Subgroups — C₂×#13117122 #16018416

Order 3328.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718344	C₂	1	1	yes	maximal, characteristic
3	#13117122	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718097	C₂	1	2	yes	maximal
5	C₂×#11718096	C₂	1	2	yes	maximal
6	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×#88124	—	2	1	no	—
8	C₂×#88123	—	2	1	no	—
9	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88058	—	2	1	no	—
12	C₂×#88058	C₂²	1	1	yes	characteristic
13	C₂×#88058	C₂²	1	2	yes	—
14	C₂×#88058	C₂²	1	1	yes	characteristic
15	C₂×#88057	—	2	1	no	—
16	#11718344	—	2	2	no	—
17	#11718344	C₂²	1	4	yes	—
18	#11718343	—	2	2	no	—
19	#11718343	C₂²	1	4	yes	—
20	C₁₃×#346	C₂²	1	4	yes	—
21	#11718097	C₂²	1	8	yes	—
22	#11718096	C₂²	1	8	yes	—
23	C₂²×Dic₂₆	—	4	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	C₂×C₄⋊Dic₁₃	—	2	1	no	—
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	—	4	1	no	—
31	C₂×#6101	—	4	1	no	—
32	C₂×#6101	—	4	1	no	—
33	#88124	—	2	4	no	—
34	#88124	—	4	2	no	—
35	#88123	—	2	4	no	—
36	#88123	—	4	2	no	—
37	C₂×C₁₃×C₁₆	Q₈	1	8	yes	complemented
38	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	C₄×C₈×C₁₃	C₂³	1	2	yes	—
41	#88058	—	2	4	no	—
42	#88058	—	2	2	no	—
43	#88058	—	2	1	no	—
44	#88058	—	2	1	no	—
45	#88058	C₂³	1	4	yes	—
46	#88058	C₂³	1	2	yes	—
47	#88058	C₂³	1	2	yes	—
48	#88057	—	2	4	no	—
49	C₂×#2255	—	13	1	no	maximal, Sylow
50	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₁₃	—	4	1	no	—
54	C₂²×Dic₁₃	—	4	1	no	—
55	C₂²×Dic₁₃	—	4	2	no	—
56	C₂×Dic₂₆	—	4	4	no	—
57	C₂×Dic₂₆	—	8	4	no	—
58	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₄²×C₁₃	D₈	1	2	yes	—
64	C₄²×C₁₃	C₂×D₄	1	2	yes	—
65	C₄⋊Dic₁₃	—	2	4	no	—
66	C₄⋊Dic₁₃	—	2	2	no	—
67	C₄⋊Dic₁₃	—	2	2	no	—
68	C₄⋊Dic₁₃	—	4	4	no	—
69	C₄⋊Dic₁₃	—	4	2	no	—
70	C₄⋊Dic₁₃	—	4	1	no	—
71	C₄⋊Dic₁₃	—	4	1	no	—
72	#6101	—	4	4	no	—
73	#6101	—	4	4	no	—
74	C₁₃×C₁₆	C₂×Q₈	1	8	yes	complemented
75	C₂×#500	—	13	1	no	—
76	C₂×#499	—	13	1	no	—
77	#2255	—	13	8	no	—
78	C₂×#367	—	13	2	no	—
79	C₂×#366	—	13	2	no	—
80	C₂×#346	D₁₃	1	1	yes	characteristic
81	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×Dic₁₃	—	4	2	no	—
91	C₂×Dic₁₃	—	4	2	no	—
92	C₂×Dic₁₃	—	4	4	no	—
93	C₂×Dic₁₃	—	8	4	no	—
94	C₂×Dic₁₃	—	8	2	no	—
95	C₂×Dic₁₃	—	8	2	no	—
96	Dic₂₆	—	8	8	no	—
97	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
98	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
99	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
100	C₂×C₄⋊Q₈	—	26	1	no	—
101	C₂×#125	—	26	1	no	—
102	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
103	#500	—	13	4	no	—
104	#500	—	26	2	no	—
105	#499	—	13	4	no	—
106	#499	—	26	2	no	—
107	C₂×#107	—	13	1	no	—
108	C₂×#107	—	13	1	no	—
109	C₂×#107	—	13	2	no	—
110	C₂×#107	—	26	1	no	—
111	C₂×#106	—	26	1	no	—
112	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
113	#367	—	13	8	no	—
114	#366	—	13	8	no	—
115	#346	C₂×D₁₃	1	4	yes	—
116	C₂²×C₁₃	#500	1	2	yes	—
117	C₂²×C₁₃	#509	1	2	yes	—
118	C₂²×C₁₃	#510	1	2	yes	—
119	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	#500	1	2	yes	—
123	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
125	Dic₁₃	—	8	4	no	—
126	Dic₁₃	—	8	2	no	—
127	Dic₁₃	—	8	2	no	—
128	C₂²×Q₈	—	52	1	no	—
129	C₂²×C₈	D₅₂	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	26	4	no	—
132	C₄⋊Q₈	—	52	2	no	—
133	#125	—	26	4	no	—
134	#125	—	52	2	no	—
135	C₂×C₄⋊C₄	—	26	1	no	—
136	C₂×C₄⋊C₄	—	26	1	no	—
137	C₂×C₄⋊C₄	—	26	2	no	—
138	C₂×C₄⋊C₄	—	52	1	no	—
139	C₂×C₄⋊C₄	—	52	1	no	—
140	C₂×C₄⋊C₄	—	52	1	no	—
141	C₂×C₄²	D₅₂	1	1	yes	characteristic
142	C₂×C₁₆	Dic₂₆	1	8	yes	complemented
143	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
144	#107	—	13	2	no	—
145	#107	—	13	4	no	—
146	#107	—	13	2	no	—
147	#107	—	26	1	no	—
148	#107	—	26	1	no	—
149	#107	—	26	2	no	—
150	#107	—	26	4	no	—
151	#106	—	26	4	no	—
152	C₄×C₈	D₅₂	1	2	yes	—
153	C₄×C₈	C₂²×D₁₃	1	2	yes	—
154	C₂×C₁₃	#2255	1	4	yes	complemented
155	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
156	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
157	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
158	C₂×Q₈	—	52	4	no	—
159	C₂×Q₈	—	104	4	no	—
160	C₂²×C₄	—	52	1	no	—
161	C₂²×C₄	—	52	1	no	—
162	C₂²×C₄	—	52	2	no	—
163	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
164	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
166	C₂×C₈	Dic₅₂	1	4	yes	—
167	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
168	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
169	C₂×C₈	C₂×D₅₂	1	2	yes	—
170	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
171	C₄⋊C₄	—	26	2	no	—
172	C₄⋊C₄	—	26	2	no	—
173	C₄⋊C₄	—	26	4	no	—
174	C₄⋊C₄	—	52	1	no	—
175	C₄⋊C₄	—	52	1	no	—
176	C₄⋊C₄	—	52	2	no	—
177	C₄⋊C₄	—	52	4	no	—
178	C₄⋊C₄	—	52	4	no	—
179	C₄⋊C₄	—	52	4	no	—
180	C₄²	D₁₀₄	1	2	yes	—
181	C₄²	C₂×D₅₂	1	2	yes	—
182	C₁₆	C₂×Dic₂₆	1	8	yes	complemented
183	C₁₃	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
184	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
185	Q₈	—	104	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	52	2	no	—
189	C₂×C₄	—	52	2	no	—
190	C₂×C₄	—	52	4	no	—
191	C₂×C₄	—	104	2	no	—
192	C₂×C₄	—	104	4	no	—
193	C₂×C₄	—	104	2	no	—
194	C₂×C₄	#88123	1	2	yes	—
195	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
196	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
197	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
199	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
200	C₈	#88123	1	2	yes	—
201	C₈	C₂×Dic₅₂	1	4	yes	—
202	C₈	C₂²×Dic₂₆	1	2	yes	—
203	C₂²	#11718343	1	2	yes	—
204	C₂²	#11718569	1	2	yes	—
205	C₂²	#11718570	1	2	yes	—
206	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	104	2	no	—
210	C₄	—	104	4	no	—
211	C₄	—	104	2	no	—
212	C₄	#11718343	1	2	yes	—
213	C₄	C₂×#88123	1	1	yes	characteristic
214	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
215	C₂	#13117122	1	4	yes	direct product factor
216	C₂	C₂×#11718343	1	1	yes	characteristic
217	C₂	C₂×#11718569	1	1	yes	characteristic
218	C₂	C₂×#11718570	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	C₂×#2255
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #13117122	C₂	direct (N × H)
37. C₂×C₁₃×C₁₆	Q₈	semidirect
74. C₁₃×C₁₆	C₂×Q₈	semidirect
142. C₂×C₁₆	Dic₂₆	semidirect
154. C₂×C₁₃	#2255	semidirect
182. C₁₆	C₂×Dic₂₆	semidirect
183. C₁₃	C₂×#2255	semidirect
215. C₂	#13117122	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×#11718343	2
C₂×#11718344	2
#13117122	2
C₂×#11718097	2
C₂×#11718096	2
C₂×C₁₃×#346	2
C₂×#2255	13

Subgroups — C2×#13117122 #16018416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117122 #16018416