Subgroups — C₂×#13116953 #16017473

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718562	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718821	C₂	1	2	yes	maximal, complemented
3	#13116953	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718098	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718155	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
8	C₂×#88186	—	2	2	no	—
9	C₂×#88155	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×D₁₆	—	2	1	no	—
11	C₂×C₁₃×D₁₆	—	2	1	no	—
12	C₂×C₁₃×D₁₆	—	2	2	no	—
13	C₂×C₁₃×D₁₆	C₂²	1	4	yes	complemented
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×#88073	—	2	2	no	—
16	#11718821	—	2	4	no	—
17	#11718821	C₂²	1	8	yes	complemented
18	C₂×#88059	C₂²	1	2	yes	—
19	C₂×#88052	C₂²	1	1	yes	characteristic
20	#11718562	—	2	2	no	—
21	#11718562	C₂²	1	4	yes	complemented
22	C₂×#88038	—	2	1	no	—
23	C₂×C₁₆×D₁₃	—	2	1	no	—
24	#11718155	C₂²	1	8	yes	complemented
25	#11718098	C₂²	1	4	yes	complemented
26	C₂²×D₄×C₁₃	—	2	2	no	—
27	C₂×#6127	D₄	1	1	yes	characteristic
28	C₂×C₁₃×D₈	—	2	2	no	—
29	C₂×C₁₃×D₈	—	2	2	no	—
30	C₂×C₁₃×D₈	—	2	2	no	—
31	C₂×C₁₃×D₈	—	2	2	no	—
32	C₂×C₁₃×D₈	—	2	2	no	—
33	C₂×C₁₃×D₈	C₂³	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×#6105	—	4	2	no	—
36	#88186	—	2	8	no	—
37	#88186	—	4	4	no	—
38	C₂×#6104	—	4	2	no	—
39	C₂×#6099	—	2	2	no	—
40	C₂×#6099	D₄	1	1	yes	characteristic
41	#88155	—	2	1	no	—
42	#88155	—	2	1	no	—
43	#88155	D₄	1	2	yes	—
44	#88155	C₂³	1	2	yes	—
45	C₂×#6094	—	2	1	no	—
46	C₂×C₈×D₁₃	—	2	1	no	—
47	C₁₃×D₁₆	—	2	4	no	—
48	C₁₃×D₁₆	—	2	4	no	—
49	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
50	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
51	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
52	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
53	#88073	—	2	8	no	—
54	#88059	—	2	2	no	—
55	#88059	C₂³	1	4	yes	—
56	#88052	—	2	1	no	—
57	#88052	C₂³	1	2	yes	—
58	C₂×#6090	—	2	1	no	—
59	#88038	—	2	4	no	—
60	C₁₆×D₁₃	—	2	4	no	—
61	C₂×#2221	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	—	4	2	no	—
63	C₂×D₄×C₁₃	—	2	4	no	—
64	C₂×D₄×C₁₃	—	4	2	no	—
65	C₂×D₄×C₁₃	—	4	2	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	4	2	no	—
68	C₂×D₄×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₃	—	2	1	no	—
71	#6127	—	2	1	no	—
72	#6127	—	2	1	no	—
73	#6127	D₈	1	2	yes	—
74	#6127	C₂×D₄	1	2	yes	—
75	C₂×D₅₂	D₈	1	1	yes	characteristic
76	C₂×C₄×D₁₃	—	2	1	no	—
77	C₂×Dic₂₆	D₈	1	1	yes	characteristic
78	C₁₃×D₈	—	2	4	no	—
79	C₁₃×D₈	—	2	4	no	—
80	C₁₃×D₈	—	4	2	no	—
81	C₁₃×D₈	—	4	2	no	—
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	#6105	—	4	8	no	—
89	#6104	—	4	8	no	—
90	#6099	—	2	1	no	—
91	#6099	—	2	4	no	—
92	#6099	—	4	2	no	—
93	#6099	C₂×D₄	1	2	yes	—
94	C₂×#1055	—	2	1	no	—
95	C₂×#1055	—	4	2	no	—
96	#6094	—	2	2	no	—
97	#6094	—	2	2	no	—
98	C₈×D₁₃	—	2	2	no	—
99	C₈×D₁₃	—	2	2	no	—
100	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
101	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
102	#6090	—	2	2	no	—
103	C₂²×D₁₆	D₁₃	1	1	yes	characteristic, complemented
104	C₂×#504	—	13	1	no	—
105	C₂×#472	—	13	2	no	—
106	#2221	—	13	8	no	—
107	C₂×#368	—	13	1	no	—
108	C₂×#361	—	13	2	no	—
109	C₂³×C₁₃	—	4	4	no	—
110	C₂³×C₁₃	—	8	2	no	—
111	C₂³×C₁₃	—	8	2	no	—
112	C₂³×C₁₃	—	16	4	no	—
113	C₂³×C₁₃	#135	1	1	yes	characteristic
114	C₂²×D₁₃	—	2	1	no	—
115	D₄×C₁₃	—	4	4	no	—
116	D₄×C₁₃	—	4	4	no	—
117	D₄×C₁₃	—	8	2	no	—
118	D₄×C₁₃	—	8	2	no	—
119	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₁₃	—	2	2	no	—
126	C₂²⋊D₁₃	—	2	2	no	—
127	C₂×Dic₁₃	—	2	1	no	—
128	D₅₂	—	2	1	no	—
129	D₅₂	C₂×D₈	1	2	yes	—
130	C₄×D₁₃	—	2	2	no	—
131	C₄×D₁₃	—	2	2	no	—
132	Dic₂₆	—	2	1	no	—
133	Dic₂₆	C₂×D₈	1	2	yes	—
134	C₈×C₁₃	#121	1	1	yes	characteristic
135	C₈×C₁₃	#121	1	1	yes	characteristic
136	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
138	#1055	—	2	2	no	—
139	#1055	—	4	4	no	—
140	C₂×AGL(1,ℤ/8)	—	26	2	no	—
141	C₂²×D₈	C₂×D₁₃	1	2	yes	—
142	C₂×#131	—	13	1	no	—
143	C₂×D₁₆	—	2	1	no	—
144	C₂×D₁₆	—	2	1	no	—
145	C₂×D₁₆	—	2	2	no	—
146	C₂×D₁₆	C₂×D₁₃	1	4	yes	complemented
147	#504	—	13	4	no	—
148	#504	—	26	2	no	—
149	C₂×M₃₂	—	13	1	no	—
150	C₂×M₃₂	—	26	1	no	—
151	C₂²×C₁₆	—	26	1	no	—
152	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
153	#472	—	13	8	no	—
154	#472	—	26	4	no	—
155	C₂×#108	—	13	2	no	—
156	C₂×#100	—	26	2	no	—
157	#368	—	13	4	no	—
158	#361	—	13	8	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	8	4	no	—
161	C₂²×C₁₃	—	16	4	no	—
162	C₂²×C₁₃	—	16	4	no	—
163	C₂²×C₁₃	—	16	4	no	—
164	C₂²×C₁₃	#466	1	1	yes	characteristic
165	C₂²×C₁₃	#466	1	1	yes	characteristic
166	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
167	C₂×D₁₃	—	2	2	no	—
168	C₂×D₁₃	—	4	2	no	—
169	C₄×C₁₃	#466	1	1	yes	characteristic
170	C₄×C₁₃	#466	1	1	yes	characteristic
171	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
172	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
173	Dic₁₃	—	2	2	no	—
174	C₂×#41	—	13	1	no	—
175	C₂²×D₄	—	2	2	no	—
176	AGL(1,ℤ/8)	—	26	8	no	—
177	AGL(1,ℤ/8)	—	52	4	no	—
178	C₂×SD₁₆	—	52	2	no	—
179	C₂×D₈	—	2	2	no	—
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	—	2	2	no	—
183	C₂×D₈	—	2	2	no	—
184	C₂×D₈	—	52	2	no	—
185	C₂×D₈	C₂²×D₁₃	1	4	yes	—
186	#131	—	13	2	no	—
187	#131	—	13	2	no	—
188	#131	—	26	1	no	—
189	#131	—	26	1	no	—
190	C₂×M₁₆	—	13	1	no	—
191	C₂×M₁₆	—	26	1	no	—
192	C₂×M₁₆	—	26	2	no	—
193	C₂²×C₈	—	26	1	no	—
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	D₁₆	—	2	4	no	—
196	D₁₆	—	2	4	no	—
197	M₃₂	—	13	2	no	—
198	M₃₂	—	26	1	no	—
199	M₃₂	—	26	4	no	—
200	C₂×C₁₆	—	26	1	no	—
201	C₂×C₁₆	—	26	4	no	—
202	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
203	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
204	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
205	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
206	#108	—	13	4	no	—
207	#108	—	26	2	no	—
208	#100	—	26	8	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	—	16	4	no	—
212	C₂×C₁₃	#2221	1	2	yes	complemented
213	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
214	D₁₃	—	4	2	no	—
215	C₂⁴	—	4	2	no	—
216	#41	—	13	2	no	—
217	#41	—	13	2	no	—
218	#41	—	26	1	no	—
219	#41	—	26	1	no	—
220	C₂×Q₈	—	13	1	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	13	1	no	—
228	C₂×D₄	—	26	1	no	—
229	C₂²×C₄	—	26	1	no	—
230	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
231	SD₁₆	—	52	8	no	—
232	D₈	—	2	4	no	—
233	D₈	—	2	4	no	—
234	D₈	—	4	2	no	—
235	D₈	—	4	2	no	—
236	D₈	—	52	8	no	—
237	M₁₆	—	13	2	no	—
238	M₁₆	—	26	1	no	—
239	M₁₆	—	26	2	no	—
240	M₁₆	—	26	2	no	—
241	M₁₆	—	26	4	no	—
242	M₁₆	—	52	2	no	—
243	C₂×C₈	—	26	1	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	—	26	2	no	—
246	C₂×C₈	—	52	2	no	—
247	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
248	C₂×C₈	#6129	1	1	yes	characteristic
249	C₂×C₈	#6129	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
253	C₁₆	—	26	2	no	—
254	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
255	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
256	C₁₃	C₂×#2221	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	4	4	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	16	4	no	—
261	C₂³	—	26	1	no	—
262	C₂³	#88166	1	1	yes	characteristic
263	Q₈	—	13	2	no	—
264	Q₈	—	26	1	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	2	no	—
268	D₄	—	8	2	no	—
269	D₄	—	13	2	no	—
270	D₄	—	26	2	no	—
271	D₄	—	26	2	no	—
272	D₄	—	26	1	no	—
273	C₂×C₄	—	26	1	no	—
274	C₂×C₄	—	26	2	no	—
275	C₂×C₄	—	26	2	no	—
276	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
277	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
278	C₂×C₄	#88166	1	1	yes	characteristic
279	C₂×C₄	#88192	1	1	yes	characteristic
280	C₂×C₄	#88192	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
282	C₈	—	26	2	no	—
283	C₈	—	52	4	no	—
284	C₈	#88192	1	1	yes	characteristic
285	C₈	#88192	1	1	yes	characteristic
286	C₈	C₂×#6129	1	1	yes	characteristic
287	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	16	4	no	—
291	C₂²	—	16	4	no	—
292	C₂²	—	16	4	no	—
293	C₂²	—	26	2	no	—
294	C₂²	—	52	2	no	—
295	C₂²	#11718818	1	1	yes	characteristic
296	C₂²	#11718818	1	1	yes	characteristic
297	C₂²	C₂×#88166	1	1	yes	characteristic
298	C₄	—	26	2	no	—
299	C₄	#11718818	1	1	yes	characteristic
300	C₄	#11718818	1	1	yes	characteristic
301	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
302	C₄	C₂×#88192	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	16	4	no	—
306	C₂	—	52	2	no	—
307	C₂	#13116953	1	2	yes	direct product factor
308	C₂	C₂×#11718818	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718818
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×D₁₆	C₂
socle	C₂²×C₁₃	#466
hypercenter	C₂²×D₁₆	D₁₃
hypocenter	C₁₃	C₂×#2221
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11718562	C₂	semidirect
2. C₂×#11718821	C₂	semidirect
3. #13116953	C₂	direct (N × H)
4. C₂×#11718098	C₂	semidirect
5. C₂×#11718155	C₂	semidirect
6. C₂²×C₁₃×D₁₆	C₂	semidirect
13. C₂×C₁₃×D₁₆	C₂²	semidirect
17. #11718821	C₂²	semidirect
21. #11718562	C₂²	semidirect
24. #11718155	C₂²	semidirect
25. #11718098	C₂²	semidirect
103. C₂²×D₁₆	D₁₃	semidirect
146. C₂×D₁₆	C₂×D₁₃	semidirect
212. C₂×C₁₃	#2221	semidirect
256. C₁₃	C₂×#2221	semidirect
307. C₂	#13116953	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×#11718562	2
C₂×#11718821	2
#13116953	2
C₂×#11718098	2
C₂×#11718155	2
C₂²×C₁₃×D₁₆	2
C₂×#2221	13

Subgroups — C2×#13116953 #16017473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116953 #16017473