Subgroups — C₂×D₁₃⋊#334 #16794900

Order 3328.

942 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊(C₄⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#11718255	C₂	1	2	yes	maximal, complemented
3	D₁₃⋊#334	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718253	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₁₃⋊C₈	—	2	2	no	—
7	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88112	—	2	2	no	—
11	C₂×C₄⋊#1057	C₂²	1	4	yes	—
12	D₁₃⋊(C₄⋊C₈)	—	2	4	no	—
13	D₁₃⋊(C₄⋊C₈)	C₂²	1	8	yes	—
14	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₄	1	1	yes	characteristic
18	C₄×C₈×D₁₃	C₄	1	4	yes	—
19	C₄×C₈×D₁₃	C₂²	1	4	yes	—
20	#11718255	C₂²	1	8	yes	complemented
21	#11718253	C₂²	1	8	yes	complemented
22	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
25	C₂²×#1057	—	2	4	no	—
26	C₂×D₁₃⋊C₈	—	2	8	no	—
27	C₂×D₁₃⋊C₈	—	4	8	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₁₃	—	2	4	no	—
37	C₂×C₈×D₁₃	C₈	1	8	yes	complemented
38	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
40	C₄²×D₁₃	C₂×C₄	1	2	yes	—
41	C₄²×D₁₃	D₄	1	2	yes	—
42	C₄²×D₁₃	Q₈	1	2	yes	—
43	C₄²×D₁₃	C₂³	1	2	yes	—
44	#88112	—	2	8	no	—
45	C₄⋊#1057	—	2	4	no	—
46	C₄⋊#1057	C₂³	1	8	yes	—
47	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
48	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₈×Dic₁₃	C₂³	1	2	yes	—
51	C₈×Dic₁₃	C₂³	1	2	yes	—
52	C₄×#1055	C₂×C₄	1	4	yes	—
53	C₂²×#334	—	13	1	no	maximal, Sylow
54	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×D₁₃	—	2	4	no	—
62	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
66	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
67	C₂×C₄×D₁₃	SD₁₆	1	8	yes	—
68	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
71	C₂×#1057	—	2	8	no	—
72	C₂×#1057	—	4	8	no	—
73	D₁₃⋊C₈	—	4	16	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	complemented
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
80	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
81	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
82	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
85	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
87	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
88	C₂×#1055	—	2	2	no	—
89	C₂×#1055	C₂×C₈	1	4	yes	complemented
90	C₂×#1055	C₂²×C₄	1	2	yes	—
91	C₂×#1055	C₂²×C₄	1	2	yes	—
92	C₈×D₁₃	—	2	8	no	—
93	C₈×D₁₃	C₂×C₈	1	16	yes	complemented
94	C₂²×C₄⋊C₈	—	13	2	no	—
95	C₂²×C₄×C₈	—	13	1	no	—
96	C₂×#334	—	13	2	no	—
97	C₂×#334	—	13	8	no	—
98	C₂×#334	—	13	2	no	—
99	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×D₁₃	C₄⋊C₈	1	4	yes	—
101	C₂²×D₁₃	#106	1	4	yes	—
102	C₂²×D₁₃	#108	1	4	yes	—
103	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	#106	1	4	yes	—
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₁₃	—	2	2	no	—
113	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
114	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×Dic₁₃	C₂×SD₁₆	1	4	yes	—
118	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₄×D₁₃	—	2	4	no	—
121	C₄×D₁₃	—	2	4	no	—
122	C₄×D₁₃	C₄⋊C₈	1	8	yes	—
123	C₄×D₁₃	#106	1	8	yes	—
124	C₄×D₁₃	C₂²×C₈	1	4	yes	—
125	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
126	C₄×D₁₃	C₂×SD₁₆	1	8	yes	—
127	#1057	—	4	8	no	—
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	C₂²×C₈	1	4	yes	complemented
130	#1055	—	2	2	no	—
131	#1055	C₂²×C₈	1	4	yes	complemented
132	C₂³×C₈	—	13	1	no	—
133	C₂³×C₈	—	13	1	no	—
134	C₂³×C₈	—	26	2	no	—
135	C₂²×C₄²	—	13	1	no	—
136	C₂×C₄⋊C₈	—	13	8	no	—
137	C₂×C₄⋊C₈	—	13	4	no	—
138	C₂×C₄⋊C₈	—	26	4	no	—
139	C₂×C₄⋊C₈	—	26	2	no	—
140	C₂×C₄×C₈	—	13	1	no	—
141	C₂×C₄×C₈	—	13	1	no	—
142	C₂×C₄×C₈	—	13	1	no	—
143	C₂×C₄×C₈	—	13	4	no	—
144	C₂×C₄×C₈	—	13	4	no	—
145	C₂×C₄×C₈	#261	1	1	yes	characteristic
146	#334	—	13	8	no	—
147	#334	—	13	8	no	—
148	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
149	C₂²×C₁₃	C₂×#106	1	2	yes	—
150	C₂²×C₁₃	C₂×#108	1	2	yes	—
151	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂×D₁₃	#334	1	16	yes	complemented
153	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
154	C₂×D₁₃	C₂×#106	1	4	yes	—
155	C₂×D₁₃	C₂×#108	1	4	yes	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
158	C₄×C₁₃	C₂×#106	1	4	yes	—
159	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
160	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
161	Dic₁₃	—	2	1	no	—
162	Dic₁₃	—	2	1	no	—
163	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
164	Dic₁₃	C₂×#106	1	2	yes	—
165	Dic₁₃	C₂²×SD₁₆	1	2	yes	—
166	C₂³×C₄	—	13	1	no	—
167	C₂³×C₄	—	13	1	no	—
168	C₂³×C₄	—	13	1	no	—
169	C₂²×C₈	—	13	1	no	—
170	C₂²×C₈	—	13	8	no	—
171	C₂²×C₈	—	13	4	no	—
172	C₂²×C₈	—	13	1	no	—
173	C₂²×C₈	—	13	4	no	—
174	C₂²×C₈	—	26	4	no	—
175	C₂²×C₈	—	26	8	no	—
176	C₂²×C₈	—	26	4	no	—
177	C₂²×C₈	—	52	8	no	—
178	C₂²×C₈	C₂×#261	1	1	yes	characteristic
179	C₂²×C₈	C₂×#261	1	1	yes	characteristic
180	C₂×C₄²	—	13	1	no	—
181	C₂×C₄²	—	13	1	no	—
182	C₂×C₄²	—	13	1	no	—
183	C₂×C₄²	—	13	2	no	—
184	C₂×C₄²	—	13	2	no	—
185	C₂×C₄²	—	13	2	no	—
186	C₂×C₄²	—	13	2	no	—
187	C₂×C₄²	C₂×#261	1	1	yes	characteristic
188	C₄⋊C₈	—	13	8	no	—
189	C₄⋊C₈	—	26	4	no	—
190	C₄⋊C₈	—	26	8	no	—
191	C₄×C₈	—	13	2	no	—
192	C₄×C₈	—	13	2	no	—
193	C₄×C₈	—	13	2	no	—
194	C₄×C₈	—	13	2	no	—
195	C₄×C₈	—	13	4	no	—
196	C₄×C₈	C₂×#261	1	4	yes	—
197	C₂×C₁₃	C₂×#334	1	4	yes	complemented
198	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
199	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
200	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
201	D₁₃	C₂×#334	1	8	yes	complemented
202	C₂⁴	—	13	1	no	—
203	C₂²×C₄	—	13	1	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	13	8	no	—
206	C₂²×C₄	—	13	4	no	—
207	C₂²×C₄	—	13	4	no	—
208	C₂²×C₄	—	13	2	no	—
209	C₂²×C₄	—	13	2	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	—	13	2	no	—
215	C₂²×C₄	—	26	4	no	—
216	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
217	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
218	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	13	4	no	—
221	C₂×C₈	—	13	2	no	—
222	C₂×C₈	—	13	16	no	—
223	C₂×C₈	—	13	2	no	—
224	C₂×C₈	—	26	8	no	—
225	C₂×C₈	—	26	2	no	—
226	C₂×C₈	—	26	8	no	—
227	C₂×C₈	—	52	16	no	—
228	C₂×C₈	—	52	8	no	—
229	C₂×C₈	D₁₃⋊C₈	1	4	yes	complemented
230	C₂×C₈	C₂²×#261	1	2	yes	—
231	C₂×C₈	C₂²×#261	1	2	yes	—
232	C₄²	—	13	2	no	—
233	C₄²	—	13	1	no	—
234	C₄²	—	13	2	no	—
235	C₄²	—	13	2	no	—
236	C₄²	—	13	1	no	—
237	C₄²	—	13	2	no	—
238	C₄²	—	13	1	no	—
239	C₄²	—	13	1	no	—
240	C₄²	C₄⋊#261	1	2	yes	—
241	C₄²	C₂²×#261	1	2	yes	—
242	C₁₃	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	13	4	no	—
244	C₂³	—	13	4	no	—
245	C₂³	—	13	4	no	—
246	C₂³	—	13	2	no	—
247	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	13	4	no	—
250	C₂×C₄	—	13	4	no	—
251	C₂×C₄	—	13	1	no	—
252	C₂×C₄	—	13	2	no	—
253	C₂×C₄	—	13	1	no	—
254	C₂×C₄	—	13	8	no	—
255	C₂×C₄	—	13	8	no	—
256	C₂×C₄	—	13	8	no	—
257	C₂×C₄	—	13	4	no	—
258	C₂×C₄	—	13	4	no	—
259	C₂×C₄	—	13	1	no	—
260	C₂×C₄	—	13	1	no	—
261	C₂×C₄	—	26	4	no	—
262	C₂×C₄	—	26	4	no	—
263	C₂×C₄	—	26	2	no	—
264	C₂×C₄	#88101	1	4	yes	—
265	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
266	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
270	C₂×C₄	C₂³×#261	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	13	4	no	—
273	C₈	—	26	2	no	—
274	C₈	—	52	8	no	—
275	C₈	C₂×D₁₃⋊C₈	1	4	yes	complemented
276	C₂²	—	13	16	no	—
277	C₂²	—	13	4	no	—
278	C₂²	—	13	4	no	—
279	C₂²	—	13	4	no	—
280	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
281	C₂²	C₂×#88101	1	2	yes	—
282	C₂²	D₁₃⋊#108	1	2	yes	—
283	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	13	4	no	—
286	C₄	—	13	2	no	—
287	C₄	—	13	2	no	—
288	C₄	—	26	1	no	—
289	C₄	—	26	1	no	—
290	C₄	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
291	C₄	C₂×#88101	1	4	yes	—
292	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
293	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
294	C₂	—	13	8	no	—
295	C₂	D₁₃⋊#334	1	4	yes	direct product factor
296	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
297	C₂	C₂²×#88101	1	1	yes	characteristic
298	C₂	C₂×D₁₃⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂²×#334
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718255	C₂	semidirect
3. D₁₃⋊#334	C₂	direct (N × H)
4. C₂×#11718253	C₂	semidirect
20. #11718255	C₂²	semidirect
21. #11718253	C₂²	semidirect
37. C₂×C₈×D₁₃	C₈	semidirect
75. C₂×C₈×C₁₃	C₂×C₈	semidirect
89. C₂×#1055	C₂×C₈	semidirect
93. C₈×D₁₃	C₂×C₈	semidirect
129. C₈×C₁₃	C₂²×C₈	semidirect
131. #1055	C₂²×C₈	semidirect
152. C₂×D₁₃	#334	semidirect
197. C₂×C₁₃	C₂×#334	semidirect
201. D₁₃	C₂×#334	semidirect
229. C₂×C₈	D₁₃⋊C₈	semidirect
242. C₁₃	C₂²×#334	semidirect
275. C₈	C₂×D₁₃⋊C₈	semidirect
295. C₂	D₁₃⋊#334	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×D₁₃⋊(C₄⋊C₈)	2
C₂×#11718255	2
D₁₃⋊#334	2
C₂×#11718253	2
C₂×C₄×C₈×D₁₃	2
C₂²×#334	13

Subgroups — C2×D13⋊#334 #16794900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₃⋊#334 #16794900