Subgroups — C₂×#13116648 #16016041

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116648	C₂	1	8	yes	maximal, direct product factor
2	C₂×#11718055	C₂	1	2	yes	maximal, complemented
3	C₂×#11718145	C₂	1	2	yes	maximal
4	C₂×#11718058	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
9	C₄×C₁₃×M₁₆	C₄	1	4	yes	—
10	C₄×C₁₃×M₁₆	C₂²	1	4	yes	—
11	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
12	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	4	yes	—
16	#11718145	C₂²	1	8	yes	—
17	#11718058	C₂²	1	8	yes	complemented
18	#11718055	C₂²	1	8	yes	complemented
19	C₂³×C₄×C₁₃	—	2	1	no	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₃	—	2	2	no	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
37	C₁₃×#97	C₂×C₄	1	4	yes	—
38	C₁₃×#97	C₂×C₄	1	2	yes	—
39	C₁₃×#97	C₂³	1	2	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₃	C₂³	1	2	yes	—
43	#88052	—	2	4	no	—
44	#88052	C₂³	1	8	yes	—
45	C₂×#6090	—	2	4	no	—
46	C₂×#2165	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	4	4	no	—
56	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₁₃×M₁₆	—	2	4	no	—
61	C₁₃×M₁₆	—	2	8	no	—
62	C₁₃×M₁₆	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
68	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₃	#41	1	2	yes	—
74	C₂×C₈×C₁₃	#41	1	2	yes	—
75	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₄²×C₁₃	—	2	1	no	—
77	C₄²×C₁₃	—	2	4	no	—
78	C₄²×C₁₃	—	2	1	no	—
79	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	#6090	—	2	8	no	—
82	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
83	#2165	—	13	8	no	—
84	C₂×#364	—	13	2	no	—
85	C₂×#349	—	13	2	no	—
86	C₂×#347	—	13	2	no	—
87	C₂³×C₁₃	—	2	1	no	—
88	C₂³×C₁₃	—	2	1	no	—
89	C₂³×C₁₃	—	4	2	no	—
90	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	4	8	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	#117	1	2	yes	—
113	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
114	C₈×C₁₃	C₂×#41	1	2	yes	—
115	C₂²×M₁₆	Dic₁₃	1	1	yes	characteristic
116	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
117	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
118	C₂×M₃₂	—	13	4	no	—
119	C₄×M₁₆	Dic₁₃	1	4	yes	—
120	C₄×M₁₆	C₂×D₁₃	1	4	yes	—
121	C₂×#97	Dic₁₃	1	1	yes	characteristic
122	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
123	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
125	#364	—	13	8	no	—
126	#349	—	13	8	no	—
127	#347	—	13	8	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	4	2	no	—
131	C₂²×C₁₃	—	4	2	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	#432	1	1	yes	characteristic
134	C₂²×C₁₃	#432	1	1	yes	characteristic
135	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	4	2	no	—
139	C₄×C₁₃	#432	1	1	yes	characteristic
140	C₄×C₁₃	#432	1	1	yes	characteristic
141	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
142	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
143	C₂³×C₄	—	2	1	no	—
144	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
150	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
151	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
152	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
153	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
159	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
161	M₃₂	—	13	8	no	—
162	M₃₂	—	26	4	no	—
163	C₂×C₁₆	—	26	4	no	—
164	#97	C₂×Dic₁₃	1	4	yes	—
165	#97	C₂×Dic₁₃	1	2	yes	—
166	#97	C₂²×D₁₃	1	2	yes	—
167	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
168	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
169	C₄×C₈	C₂²×D₁₃	1	2	yes	—
170	C₂×C₁₃	—	2	1	no	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	4	2	no	—
173	C₂×C₁₃	#2165	1	2	yes	complemented
174	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
175	C₂⁴	#6099	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	4	4	no	—
184	C₂²×C₄	#6099	1	1	yes	characteristic
185	C₂²×C₄	#6099	1	2	yes	—
186	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
187	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
188	M₁₆	—	2	8	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	#6099	1	2	yes	—
196	C₂×C₈	#6099	1	2	yes	—
197	C₂×C₈	#6127	1	2	yes	—
198	C₂×C₈	#6127	1	2	yes	—
199	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
200	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
201	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
202	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
204	C₄²	—	2	1	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	1	no	—
207	C₄²	C₂²×Dic₁₃	1	2	yes	—
208	C₄²	C₂²×Dic₁₃	1	2	yes	—
209	C₁₆	—	26	8	no	—
210	C₁₃	C₂×#2165	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₂×#6099	1	1	yes	characteristic
215	C₂³	C₂×#6099	1	2	yes	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×#6099	1	2	yes	—
228	C₂×C₄	C₂×#6099	1	1	yes	characteristic
229	C₂×C₄	C₂×#6099	1	1	yes	characteristic
230	C₂×C₄	C₂×#6099	1	1	yes	characteristic
231	C₂×C₄	#88180	1	1	yes	characteristic
232	C₂×C₄	#88180	1	1	yes	characteristic
233	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	C₂×#6099	1	4	yes	—
237	C₈	#88180	1	2	yes	—
238	C₈	C₂×#6127	1	2	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	#11718587	1	1	yes	characteristic
245	C₂²	#11718587	1	1	yes	characteristic
246	C₂²	C₂²×#6099	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	2	no	—
250	C₄	#11718587	1	1	yes	characteristic
251	C₄	#11718587	1	1	yes	characteristic
252	C₄	C₂²×#6099	1	1	yes	characteristic
253	C₄	C₂×#88180	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	#13116648	1	2	yes	direct product factor
258	C₂	C₂×#11718587	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88180
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	#432
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	C₂×#2165
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #13116648	C₂	direct (N × H)
2. C₂×#11718055	C₂	semidirect
4. C₂×#11718058	C₂	semidirect
17. #11718058	C₂²	semidirect
18. #11718055	C₂²	semidirect
173. C₂×C₁₃	#2165	semidirect
210. C₁₃	C₂×#2165	semidirect
257. C₂	#13116648	direct (N × H)

Maximal subgroups

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

subgroup	index
#13116648	2
C₂×#11718055	2
C₂×#11718145	2
C₂×#11718058	2
C₂×C₄×C₁₃×M₁₆	2
C₂×#2165	13

Subgroups — C2×#13116648 #16016041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116648 #16016041