Subgroups — #16794916

Order 3328.

778 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊#431	C₂	1	2	yes	maximal
2	#13130388	C₂	1	8	yes	maximal, complemented
3	#13130385	C₂	1	2	yes	maximal, complemented
4	#13130383	C₂	1	2	yes	maximal, complemented
5	C₂×#11718368	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6094	C₄	1	1	yes	characteristic
7	C₂²×#6094	C₄	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	#11719125	—	2	2	no	—
10	D₁₃⋊(C₂²⋊C₈)	—	2	2	no	—
11	#11719119	C₂²	1	4	yes	—
12	#11719051	C₂²	1	4	yes	—
13	D₁₃⋊(C₄⋊C₈)	C₂²	1	4	yes	—
14	#11719049	—	2	2	no	—
15	C₄×D₁₃⋊C₈	—	2	2	no	—
16	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88043	C₄	1	1	yes	characteristic
20	#11718368	C₄	1	4	yes	—
21	#11718368	C₂²	1	4	yes	—
22	#11718256	C₂²	1	4	yes	complemented
23	#11718255	C₂²	1	4	yes	complemented
24	#11718254	C₂²	1	4	yes	complemented
25	#11718253	C₂²	1	4	yes	complemented
26	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
29	C₂×D₁₃⋊C₈	—	2	4	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂×#6094	—	2	4	no	—
39	C₂×#6094	C₈	1	8	yes	complemented
40	C₂×#6094	C₂×C₄	1	4	yes	—
41	C₂×#6094	C₂×C₄	1	4	yes	—
42	C₄²×D₁₃	C₂×C₄	1	2	yes	—
43	C₄²×D₁₃	D₄	1	2	yes	—
44	C₄²×D₁₃	Q₈	1	2	yes	—
45	C₄²×D₁₃	C₂³	1	2	yes	—
46	C₂²⋊#1057	—	2	4	no	—
47	#88112	—	2	4	no	—
48	#88111	—	2	4	no	—
49	C₄⋊#1057	C₂³	1	8	yes	—
50	C₄×#1057	—	2	4	no	—
51	C₄×#1057	—	2	4	no	—
52	C₁₃×#97	C₂×C₄	1	4	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₃	C₂³	1	2	yes	—
56	C₈×Dic₁₃	C₂³	1	2	yes	—
57	#88043	C₂×C₄	1	4	yes	—
58	#17787	—	13	1	no	maximal, Sylow
59	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄×D₁₃	—	4	2	no	—
67	C₂×C₄×D₁₃	—	4	2	no	—
68	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
72	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
73	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
76	C₂×#1057	—	2	8	no	—
77	D₁₃⋊C₈	—	4	8	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	complemented
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
84	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
85	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
86	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
89	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
90	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
91	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
92	C₂×#1055	—	2	2	no	—
93	C₂×#1055	C₂×C₈	1	4	yes	complemented
94	C₂×#1055	C₂²×C₄	1	2	yes	—
95	C₂×#1055	C₂²×C₄	1	2	yes	—
96	#6094	—	2	8	no	—
97	#6094	C₂×C₈	1	16	yes	complemented
98	C₂×#431	—	13	2	no	—
99	C₂×C₄×M₁₆	—	13	1	no	—
100	#1565	—	13	8	no	—
101	#1564	—	13	2	no	—
102	#1564	—	13	2	no	—
103	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×D₁₃	—	4	1	no	—
105	C₂²×D₁₃	—	4	1	no	—
106	C₂²×D₁₃	C₄⋊C₈	1	4	yes	—
107	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×Dic₁₃	—	2	2	no	—
117	C₂×Dic₁₃	—	2	2	no	—
118	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
119	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
124	C₄×D₁₃	—	2	4	no	—
125	C₄×D₁₃	—	2	4	no	—
126	C₄×D₁₃	—	4	2	no	—
127	C₄×D₁₃	—	4	2	no	—
128	C₄×D₁₃	C₄⋊C₈	1	8	yes	—
129	C₄×D₁₃	C₂²×C₈	1	4	yes	—
130	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
131	#1057	—	4	8	no	—
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	C₂²×C₈	1	4	yes	—
134	#1055	—	2	2	no	—
135	#1055	C₂²×C₈	1	4	yes	—
136	C₂²×M₁₆	—	13	1	no	—
137	C₂²×M₁₆	—	13	1	no	—
138	C₂²×C₄²	—	13	1	no	—
139	#431	—	13	4	no	—
140	#431	—	13	4	no	—
141	#431	—	26	2	no	—
142	#431	—	26	2	no	—
143	C₂×C₄⋊C₈	—	13	4	no	—
144	C₂×C₂²⋊C₈	—	26	2	no	—
145	C₄×M₁₆	—	13	4	no	—
146	C₄×M₁₆	—	13	4	no	—
147	C₂×#97	—	13	1	no	—
148	C₂×#97	#261	1	1	yes	characteristic
149	C₂×C₄×C₈	—	13	1	no	—
150	C₂×C₄×C₈	—	13	1	no	—
151	C₂×C₄×C₈	—	26	2	no	—
152	#335	—	13	4	no	—
153	#335	—	13	4	no	—
154	#334	—	13	4	no	—
155	#334	—	13	4	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
159	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂×D₁₃	—	2	2	no	—
161	C₂×D₁₃	—	2	2	no	—
162	C₂×D₁₃	—	4	4	no	—
163	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
167	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
168	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
169	Dic₁₃	—	2	1	no	—
170	Dic₁₃	—	2	1	no	—
171	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
172	Dic₁₃	#430	1	2	yes	—
173	Dic₁₃	#575	1	2	yes	—
174	C₂³×C₄	—	13	1	no	—
175	C₂³×C₄	—	13	1	no	—
176	C₂³×C₄	—	13	1	no	—
177	C₂×M₁₆	—	13	8	no	—
178	C₂×M₁₆	—	13	4	no	—
179	C₂×M₁₆	—	13	4	no	—
180	C₂×M₁₆	—	26	4	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₂²×C₈	—	13	1	no	—
183	C₂²×C₈	—	26	4	no	—
184	C₂²×C₈	C₂×#261	1	1	yes	characteristic
185	C₂²×C₈	C₂×#261	1	1	yes	characteristic
186	C₂×C₄²	—	13	1	no	—
187	C₂×C₄²	—	13	1	no	—
188	C₂×C₄²	—	13	1	no	—
189	C₂×C₄²	—	13	2	no	—
190	C₂×C₄²	—	13	2	no	—
191	C₂×C₄²	—	13	2	no	—
192	C₂×C₄²	—	13	2	no	—
193	C₂×C₄²	C₂×#261	1	1	yes	characteristic
194	C₄⋊C₈	—	13	8	no	—
195	C₄⋊C₈	—	26	4	no	—
196	C₂²⋊C₈	—	26	4	no	—
197	C₂²⋊C₈	—	26	4	no	—
198	#97	—	13	4	no	—
199	#97	C₂×#261	1	4	yes	—
200	C₄×C₈	—	13	2	no	—
201	C₄×C₈	—	13	2	no	—
202	C₄×C₈	—	13	2	no	—
203	C₄×C₈	—	13	2	no	—
204	C₄×C₈	—	26	4	no	—
205	C₄×C₈	—	26	4	no	—
206	C₂×C₁₃	—	2	2	no	—
207	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
208	C₂×C₁₃	#2911	1	1	yes	characteristic
209	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
210	D₁₃	—	2	4	no	—
211	C₂⁴	—	13	1	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	4	no	—
214	C₂²×C₄	—	13	4	no	—
215	C₂²×C₄	—	13	2	no	—
216	C₂²×C₄	—	13	2	no	—
217	C₂²×C₄	—	13	1	no	—
218	C₂²×C₄	—	13	1	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	52	2	no	—
224	C₂²×C₄	—	52	2	no	—
225	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
226	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
227	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
228	M₁₆	—	13	16	no	—
229	M₁₆	—	26	8	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	13	4	no	—
232	C₂×C₈	—	13	2	no	—
233	C₂×C₈	—	13	2	no	—
234	C₂×C₈	—	26	2	no	—
235	C₂×C₈	—	26	8	no	—
236	C₂×C₈	—	52	8	no	—
237	C₂×C₈	D₁₃⋊C₈	1	4	yes	complemented
238	C₂×C₈	C₂²×#261	1	2	yes	—
239	C₂×C₈	C₂²×#261	1	2	yes	—
240	C₄²	—	13	2	no	—
241	C₄²	—	13	2	no	—
242	C₄²	—	13	2	no	—
243	C₄²	—	13	2	no	—
244	C₄²	—	13	1	no	—
245	C₄²	—	13	1	no	—
246	C₄²	—	13	1	no	—
247	C₄²	—	13	1	no	—
248	C₄²	C₄⋊#261	1	2	yes	—
249	C₄²	C₂²×#261	1	2	yes	—
250	C₁₃	#17787	1	1	yes	characteristic, complemented, Sylow
251	C₂³	—	13	2	no	—
252	C₂³	—	13	4	no	—
253	C₂³	—	52	1	no	—
254	C₂³	—	52	1	no	—
255	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	13	4	no	—
259	C₂×C₄	—	13	1	no	—
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	13	1	no	—
262	C₂×C₄	—	13	8	no	—
263	C₂×C₄	—	13	4	no	—
264	C₂×C₄	—	13	4	no	—
265	C₂×C₄	—	13	1	no	—
266	C₂×C₄	—	13	1	no	—
267	C₂×C₄	—	26	4	no	—
268	C₂×C₄	—	26	4	no	—
269	C₂×C₄	—	26	2	no	—
270	C₂×C₄	—	26	2	no	—
271	C₂×C₄	—	52	2	no	—
272	C₂×C₄	—	52	2	no	—
273	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
274	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
276	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
277	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
278	C₂×C₄	C₂³×#261	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	13	4	no	—
281	C₈	—	26	2	no	—
282	C₈	—	52	8	no	—
283	C₈	C₂×D₁₃⋊C₈	1	4	yes	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	13	4	no	—
287	C₂²	—	26	2	no	—
288	C₂²	—	26	2	no	—
289	C₂²	—	52	4	no	—
290	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
291	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	13	4	no	—
295	C₄	—	13	2	no	—
296	C₄	—	13	2	no	—
297	C₄	—	26	1	no	—
298	C₄	—	26	1	no	—
299	C₄	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
300	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
301	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	26	4	no	—
304	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
305	C₂	#13131489	1	1	yes	characteristic
306	C₂	D₁₃⋊#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#97	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#17787
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #13130388	C₂	semidirect
3. #13130385	C₂	semidirect
4. #13130383	C₂	semidirect
22. #11718256	C₂²	semidirect
23. #11718255	C₂²	semidirect
24. #11718254	C₂²	semidirect
25. #11718253	C₂²	semidirect
39. C₂×#6094	C₈	semidirect
79. C₂×C₈×C₁₃	C₂×C₈	semidirect
93. C₂×#1055	C₂×C₈	semidirect
97. #6094	C₂×C₈	semidirect
237. C₂×C₈	D₁₃⋊C₈	semidirect
250. C₁₃	#17787	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊#431	2
#13130388	2
#13130385	2
#13130383	2
C₂×#11718368	2
#17787	13