Subgroups — C₂×#13123447 #16726617

Order 3328.

2586 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719524	C₂	1	2	yes	maximal
2	#13123447	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718786	C₂	1	6	yes	maximal, complemented
4	C₂²×#88059	C₂	1	6	yes	maximal, complemented
5	C₂²×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#131	C₄	1	8	yes	—
8	C₂×C₁₃×#131	C₄	1	8	yes	—
9	C₂×C₁₃×#131	C₂²	1	8	yes	—
10	C₂²×C₁₃×M₁₆	C₂²	1	3	yes	—
11	C₂³×C₈×C₁₃	C₂²	1	3	yes	—
12	#11719524	—	2	8	no	—
13	#11719524	C₂²	1	16	yes	—
14	C₂×#88202	—	2	8	no	—
15	C₂²×#6099	C₂²	1	12	yes	—
16	#11718786	C₂²	1	48	yes	complemented
17	C₂×#88059	C₂²	1	4	yes	—
18	C₂×#88059	C₂²	1	12	yes	—
19	C₂×#88059	C₂²	1	48	yes	complemented
20	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
21	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
22	C₂×C₁₃×#41	D₄	1	4	yes	—
23	C₂×C₁₃×#41	D₄	1	4	yes	—
24	C₂×C₁₃×#41	Q₈	1	4	yes	—
25	C₂×C₁₃×#41	C₂³	1	4	yes	—
26	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
27	C₂²×D₄×C₁₃	Q₈	1	3	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
29	C₁₃×#131	C₂×C₄	1	64	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	12	yes	—
31	C₂×C₁₃×M₁₆	C₂×C₄	1	12	yes	—
32	C₂×C₁₃×M₁₆	C₂³	1	12	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	12	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	12	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
38	#88202	—	2	64	no	—
39	C₂×#6099	—	2	12	no	—
40	C₂×#6099	—	2	24	no	—
41	C₂×#6099	C₂³	1	8	yes	—
42	C₂×#6099	C₂³	1	48	yes	—
43	C₂²×#1055	—	2	12	no	—
44	#88059	C₂³	1	16	yes	—
45	#88059	C₂³	1	48	yes	—
46	C₂×#2916	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	C₂×Q₈	1	3	yes	—
48	C₁₃×#41	C₄⋊C₄	1	16	yes	—
49	C₁₃×#41	C₄⋊C₄	1	16	yes	—
50	C₁₃×#41	C₂²×C₄	1	16	yes	—
51	C₁₃×#41	C₂×D₄	1	16	yes	—
52	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
53	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
54	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
55	C₂×D₄×C₁₃	C₄⋊C₄	1	12	yes	—
56	C₂×D₄×C₁₃	C₄⋊C₄	1	12	yes	—
57	C₂×D₄×C₁₃	C₂×Q₈	1	12	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
63	C₂²×C₄×C₁₃	C₂×Q₈	1	6	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
66	C₁₃×M₁₆	C₂²×C₄	1	48	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	48	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
73	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
74	#6099	—	2	8	no	—
75	#6099	—	2	48	no	—
76	#6099	C₂⁴	1	16	yes	—
77	C₂×#1055	—	2	8	no	—
78	C₂×#1055	—	2	48	no	—
79	C₂×#568	—	13	2	no	—
80	C₂²×#131	D₁₃	1	1	yes	characteristic
81	#2916	—	13	16	no	—
82	C₂×#430	—	13	6	no	—
83	C₂²×#108	—	13	6	no	—
84	C₂³×C₁₃	—	4	6	no	—
85	C₂³×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
86	C₂³×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
87	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
88	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
89	Q₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
90	D₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
94	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
95	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
98	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
99	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
103	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
104	C₈×C₁₃	C₂³×C₄	1	4	yes	—
105	C₈×C₁₃	C₂³×C₄	1	12	yes	—
106	#1055	—	2	16	no	—
107	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
108	#568	—	13	16	no	—
109	#568	—	26	8	no	—
110	C₂×#131	—	26	8	no	—
111	C₂×#131	Dic₁₃	1	8	yes	—
112	C₂×#131	Dic₁₃	1	8	yes	—
113	C₂×#131	C₂×D₁₃	1	8	yes	—
114	C₂²×M₁₆	—	13	12	no	—
115	C₂²×M₁₆	C₂×D₁₃	1	3	yes	—
116	C₂³×C₈	C₂×D₁₃	1	3	yes	—
117	#430	—	13	48	no	—
118	C₂×#108	—	13	4	no	—
119	C₂×#108	—	13	12	no	—
120	C₂×#108	—	13	48	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	—	2	12	no	—
123	C₂²×C₁₃	—	4	6	no	—
124	C₂²×C₁₃	—	4	6	no	—
125	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
127	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
129	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
132	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
133	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
134	C₂×#41	Dic₂₆	1	4	yes	—
135	C₂×#41	D₅₂	1	4	yes	—
136	C₂×#41	D₅₂	1	4	yes	—
137	C₂×#41	C₂×Dic₁₃	1	4	yes	—
138	C₂×#41	C₂×Dic₁₃	1	4	yes	—
139	C₂×#41	C₂²×D₁₃	1	4	yes	—
140	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
141	C₂²×D₄	Dic₂₆	1	3	yes	—
142	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
143	#131	—	26	64	no	—
144	#131	C₂×Dic₁₃	1	64	yes	—
145	C₂×M₁₆	—	13	8	no	—
146	C₂×M₁₆	—	13	48	no	—
147	C₂×M₁₆	—	26	12	no	—
148	C₂×M₁₆	—	26	24	no	—
149	C₂×M₁₆	C₂×Dic₁₃	1	12	yes	—
150	C₂×M₁₆	C₂×Dic₁₃	1	12	yes	—
151	C₂×M₁₆	C₂²×D₁₃	1	12	yes	—
152	C₂²×C₈	—	26	12	no	—
153	C₂²×C₈	C₂×Dic₁₃	1	12	yes	—
154	C₂²×C₈	C₂×Dic₁₃	1	12	yes	—
155	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
157	C₂²×C₈	C₂²×D₁₃	1	12	yes	—
158	#108	—	13	16	no	—
159	#108	—	13	48	no	—
160	C₂×C₁₃	—	2	1	no	—
161	C₂×C₁₃	—	2	1	no	—
162	C₂×C₁₃	—	2	12	no	—
163	C₂×C₁₃	#2916	1	2	yes	complemented
164	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
165	C₂⁴	C₂×Dic₂₆	1	3	yes	—
166	#41	C₄⋊Dic₁₃	1	16	yes	—
167	#41	C₄⋊Dic₁₃	1	16	yes	—
168	#41	C₂×D₅₂	1	16	yes	—
169	#41	C₂²×Dic₁₃	1	16	yes	—
170	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
171	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
172	C₂×Q₈	C₂×Dic₂₆	1	4	yes	—
173	C₂×D₄	C₄⋊Dic₁₃	1	12	yes	—
174	C₂×D₄	C₄⋊Dic₁₃	1	12	yes	—
175	C₂×D₄	C₂×Dic₂₆	1	12	yes	—
176	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
177	C₂²×C₄	C₂×Dic₂₆	1	6	yes	—
178	C₂²×C₄	C₂×D₅₂	1	6	yes	—
179	C₂²×C₄	C₂×D₅₂	1	6	yes	—
180	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
181	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
182	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
183	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
184	M₁₆	—	13	16	no	—
185	M₁₆	—	26	8	no	—
186	M₁₆	—	26	48	no	—
187	M₁₆	C₂²×Dic₁₃	1	48	yes	—
188	C₂×C₈	—	26	48	no	—
189	C₂×C₈	—	26	8	no	—
190	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
191	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
192	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
193	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
194	C₂×C₈	C₂²×Dic₁₃	1	48	yes	—
195	C₂×C₈	C₂³×D₁₃	1	2	yes	—
196	C₂×C₈	C₂³×D₁₃	1	6	yes	—
197	C₁₃	C₂×#2916	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	4	6	no	—
199	C₂³	C₂×C₄⋊Dic₁₃	1	6	yes	—
200	C₂³	C₂×C₄⋊Dic₁₃	1	6	yes	—
201	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
202	C₂³	C₂²×Dic₂₆	1	6	yes	—
203	Q₈	C₂×C₄⋊Dic₁₃	1	16	yes	—
204	D₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
209	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
211	C₂×C₄	C₂²×D₅₂	1	12	yes	—
212	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
214	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
215	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
217	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
218	C₈	—	26	16	no	—
219	C₈	C₂³×Dic₁₃	1	4	yes	—
220	C₈	C₂³×Dic₁₃	1	12	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	12	no	—
223	C₂²	—	4	6	no	—
224	C₂²	—	4	6	no	—
225	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
226	C₂²	C₂²×C₄⋊Dic₁₃	1	12	yes	—
227	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
228	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
229	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
230	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
231	C₄	C₂²×C₄⋊Dic₁₃	1	12	yes	—
232	C₄	C₂³×D₅₂	1	1	yes	characteristic
233	C₄	C₂⁴×Dic₁₃	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	2	12	no	—
237	C₂	#13123447	1	2	yes	direct product factor
238	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂²×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×#131	D₁₃
hypocenter	C₁₃	C₂×#2916
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #13123447	C₂	direct (N × H)
3. C₂×#11718786	C₂	semidirect
4. C₂²×#88059	C₂	semidirect
16. #11718786	C₂²	semidirect
19. C₂×#88059	C₂²	semidirect
163. C₂×C₁₃	#2916	semidirect
197. C₁₃	C₂×#2916	semidirect
237. C₂	#13123447	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×#11719524	2
#13123447	2
C₂×#11718786	2
C₂²×#88059	2
C₂²×C₁₃×#131	2
C₂×#2916	13

Subgroups — C2×#13123447 #16726617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123447 #16726617