Subgroups — C₂×C₄×#88240 #16808016

Order 3328.

3366 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88240	C₂	1	2	yes	maximal
2	C₄×#88240	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11719072	C₂	1	6	yes	maximal, complemented
4	C₂×C₄×#6118	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×D₁₃⋊C₈	C₂	1	3	yes	maximal, complemented
6	C₂×C₄×#6131	C₂	1	1	yes	maximal, characteristic
7	C₂²×#6131	C₂²	1	1	yes	characteristic
8	C₂×#88240	C₄	1	32	yes	direct product factor
9	C₂×#88240	C₂²	1	16	yes	—
10	C₂²×#6118	C₂²	1	6	yes	—
11	C₂²×D₁₃⋊C₈	C₂²	1	6	yes	—
12	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
13	C₂×Q₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₄×#6131	C₄	1	8	yes	—
15	C₄×#6131	C₂²	1	8	yes	—
16	C₂×#88147	C₄	1	3	yes	—
17	C₂×#88146	C₂²	1	3	yes	—
18	C₂×C₄×D₅₂	C₄	1	3	yes	—
19	C₂×C₄²×D₁₃	C₂²	1	3	yes	—
20	C₂×#88110	C₂²	1	6	yes	—
21	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
22	C₂×C₄×#1057	C₂²	1	3	yes	—
23	C₂×C₄×#1057	C₂²	1	6	yes	—
24	#11719072	C₂²	1	48	yes	complemented
25	C₄×#6118	C₂²	1	24	yes	complemented
26	C₄×D₁₃⋊C₈	C₂²	1	24	yes	complemented
27	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#6131	C₂×C₄	1	16	yes	—
29	C₂×#6131	C₂×C₄	1	4	yes	—
30	C₂×#6131	C₂³	1	4	yes	—
31	C₂²×D₅₂	C₂×C₄	1	3	yes	—
32	C₂²×C₄×D₁₃	C₂×C₄	1	3	yes	—
33	C₂²×C₄×D₁₃	C₂×C₄	1	3	yes	—
34	C₂²×C₄×D₁₃	C₂³	1	3	yes	—
35	C₂²×#1057	C₂³	1	2	yes	—
36	C₂²×#1057	C₂³	1	6	yes	—
37	#88240	C₂×C₄	1	128	yes	direct product factor
38	C₂×#6118	C₂×C₄	1	48	yes	complemented
39	C₂×#6118	C₂³	1	24	yes	—
40	C₂×D₁₃⋊C₈	C₂×C₄	1	48	yes	complemented
41	C₂×D₁₃⋊C₈	C₂³	1	24	yes	—
42	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
44	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
45	Q₈×Dic₁₃	C₂×C₄	1	8	yes	—
46	C₂×#6103	C₂×C₄	1	6	yes	—
47	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
49	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
51	#88147	C₂×C₄	1	24	yes	—
52	#88146	C₂×C₄	1	12	yes	—
53	#88146	C₂³	1	12	yes	—
54	C₄×D₅₂	C₂×C₄	1	24	yes	—
55	C₄²×D₁₃	C₂×C₄	1	12	yes	—
56	C₄²×D₁₃	C₂³	1	12	yes	—
57	#88110	C₂³	1	24	yes	—
58	C₄×#1057	C₂³	1	4	yes	—
59	C₄×#1057	C₂³	1	12	yes	—
60	C₄×#1057	C₂³	1	24	yes	—
61	C₂×C₄×#131	—	13	1	no	maximal, Sylow
62	C₂³×D₁₃	C₂²×C₄	1	3	yes	—
63	C₂×Q₈×C₁₃	C₄²	1	8	yes	—
64	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
69	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
71	#6131	C₄²	1	32	yes	—
72	#6131	C₂²×C₄	1	32	yes	—
73	C₂×D₅₂	C₄²	1	24	yes	—
74	C₂×D₅₂	C₂²×C₄	1	12	yes	—
75	C₂×C₄×D₁₃	—	2	12	no	—
76	C₂×C₄×D₁₃	—	2	12	no	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
78	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
79	C₂×C₄×D₁₃	C₂²×C₄	1	6	yes	—
80	C₂×C₄×D₁₃	C₂²×C₄	1	24	yes	—
81	C₂×C₄×D₁₃	C₂⁴	1	6	yes	—
82	C₂×#1057	C₂²×C₄	1	8	yes	—
83	C₂×#1057	C₂²×C₄	1	24	yes	—
84	C₂×#1057	C₂⁴	1	4	yes	—
85	C₂×#1057	C₂⁴	1	12	yes	—
86	#6118	C₂²×C₄	1	96	yes	complemented
87	D₁₃⋊C₈	C₂²×C₄	1	96	yes	complemented
88	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
89	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
90	#6103	C₂²×C₄	1	24	yes	—
91	C₄⋊Dic₁₃	C₂²×C₄	1	12	yes	—
92	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
93	C₄×Dic₁₃	C₂²×C₄	1	6	yes	—
94	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
95	C₄×Dic₁₃	C₂⁴	1	2	yes	—
96	C₄×Dic₁₃	C₂⁴	1	6	yes	—
97	C₂²×#131	—	13	2	no	—
98	C₂×C₄×#41	—	13	1	no	—
99	C₄×#131	—	13	16	no	—
100	C₂×#405	—	13	6	no	—
101	C₂×C₄×M₁₆	—	13	3	no	—
102	C₂²×C₄×C₈	—	13	3	no	—
103	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
104	C₂²×D₁₃	—	2	6	no	—
105	C₂²×D₁₃	—	2	6	no	—
106	C₂²×D₁₃	C₂×C₄²	1	12	yes	—
107	C₂²×D₁₃	C₂³×C₄	1	6	yes	—
108	Q₈×C₁₃	C₂×C₄²	1	16	yes	—
109	C₂×C₄×C₁₃	—	2	6	no	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	12	yes	—
111	C₂×C₄×C₁₃	#131	1	4	yes	complemented
112	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
115	C₂×Dic₁₃	—	2	6	no	—
116	C₂×Dic₁₃	#131	1	4	yes	complemented
117	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
118	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
119	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
120	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂⁵	1	1	yes	characteristic
122	D₅₂	C₂×C₄²	1	48	yes	—
123	C₄×D₁₃	—	2	24	no	—
124	C₄×D₁₃	—	2	24	no	—
125	C₄×D₁₃	C₂×C₄²	1	24	yes	—
126	C₄×D₁₃	C₂³×C₄	1	24	yes	—
127	#1057	C₂³×C₄	1	8	yes	—
128	#1057	C₂³×C₄	1	24	yes	—
129	C₂²×#41	—	13	1	no	—
130	C₂×#131	—	13	16	no	—
131	C₂×#131	—	13	32	no	—
132	C₂²×M₁₆	—	13	6	no	—
133	C₂³×C₈	—	13	6	no	—
134	C₄×#41	—	13	8	no	—
135	C₄×#41	—	13	8	no	—
136	C₂×C₄×Q₈	—	13	1	no	—
137	C₂×C₄×Q₈	#261	1	1	yes	characteristic
138	C₂×C₄×D₄	—	13	3	no	—
139	C₂×C₄×D₄	—	13	3	no	—
140	C₂×#117	—	13	3	no	—
141	C₂²×C₄²	—	13	3	no	—
142	#405	—	13	48	no	—
143	C₄×M₁₆	—	13	24	no	—
144	C₂×#97	—	13	6	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	13	3	no	—
147	C₂×C₄×C₈	—	13	6	no	—
148	C₂×C₄×C₈	—	13	24	no	—
149	C₂²×C₁₃	C₂²×C₄²	1	2	yes	—
150	C₂²×C₁₃	C₂×#131	1	2	yes	—
151	C₂²×C₁₃	C₂×#131	1	2	yes	—
152	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
153	C₂×D₁₃	—	2	24	no	—
154	C₂×D₁₃	—	2	6	no	—
155	C₂×D₁₃	—	2	6	no	—
156	C₂×D₁₃	C₂²×C₄²	1	12	yes	—
157	C₄×C₁₃	—	2	6	no	—
158	C₄×C₁₃	C₂²×C₄²	1	12	yes	—
159	C₄×C₁₃	C₂×#131	1	4	yes	complemented
160	Dic₁₃	—	2	6	no	—
161	Dic₁₃	C₂²×C₄²	1	2	yes	—
162	Dic₁₃	C₂×#131	1	4	yes	complemented
163	Dic₁₃	C₂⁴×C₄	1	2	yes	—
164	C₂×#41	—	13	16	no	—
165	C₂×#41	—	13	4	no	—
166	C₂×#41	—	13	4	no	—
167	C₂²×Q₈	C₂×#261	1	1	yes	characteristic
168	C₂²×D₄	—	13	3	no	—
169	C₂³×C₄	—	13	3	no	—
170	C₂³×C₄	—	13	3	no	—
171	C₂³×C₄	—	13	3	no	—
172	#131	—	13	128	no	—
173	C₂×M₁₆	—	13	24	no	—
174	C₂×M₁₆	—	13	48	no	—
175	C₂²×C₈	—	13	2	no	—
176	C₂²×C₈	—	13	6	no	—
177	C₂²×C₈	—	13	24	no	—
178	C₂²×C₈	—	13	48	no	—
179	C₄×Q₈	—	13	8	no	—
180	C₄×Q₈	C₂×#261	1	8	yes	—
181	C₄×D₄	—	13	24	no	—
182	C₄×D₄	—	13	24	no	—
183	#117	—	13	12	no	—
184	#117	—	13	12	no	—
185	C₂×C₄⋊C₄	—	13	3	no	—
186	C₂×C₄⋊C₄	C₂×#261	1	3	yes	—
187	C₂×C₂²⋊C₄	—	13	6	no	—
188	C₂×C₄²	—	13	3	no	—
189	C₂×C₄²	—	13	3	no	—
190	C₂×C₄²	—	13	1	no	—
191	C₂×C₄²	—	13	12	no	—
192	C₂×C₄²	—	13	12	no	—
193	C₂×C₄²	C₂×#261	1	3	yes	—
194	#97	—	13	24	no	—
195	C₄×C₈	—	13	4	no	—
196	C₄×C₈	—	13	12	no	—
197	C₄×C₈	—	13	24	no	—
198	C₂×C₁₃	C₄×#131	1	4	yes	complemented
199	C₂×C₁₃	C₂³×C₄²	1	1	yes	characteristic
200	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
201	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
202	D₁₃	—	2	12	no	—
203	C₂⁴	—	13	3	no	—
204	#41	—	13	32	no	—
205	#41	—	13	32	no	—
206	C₂×Q₈	C₄×#261	1	8	yes	—
207	C₂×Q₈	C₂²×#261	1	4	yes	—
208	C₂×D₄	—	13	24	no	—
209	C₂×D₄	—	13	12	no	—
210	C₂²×C₄	—	13	1	no	—
211	C₂²×C₄	—	13	1	no	—
212	C₂²×C₄	—	13	24	no	—
213	C₂²×C₄	—	13	6	no	—
214	C₂²×C₄	—	13	6	no	—
215	C₂²×C₄	—	13	3	no	—
216	C₂²×C₄	—	13	12	no	—
217	C₂²×C₄	—	13	12	no	—
218	C₂²×C₄	—	26	12	no	—
219	C₂²×C₄	—	26	12	no	—
220	C₂²×C₄	C₂²×#261	1	3	yes	—
221	C₂²×C₄	C₂²×#261	1	3	yes	—
222	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
223	M₁₆	—	13	96	no	—
224	C₂×C₈	—	13	4	no	—
225	C₂×C₈	—	13	12	no	—
226	C₂×C₈	—	13	8	no	—
227	C₂×C₈	—	13	24	no	—
228	C₂×C₈	—	13	96	no	—
229	C₄⋊C₄	—	13	12	no	—
230	C₄⋊C₄	C₂²×#261	1	12	yes	—
231	C₂²⋊C₄	—	13	24	no	—
232	C₄²	—	13	2	no	—
233	C₄²	—	13	6	no	—
234	C₄²	—	13	2	no	—
235	C₄²	—	13	6	no	—
236	C₄²	—	13	12	no	—
237	C₄²	C₂²×#261	1	12	yes	—
238	C₁₃	C₂×C₄×#131	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	13	12	no	—
240	C₂³	—	13	6	no	—
241	C₂³	—	26	6	no	—
242	C₂³	—	26	6	no	—
243	C₂³	C₂³×#261	1	1	yes	characteristic
244	Q₈	C₂×C₄×#261	1	16	yes	—
245	D₄	—	13	48	no	—
246	C₂×C₄	—	2	6	no	—
247	C₂×C₄	—	13	4	no	—
248	C₂×C₄	—	13	4	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	13	2	no	—
252	C₂×C₄	—	13	24	no	—
253	C₂×C₄	—	13	24	no	—
254	C₂×C₄	—	13	6	no	—
255	C₂×C₄	—	26	24	no	—
256	C₂×C₄	—	26	6	no	—
257	C₂×C₄	—	26	24	no	—
258	C₂×C₄	C₂×C₄×#261	1	12	yes	—
259	C₂×C₄	#88240	1	4	yes	direct product factor
260	C₂×C₄	C₂³×#261	1	6	yes	—
261	C₂×C₄	C₂³×#261	1	6	yes	—
262	C₂×C₄	C₂³×#261	1	2	yes	—
263	C₈	—	13	8	no	—
264	C₈	—	13	24	no	—
265	C₂²	—	13	12	no	—
266	C₂²	—	26	6	no	—
267	C₂²	—	26	6	no	—
268	C₂²	—	26	24	no	—
269	C₂²	C₂²×C₄×#261	1	2	yes	—
270	C₂²	C₂×#88240	1	2	yes	—
271	C₂²	D₁₃⋊#131	1	2	yes	—
272	C₂²	C₂⁴×#261	1	1	yes	characteristic
273	C₄	—	2	6	no	—
274	C₄	—	13	2	no	—
275	C₄	—	13	2	no	—
276	C₄	—	13	4	no	—
277	C₄	—	26	6	no	—
278	C₄	C₂²×C₄×#261	1	12	yes	—
279	C₄	C₂×#88240	1	4	yes	direct product factor
280	C₂	—	26	12	no	—
281	C₂	C₄×#88240	1	4	yes	direct product factor
282	C₂	C₂³×C₄×#261	1	1	yes	characteristic
283	C₂	C₂×D₁₃⋊#131	1	1	yes	characteristic
284	C₂	C₂²×#88240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂³×C₄²
Frattini	C₂²	C₂⁴×#261
Fitting	C₂×C₄×Q₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂²×#6131	C₂²
cosocle	C₂×Dic₁₃	C₂⁵
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	C₂×C₄×#131
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×#88240	C₂	direct (N × H)
3. C₂×#11719072	C₂	semidirect
4. C₂×C₄×#6118	C₂	semidirect
5. C₂×C₄×D₁₃⋊C₈	C₂	semidirect
8. C₂×#88240	C₄	direct (N × H)
24. #11719072	C₂²	semidirect
25. C₄×#6118	C₂²	semidirect
26. C₄×D₁₃⋊C₈	C₂²	semidirect
37. #88240	C₂×C₄	direct (N × H)
38. C₂×#6118	C₂×C₄	semidirect
40. C₂×D₁₃⋊C₈	C₂×C₄	semidirect
86. #6118	C₂²×C₄	semidirect
87. D₁₃⋊C₈	C₂²×C₄	semidirect
111. C₂×C₄×C₁₃	#131	semidirect
116. C₂×Dic₁₃	#131	semidirect
159. C₄×C₁₃	C₂×#131	semidirect
162. Dic₁₃	C₂×#131	semidirect
198. C₂×C₁₃	C₄×#131	semidirect
238. C₁₃	C₂×C₄×#131	semidirect
259. C₂×C₄	#88240	direct (N × H)
279. C₄	C₂×#88240	direct (N × H)
281. C₂	C₄×#88240	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×#88240	2
C₄×#88240	2
C₂×#11719072	2
C₂×C₄×#6118	2
C₂×C₄×D₁₃⋊C₈	2
C₂×C₄×#6131	2
C₂×C₄×#131	13

Subgroups — C2×C4×#88240 #16808016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88240 #16808016