Subgroups — C₂×#13116876 #16017163

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88068	C₂	1	2	yes	maximal, complemented
2	C₂×#11718356	C₂	1	1	yes	maximal, characteristic
3	#13116876	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718083	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
9	C₂²×D₁₀₄	—	2	1	no	—
10	C₂²×D₁₀₄	C₂²	1	2	yes	—
11	C₂×#88128	—	2	1	no	—
12	C₄×C₁₃×Q₁₆	—	2	2	no	—
13	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#103	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88068	—	2	8	no	—
18	C₂×#88068	C₂²	1	8	yes	complemented
19	C₂²×#6090	C₂²	1	2	yes	—
20	#11718356	—	2	2	no	—
21	#11718356	C₂²	1	4	yes	—
22	#11718083	C₂²	1	8	yes	complemented
23	C₄⋊#6090	C₂²	1	4	yes	complemented
24	C₂²×Q₈×C₁₃	—	2	1	no	—
25	C₂²×D₅₂	—	2	2	no	—
26	C₂²×D₅₂	—	4	1	no	—
27	C₂²×D₅₂	—	4	1	no	—
28	C₂×C₁₃×Q₁₆	—	2	1	no	—
29	C₂×C₁₃×Q₁₆	—	2	2	no	—
30	C₂×C₁₃×Q₁₆	D₄	1	4	yes	complemented
31	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₁₃	—	2	4	no	—
35	C₄×Q₈×C₁₃	—	4	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
38	C₂×C₄²×C₁₃	—	4	1	no	—
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×D₁₀₄	—	2	4	no	—
41	C₂×D₁₀₄	—	2	4	no	—
42	C₂×D₁₀₄	—	2	4	no	—
43	C₂×D₁₀₄	—	2	2	no	—
44	C₂×D₁₀₄	—	4	4	no	—
45	C₂×D₁₀₄	C₂³	1	4	yes	—
46	#88128	—	2	4	no	—
47	#88128	—	4	2	no	—
48	C₁₃×#107	—	2	1	no	—
49	C₁₃×#107	C₂³	1	2	yes	—
50	C₁₃×#103	—	2	4	no	—
51	C₄×C₈×C₁₃	D₄	1	2	yes	—
52	C₄×C₈×C₁₃	C₂³	1	2	yes	—
53	#88068	—	2	16	no	—
54	C₂×#6090	—	2	4	no	—
55	C₂×#6090	C₂³	1	4	yes	—
56	C₂×#2207	—	13	1	no	maximal, Sylow
57	C₂³×D₁₃	—	4	2	no	—
58	C₂×Q₈×C₁₃	—	2	4	no	—
59	C₂×Q₈×C₁₃	—	2	2	no	—
60	C₂×Q₈×C₁₃	—	4	1	no	—
61	C₂×Q₈×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×D₅₂	—	2	4	no	—
68	C₂×D₅₂	—	4	4	no	—
69	C₂×D₅₂	—	4	4	no	—
70	C₂×D₅₂	—	4	4	no	—
71	C₂×D₅₂	—	4	4	no	—
72	C₂×D₅₂	—	4	2	no	—
73	C₂×D₅₂	—	8	4	no	—
74	C₂×D₅₂	—	8	4	no	—
75	C₁₃×Q₁₆	—	2	4	no	—
76	C₁₃×Q₁₆	—	2	2	no	—
77	C₁₃×Q₁₆	C₂×D₄	1	4	yes	complemented
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃	#41	1	2	yes	—
83	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	4	1	no	—
86	C₁₃×C₄⋊C₄	—	4	4	no	—
87	C₄²×C₁₃	—	4	4	no	—
88	C₄²×C₁₃	D₈	1	2	yes	—
89	C₄²×C₁₃	C₂×D₄	1	2	yes	—
90	D₁₀₄	—	2	4	no	—
91	D₁₀₄	—	2	4	no	—
92	D₁₀₄	—	4	8	no	—
93	D₁₀₄	—	4	4	no	—
94	#6090	—	2	4	no	—
95	C₂²×SD₃₂	—	13	2	no	—
96	C₂×#493	—	13	1	no	—
97	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
98	#2207	—	13	8	no	—
99	C₂×C₄⋊C₁₆	—	13	1	no	—
100	C₂×#357	—	13	2	no	—
101	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₁₃	—	4	4	no	—
103	C₂²×D₁₃	—	8	4	no	—
104	C₂²×D₁₃	—	8	4	no	—
105	C₂²×D₁₃	—	8	4	no	—
106	Q₈×C₁₃	—	2	4	no	—
107	Q₈×C₁₃	—	4	2	no	—
108	Q₈×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	8	2	no	—
114	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
115	C₂×C₄×C₁₃	#121	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	D₅₂	—	4	4	no	—
122	D₅₂	—	4	4	no	—
123	D₅₂	—	8	8	no	—
124	D₅₂	—	8	4	no	—
125	D₅₂	—	8	8	no	—
126	C₈×C₁₃	—	2	2	no	—
127	C₈×C₁₃	#121	1	2	yes	—
128	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
131	C₂²×D₈	—	13	2	no	—
132	C₂²×D₈	—	26	1	no	—
133	C₂×#127	—	26	1	no	—
134	C₂×C₄×Q₈	—	2	1	no	—
135	C₂×SD₃₂	—	13	8	no	—
136	C₂×SD₃₂	—	26	8	no	—
137	C₂²×C₁₆	—	13	2	no	—
138	#493	—	13	4	no	—
139	#493	—	26	2	no	—
140	C₄×Q₁₆	—	2	2	no	—
141	C₄×Q₁₆	C₂×D₁₃	1	4	yes	complemented
142	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
143	C₂×#103	—	2	1	no	—
144	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
145	C₄⋊C₁₆	—	13	4	no	—
146	#357	—	13	8	no	—
147	C₂²×C₁₃	#459	1	2	yes	—
148	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
149	C₂²×C₁₃	#509	1	2	yes	—
150	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₁₃	—	8	4	no	—
152	C₂×D₁₃	—	8	4	no	—
153	C₂×D₁₃	—	8	4	no	—
154	C₂×D₁₃	—	16	8	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	4	4	no	—
157	C₄×C₁₃	—	8	2	no	—
158	C₄×C₁₃	#459	1	2	yes	—
159	C₄×C₁₃	C₂×SD₃₂	1	4	yes	complemented
160	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
161	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×D₄	—	26	2	no	—
164	C₂²×D₄	—	52	1	no	—
165	C₂²×D₄	—	52	1	no	—
166	C₂×Q₁₆	—	2	2	no	—
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	D₅₂	1	4	yes	complemented
169	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
170	C₂×D₈	—	13	4	no	—
171	C₂×D₈	—	26	4	no	—
172	C₂×D₈	—	26	4	no	—
173	C₂×D₈	—	26	2	no	—
174	C₂×D₈	—	26	4	no	—
175	C₂×D₈	—	52	4	no	—
176	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	#127	—	26	4	no	—
179	#127	—	52	2	no	—
180	C₄×Q₈	—	2	4	no	—
181	C₄×Q₈	—	4	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	4	1	no	—
184	C₂×C₄²	—	4	1	no	—
185	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
186	SD₃₂	—	26	16	no	—
187	C₂×C₁₆	—	13	4	no	—
188	C₂×C₁₆	—	26	4	no	—
189	#107	—	2	1	no	—
190	#107	C₂²×D₁₃	1	2	yes	—
191	#103	—	2	4	no	—
192	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
193	C₄×C₈	C₂²×D₁₃	1	2	yes	—
194	C₂×C₁₃	#2207	1	4	yes	complemented
195	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
196	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
197	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
198	D₁₃	—	16	4	no	—
199	C₂⁴	—	52	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂×D₄	—	26	4	no	—
205	C₂×D₄	—	52	4	no	—
206	C₂×D₄	—	52	4	no	—
207	C₂×D₄	—	52	4	no	—
208	C₂×D₄	—	52	2	no	—
209	C₂×D₄	—	52	4	no	—
210	C₂×D₄	—	104	4	no	—
211	C₂×D₄	—	104	4	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	#6104	1	1	yes	characteristic
215	C₂²×C₄	#6104	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	Q₁₆	—	2	2	no	—
219	Q₁₆	C₂×D₅₂	1	4	yes	complemented
220	D₈	—	26	4	no	—
221	D₈	—	26	4	no	—
222	D₈	—	52	8	no	—
223	D₈	—	52	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×D₅₂	1	2	yes	—
226	C₂×C₈	#6127	1	2	yes	—
227	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
228	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄²	—	4	4	no	—
234	C₄²	#6104	1	2	yes	—
235	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
236	C₁₆	—	26	4	no	—
237	C₁₃	C₂×#2207	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	52	4	no	—
239	C₂³	—	104	4	no	—
240	C₂³	—	104	4	no	—
241	C₂³	—	104	4	no	—
242	C₂³	C₂×#6104	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	4	4	no	—
245	Q₈	—	4	2	no	—
246	D₄	—	52	4	no	—
247	D₄	—	52	4	no	—
248	D₄	—	104	8	no	—
249	D₄	—	104	4	no	—
250	D₄	—	104	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	#88068	1	4	yes	complemented
257	C₂×C₄	#88184	1	2	yes	—
258	C₂×C₄	C₂×#6104	1	2	yes	—
259	C₂×C₄	C₂×#6104	1	1	yes	characteristic
260	C₂×C₄	C₂×#6104	1	2	yes	—
261	C₂×C₄	#88203	1	2	yes	—
262	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	#88184	1	2	yes	—
265	C₈	C₂²×D₅₂	1	1	yes	characteristic
266	C₈	C₂×#6127	1	1	yes	characteristic
267	C₂²	—	104	4	no	—
268	C₂²	—	104	4	no	—
269	C₂²	—	104	4	no	—
270	C₂²	—	208	8	no	—
271	C₂²	#11718676	1	2	yes	—
272	C₂²	C₂×#88068	1	2	yes	—
273	C₂²	#11718854	1	2	yes	—
274	C₂²	C₂²×#6104	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	8	2	no	—
278	C₄	#11718676	1	2	yes	—
279	C₄	C₂×#88068	1	4	yes	complemented
280	C₄	C₂×#88184	1	1	yes	characteristic
281	C₄	C₂×#88203	1	1	yes	characteristic
282	C₂	—	208	4	no	—
283	C₂	#13116876	1	4	yes	direct product factor
284	C₂	C₂×#11718676	1	1	yes	characteristic
285	C₂	C₂²×#88068	1	1	yes	characteristic
286	C₂	C₂×#11718854	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂×#11718356	C₂
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	C₂×#2207
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#88068	C₂	semidirect
3. #13116876	C₂	direct (N × H)
4. C₂×C₄⋊#6090	C₂	semidirect
5. C₂×#11718083	C₂	semidirect
6. C₂×C₄×C₁₃×Q₁₆	C₂	semidirect
13. C₄×C₁₃×Q₁₆	C₂²	semidirect
18. C₂×#88068	C₂²	semidirect
22. #11718083	C₂²	semidirect
23. C₄⋊#6090	C₂²	semidirect
30. C₂×C₁₃×Q₁₆	D₄	semidirect
77. C₁₃×Q₁₆	C₂×D₄	semidirect
97. C₂×C₄×Q₁₆	D₁₃	semidirect
114. C₂×C₄×C₁₃	SD₃₂	semidirect
141. C₄×Q₁₆	C₂×D₁₃	semidirect
159. C₄×C₁₃	C₂×SD₃₂	semidirect
168. C₂×Q₁₆	D₅₂	semidirect
194. C₂×C₁₃	#2207	semidirect
219. Q₁₆	C₂×D₅₂	semidirect
237. C₁₃	C₂×#2207	semidirect
256. C₂×C₄	#88068	semidirect
279. C₄	C₂×#88068	semidirect
283. C₂	#13116876	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88068	2
C₂×#11718356	2
#13116876	2
C₂×C₄⋊#6090	2
C₂×#11718083	2
C₂×C₄×C₁₃×Q₁₆	2
C₂×#2207	13

Subgroups — C2×#13116876 #16017163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116876 #16017163