Subgroups — C₂×#13116673 #16016147

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718801	C₂	1	2	yes	maximal, complemented
2	#13116673	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718055	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718153	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#135	C₄	1	1	yes	characteristic
8	C₂×C₁₃×#131	C₂²	1	2	yes	—
9	C₁₃×#443	C₄	1	4	yes	—
10	C₁₃×#443	C₂²	1	4	yes	—
11	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
12	C₂×C₁₃×#104	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718801	—	2	4	no	—
15	#11718801	C₂²	1	8	yes	complemented
16	C₂×#88052	—	2	2	no	—
17	C₂×#88052	C₂²	1	2	yes	—
18	C₂²×#6090	—	2	2	no	—
19	C₂²×#6090	C₂²	1	1	yes	characteristic
20	C₂²×#6090	C₂²	1	1	yes	characteristic
21	#11718153	C₂²	1	8	yes	complemented
22	#11718055	C₂²	1	4	yes	complemented
23	C₄×#6090	C₂²	1	4	yes	complemented
24	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
25	C₁₃×#135	C₈	1	4	yes	—
26	C₁₃×#135	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×Q₁₆	C₈	1	1	yes	characteristic
28	C₂×C₁₃×SD₁₆	C₈	1	2	yes	—
29	C₂×C₁₃×D₈	C₈	1	1	yes	characteristic
30	C₁₃×#131	—	2	2	no	—
31	C₁₃×#131	—	2	2	no	—
32	C₁₃×#131	C₂×C₄	1	4	yes	—
33	C₁₃×#131	C₂³	1	4	yes	—
34	C₂×C₁₃×M₁₆	—	2	2	no	—
35	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₁₃×#108	C₂×C₄	1	4	yes	—
41	C₁₃×#104	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₃	C₂³	1	2	yes	—
44	#88052	—	2	2	no	—
45	#88052	—	2	8	no	—
46	#88052	C₂³	1	4	yes	—
47	C₂×#6090	—	2	2	no	—
48	C₂×#6090	—	2	1	no	—
49	C₂×#6090	—	2	1	no	—
50	C₂×#6090	—	2	8	no	—
51	C₂×#6090	D₄	1	1	yes	characteristic
52	C₂×#6090	D₄	1	1	yes	characteristic
53	C₂×#6090	D₄	1	2	yes	—
54	C₂×#6090	C₂³	1	2	yes	—
55	C₂×#6090	C₂³	1	2	yes	—
56	C₂×#2170	—	13	1	no	maximal, Sylow
57	C₁₃×#41	—	2	2	no	—
58	C₁₃×#41	—	2	2	no	—
59	C₁₃×#41	C₂×C₈	1	4	yes	—
60	C₁₃×#41	C₂²×C₄	1	4	yes	—
61	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₁₃×Q₁₆	C₂×C₈	1	4	yes	—
68	C₁₃×SD₁₆	C₂×C₈	1	8	yes	—
69	C₁₃×D₈	C₂×C₈	1	4	yes	—
70	C₁₃×M₁₆	—	2	2	no	—
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₁₃	—	2	1	no	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	#6090	—	2	4	no	—
90	#6090	—	2	2	no	—
91	#6090	C₂×D₄	1	2	yes	—
92	#6090	C₂×D₄	1	2	yes	—
93	C₂×#504	—	13	2	no	—
94	C₂×#443	D₁₃	1	1	yes	characteristic
95	#2170	—	13	8	no	—
96	C₂×#364	—	13	1	no	—
97	C₂×#350	—	13	2	no	—
98	C₂×C₄×C₁₆	—	13	1	no	—
99	C₂³×C₁₃	—	2	2	no	—
100	C₂³×C₁₃	#131	1	1	yes	characteristic
101	Q₈×C₁₃	—	2	2	no	—
102	Q₈×C₁₃	C₂²×C₈	1	4	yes	—
103	D₄×C₁₃	—	2	4	no	—
104	D₄×C₁₃	—	2	4	no	—
105	D₄×C₁₃	—	2	2	no	—
106	D₄×C₁₃	C₂²×C₈	1	4	yes	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₁₃	—	2	4	no	—
119	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₈×C₁₃	C₂²×C₈	1	2	yes	—
122	C₈×C₁₃	C₂²×C₈	1	2	yes	—
123	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×#135	Dic₁₃	1	1	yes	characteristic
126	C₂×#131	C₂×D₁₃	1	2	yes	—
127	#504	—	13	8	no	—
128	#504	—	26	4	no	—
129	C₂×M₃₂	—	13	2	no	—
130	C₂×M₃₂	—	26	2	no	—
131	C₂²×C₁₆	—	13	1	no	—
132	C₂²×C₁₆	—	13	1	no	—
133	C₂²×C₁₆	—	26	2	no	—
134	#443	Dic₁₃	1	4	yes	—
135	#443	C₂×D₁₃	1	4	yes	—
136	C₂×#108	Dic₁₃	1	1	yes	characteristic
137	C₂×#104	Dic₁₃	1	2	yes	—
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#364	—	13	4	no	—
140	#350	—	13	8	no	—
141	C₄×C₁₆	—	13	4	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	4	no	—
144	C₂²×C₁₃	—	4	4	no	—
145	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
146	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	2	4	no	—
150	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
151	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
152	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
154	C₂×#41	C₂×Dic₁₃	1	2	yes	—
155	#135	#1055	1	4	yes	—
156	#135	C₂×Dic₁₃	1	4	yes	—
157	C₂×Q₁₆	#1055	1	1	yes	characteristic
158	C₂×SD₁₆	#1055	1	2	yes	—
159	C₂×D₈	#1055	1	1	yes	characteristic
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	C₂×Dic₁₃	1	4	yes	—
163	#131	C₂²×D₁₃	1	4	yes	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
170	M₃₂	—	13	4	no	—
171	M₃₂	—	26	8	no	—
172	M₃₂	—	26	2	no	—
173	C₂×C₁₆	—	13	2	no	—
174	C₂×C₁₆	—	13	1	no	—
175	C₂×C₁₆	—	13	1	no	—
176	C₂×C₁₆	—	13	2	no	—
177	C₂×C₁₆	—	13	2	no	—
178	C₂×C₁₆	—	26	8	no	—
179	C₂×C₁₆	—	26	2	no	—
180	C₂×C₁₆	—	26	1	no	—
181	C₂×C₁₆	—	26	1	no	—
182	#108	C₂×Dic₁₃	1	4	yes	—
183	#104	C₂×Dic₁₃	1	8	yes	—
184	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
185	C₄×C₈	C₂²×D₁₃	1	2	yes	—
186	C₂×C₁₃	—	2	1	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	4	4	no	—
189	C₂×C₁₃	#2170	1	2	yes	complemented
190	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	C₂×#1055	1	4	yes	—
194	#41	C₂²×Dic₁₃	1	4	yes	—
195	C₂×Q₈	C₂×#1055	1	2	yes	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	C₂×#1055	1	2	yes	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
201	Q₁₆	C₂×#1055	1	4	yes	—
202	SD₁₆	C₂×#1055	1	8	yes	—
203	D₈	C₂×#1055	1	4	yes	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	C₂²×Dic₁₃	1	4	yes	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂×#1055	1	1	yes	characteristic
212	C₂×C₈	C₂×#1055	1	1	yes	characteristic
213	C₂×C₈	C₂×#1055	1	2	yes	—
214	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
215	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
216	C₂×C₈	#6129	1	1	yes	characteristic
217	C₂×C₈	#6129	1	1	yes	characteristic
218	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
219	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
221	C₄²	—	2	1	no	—
222	C₄²	C₂²×Dic₁₃	1	2	yes	—
223	C₁₆	—	13	2	no	—
224	C₁₆	—	13	2	no	—
225	C₁₆	—	26	4	no	—
226	C₁₆	—	26	2	no	—
227	C₁₃	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	2	2	no	—
229	C₂³	#88202	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	C₂²×#1055	1	4	yes	—
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	2	2	no	—
235	D₄	C₂²×#1055	1	4	yes	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
242	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
244	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
245	C₂×C₄	#88202	1	1	yes	characteristic
246	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	C₂²×#1055	1	2	yes	—
249	C₈	C₂²×#1055	1	2	yes	—
250	C₈	D₄×Dic₁₃	1	1	yes	characteristic
251	C₈	D₄×Dic₁₃	1	1	yes	characteristic
252	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
253	C₈	C₂×#6129	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	D₄×#1055	1	1	yes	characteristic
258	C₂²	D₄×#1055	1	1	yes	characteristic
259	C₂²	C₂×#88202	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	D₄×#1055	1	1	yes	characteristic
263	C₄	D₄×#1055	1	1	yes	characteristic
264	C₄	C₂³×#1055	1	1	yes	characteristic
265	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	4	4	no	—
269	C₂	#13116673	1	2	yes	direct product factor
270	C₂	C₂×D₄×#1055	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#443	C₂
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×C₁₃×#135	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#443	D₁₃
hypocenter	C₁₃	C₂×#2170
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#11718801	C₂	semidirect
2. #13116673	C₂	direct (N × H)
3. C₂×#11718055	C₂	semidirect
4. C₂×#11718153	C₂	semidirect
5. C₂×C₄×#6090	C₂	semidirect
15. #11718801	C₂²	semidirect
21. #11718153	C₂²	semidirect
22. #11718055	C₂²	semidirect
23. C₄×#6090	C₂²	semidirect
189. C₂×C₁₃	#2170	semidirect
227. C₁₃	C₂×#2170	semidirect
269. C₂	#13116673	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×#11718801	2
#13116673	2
C₂×#11718055	2
C₂×#11718153	2
C₂×C₄×#6090	2
C₂×C₁₃×#443	2
C₂×#2170	13

Subgroups — C2×#13116673 #16016147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116673 #16016147