1	no	—
118	C₂×C₁₃	—	2	3	no	—
119	C₂×C₁₃	C₂×#364	1	6	yes	complemented
120	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
121	C₂⁴	#6099	1	1	yes	characteristic
122	C₂²×C₄	—	2	8	no	—
123	C₂²×C₄	—	2	6	no	—
124	C₂²×C₄	C₂×#1055	1	3	yes	—
125	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
126	C₂²×C₄	#6099	1	3	yes	—
127	C₂²×C₄	C₄⋊Dic₁₃	1	3	yes	—
128	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
129	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
130	C₂×C₈	C₂×#1055	1	12	yes	—
131	C₂×C₈	C₂×#1055	1	12	yes	—
132	C₂×C₈	C₄⋊Dic₁₃	1	3	yes	—
133	C₂×C₈	C₄⋊Dic₁₃	1	6	yes	—
134	C₂×C₈	C₄⋊Dic₁₃	1	3	yes	—
135	C₂×C₈	C₂×Dic₂₆	1	3	yes	—
136	C₂×C₈	C₂×Dic₂₆	1	3	yes	—
137	C₂×C₈	C₂×D₅₂	1	3	yes	—
138	C₂×C₈	C₂×D₅₂	1	3	yes	—
139	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
140	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
141	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
142	C₄²	—	2	6	no	—
143	C₄²	C₂²×Dic₁₃	1	4	yes	—
144	C₁₆	—	26	8	no	—
145	C₁₃	C₂²×#364	1	1	yes	characteristic, complemented, Sylow
146	C₂³	—	2	4	no	—
147	C₂³	C₄⋊#1055	1	3	yes	—
148	C₂³	C₄⋊#1055	1	1	yes	characteristic
149	C₂³	C₂×#6099	1	3	yes	—
150	C₂×C₄	—	2	24	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	C₄⋊#1055	1	6	yes	—
153	C₂×C₄	C₄⋊#1055	1	3	yes	—
154	C₂×C₄	C₄⋊#1055	1	3	yes	—
155	C₂×C₄	C₂²×#1055	1	3	yes	—
156	C₂×C₄	C₂²×#1055	1	3	yes	—
157	C₂×C₄	C₂×#6099	1	3	yes	—
158	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
159	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
160	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
161	C₈	C₂²×#1055	1	8	yes	—
162	C₈	C₂×C₄⋊Dic₁₃	1	3	yes	—
163	C₈	C₂×C₄⋊Dic₁₃	1	3	yes	—
164	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
165	C₈	C₂²×D₅₂	1	1	yes	characteristic
166	C₂²	—	2	6	no	—
167	C₂²	—	2	6	no	—
168	C₂²	#11718055	1	4	yes	direct product factor
169	C₂²	C₂×C₄⋊#1055	1	3	yes	—
170	C₂²	C₂×C₄⋊#1055	1	3	yes	—
171	C₂²	C₂²×#6099	1	1	yes	characteristic
172	C₄	—	2	8	no	—
173	C₄	C₂×C₄⋊#1055	1	3	yes	—
174	C₄	C₂×C₄⋊#1055	1	3	yes	—
175	C₄	C₂³×#1055	1	1	yes	characteristic
176	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	3	no	—
179	C₂	C₂×#11718055	1	6	yes	direct product factor
180	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₅₂
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₄⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×#364
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718055	C₂	direct (N × H)
12. #11718055	C₂²	direct (N × H)
90. C₂²×C₁₃	#364	semidirect
119. C₂×C₁₃	C₂×#364	semidirect
145. C₁₃	C₂²×#364	semidirect
168. C₂²	#11718055	direct (N × H)
179. C₂	C₂×#11718055	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×#88052	2
C₂×#11718055	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#364	13

Subgroups — C22×#11718055 #16015915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718055 #16015915