Subgroups — C₂×#13129865 #16788673

Order 3328.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6092	C₂	1	2	yes	maximal
2	#13129865	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6092	—	2	2	no	—
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×#1055	C₄	1	1	yes	characteristic
7	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
8	C₄⋊#6092	—	2	2	no	—
9	C₄⋊#6092	C₂²	1	4	yes	—
10	C₈×#1055	C₄	1	2	yes	—
11	C₈×#1055	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×#1055	—	2	1	no	—
16	C₂²×#1055	D₄	1	1	yes	characteristic
17	C₂²×#1055	Q₈	1	1	yes	characteristic
18	C₂×#6092	—	2	4	no	—
19	C₂×#6092	—	4	4	no	—
20	C₄×C₈×C₁₃	C₈	1	2	yes	—
21	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₄×#1055	—	2	1	no	—
23	C₄×#1055	C₂×C₄	1	2	yes	—
24	C₄×#1055	C₂×C₄	1	1	yes	characteristic
25	C₄×#1055	D₄	1	1	yes	characteristic
26	C₄×#1055	Q₈	1	1	yes	characteristic
27	C₄×#1055	C₂³	1	1	yes	characteristic
28	C₂×#1370	—	13	1	no	maximal, Sylow
29	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	C₁₆	1	4	yes	complemented
34	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
36	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
39	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×#1055	—	2	4	no	—
41	C₂×#1055	—	2	2	no	—
42	C₂×#1055	—	2	2	no	—
43	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#1055	D₈	1	2	yes	—
46	C₂×#1055	Q₁₆	1	2	yes	—
47	C₂×#1055	C₂×D₄	1	1	yes	characteristic
48	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
49	#6092	—	4	4	no	—
50	C₂×C₄⋊C₁₆	—	13	2	no	—
51	C₂×C₈²	—	13	1	no	—
52	#1370	—	13	4	no	—
53	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#107	1	2	yes	—
59	C₂×C₄×C₁₃	#108	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₁₃	—	2	2	no	—
66	C₈×C₁₃	C₂×C₁₆	1	4	yes	complemented
67	#1055	—	2	4	no	—
68	#1055	—	2	1	no	—
69	#1055	—	2	1	no	—
70	#1055	#107	1	2	yes	—
71	#1055	C₂×D₈	1	1	yes	characteristic
72	#1055	C₂×Q₁₆	1	1	yes	characteristic
73	C₂²×C₁₆	—	26	2	no	—
74	C₂×C₄×C₈	—	13	1	no	—
75	C₂×C₄×C₈	—	13	1	no	—
76	C₂×C₄×C₈	#261	1	1	yes	characteristic
77	C₄⋊C₁₆	—	13	4	no	—
78	C₄⋊C₁₆	—	26	2	no	—
79	C₈²	—	13	2	no	—
80	C₈²	—	13	2	no	—
81	C₂²×C₁₃	#335	1	2	yes	—
82	C₂²×C₁₃	C₄⋊C₁₆	1	2	yes	—
83	C₂²×C₁₃	#364	1	2	yes	—
84	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	—	2	1	no	—
87	C₄×C₁₃	#335	1	2	yes	—
88	C₄×C₁₃	C₄⋊C₁₆	1	2	yes	—
89	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
90	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
91	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
92	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
93	C₂²×C₈	—	13	1	no	—
94	C₂²×C₈	—	13	1	no	—
95	C₂²×C₈	—	26	1	no	—
96	C₂²×C₈	#1057	1	1	yes	characteristic
97	C₂²×C₈	#1057	1	1	yes	characteristic
98	C₂×C₄²	C₂×#261	1	1	yes	characteristic
99	C₂×C₁₆	—	26	4	no	—
100	C₂×C₁₆	—	52	4	no	—
101	C₄×C₈	—	13	1	no	—
102	C₄×C₈	—	13	1	no	—
103	C₄×C₈	—	13	1	no	—
104	C₄×C₈	—	13	1	no	—
105	C₄×C₈	—	13	2	no	—
106	C₄×C₈	—	26	1	no	—
107	C₄×C₈	#1057	1	2	yes	—
108	C₄×C₈	C₂×#261	1	2	yes	—
109	C₂×C₁₃	#1370	1	4	yes	complemented
110	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
111	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
112	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
113	C₂²×C₄	#6118	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
115	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	13	2	no	—
118	C₂×C₈	—	13	2	no	—
119	C₂×C₈	—	13	1	no	—
120	C₂×C₈	—	13	1	no	—
121	C₂×C₈	—	13	1	no	—
122	C₂×C₈	—	13	1	no	—
123	C₂×C₈	—	26	4	no	—
124	C₂×C₈	—	26	2	no	—
125	C₂×C₈	—	26	2	no	—
126	C₂×C₈	#6092	1	4	yes	complemented
127	C₂×C₈	C₂×#1057	1	2	yes	—
128	C₂×C₈	C₂×#1057	1	2	yes	—
129	C₄²	#6118	1	1	yes	characteristic
130	C₄²	C₄⋊#261	1	1	yes	characteristic
131	C₄²	C₂×#1057	1	1	yes	characteristic
132	C₄²	C₂²×#261	1	1	yes	characteristic
133	C₁₆	—	52	4	no	—
134	C₁₃	C₂×#1370	1	1	yes	characteristic, complemented, Sylow
135	C₂³	C₄⋊#1057	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#88102	1	2	yes	—
138	C₂×C₄	#88104	1	2	yes	—
139	C₂×C₄	C₂×#6092	1	2	yes	—
140	C₂×C₄	#88106	1	2	yes	—
141	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
142	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
143	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
144	C₂×C₄	C₂×#6118	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
146	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	13	1	no	—
149	C₈	—	13	2	no	—
150	C₈	—	13	1	no	—
151	C₈	—	26	4	no	—
152	C₈	—	26	1	no	—
153	C₈	—	26	1	no	—
154	C₈	C₂×#6092	1	4	yes	complemented
155	C₂²	C₄⋊#6092	1	2	yes	—
156	C₂²	#11718254	1	2	yes	—
157	C₂²	#11718261	1	2	yes	—
158	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	C₄⋊#6092	1	2	yes	—
162	C₄	#11718254	1	2	yes	—
163	C₄	C₂×#88102	1	1	yes	characteristic
164	C₄	C₂×#88104	1	1	yes	characteristic
165	C₄	C₂²×#6092	1	1	yes	characteristic
166	C₄	C₂×#88106	1	1	yes	characteristic
167	C₂	#13129865	1	4	yes	direct product factor
168	C₂	C₂×#11718254	1	1	yes	characteristic
169	C₂	C₂×C₄⋊#6092	1	1	yes	characteristic
170	C₂	C₂×#11718261	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#261
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1370
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13129865	C₂	direct (N × H)
33. C₂×C₈×C₁₃	C₁₆	semidirect
66. C₈×C₁₃	C₂×C₁₆	semidirect
109. C₂×C₁₃	#1370	semidirect
126. C₂×C₈	#6092	semidirect
134. C₁₃	C₂×#1370	semidirect
154. C₈	C₂×#6092	semidirect
167. C₂	#13129865	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6092	2
#13129865	2
C₂×C₈×#1055	2
C₂×#1370	13

Subgroups — C2×#13129865 #16788673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13129865 #16788673