Subgroups — C₂×#13117061 #16018204

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718356	C₂	1	1	yes	maximal, characteristic
2	#13117061	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718083	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
7	C₂²×D₁₀₄	—	2	1	no	—
8	C₂²×D₁₀₄	C₂²	1	2	yes	—
9	C₂×#88128	—	2	1	no	—
10	C₁₃×#500	—	2	2	no	—
11	C₁₃×#500	C₂²	1	4	yes	complemented
12	C₂×C₁₃×#107	—	2	1	no	—
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×#6090	C₂²	1	2	yes	—
16	#11718356	—	2	2	no	—
17	#11718356	C₂²	1	4	yes	—
18	#11718083	C₂²	1	16	yes	complemented
19	C₄×#6090	C₂²	1	4	yes	complemented
20	C₂²×Q₈×C₁₃	—	4	1	no	—
21	C₂²×D₅₂	—	2	2	no	—
22	C₂²×D₅₂	—	4	1	no	—
23	C₂²×D₅₂	—	4	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₁₃×C₄⋊Q₈	—	2	4	no	—
27	C₁₃×C₄⋊Q₈	—	4	2	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×D₁₀₄	—	2	8	no	—
32	C₂×D₁₀₄	—	2	4	no	—
33	C₂×D₁₀₄	—	2	2	no	—
34	C₂×D₁₀₄	—	4	4	no	—
35	C₂×D₁₀₄	C₂³	1	4	yes	—
36	#88128	—	2	4	no	—
37	#88128	—	4	2	no	—
38	C₁₃×#107	—	2	4	no	—
39	C₁₃×#107	—	2	2	no	—
40	C₁₃×#107	C₂³	1	4	yes	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	C₄×C₈×C₁₃	C₂³	1	2	yes	—
43	C₂×#6090	—	2	4	no	—
44	C₂×#6090	C₂³	1	4	yes	—
45	C₂×#2240	—	13	1	no	maximal, Sylow
46	C₂³×D₁₃	—	4	2	no	—
47	C₂×Q₈×C₁₃	—	4	4	no	—
48	C₂×Q₈×C₁₃	—	8	4	no	—
49	C₂²×C₄×C₁₃	—	4	2	no	—
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	C₂×D₅₂	—	2	4	no	—
54	C₂×D₅₂	—	4	4	no	—
55	C₂×D₅₂	—	4	4	no	—
56	C₂×D₅₂	—	4	4	no	—
57	C₂×D₅₂	—	4	4	no	—
58	C₂×D₅₂	—	4	2	no	—
59	C₂×D₅₂	—	8	4	no	—
60	C₂×D₅₂	—	8	4	no	—
61	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₁₃	#41	1	4	yes	—
65	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	—	2	4	no	—
67	C₁₃×C₄⋊C₄	—	4	2	no	—
68	C₁₃×C₄⋊C₄	—	4	4	no	—
69	C₄²×C₁₃	D₈	1	2	yes	—
70	C₄²×C₁₃	C₂×D₄	1	2	yes	—
71	D₁₀₄	—	2	8	no	—
72	D₁₀₄	—	4	4	no	—
73	D₁₀₄	—	4	8	no	—
74	#6090	—	2	4	no	—
75	C₂×#500	D₁₃	1	1	yes	characteristic, complemented
76	C₂×#493	—	13	1	no	—
77	#2240	—	13	8	no	—
78	C₂×#357	—	13	4	no	—
79	C₂×C₄×C₁₆	—	13	1	no	—
80	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₁₃	—	4	4	no	—
82	C₂²×D₁₃	—	8	4	no	—
83	C₂²×D₁₃	—	8	4	no	—
84	C₂²×D₁₃	—	8	4	no	—
85	Q₈×C₁₃	—	8	8	no	—
86	C₂×C₄×C₁₃	—	4	4	no	—
87	C₂×C₄×C₁₃	—	8	4	no	—
88	C₂×C₄×C₁₃	D₁₆	1	4	yes	—
89	C₂×C₄×C₁₃	SD₃₂	1	4	yes	—
90	C₂×C₄×C₁₃	#124	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
96	D₅₂	—	4	4	no	—
97	D₅₂	—	4	4	no	—
98	D₅₂	—	8	4	no	—
99	D₅₂	—	8	8	no	—
100	D₅₂	—	8	8	no	—
101	C₈×C₁₃	#124	1	2	yes	—
102	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
103	C₈×C₁₃	C₂×#41	1	2	yes	—
104	C₂²×D₈	—	13	2	no	—
105	C₂²×D₈	—	26	1	no	—
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	C₂×#127	—	26	1	no	—
108	C₂²×C₁₆	—	13	2	no	—
109	#500	—	2	2	no	—
110	#500	C₂×D₁₃	1	4	yes	complemented
111	#493	—	13	4	no	—
112	#493	—	26	2	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	C₂×D₁₃	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
116	#357	—	13	16	no	—
117	C₄×C₁₆	—	13	4	no	—
118	C₂²×C₁₃	#486	1	2	yes	—
119	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
120	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
121	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
122	C₂×D₁₃	—	8	4	no	—
123	C₂×D₁₃	—	8	4	no	—
124	C₂×D₁₃	—	8	4	no	—
125	C₂×D₁₃	—	16	8	no	—
126	C₄×C₁₃	—	8	4	no	—
127	C₄×C₁₃	#486	1	2	yes	—
128	C₄×C₁₃	C₂×D₁₆	1	4	yes	—
129	C₄×C₁₃	C₂×SD₃₂	1	4	yes	—
130	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
131	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
132	C₂²×Q₈	—	4	1	no	—
133	C₂²×D₄	—	26	2	no	—
134	C₂²×D₄	—	52	1	no	—
135	C₂²×D₄	—	52	1	no	—
136	C₂×D₈	—	13	4	no	—
137	C₂×D₈	—	26	8	no	—
138	C₂×D₈	—	26	2	no	—
139	C₂×D₈	—	26	4	no	—
140	C₂×D₈	—	52	4	no	—
141	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	—	4	2	no	—
145	#127	—	26	4	no	—
146	#127	—	52	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	4	1	no	—
149	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂×C₁₆	—	13	4	no	—
151	C₂×C₁₆	—	26	4	no	—
152	#107	—	2	4	no	—
153	#107	—	2	2	no	—
154	#107	C₂²×D₁₃	1	4	yes	—
155	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
156	C₄×C₈	C₂²×D₁₃	1	2	yes	—
157	C₂×C₁₃	#2240	1	4	yes	complemented
158	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
159	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
160	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
161	D₁₃	—	16	4	no	—
162	C₂⁴	—	52	2	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	8	4	no	—
165	C₂×D₄	—	26	4	no	—
166	C₂×D₄	—	52	4	no	—
167	C₂×D₄	—	52	4	no	—
168	C₂×D₄	—	52	2	no	—
169	C₂×D₄	—	52	4	no	—
170	C₂×D₄	—	52	4	no	—
171	C₂×D₄	—	104	4	no	—
172	C₂×D₄	—	104	4	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	#6104	1	1	yes	characteristic
175	C₂²×C₄	#6104	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
177	D₈	—	26	8	no	—
178	D₈	—	52	8	no	—
179	D₈	—	52	4	no	—
180	C₂×C₈	#6106	1	4	yes	—
181	C₂×C₈	#6131	1	4	yes	—
182	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
183	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄²	#6104	1	2	yes	—
189	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
190	C₁₆	—	26	4	no	—
191	C₁₃	C₂×#2240	1	1	yes	characteristic, complemented, Sylow
192	C₂³	—	52	4	no	—
193	C₂³	—	104	4	no	—
194	C₂³	—	104	4	no	—
195	C₂³	—	104	4	no	—
196	C₂³	C₂×#6104	1	1	yes	characteristic
197	Q₈	—	8	8	no	—
198	D₄	—	52	4	no	—
199	D₄	—	52	4	no	—
200	D₄	—	104	4	no	—
201	D₄	—	104	8	no	—
202	D₄	—	104	8	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	#88066	1	4	yes	—
206	C₂×C₄	#88068	1	4	yes	—
207	C₂×C₄	C₂×#6104	1	1	yes	characteristic
208	C₂×C₄	C₂×#6104	1	2	yes	—
209	C₂×C₄	C₂×#6104	1	2	yes	—
210	C₂×C₄	C₂×#6106	1	2	yes	—
211	C₂×C₄	#88201	1	2	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
213	C₈	C₂×#6106	1	4	yes	—
214	C₈	#88201	1	2	yes	—
215	C₈	C₂×#6131	1	2	yes	—
216	C₂²	—	104	4	no	—
217	C₂²	—	104	4	no	—
218	C₂²	—	104	4	no	—
219	C₂²	—	208	8	no	—
220	C₂²	#11718749	1	2	yes	—
221	C₂²	C₂×#88066	1	2	yes	—
222	C₂²	C₂×#88068	1	2	yes	—
223	C₂²	C₂²×#6104	1	1	yes	characteristic
224	C₄	—	8	4	no	—
225	C₄	#11718749	1	2	yes	—
226	C₄	C₂×#88066	1	4	yes	—
227	C₄	C₂×#88068	1	4	yes	—
228	C₄	C₂²×#6106	1	1	yes	characteristic
229	C₄	C₂×#88201	1	1	yes	characteristic
230	C₂	—	208	4	no	—
231	C₂	#13117061	1	4	yes	direct product factor
232	C₂	C₂×#11718749	1	1	yes	characteristic
233	C₂	C₂²×#88066	1	1	yes	characteristic
234	C₂	C₂²×#88068	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂×#11718356	C₂
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	C₂×#2240
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #13117061	C₂	direct (N × H)
3. C₂×#11718083	C₂	semidirect
4. C₂×C₄×#6090	C₂	semidirect
5. C₂×C₁₃×#500	C₂	semidirect
11. C₁₃×#500	C₂²	semidirect
18. #11718083	C₂²	semidirect
19. C₄×#6090	C₂²	semidirect
75. C₂×#500	D₁₃	semidirect
110. #500	C₂×D₁₃	semidirect
157. C₂×C₁₃	#2240	semidirect
191. C₁₃	C₂×#2240	semidirect
231. C₂	#13117061	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×#11718356	2
#13117061	2
C₂×#11718083	2
C₂×C₄×#6090	2
C₂×C₁₃×#500	2
C₂×#2240	13

Subgroups — C2×#13117061 #16018204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117061 #16018204