Subgroups — C₂×#13123770 #16727284

Order 3328.

1982 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719175	C₂	1	1	yes	maximal, characteristic
2	#13123770	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718340	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×Dic₂₆	C₂	1	4	yes	maximal, complemented
5	C₂×#11718763	C₂	1	2	yes	maximal
6	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
7	C₂×#11718762	C₂	1	2	yes	maximal
8	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88180	C₂²	1	2	yes	—
13	C₂×#88178	C₄	1	4	yes	—
14	C₂²×#6099	C₂²	1	2	yes	—
15	#11719175	C₄	1	8	yes	—
16	#11719175	C₂²	1	8	yes	—
17	C₂×#88124	C₄	1	2	yes	—
18	C₂×#88123	C₄	1	2	yes	—
19	C₂×C₄×Dic₂₆	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	8	yes	—
23	#11718763	C₂²	1	16	yes	—
24	#11718762	C₂²	1	16	yes	—
25	C₂×#88055	C₂²	1	4	yes	—
26	C₂×#88054	C₂²	1	8	yes	—
27	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
28	#11718585	C₂²	1	8	yes	—
29	C₂×C₄⋊#1055	C₂²	1	4	yes	—
30	#11718340	C₂²	1	32	yes	complemented
31	C₈×Dic₂₆	C₂²	1	32	yes	complemented
32	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₂₆	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₁₃	—	2	4	no	—
36	C₂²×C₈×C₁₃	Q₈	1	8	yes	complemented
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
47	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
48	#88180	C₂×C₄	1	8	yes	—
49	#88180	C₂³	1	8	yes	—
50	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
52	#88178	C₂×C₄	1	32	yes	—
53	C₂×#6101	C₂×C₄	1	8	yes	—
54	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
55	C₂×#6099	—	2	8	no	—
56	C₂×#6099	C₂³	1	8	yes	—
57	C₂²×#1055	C₂³	1	4	yes	—
58	#88124	C₂×C₄	1	16	yes	—
59	#88123	C₂×C₄	1	16	yes	—
60	C₄×Dic₂₆	C₂×C₄	1	16	yes	—
61	C₄×Dic₂₆	C₂³	1	16	yes	—
62	C₄×C₈×C₁₃	C₂³	1	8	yes	—
63	C₄×C₈×C₁₃	C₂³	1	8	yes	—
64	#88055	C₂³	1	16	yes	—
65	#88054	C₂³	1	32	yes	—
66	C₈×Dic₁₃	C₂³	1	16	yes	—
67	C₄⋊#1055	C₂³	1	16	yes	—
68	C₂×#2960	—	13	1	no	maximal, Sylow
69	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	2	8	no	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₁₃	#41	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
80	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
81	C₂×Dic₂₆	—	2	16	no	—
82	C₂×Dic₂₆	C₂²×C₄	1	16	yes	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	8	no	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
86	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
87	C₂×C₈×C₁₃	C₂×Q₈	1	16	yes	complemented
88	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
89	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
90	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
93	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
94	C₄²×C₁₃	C₂⁴	1	4	yes	—
95	C₂²⋊Dic₁₃	C₂²×C₄	1	16	yes	—
96	C₄⋊Dic₁₃	C₂²×C₄	1	16	yes	—
97	C₄⋊Dic₁₃	C₂²×C₄	1	16	yes	—
98	#6101	C₂²×C₄	1	32	yes	—
99	C₄×Dic₁₃	C₂²×C₄	1	8	yes	—
100	C₄×Dic₁₃	C₂⁴	1	8	yes	—
101	#6099	—	2	16	no	—
102	C₂×#1055	—	2	8	no	—
103	C₂×#1055	C₂⁴	1	8	yes	—
104	C₂×#533	—	13	1	no	—
105	#2960	—	13	16	no	—
106	C₂×#446	—	13	4	no	—
107	C₂×C₈×Q₈	—	13	4	no	—
108	C₂×#424	—	13	2	no	—
109	C₂×#423	—	13	1	no	—
110	C₂×#405	—	13	2	no	—
111	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	—	2	4	no	—
113	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
114	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
115	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	16	no	—
118	C₂×C₄×C₁₃	C₄×Q₈	1	8	yes	—
119	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
120	C₂×C₄×C₁₃	#131	1	8	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
125	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
129	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
130	C₂×Dic₁₃	—	2	16	no	—
131	C₂×Dic₁₃	C₂³×C₄	1	16	yes	—
132	Dic₂₆	—	2	32	no	—
133	C₈×C₁₃	—	2	4	no	—
134	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
135	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
136	#1055	—	2	8	no	—
137	C₂²×M₁₆	—	13	2	no	—
138	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
140	#533	—	13	8	no	—
141	#533	—	13	8	no	—
142	C₂×C₄⋊Q₈	—	13	2	no	—
143	C₂×#125	—	13	2	no	—
144	C₂×C₂²⋊Q₈	—	13	4	no	—
145	C₂×C₄×Q₈	—	13	4	no	—
146	C₂×#117	—	13	2	no	—
147	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
148	#446	—	13	32	no	—
149	C₈×Q₈	—	13	32	no	—
150	#424	—	13	16	no	—
151	#423	—	13	8	no	—
152	C₂×C₄⋊C₈	—	13	8	no	—
153	C₂×C₄⋊C₈	—	13	4	no	—
154	#405	—	13	16	no	—
155	C₂×#97	—	13	4	no	—
156	C₂×C₄×C₈	—	13	4	no	—
157	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
158	C₂×C₄×C₈	C₂×D₁₃	1	8	yes	—
159	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
160	C₂²×C₁₃	—	2	8	no	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	C₂×C₄×Q₈	1	4	yes	—
163	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
164	C₂²×C₁₃	C₂×#131	1	4	yes	—
165	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
166	C₄×C₁₃	—	2	4	no	—
167	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
168	C₄×C₁₃	C₂×C₄×Q₈	1	4	yes	—
169	C₄×C₁₃	C₂×#131	1	8	yes	—
170	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
171	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
172	Dic₁₃	—	2	16	no	—
173	C₂²×Q₈	—	13	4	no	—
174	C₂³×C₄	C₄×D₁₃	1	2	yes	—
175	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
176	C₂×M₁₆	—	13	8	no	—
177	C₂×M₁₆	—	26	8	no	—
178	C₂²×C₈	—	2	4	no	—
179	C₂²×C₈	—	13	4	no	—
180	C₂²×C₈	Dic₂₆	1	8	yes	complemented
181	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
182	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
183	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
185	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₄⋊Q₈	—	13	16	no	—
187	#125	—	13	16	no	—
188	C₂²⋊Q₈	—	13	32	no	—
189	C₄×Q₈	—	13	16	no	—
190	C₄×Q₈	—	13	16	no	—
191	#117	—	13	8	no	—
192	#117	—	13	8	no	—
193	C₂×C₄⋊C₄	—	13	4	no	—
194	C₂×C₄⋊C₄	—	13	8	no	—
195	C₂×C₄⋊C₄	—	13	4	no	—
196	C₂×C₂²⋊C₄	—	13	4	no	—
197	C₂×C₄²	—	13	4	no	—
198	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₁₃	1	4	yes	—
201	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
202	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
203	C₄⋊C₈	—	13	32	no	—
204	C₄⋊C₈	—	13	16	no	—
205	#97	—	13	16	no	—
206	C₄×C₈	—	13	16	no	—
207	C₄×C₈	C₂²×D₁₃	1	8	yes	—
208	C₄×C₈	C₂²×D₁₃	1	8	yes	—
209	C₂×C₁₃	—	2	4	no	—
210	C₂×C₁₃	#2960	1	4	yes	complemented
211	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
212	C₂×C₁₃	C₂²×#131	1	2	yes	—
213	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂×Q₈	—	13	16	no	—
215	C₂×Q₈	—	26	16	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	13	8	no	—
218	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
219	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
221	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
224	C₂²×C₄	#6127	1	4	yes	—
225	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
227	M₁₆	—	26	16	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	8	no	—
230	C₂×C₈	—	13	8	no	—
231	C₂×C₈	—	26	8	no	—
232	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
233	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
234	C₂×C₈	C₂×Dic₂₆	1	16	yes	complemented
235	C₂×C₈	C₂³×D₁₃	1	4	yes	—
236	C₂×C₈	C₂³×D₁₃	1	2	yes	—
237	C₂×C₈	C₂³×D₁₃	1	2	yes	—
238	C₄⋊C₄	—	13	16	no	—
239	C₄⋊C₄	—	13	32	no	—
240	C₄⋊C₄	—	13	16	no	—
241	C₂²⋊C₄	—	13	16	no	—
242	C₄²	—	13	8	no	—
243	C₄²	—	13	8	no	—
244	C₄²	C₂×C₄×D₁₃	1	4	yes	—
245	C₄²	C₂×C₄×D₁₃	1	4	yes	—
246	C₄²	C₂×C₄×D₁₃	1	4	yes	—
247	C₄²	C₂³×D₁₃	1	4	yes	—
248	C₁₃	C₂×#2960	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	2	4	no	—
250	C₂³	C₄×Dic₂₆	1	4	yes	—
251	C₂³	C₂²×C₄×D₁₃	1	2	yes	—
252	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
253	Q₈	—	26	32	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	13	16	no	—
257	C₂×C₄	—	26	16	no	—
258	C₂×C₄	C₄×Dic₂₆	1	8	yes	—
259	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
260	C₂×C₄	#88155	1	8	yes	—
261	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
262	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
264	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
266	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
267	C₂×C₄	C₂×#6127	1	2	yes	—
268	C₂×C₄	C₂×#6127	1	4	yes	—
269	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	—	26	8	no	—
272	C₈	C₂²×Dic₂₆	1	4	yes	—
273	C₈	C₂²×Dic₂₆	1	4	yes	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	8	no	—
276	C₂²	C₂×C₄×Dic₂₆	1	4	yes	—
277	C₂²	C₂×C₄×Dic₂₆	1	2	yes	—
278	C₂²	C₂×#88155	1	4	yes	—
279	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	26	16	no	—
282	C₄	C₂×C₄×Dic₂₆	1	2	yes	—
283	C₄	C₂×C₄×Dic₂₆	1	4	yes	—
284	C₄	C₂×#88155	1	8	yes	—
285	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
286	C₄	C₂²×#6127	1	1	yes	characteristic
287	C₂	—	2	4	no	—
288	C₂	#13123770	1	4	yes	direct product factor
289	C₂	C₂²×C₄×Dic₂₆	1	1	yes	characteristic
290	C₂	C₂²×#88155	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#2960
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13123770	C₂	direct (N × H)
3. C₂×#11718340	C₂	semidirect
4. C₂×C₈×Dic₂₆	C₂	semidirect
30. #11718340	C₂²	semidirect
31. C₈×Dic₂₆	C₂²	semidirect
36. C₂²×C₈×C₁₃	Q₈	semidirect
87. C₂×C₈×C₁₃	C₂×Q₈	semidirect
180. C₂²×C₈	Dic₂₆	semidirect
210. C₂×C₁₃	#2960	semidirect
234. C₂×C₈	C₂×Dic₂₆	semidirect
248. C₁₃	C₂×#2960	semidirect
288. C₂	#13123770	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×#11719175	2
#13123770	2
C₂×#11718340	2
C₂×C₈×Dic₂₆	2
C₂×#11718763	2
C₂×#11718585	2
C₂×#11718762	2
C₂²×C₄×C₈×C₁₃	2
C₂×#2960	13

Subgroups — C2×#13123770 #16727284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123770 #16727284