Subgroups — C₂×#13112801 #15772655

Order 3328.

406 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	#13112801	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₂₆	—	2	1	no	—
7	C₂²×C₁₃×C₁₆	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	C₄	1	1	yes	characteristic
10	C₂×#88056	—	2	1	no	—
11	C₂×C₄⋊#1055	—	2	1	no	—
12	C₄×Dic₅₂	—	2	1	no	—
13	C₄×Dic₅₂	—	2	1	no	—
14	C₄×Dic₅₂	C₄	1	2	yes	—
15	C₄×Dic₅₂	C₂²	1	2	yes	—
16	C₄×C₁₃×C₁₆	—	2	1	no	—
17	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
18	#11718050	—	2	1	no	—
19	#11718050	C₂²	1	2	yes	—
20	C₂²×Dic₂₆	—	2	1	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	1	no	—
25	C₂×#6101	—	4	1	no	—
26	C₂×C₄×Dic₁₃	—	4	1	no	—
27	C₂²×#1055	—	4	1	no	—
28	C₂×Dic₅₂	—	2	1	no	—
29	C₂×Dic₅₂	—	2	2	no	—
30	C₂×Dic₅₂	C₈	1	4	yes	complemented
31	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
32	C₄×Dic₂₆	—	2	2	no	—
33	C₄×Dic₂₆	—	2	2	no	—
34	C₄×Dic₂₆	—	4	1	no	—
35	C₄×Dic₂₆	—	4	1	no	—
36	C₂×C₁₃×C₁₆	—	2	4	no	—
37	C₂×C₁₃×C₁₆	—	2	2	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	#88058	—	2	1	no	—
43	#88058	C₂×C₄	1	2	yes	—
44	#88056	—	2	2	no	—
45	#88056	—	2	2	no	—
46	C₄⋊#1055	—	2	2	no	—
47	C₄⋊#1055	—	4	1	no	—
48	C₂×#1332	—	13	1	no	maximal, Sylow
49	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	4	1	no	—
53	C₂²×Dic₁₃	—	4	1	no	—
54	C₂×Dic₂₆	—	2	4	no	—
55	C₂×Dic₂₆	—	2	2	no	—
56	C₂×Dic₂₆	—	4	1	no	—
57	C₂×Dic₂₆	—	4	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
63	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₃	D₈	1	1	yes	characteristic
67	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
68	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₄⋊Dic₁₃	—	2	2	no	—
70	C₄⋊Dic₁₃	—	4	1	no	—
71	#6101	—	4	2	no	—
72	#6101	—	4	2	no	—
73	C₄×Dic₁₃	—	4	2	no	—
74	C₄×Dic₁₃	—	4	2	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#1055	—	8	2	no	—
77	Dic₅₂	—	2	4	no	—
78	Dic₅₂	—	2	2	no	—
79	Dic₅₂	C₂×C₈	1	4	yes	complemented
80	C₁₃×C₁₆	—	2	4	no	—
81	C₂×C₄×Q₁₆	—	13	1	no	—
82	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
83	C₂×#335	—	13	1	no	—
84	#1332	—	13	4	no	—
85	C₂³×C₁₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
88	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#104	1	2	yes	—
92	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
93	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×Dic₁₃	—	4	4	no	—
98	C₂×Dic₁₃	—	4	2	no	—
99	C₂×Dic₁₃	—	4	2	no	—
100	C₂×Dic₁₃	—	8	2	no	—
101	Dic₂₆	—	2	4	no	—
102	Dic₂₆	—	4	4	no	—
103	Dic₂₆	—	4	2	no	—
104	C₈×C₁₃	—	2	1	no	—
105	C₈×C₁₃	—	2	1	no	—
106	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
107	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
108	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	#1055	—	8	2	no	—
110	C₂²×Q₁₆	—	13	1	no	—
111	C₂×C₄×Q₈	—	26	1	no	—
112	C₂²×C₁₆	—	2	1	no	—
113	C₄×Q₁₆	—	13	2	no	—
114	C₄×Q₁₆	—	13	2	no	—
115	C₄×Q₁₆	—	26	1	no	—
116	C₄×Q₁₆	—	26	1	no	—
117	C₂×#107	—	13	1	no	—
118	C₂×C₄⋊C₈	—	26	1	no	—
119	C₂×#103	—	26	1	no	—
120	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
121	C₄×C₁₆	—	2	1	no	—
122	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
123	#335	—	13	2	no	—
124	#335	—	26	1	no	—
125	C₂²×C₁₃	D₄⋊C₈	1	2	yes	—
126	C₂²×C₁₃	#358	1	2	yes	—
127	C₂²×C₁₃	#359	1	2	yes	—
128	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	D₄⋊C₈	1	2	yes	—
132	C₄×C₁₃	#358	1	2	yes	—
133	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
135	C₄×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
136	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
137	Dic₁₃	—	4	4	no	—
138	Dic₁₃	—	8	2	no	—
139	C₂²×Q₈	—	26	1	no	—
140	C₂×Q₁₆	—	13	4	no	—
141	C₂×Q₁₆	—	13	2	no	—
142	C₂×Q₁₆	—	26	2	no	—
143	C₂×Q₁₆	—	26	1	no	—
144	C₂²×C₈	—	52	1	no	—
145	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
147	C₄×Q₈	—	26	2	no	—
148	C₄×Q₈	—	26	2	no	—
149	C₄×Q₈	—	52	1	no	—
150	C₄×Q₈	—	52	1	no	—
151	C₂×C₄⋊C₄	—	26	1	no	—
152	C₂×C₄⋊C₄	—	52	1	no	—
153	C₂×C₄²	—	52	1	no	—
154	C₂×C₄²	D₅₂	1	1	yes	characteristic
155	C₂×C₁₆	—	2	4	no	—
156	C₂×C₁₆	—	2	2	no	—
157	#107	—	13	2	no	—
158	#107	—	26	1	no	—
159	C₄⋊C₈	—	26	2	no	—
160	C₄⋊C₈	—	52	1	no	—
161	#103	—	26	2	no	—
162	#103	—	26	2	no	—
163	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
164	C₄×C₈	D₅₂	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
166	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
167	C₂×C₁₃	#1332	1	4	yes	complemented
168	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
169	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#359	1	1	yes	characteristic
171	C₂×Q₈	—	26	2	no	—
172	C₂×Q₈	—	26	4	no	—
173	C₂×Q₈	—	52	2	no	—
174	C₂×Q₈	—	52	1	no	—
175	C₂²×C₄	—	52	1	no	—
176	C₂²×C₄	—	52	1	no	—
177	C₂²×C₄	#6095	1	1	yes	characteristic
178	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
179	C₂²×C₄	#6103	1	1	yes	characteristic
180	Q₁₆	—	13	4	no	—
181	Q₁₆	—	26	2	no	—
182	Q₁₆	—	26	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	52	2	no	—
185	C₂×C₈	—	104	2	no	—
186	C₂×C₈	C₈×D₁₃	1	2	yes	—
187	C₂×C₈	#6094	1	2	yes	—
188	C₂×C₈	#6103	1	1	yes	characteristic
189	C₂×C₈	#6103	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
192	C₄⋊C₄	—	26	2	no	—
193	C₄⋊C₄	—	52	2	no	—
194	C₄⋊C₄	—	52	1	no	—
195	C₄⋊C₄	—	52	2	no	—
196	C₄²	—	52	2	no	—
197	C₄²	—	52	2	no	—
198	C₄²	#6095	1	1	yes	characteristic
199	C₄²	D₁₀₄	1	1	yes	characteristic
200	C₄²	#6103	1	1	yes	characteristic
201	C₄²	C₂×D₅₂	1	1	yes	characteristic
202	C₁₆	—	2	4	no	—
203	C₁₃	C₂×#1332	1	1	yes	characteristic, complemented, Sylow
204	C₂³	#88061	1	1	yes	characteristic
205	Q₈	—	26	4	no	—
206	Q₈	—	52	4	no	—
207	Q₈	—	52	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	52	2	no	—
210	C₂×C₄	—	52	4	no	—
211	C₂×C₄	—	52	2	no	—
212	C₂×C₄	—	104	2	no	—
213	C₂×C₄	#88040	1	2	yes	—
214	C₂×C₄	Dic₁₀₄	1	2	yes	—
215	C₂×C₄	#88045	1	2	yes	—
216	C₂×C₄	#88060	1	2	yes	—
217	C₂×C₄	#88061	1	1	yes	characteristic
218	C₂×C₄	#88061	1	1	yes	characteristic
219	C₂×C₄	#88061	1	1	yes	characteristic
220	C₂×C₄	C₂×#6095	1	1	yes	characteristic
221	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
222	C₂×C₄	C₂×#6103	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	104	2	no	—
226	C₈	#88060	1	2	yes	—
227	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
228	C₈	C₂×#6094	1	1	yes	characteristic
229	C₂²	#11718051	1	2	yes	—
230	C₂²	#11718095	1	2	yes	—
231	C₂²	#11718102	1	2	yes	—
232	C₂²	C₂×#88061	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	52	4	no	—
236	C₄	—	104	2	no	—
237	C₄	#11718051	1	2	yes	—
238	C₄	#11718095	1	2	yes	—
239	C₄	C₂×#88040	1	1	yes	characteristic
240	C₄	C₂×Dic₁₀₄	1	1	yes	characteristic
241	C₄	C₂×#88045	1	1	yes	characteristic
242	C₄	C₂×#88060	1	1	yes	characteristic
243	C₂	#13112801	1	4	yes	direct product factor
244	C₂	C₂×#11718051	1	1	yes	characteristic
245	C₂	C₂×#11718095	1	1	yes	characteristic
246	C₂	C₂×#11718102	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₁₀₄
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	#102
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	C₂×#1332
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #13112801	C₂	direct (N × H)
30. C₂×Dic₅₂	C₈	semidirect
79. Dic₅₂	C₂×C₈	semidirect
167. C₂×C₁₃	#1332	semidirect
203. C₁₃	C₂×#1332	semidirect
243. C₂	#13112801	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×Dic₅₂	2
C₂×#11718050	2
#13112801	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#1332	13

Subgroups — C2×#13112801 #15772655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13112801 #15772655