Subgroups — C₂×#13116881 #16017149

Order 3328.

766 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88067	C₂	1	2	yes	maximal
2	C₂×#11718360	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116881	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718084	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃	—	2	1	no	—
9	C₂²×Dic₅₂	—	2	1	no	—
10	C₂²×Dic₅₂	C₂²	1	2	yes	—
11	C₂×#88123	—	2	1	no	—
12	C₄×C₁₃×D₈	—	2	2	no	—
13	C₄×C₁₃×D₈	C₂²	1	4	yes	—
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×#88067	—	2	8	no	—
18	C₂×#88067	C₂²	1	8	yes	—
19	C₂²×#6090	C₂²	1	2	yes	—
20	#11718360	—	2	2	no	—
21	#11718360	C₂²	1	4	yes	complemented
22	#11718084	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×Dic₂₆	—	2	2	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	2	no	—
29	C₂×C₁₃×D₈	D₄	1	4	yes	—
30	C₂×C₁₃×D₈	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₄×D₄×C₁₃	—	2	4	no	—
34	C₄×D₄×C₁₃	—	4	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃	—	4	1	no	—
39	C₂×C₄⋊Dic₁₃	—	4	1	no	—
40	C₂×Dic₅₂	—	2	4	no	—
41	C₂×Dic₅₂	—	2	4	no	—
42	C₂×Dic₅₂	—	2	4	no	—
43	C₂×Dic₅₂	—	2	2	no	—
44	C₂×Dic₅₂	—	4	4	no	—
45	C₂×Dic₅₂	C₂³	1	4	yes	—
46	#88123	—	2	4	no	—
47	#88123	—	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	#88067	—	2	16	no	—
54	C₂×#6090	—	2	4	no	—
55	C₂×#6090	C₂³	1	4	yes	—
56	C₂×#2208	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	2	4	no	—
59	C₂×D₄×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	—	4	1	no	—
61	C₂×D₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	4	no	—
64	C₂²×C₄×C₁₃	—	8	4	no	—
65	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	4	2	no	—
69	C₂×Dic₂₆	—	2	4	no	—
70	C₂×Dic₂₆	—	4	4	no	—
71	C₂×Dic₂₆	—	4	4	no	—
72	C₂×Dic₂₆	—	4	2	no	—
73	C₁₃×D₈	—	2	4	no	—
74	C₁₃×D₈	—	2	2	no	—
75	C₁₃×D₈	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃	#41	1	2	yes	—
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	4	1	no	—
84	C₁₃×C₂²⋊C₄	—	4	4	no	—
85	C₄²×C₁₃	D₈	1	2	yes	—
86	C₄²×C₁₃	C₂×D₄	1	2	yes	—
87	C₄⋊Dic₁₃	—	4	4	no	—
88	C₄⋊Dic₁₃	—	4	4	no	—
89	Dic₅₂	—	2	4	no	—
90	Dic₅₂	—	2	4	no	—
91	Dic₅₂	—	4	8	no	—
92	Dic₅₂	—	4	4	no	—
93	#6090	—	2	4	no	—
94	C₂²×SD₃₂	—	13	2	no	—
95	C₂×#494	—	13	1	no	—
96	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
97	#2208	—	13	8	no	—
98	C₂×C₄⋊C₁₆	—	13	1	no	—
99	C₂×#358	—	13	2	no	—
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	—	4	4	no	—
102	C₂³×C₁₃	—	8	2	no	—
103	C₂³×C₁₃	—	8	2	no	—
104	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
105	D₄×C₁₃	—	2	4	no	—
106	D₄×C₁₃	—	4	2	no	—
107	D₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	8	2	no	—
111	C₂×C₄×C₁₃	—	8	8	no	—
112	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
113	C₂×C₄×C₁₃	#121	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃	—	4	4	no	—
120	C₂×Dic₁₃	—	8	4	no	—
121	Dic₂₆	—	4	4	no	—
122	Dic₂₆	—	4	4	no	—
123	Dic₂₆	—	8	4	no	—
124	C₈×C₁₃	—	2	2	no	—
125	C₈×C₁₃	#121	1	2	yes	—
126	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂²×Q₁₆	—	13	2	no	—
129	C₂²×Q₁₆	—	26	1	no	—
130	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₄⋊Q₈	—	26	1	no	—
132	C₂×C₄×D₄	—	2	1	no	—
133	C₂×SD₃₂	—	13	8	no	—
134	C₂×SD₃₂	—	26	8	no	—
135	C₂²×C₁₆	—	13	2	no	—
136	#494	—	13	4	no	—
137	#494	—	26	2	no	—
138	C₄×D₈	—	2	2	no	—
139	C₄×D₈	C₂×D₁₃	1	4	yes	—
140	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
141	C₂×#102	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
143	C₄⋊C₁₆	—	13	4	no	—
144	#358	—	13	8	no	—
145	C₂²×C₁₃	—	4	4	no	—
146	C₂²×C₁₃	—	8	2	no	—
147	C₂²×C₁₃	—	8	2	no	—
148	C₂²×C₁₃	—	8	8	no	—
149	C₂²×C₁₃	#459	1	2	yes	—
150	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
151	C₂²×C₁₃	#510	1	2	yes	—
152	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	8	2	no	—
155	C₄×C₁₃	#459	1	2	yes	—
156	C₄×C₁₃	C₂×SD₃₂	1	4	yes	complemented
157	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
158	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
159	Dic₁₃	—	8	4	no	—
160	C₂²×Q₈	—	26	2	no	—
161	C₂²×D₄	—	2	1	no	—
162	C₂³×C₄	—	4	1	no	—
163	C₂×Q₁₆	—	13	4	no	—
164	C₂×Q₁₆	—	26	4	no	—
165	C₂×Q₁₆	—	26	4	no	—
166	C₂×Q₁₆	—	26	2	no	—
167	C₂×Q₁₆	—	26	4	no	—
168	C₂×Q₁₆	—	52	4	no	—
169	C₂×D₈	—	2	2	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	D₅₂	1	4	yes	—
172	C₂×D₈	C₂²×D₁₃	1	2	yes	—
173	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
175	C₄⋊Q₈	—	26	4	no	—
176	C₄⋊Q₈	—	52	2	no	—
177	C₄×D₄	—	2	4	no	—
178	C₄×D₄	—	4	2	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	52	1	no	—
181	C₂×C₄⋊C₄	—	52	1	no	—
182	C₂×C₂²⋊C₄	—	4	1	no	—
183	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
184	SD₃₂	—	26	16	no	—
185	C₂×C₁₆	—	13	4	no	—
186	C₂×C₁₆	—	26	4	no	—
187	#107	—	2	1	no	—
188	#107	C₂²×D₁₃	1	2	yes	—
189	#102	—	2	4	no	—
190	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
191	C₄×C₈	C₂²×D₁₃	1	2	yes	—
192	C₂×C₁₃	—	8	4	no	—
193	C₂×C₁₃	#2208	1	4	yes	complemented
194	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
195	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
196	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
197	C₂⁴	—	4	1	no	—
198	C₂×Q₈	—	26	4	no	—
199	C₂×Q₈	—	52	2	no	—
200	C₂×Q₈	—	52	4	no	—
201	C₂×Q₈	—	52	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	8	4	no	—
209	C₂²×C₄	—	52	2	no	—
210	C₂²×C₄	#6104	1	1	yes	characteristic
211	C₂²×C₄	#6104	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
213	Q₁₆	—	26	4	no	—
214	Q₁₆	—	26	4	no	—
215	Q₁₆	—	52	8	no	—
216	Q₁₆	—	52	4	no	—
217	D₈	—	2	4	no	—
218	D₈	—	2	2	no	—
219	D₈	C₂×D₅₂	1	4	yes	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂×D₅₂	1	2	yes	—
222	C₂×C₈	#6127	1	2	yes	—
223	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	52	4	no	—
229	C₄⋊C₄	—	52	4	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₄²	#6104	1	2	yes	—
232	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
233	C₁₆	—	26	4	no	—
234	C₁₃	C₂×#2208	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	4	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₂×#6104	1	1	yes	characteristic
240	Q₈	—	52	4	no	—
241	Q₈	—	52	4	no	—
242	Q₈	—	104	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	—	8	8	no	—
250	C₂×C₄	—	52	4	no	—
251	C₂×C₄	—	104	4	no	—
252	C₂×C₄	#88067	1	4	yes	complemented
253	C₂×C₄	#88184	1	2	yes	—
254	C₂×C₄	C₂×#6104	1	2	yes	—
255	C₂×C₄	C₂×#6104	1	1	yes	characteristic
256	C₂×C₄	C₂×#6104	1	2	yes	—
257	C₂×C₄	#88203	1	2	yes	—
258	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	#88184	1	2	yes	—
261	C₈	C₂²×D₅₂	1	1	yes	characteristic
262	C₈	C₂×#6127	1	1	yes	characteristic
263	C₂²	—	4	4	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	8	no	—
267	C₂²	#11718676	1	2	yes	—
268	C₂²	C₂×#88067	1	2	yes	—
269	C₂²	#11718856	1	2	yes	—
270	C₂²	C₂²×#6104	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	8	2	no	—
273	C₄	—	104	4	no	—
274	C₄	#11718676	1	2	yes	—
275	C₄	C₂×#88067	1	4	yes	complemented
276	C₄	C₂×#88184	1	1	yes	characteristic
277	C₄	C₂×#88203	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	#13116881	1	4	yes	direct product factor
280	C₂	C₂×#11718676	1	1	yes	characteristic
281	C₂	C₂²×#88067	1	1	yes	characteristic
282	C₂	C₂×#11718856	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
prime radical	C₂²×C₁₃×D₈	C₂²
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	C₂×#2208
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
2. C₂×#11718360	C₂	semidirect
3. #13116881	C₂	direct (N × H)
4. C₂×C₄⋊#6090	C₂	semidirect
5. C₂×#11718084	C₂	semidirect
21. #11718360	C₂²	semidirect
22. #11718084	C₂²	semidirect
23. C₄⋊#6090	C₂²	semidirect
112. C₂×C₄×C₁₃	SD₃₂	semidirect
156. C₄×C₁₃	C₂×SD₃₂	semidirect
193. C₂×C₁₃	#2208	semidirect
234. C₁₃	C₂×#2208	semidirect
252. C₂×C₄	#88067	semidirect
275. C₄	C₂×#88067	semidirect
279. C₂	#13116881	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×#88067	2
C₂×#11718360	2
#13116881	2
C₂×C₄⋊#6090	2
C₂×#11718084	2
C₂×C₄×C₁₃×D₈	2
C₂×#2208	13

Subgroups — C2×#13116881 #16017149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116881 #16017149