Subgroups — #15833579

Order 3328.

640 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	#13123912	C₂	1	2	yes	maximal, complemented
2	#13113750	C₂	1	2	yes	maximal, complemented
3	#13113721	C₂	1	2	yes	maximal, complemented
4	#13113660	C₂	1	2	yes	maximal, complemented
5	D₈×#1055	C₂	1	2	yes	maximal, complemented
6	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113305	C₂	1	2	yes	maximal, complemented
8	#13113119	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#532	C₂²	1	2	yes	—
11	C₁₃×#500	C₄	1	1	yes	characteristic
12	C₁₃×#493	C₄	1	1	yes	characteristic
13	C₁₃×#486	C₄	1	2	yes	—
14	C₁₃×#466	C₄	1	2	yes	—
15	C₁₃×#466	C₄	1	2	yes	—
16	C₄×C₁₃×D₈	C₂²	1	2	yes	—
17	C₄×C₁₃×D₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	#11718895	—	2	4	no	—
20	#11718673	C₂²	1	4	yes	—
21	#11718672	—	2	4	no	—
22	D₄×#1055	C₂²	1	4	yes	—
23	#11718587	C₂²	1	2	yes	—
24	#11718586	C₂²	1	1	yes	characteristic
25	#11718586	C₂²	1	1	yes	characteristic
26	D₄⋊#1055	C₂²	1	4	yes	—
27	D₄⋊#1055	C₂²	1	4	yes	—
28	#11718050	C₂²	1	2	yes	—
29	#11718049	C₂²	1	2	yes	—
30	#11718047	C₂²	1	1	yes	characteristic
31	#11718047	C₂²	1	1	yes	characteristic
32	C₈×#1055	C₂²	1	1	yes	characteristic
33	C₈×#1055	C₂²	1	1	yes	characteristic
34	C₂×C₁₃×#41	—	2	2	no	—
35	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₁₃×#127	C₂×C₄	1	2	yes	—
41	C₁₃×#124	—	2	2	no	—
42	C₁₃×#121	—	2	2	no	—
43	C₁₃×#121	C₂×C₄	1	4	yes	—
44	C₄×D₄×C₁₃	—	2	2	no	—
45	C₄×D₄×C₁₃	C₂³	1	4	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×#6099	—	2	4	no	—
48	C₂²×#1055	—	2	4	no	—
49	C₁₃×#107	C₂×C₄	1	2	yes	—
50	C₁₃×#107	C₂×C₄	1	2	yes	—
51	C₁₃×#102	C₂×C₄	1	4	yes	—
52	C₁₃×#102	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²⋊#1055	—	2	2	no	—
58	C₂²⋊#1055	—	2	1	no	—
59	C₂²⋊#1055	—	2	4	no	—
60	C₂²⋊#1055	—	2	4	no	—
61	C₂²⋊#1055	D₄	1	2	yes	—
62	C₄⋊#1055	—	2	1	no	—
63	C₄⋊#1055	D₄	1	2	yes	—
64	C₄⋊#1055	C₂³	1	4	yes	—
65	#88043	—	2	2	no	—
66	C₄×#1055	C₂³	1	1	yes	characteristic
67	C₄×#1055	C₂³	1	1	yes	characteristic
68	C₄×#1055	C₂³	1	1	yes	characteristic
69	C₄×#1055	C₂³	1	1	yes	characteristic
70	#17970	—	13	1	no	maximal, Sylow
71	C₁₃×#41	—	8	4	no	—
72	C₂×Q₈×C₁₃	—	2	2	no	—
73	C₂×D₄×C₁₃	—	2	4	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
81	C₁₃×D₈	—	2	4	no	—
82	C₁₃×D₈	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
91	C₁₃×C₂²⋊C₄	—	2	4	no	—
92	C₁₃×C₂²⋊C₄	—	2	4	no	—
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₃	#41	1	1	yes	characteristic
96	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
97	#6099	—	8	4	no	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	—	2	1	no	—
100	C₂×#1055	—	2	1	no	—
101	C₂×#1055	—	8	4	no	—
102	C₂×#1055	C₂×D₄	1	2	yes	—
103	C₂×#1055	C₂×D₄	1	2	yes	—
104	#3169	D₁₃	1	1	yes	characteristic
105	#2981	—	13	2	no	—
106	#1594	—	13	2	no	—
107	#1589	—	13	2	no	—
108	#1580	—	13	2	no	—
109	C₈×D₈	—	13	2	no	—
110	#1568	—	13	1	no	—
111	#1496	—	13	2	no	—
112	#1450	—	13	1	no	—
113	C₂³×C₁₃	—	2	4	no	—
114	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
115	Q₈×C₁₃	—	8	2	no	—
116	D₄×C₁₃	—	2	4	no	—
117	D₄×C₁₃	—	2	4	no	—
118	D₄×C₁₃	—	8	2	no	—
119	D₄×C₁₃	—	8	2	no	—
120	D₄×C₁₃	—	8	4	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	#131	1	2	yes	—
137	C₈×C₁₃	#131	1	2	yes	—
138	#1055	—	2	2	no	—
139	#1055	—	2	2	no	—
140	#1055	—	4	1	no	—
141	#1055	—	4	1	no	—
142	#1055	—	4	4	no	—
143	#532	C₂×D₁₃	1	2	yes	—
144	#500	Dic₁₃	1	1	yes	characteristic
145	#493	Dic₁₃	1	1	yes	characteristic
146	#486	Dic₁₃	1	2	yes	—
147	#466	Dic₁₃	1	2	yes	—
148	#466	Dic₁₃	1	2	yes	—
149	C₄×D₈	C₂×D₁₃	1	2	yes	—
150	C₄×D₈	C₂×D₁₃	1	2	yes	—
151	#436	—	13	4	no	—
152	#435	—	26	4	no	—
153	C₈×D₄	—	13	4	no	—
154	#432	—	13	2	no	—
155	#431	—	13	1	no	—
156	#431	—	13	1	no	—
157	#408	—	26	4	no	—
158	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
159	#335	—	13	2	no	—
160	#334	—	13	2	no	—
161	D₄⋊C₈	—	13	4	no	—
162	D₄⋊C₈	—	13	4	no	—
163	#322	—	13	1	no	—
164	#322	—	13	1	no	—
165	C₈²	—	13	1	no	—
166	C₈²	—	13	1	no	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	—	4	4	no	—
172	C₂²×C₁₃	—	4	4	no	—
173	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
180	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
181	C₄×C₁₃	#576	1	2	yes	—
182	C₂×#41	—	2	2	no	—
183	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
184	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
185	C₂×M₁₆	—	26	4	no	—
186	C₂²×C₈	—	26	4	no	—
187	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
189	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
190	#127	C₂×Dic₁₃	1	2	yes	—
191	#124	—	2	2	no	—
192	#121	—	2	2	no	—
193	#121	C₂×Dic₁₃	1	4	yes	—
194	C₄×D₄	—	2	2	no	—
195	C₄×D₄	C₂²×D₁₃	1	4	yes	—
196	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
197	#107	C₂×Dic₁₃	1	2	yes	—
198	#107	C₂×Dic₁₃	1	2	yes	—
199	C₄⋊C₈	—	13	2	no	—
200	C₄⋊C₈	—	13	4	no	—
201	C₄⋊C₈	—	26	1	no	—
202	#102	C₂×Dic₁₃	1	4	yes	—
203	#102	C₂×Dic₁₃	1	4	yes	—
204	C₂²⋊C₈	—	13	2	no	—
205	C₂²⋊C₈	—	26	4	no	—
206	C₂²⋊C₈	—	26	4	no	—
207	C₂²⋊C₈	—	26	2	no	—
208	C₂²⋊C₈	—	26	1	no	—
209	#97	—	26	2	no	—
210	C₄×C₈	—	13	1	no	—
211	C₄×C₈	—	13	1	no	—
212	C₄×C₈	—	13	1	no	—
213	C₄×C₈	—	13	1	no	—
214	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
215	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
218	C₂×C₁₃	—	4	1	no	—
219	C₂×C₁₃	—	8	4	no	—
220	C₂×C₁₃	#2931	1	1	yes	characteristic
221	C₂×C₁₃	#2948	1	1	yes	characteristic
222	C₂×C₁₃	#2957	1	1	yes	characteristic
223	#41	—	8	4	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂²×Dic₁₃	1	4	yes	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	#6129	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
233	D₈	—	2	4	no	—
234	D₈	—	2	4	no	—
235	M₁₆	—	104	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	13	2	no	—
239	C₂×C₈	—	13	2	no	—
240	C₂×C₈	—	26	4	no	—
241	C₂×C₈	—	26	1	no	—
242	C₂×C₈	—	26	1	no	—
243	C₂×C₈	—	104	4	no	—
244	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₄²	D₄×D₁₃	1	1	yes	characteristic
253	C₄²	#6129	1	1	yes	characteristic
254	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
255	C₄²	C₂³×D₁₃	1	1	yes	characteristic
256	C₁₃	#17970	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	2	4	no	—
258	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
259	Q₈	—	8	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
273	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
274	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#6129	1	1	yes	characteristic
277	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	26	2	no	—
281	C₈	—	26	2	no	—
282	C₈	—	52	1	no	—
283	C₈	—	52	1	no	—
284	C₈	—	52	4	no	—
285	C₈	#88202	1	2	yes	—
286	C₈	#88202	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	4	no	—
299	C₄	#11719462	1	2	yes	—
300	C₄	C₂×#88202	1	1	yes	characteristic
301	C₄	C₂×#88202	1	1	yes	characteristic
302	C₂	—	4	1	no	—
303	C₂	—	8	4	no	—
304	C₂	#13123548	1	1	yes	characteristic
305	C₂	#13123688	1	1	yes	characteristic
306	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3169	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#17970
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13123912	C₂	semidirect
2. #13113750	C₂	semidirect
3. #13113721	C₂	semidirect
4. #13113660	C₂	semidirect
5. D₈×#1055	C₂	semidirect
6. #13113599	C₂	semidirect
7. #13113305	C₂	semidirect
8. #13113119	C₂	semidirect
256. C₁₃	#17970	semidirect

Maximal subgroups

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

subgroup	index
#13123912	2
#13113750	2
#13113721	2
#13113660	2
D₈×#1055	2
#13113599	2
#13113305	2
#13113119	2
C₁₃×#3169	2
#17970	13