Subgroups — C₂×#13116871 #16017143

Order 3328.

870 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88066	C₂	1	2	yes	maximal, complemented
2	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116871	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×D₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×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×D₈	—	2	2	no	—
13	C₄×C₁₃×D₈	C₂²	1	4	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#102	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88066	—	2	8	no	—
18	C₂×#88066	C₂²	1	8	yes	complemented
19	C₂²×#6090	C₂²	1	2	yes	—
20	#11718356	—	2	2	no	—
21	#11718356	C₂²	1	4	yes	complemented
22	#11718083	C₂²	1	8	yes	complemented
23	C₄⋊#6090	C₂²	1	4	yes	complemented
24	C₂²×D₄×C₁₃	—	2	1	no	—
25	C₂³×C₄×C₁₃	—	4	1	no	—
26	C₂²×D₅₂	—	2	2	no	—
27	C₂²×D₅₂	—	4	1	no	—
28	C₂²×D₅₂	—	4	1	no	—
29	C₂×C₁₃×D₈	—	2	1	no	—
30	C₂×C₁₃×D₈	—	2	2	no	—
31	C₂×C₁₃×D₈	D₄	1	4	yes	complemented
32	C₂×C₁₃×D₈	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₁₃	—	2	4	no	—
36	C₄×D₄×C₁₃	—	4	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	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×#102	—	2	4	no	—
51	C₄×C₈×C₁₃	D₄	1	2	yes	—
52	C₄×C₈×C₁₃	C₂³	1	2	yes	—
53	#88066	—	2	16	no	—
54	C₂×#6090	—	2	4	no	—
55	C₂×#6090	C₂³	1	4	yes	—
56	C₂×#2206	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	—	4	1	no	—
58	C₂³×D₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	2	4	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂×D₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	4	no	—
65	C₂²×C₄×C₁₃	—	8	4	no	—
66	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×D₅₂	—	2	4	no	—
70	C₂×D₅₂	—	4	4	no	—
71	C₂×D₅₂	—	4	4	no	—
72	C₂×D₅₂	—	4	4	no	—
73	C₂×D₅₂	—	4	4	no	—
74	C₂×D₅₂	—	4	2	no	—
75	C₂×D₅₂	—	8	4	no	—
76	C₂×D₅₂	—	8	4	no	—
77	C₁₃×D₈	—	2	4	no	—
78	C₁₃×D₈	—	2	2	no	—
79	C₁₃×D₈	C₂×D₄	1	4	yes	complemented
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	#41	1	2	yes	—
85	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	4	1	no	—
88	C₁₃×C₂²⋊C₄	—	4	4	no	—
89	C₄²×C₁₃	D₈	1	2	yes	—
90	C₄²×C₁₃	C₂×D₄	1	2	yes	—
91	D₁₀₄	—	2	4	no	—
92	D₁₀₄	—	2	4	no	—
93	D₁₀₄	—	4	8	no	—
94	D₁₀₄	—	4	4	no	—
95	#6090	—	2	4	no	—
96	C₂²×D₁₆	—	13	2	no	—
97	C₂×#493	—	13	1	no	—
98	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic, complemented
99	#2206	—	13	8	no	—
100	C₂×C₄⋊C₁₆	—	13	1	no	—
101	C₂×#357	—	13	2	no	—
102	C₂³×C₁₃	—	4	2	no	—
103	C₂³×C₁₃	—	4	4	no	—
104	C₂³×C₁₃	—	8	2	no	—
105	C₂³×C₁₃	—	8	2	no	—
106	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₁₃	—	4	4	no	—
108	C₂²×D₁₃	—	8	4	no	—
109	C₂²×D₁₃	—	8	4	no	—
110	C₂²×D₁₃	—	8	4	no	—
111	D₄×C₁₃	—	2	4	no	—
112	D₄×C₁₃	—	4	2	no	—
113	D₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	8	2	no	—
117	C₂×C₄×C₁₃	—	8	8	no	—
118	C₂×C₄×C₁₃	D₁₆	1	4	yes	complemented
119	C₂×C₄×C₁₃	#121	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	D₅₂	—	4	4	no	—
126	D₅₂	—	4	4	no	—
127	D₅₂	—	8	8	no	—
128	D₅₂	—	8	4	no	—
129	D₅₂	—	8	8	no	—
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	#121	1	2	yes	—
132	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂²×D₈	—	13	2	no	—
135	C₂²×D₈	—	26	1	no	—
136	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×#127	—	26	1	no	—
138	C₂×C₄×D₄	—	2	1	no	—
139	C₂×D₁₆	—	13	8	no	—
140	C₂×D₁₆	—	26	8	no	—
141	C₂²×C₁₆	—	13	2	no	—
142	#493	—	13	4	no	—
143	#493	—	26	2	no	—
144	C₄×D₈	—	2	2	no	—
145	C₄×D₈	C₂×D₁₃	1	4	yes	complemented
146	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
147	C₂×#102	—	2	1	no	—
148	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
149	C₄⋊C₁₆	—	13	4	no	—
150	#357	—	13	8	no	—
151	C₂²×C₁₃	—	4	4	no	—
152	C₂²×C₁₃	—	8	2	no	—
153	C₂²×C₁₃	—	8	2	no	—
154	C₂²×C₁₃	—	8	8	no	—
155	C₂²×C₁₃	#459	1	2	yes	—
156	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
157	C₂²×C₁₃	#509	1	2	yes	—
158	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₁₃	—	8	4	no	—
160	C₂×D₁₃	—	8	4	no	—
161	C₂×D₁₃	—	8	4	no	—
162	C₂×D₁₃	—	16	8	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	8	2	no	—
165	C₄×C₁₃	#459	1	2	yes	—
166	C₄×C₁₃	C₂×D₁₆	1	4	yes	complemented
167	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
168	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	26	2	no	—
171	C₂²×D₄	—	52	1	no	—
172	C₂²×D₄	—	52	1	no	—
173	C₂³×C₄	—	4	1	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	13	4	no	—
177	C₂×D₈	—	26	4	no	—
178	C₂×D₈	—	26	4	no	—
179	C₂×D₈	—	26	2	no	—
180	C₂×D₈	—	26	4	no	—
181	C₂×D₈	—	52	4	no	—
182	C₂×D₈	D₅₂	1	4	yes	complemented
183	C₂×D₈	C₂²×D₁₃	1	2	yes	—
184	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	#127	—	26	4	no	—
187	#127	—	52	2	no	—
188	C₄×D₄	—	2	4	no	—
189	C₄×D₄	—	4	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₂²⋊C₄	—	4	1	no	—
192	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
193	D₁₆	—	26	16	no	—
194	C₂×C₁₆	—	13	4	no	—
195	C₂×C₁₆	—	26	4	no	—
196	#107	—	2	1	no	—
197	#107	C₂²×D₁₃	1	2	yes	—
198	#102	—	2	4	no	—
199	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
200	C₄×C₈	C₂²×D₁₃	1	2	yes	—
201	C₂×C₁₃	—	8	4	no	—
202	C₂×C₁₃	#2206	1	4	yes	complemented
203	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
204	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
206	D₁₃	—	16	4	no	—
207	C₂⁴	—	4	1	no	—
208	C₂⁴	—	52	2	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	26	4	no	—
214	C₂×D₄	—	52	4	no	—
215	C₂×D₄	—	52	4	no	—
216	C₂×D₄	—	52	4	no	—
217	C₂×D₄	—	52	2	no	—
218	C₂×D₄	—	52	4	no	—
219	C₂×D₄	—	104	4	no	—
220	C₂×D₄	—	104	4	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	8	4	no	—
224	C₂²×C₄	#6104	1	1	yes	characteristic
225	C₂²×C₄	#6104	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
227	D₈	—	2	4	no	—
228	D₈	—	2	2	no	—
229	D₈	—	26	4	no	—
230	D₈	—	26	4	no	—
231	D₈	—	52	8	no	—
232	D₈	—	52	4	no	—
233	D₈	C₂×D₅₂	1	4	yes	complemented
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂×D₅₂	1	2	yes	—
236	C₂×C₈	#6127	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₄²	#6104	1	2	yes	—
244	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
245	C₁₆	—	26	4	no	—
246	C₁₃	C₂×#2206	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	4	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	52	4	no	—
252	C₂³	—	104	4	no	—
253	C₂³	—	104	4	no	—
254	C₂³	—	104	4	no	—
255	C₂³	C₂×#6104	1	1	yes	characteristic
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	52	4	no	—
260	D₄	—	52	4	no	—
261	D₄	—	104	8	no	—
262	D₄	—	104	4	no	—
263	D₄	—	104	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	#88066	1	4	yes	complemented
269	C₂×C₄	#88184	1	2	yes	—
270	C₂×C₄	C₂×#6104	1	2	yes	—
271	C₂×C₄	C₂×#6104	1	1	yes	characteristic
272	C₂×C₄	C₂×#6104	1	2	yes	—
273	C₂×C₄	#88203	1	2	yes	—
274	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	#88184	1	2	yes	—
277	C₈	C₂²×D₅₂	1	1	yes	characteristic
278	C₈	C₂×#6127	1	1	yes	characteristic
279	C₂²	—	4	4	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	8	no	—
283	C₂²	—	104	4	no	—
284	C₂²	—	104	4	no	—
285	C₂²	—	104	4	no	—
286	C₂²	—	208	8	no	—
287	C₂²	#11718676	1	2	yes	—
288	C₂²	C₂×#88066	1	2	yes	—
289	C₂²	#11718854	1	2	yes	—
290	C₂²	C₂²×#6104	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	8	2	no	—
293	C₄	#11718676	1	2	yes	—
294	C₄	C₂×#88066	1	4	yes	complemented
295	C₄	C₂×#88184	1	1	yes	characteristic
296	C₄	C₂×#88203	1	1	yes	characteristic
297	C₂	—	8	4	no	—
298	C₂	—	208	4	no	—
299	C₂	#13116871	1	4	yes	direct product factor
300	C₂	C₂×#11718676	1	1	yes	characteristic
301	C₂	C₂²×#88066	1	1	yes	characteristic
302	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×D₈	C₂
socle	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	C₂×#2206
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#88066	C₂	semidirect
2. C₂×#11718356	C₂	semidirect
3. #13116871	C₂	direct (N × H)
4. C₂×C₄⋊#6090	C₂	semidirect
5. C₂×#11718083	C₂	semidirect
6. C₂×C₄×C₁₃×D₈	C₂	semidirect
13. C₄×C₁₃×D₈	C₂²	semidirect
18. C₂×#88066	C₂²	semidirect
21. #11718356	C₂²	semidirect
22. #11718083	C₂²	semidirect
23. C₄⋊#6090	C₂²	semidirect
31. C₂×C₁₃×D₈	D₄	semidirect
79. C₁₃×D₈	C₂×D₄	semidirect
98. C₂×C₄×D₈	D₁₃	semidirect
118. C₂×C₄×C₁₃	D₁₆	semidirect
145. C₄×D₈	C₂×D₁₃	semidirect
166. C₄×C₁₃	C₂×D₁₆	semidirect
182. C₂×D₈	D₅₂	semidirect
202. C₂×C₁₃	#2206	semidirect
233. D₈	C₂×D₅₂	semidirect
246. C₁₃	C₂×#2206	semidirect
268. C₂×C₄	#88066	semidirect
294. C₄	C₂×#88066	semidirect
299. C₂	#13116871	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×#88066	2
C₂×#11718356	2
#13116871	2
C₂×C₄⋊#6090	2
C₂×#11718083	2
C₂×C₄×C₁₃×D₈	2
C₂×#2206	13

Subgroups — C2×#13116871 #16017143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116871 #16017143