Subgroups — C₂×#13123942 #16727888

Order 3328.

1706 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123942	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11718685	C₂	1	8	yes	maximal, complemented
3	C₂×#11718661	C₂	1	4	yes	maximal
4	C₂×#11718587	C₂	1	2	yes	maximal
5	C₂×C₁₃×#533	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₄	1	8	yes	—
7	C₁₃×#533	C₂²	1	8	yes	—
8	C₂×C₁₃×C₄⋊Q₈	C₄	1	2	yes	—
9	C₂×C₁₃×#125	C₄	1	2	yes	—
10	C₂×C₁₃×C₂²⋊Q₈	C₄	1	4	yes	—
11	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
12	C₂×C₁₃×#117	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
15	#11718685	C₂²	1	64	yes	complemented
16	#11718661	C₂²	1	32	yes	—
17	#11718587	C₂²	1	16	yes	—
18	C₂×C₄⋊#1055	C₂²	1	4	yes	—
19	C₂×C₄⋊#1055	C₂²	1	8	yes	—
20	C₂×#88043	C₂²	1	4	yes	—
21	C₂×#88043	C₂²	1	4	yes	—
22	C₂×C₄×#1055	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₁₃×C₄⋊Q₈	C₂×C₄	1	16	yes	—
27	C₁₃×#125	C₂×C₄	1	16	yes	—
28	C₁₃×C₂²⋊Q₈	C₂×C₄	1	32	yes	—
29	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	—
30	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
31	C₁₃×#117	C₂×C₄	1	8	yes	—
32	C₁₃×#117	C₂³	1	8	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
43	C₂²×#1055	C₂³	1	8	yes	—
44	C₂²⋊#1055	C₂³	1	16	yes	—
45	C₄⋊#1055	C₂³	1	32	yes	—
46	C₄⋊#1055	C₂³	1	16	yes	—
47	#88043	C₂³	1	16	yes	—
48	#88043	C₂³	1	16	yes	—
49	C₄×#1055	C₂³	1	16	yes	—
50	C₂×#2987	—	13	1	no	maximal, Sylow
51	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×Q₈×C₁₃	—	4	8	no	—
53	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	4	4	no	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	C₂²×C₄	1	16	yes	—
65	C₁₃×C₄⋊C₄	C₂²×C₄	1	32	yes	—
66	C₁₃×C₄⋊C₄	C₂²×C₄	1	16	yes	—
67	C₁₃×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
70	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
71	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
72	C₄²×C₁₃	C₂⁴	1	4	yes	—
73	C₄²×C₁₃	C₂⁴	1	8	yes	—
74	C₂×#1055	—	2	16	no	—
75	C₂×#1055	C₂⁴	1	16	yes	—
76	C₂×#533	D₁₃	1	1	yes	characteristic
77	#2987	—	13	16	no	—
78	C₂×#446	—	13	8	no	—
79	C₂×#433	—	13	4	no	—
80	C₂×#432	—	13	2	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	—	2	4	no	—
83	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
84	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
85	Q₈×C₁₃	—	4	16	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	4	no	—
89	C₂×C₄×C₁₃	—	2	16	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	4	8	no	—
93	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
95	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₁₃	C₂³×C₄	1	16	yes	—
97	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
99	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
100	#1055	—	2	16	no	—
101	#533	Dic₁₃	1	8	yes	—
102	#533	C₂×D₁₃	1	8	yes	—
103	C₂×C₄⋊Q₈	Dic₁₃	1	2	yes	—
104	C₂×#125	Dic₁₃	1	2	yes	—
105	C₂×C₂²⋊Q₈	Dic₁₃	1	4	yes	—
106	C₂×C₄×Q₈	C₂×D₁₃	1	4	yes	—
107	C₂×#117	C₂×D₁₃	1	2	yes	—
108	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
109	#446	—	13	64	no	—
110	#433	—	13	32	no	—
111	#432	—	13	16	no	—
112	C₂×C₄⋊C₈	—	13	4	no	—
113	C₂×C₄⋊C₈	—	13	8	no	—
114	C₂×C₂²⋊C₈	—	13	4	no	—
115	C₂×#97	—	13	4	no	—
116	C₂×#97	—	13	4	no	—
117	C₂×C₄×C₈	—	13	4	no	—
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	—	2	4	no	—
120	C₂²×C₁₃	—	4	4	no	—
121	C₂²×C₁₃	#519	1	2	yes	—
122	C₂²×C₁₃	#568	1	4	yes	—
123	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	2	4	no	—
126	C₄×C₁₃	—	2	4	no	—
127	C₄×C₁₃	—	2	16	no	—
128	C₄×C₁₃	#519	1	2	yes	—
129	C₄×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
130	C₂²×Q₈	C₂×Dic₁₃	1	4	yes	—
131	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
132	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
133	C₂²×C₈	—	13	8	no	—
134	C₄⋊Q₈	C₂×Dic₁₃	1	16	yes	—
135	#125	C₂×Dic₁₃	1	16	yes	—
136	C₂²⋊Q₈	C₂×Dic₁₃	1	32	yes	—
137	C₄×Q₈	C₂×Dic₁₃	1	16	yes	—
138	C₄×Q₈	C₂²×D₁₃	1	16	yes	—
139	#117	C₂×Dic₁₃	1	8	yes	—
140	#117	C₂²×D₁₃	1	8	yes	—
141	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
142	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
143	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
144	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
145	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
148	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
149	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
150	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
151	C₄⋊C₈	—	13	16	no	—
152	C₄⋊C₈	—	13	32	no	—
153	C₂²⋊C₈	—	13	16	no	—
154	#97	—	13	16	no	—
155	#97	—	13	16	no	—
156	C₄×C₈	—	13	16	no	—
157	C₂×C₁₃	—	4	2	no	—
158	C₂×C₁₃	#2987	1	4	yes	complemented
159	C₂×C₁₃	C₂×#519	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#568	1	2	yes	—
161	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
162	C₂×Q₈	—	4	8	no	—
163	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
168	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
171	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
172	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
173	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
174	C₂×C₈	—	13	16	no	—
175	C₂×C₈	—	26	16	no	—
176	C₄⋊C₄	C₂²×Dic₁₃	1	16	yes	—
177	C₄⋊C₄	C₂²×Dic₁₃	1	32	yes	—
178	C₄⋊C₄	C₂²×Dic₁₃	1	16	yes	—
179	C₂²⋊C₄	C₂²×Dic₁₃	1	16	yes	—
180	C₄²	C₂²×Dic₁₃	1	4	yes	—
181	C₄²	C₂²×Dic₁₃	1	4	yes	—
182	C₄²	C₂²×Dic₁₃	1	4	yes	—
183	C₄²	C₂²×Dic₁₃	1	8	yes	—
184	C₄²	C₂³×D₁₃	1	8	yes	—
185	C₄²	C₂³×D₁₃	1	4	yes	—
186	C₁₃	C₂×#2987	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	C₂³×Dic₁₃	1	2	yes	—
190	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
191	Q₈	—	4	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	#88254	1	4	yes	—
200	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
201	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
202	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
203	C₂×C₄	C₂³×Dic₁₃	1	16	yes	—
204	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
205	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
206	C₈	—	26	16	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	4	4	no	—
210	C₂²	#11719517	1	2	yes	—
211	C₂²	#11719524	1	4	yes	—
212	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	2	4	no	—
215	C₄	—	2	16	no	—
216	C₄	—	2	2	no	—
217	C₄	#11719517	1	2	yes	—
218	C₄	C₂×#88254	1	2	yes	—
219	C₂	—	4	2	no	—
220	C₂	#13123942	1	4	yes	direct product factor
221	C₂	C₂×#11719517	1	1	yes	characteristic
222	C₂	C₂×#11719524	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#533	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂×#533	D₁₃
hypocenter	C₁₃	C₂×#2987
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #13123942	C₂	direct (N × H)
2. C₂×#11718685	C₂	semidirect
15. #11718685	C₂²	semidirect
158. C₂×C₁₃	#2987	semidirect
186. C₁₃	C₂×#2987	semidirect
220. C₂	#13123942	direct (N × H)

Maximal subgroups

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

subgroup	index
#13123942	2
C₂×#11718685	2
C₂×#11718661	2
C₂×#11718587	2
C₂×C₁₃×#533	2
C₂×#2987	13

Subgroups — C2×#13123942 #16727888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123942 #16727888