Subgroups — #16795818

Order 3328.

750 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131561	C₂	1	2	yes	maximal, complemented
2	#13130468	C₂	1	4	yes	maximal, complemented
3	#13130417	C₂	1	4	yes	maximal, complemented
4	#13130394	C₂	1	1	yes	maximal, characteristic, complemented
5	#13130183	C₂	1	2	yes	maximal, complemented
6	C₈×D₁₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#13125416	C₂	1	1	yes	maximal, characteristic
8	C₂×#88240	—	2	2	no	—
9	#11719430	C₂²	1	2	yes	—
10	#11719159	—	2	2	no	—
11	#11719076	—	2	2	no	—
12	#11719076	C₂²	1	4	yes	—
13	#11719073	—	2	2	no	—
14	#11719073	C₂²	1	4	yes	—
15	#11719052	C₂²	1	2	yes	—
16	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₁₃×#500	C₄	1	1	yes	characteristic
19	#11718749	C₄	1	2	yes	—
20	#11718544	C₄	1	4	yes	—
21	#11718533	C₂²	1	4	yes	—
22	#11718356	C₄	1	1	yes	characteristic
23	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
24	#11718254	C₂²	1	4	yes	—
25	C₈×#1057	C₂²	1	2	yes	—
26	C₈×#1057	C₂²	1	2	yes	—
27	#11718243	C₂²	1	8	yes	—
28	C₂×#6131	—	2	2	no	—
29	#88240	—	4	8	no	—
30	C₂×#6118	—	2	2	no	—
31	C₂×#6118	—	2	4	no	—
32	C₂×D₁₃⋊C₈	—	2	1	no	—
33	C₂×D₁₃⋊C₈	—	2	4	no	—
34	C₂×D₁₃⋊C₈	D₄	1	1	yes	characteristic
35	C₂×D₁₃⋊C₈	D₄	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	#88201	—	2	2	no	—
38	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	#88149	—	2	2	no	—
41	#88149	C₂×C₄	1	4	yes	—
42	#88147	—	2	2	no	—
43	#88147	C₂³	1	4	yes	—
44	#88128	C₂×C₄	1	2	yes	—
45	C₄²×D₁₃	C₂³	1	1	yes	characteristic
46	#88111	—	2	4	no	—
47	#88111	—	2	4	no	—
48	C₄⋊#1057	—	2	2	no	—
49	C₄⋊#1057	C₂³	1	4	yes	—
50	C₄×#1057	—	2	1	no	—
51	C₄×#1057	D₄	1	1	yes	characteristic
52	C₄×#1057	D₄	1	1	yes	characteristic
53	C₄×#1057	C₂³	1	2	yes	—
54	C₄×#1057	C₂³	1	1	yes	characteristic
55	C₄×#1057	C₂³	1	1	yes	characteristic
56	C₁₃×#107	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₁₃	C₂³	1	2	yes	—
59	#88049	C₂×C₄	1	8	yes	—
60	C₄×#1055	C₂×C₄	1	1	yes	characteristic
61	#17956	—	13	1	no	maximal, Sylow
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	#6131	—	4	4	no	—
64	#6131	—	4	4	no	—
65	C₂×D₅₂	—	2	4	no	—
66	C₂×D₅₂	—	2	2	no	—
67	C₂×D₅₂	—	2	2	no	—
68	C₂×D₅₂	C₂²×C₄	1	4	yes	—
69	C₂×C₄×D₁₃	—	2	4	no	—
70	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
73	C₂×#1057	—	2	2	no	—
74	C₂×#1057	—	2	4	no	—
75	C₂×#1057	C₂×D₄	1	1	yes	characteristic
76	C₂×#1057	C₂×D₄	1	1	yes	characteristic
77	C₂×#1057	C₂×D₄	1	1	yes	characteristic
78	C₂×#1057	C₂×D₄	1	1	yes	characteristic
79	#6118	—	4	4	no	—
80	#6118	—	4	8	no	—
81	D₁₃⋊C₈	—	2	4	no	—
82	D₁₃⋊C₈	—	2	2	no	—
83	D₁₃⋊C₈	—	4	8	no	—
84	D₁₃⋊C₈	D₈	1	4	yes	complemented
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	#6103	—	2	4	no	—
90	#6103	—	2	4	no	—
91	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄×Dic₁₃	#41	1	1	yes	characteristic
93	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
94	C₂×#1055	C₂²×C₄	1	2	yes	—
95	D₁₀₄	—	2	8	no	—
96	C₈×D₁₃	—	2	4	no	—
97	#3169	—	13	1	no	—
98	#2931	—	13	2	no	—
99	#1589	—	13	4	no	—
100	C₈×D₈	—	13	4	no	—
101	#1567	—	13	1	no	—
102	#1492	—	13	2	no	—
103	C₂×C₈²	—	13	1	no	—
104	C₂²×D₁₃	—	2	4	no	—
105	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
114	D₅₂	—	2	8	no	—
115	D₅₂	—	4	8	no	—
116	D₅₂	—	4	4	no	—
117	D₅₂	—	4	4	no	—
118	C₄×D₁₃	—	2	2	no	—
119	C₄×D₁₃	—	2	2	no	—
120	C₄×D₁₃	—	4	4	no	—
121	C₄×D₁₃	—	4	4	no	—
122	C₄×D₁₃	C₂×D₈	1	2	yes	—
123	C₄×D₁₃	#135	1	2	yes	—
124	#1057	—	2	4	no	—
125	#1057	—	2	1	no	—
126	#1057	—	2	1	no	—
127	#1057	—	4	4	no	—
128	#1057	C₂×D₈	1	2	yes	—
129	#1057	C₂×D₈	1	2	yes	—
130	C₈×C₁₃	#131	1	4	yes	complemented
131	#1055	—	2	2	no	—
132	C₂×#131	—	26	2	no	—
133	#532	—	13	2	no	—
134	#500	#261	1	1	yes	characteristic
135	#493	—	13	1	no	—
136	#486	—	13	2	no	—
137	#466	—	13	4	no	—
138	C₄×D₈	—	13	4	no	—
139	#436	—	13	4	no	—
140	#436	—	26	2	no	—
141	C₈×D₄	—	13	4	no	—
142	C₈×D₄	—	26	2	no	—
143	#423	—	13	2	no	—
144	#408	—	26	2	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	13	1	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	#335	—	13	4	no	—
149	D₄⋊C₈	—	13	8	no	—
150	C₈²	—	13	2	no	—
151	C₈²	—	13	2	no	—
152	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×D₁₃	—	2	1	no	—
154	C₂×D₁₃	—	2	1	no	—
155	C₂×D₁₃	—	4	4	no	—
156	C₂×D₁₃	—	4	4	no	—
157	C₂×D₁₃	—	4	4	no	—
158	C₂×D₁₃	C₄×D₈	1	2	yes	—
159	C₄×C₁₃	—	4	4	no	—
160	C₄×C₁₃	C₄×D₈	1	2	yes	—
161	C₄×C₁₃	#443	1	2	yes	—
162	C₄×C₁₃	C₂×#131	1	2	yes	—
163	Dic₁₃	—	2	1	no	—
164	Dic₁₃	—	2	1	no	—
165	Dic₁₃	C₂²×D₈	1	1	yes	characteristic
166	Dic₁₃	C₂×#135	1	1	yes	characteristic
167	C₂×#41	—	26	2	no	—
168	C₂×D₈	—	13	4	no	—
169	#131	—	52	8	no	—
170	C₂×M₁₆	—	26	4	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂²×C₈	—	13	1	no	—
173	C₂²×C₈	—	13	1	no	—
174	C₂²×C₈	—	13	2	no	—
175	C₂²×C₈	—	26	4	no	—
176	C₂²×C₈	—	26	1	no	—
177	C₄⋊Q₈	C₂×#261	1	2	yes	—
178	#127	—	13	2	no	—
179	#124	—	26	2	no	—
180	#121	—	13	4	no	—
181	#121	—	26	2	no	—
182	C₄×D₄	—	13	4	no	—
183	C₄×D₄	—	26	2	no	—
184	C₂×C₄²	—	13	1	no	—
185	#107	C₂×#261	1	4	yes	—
186	C₄⋊C₈	—	13	4	no	—
187	C₄⋊C₈	—	26	2	no	—
188	#102	—	13	8	no	—
189	C₂²⋊C₈	—	26	4	no	—
190	C₂²⋊C₈	—	26	4	no	—
191	C₄×C₈	—	13	1	no	—
192	C₄×C₈	—	13	1	no	—
193	C₄×C₈	—	13	1	no	—
194	C₄×C₈	—	13	1	no	—
195	C₄×C₈	—	13	2	no	—
196	C₄×C₈	—	13	2	no	—
197	C₄×C₈	—	13	1	no	—
198	C₄×C₈	—	26	1	no	—
199	C₄×C₈	C₂×#261	1	1	yes	characteristic
200	C₂×C₁₃	#2931	1	1	yes	characteristic
201	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
202	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
203	D₁₃	—	2	2	no	—
204	D₁₃	—	8	4	no	—
205	#41	—	52	4	no	—
206	#41	—	52	4	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×D₄	—	13	4	no	—
209	C₂×D₄	—	26	4	no	—
210	C₂×D₄	—	26	2	no	—
211	C₂×D₄	—	26	2	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	26	4	no	—
216	D₈	—	26	8	no	—
217	M₁₆	—	52	8	no	—
218	M₁₆	—	52	4	no	—
219	C₂×C₈	—	13	1	no	—
220	C₂×C₈	—	13	1	no	—
221	C₂×C₈	—	13	4	no	—
222	C₂×C₈	—	13	1	no	—
223	C₂×C₈	—	13	1	no	—
224	C₂×C₈	—	13	2	no	—
225	C₂×C₈	—	26	2	no	—
226	C₂×C₈	—	26	2	no	—
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	—	52	8	no	—
231	C₂×C₈	C₂²×#261	1	2	yes	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	C₂²×#261	1	4	yes	—
234	C₂²⋊C₄	—	26	4	no	—
235	C₂²⋊C₄	—	26	4	no	—
236	C₄²	—	13	1	no	—
237	C₄²	—	13	1	no	—
238	C₄²	—	13	1	no	—
239	C₄²	C₂²×#261	1	1	yes	characteristic
240	C₁₃	#17956	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	13	1	no	—
242	C₂³	—	26	4	no	—
243	Q₈	—	4	4	no	—
244	D₄	—	26	8	no	—
245	D₄	—	52	8	no	—
246	D₄	—	52	4	no	—
247	D₄	—	52	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	2	no	—
252	C₂×C₄	—	13	1	no	—
253	C₂×C₄	—	13	1	no	—
254	C₂×C₄	—	26	2	no	—
255	C₂×C₄	—	26	2	no	—
256	C₂×C₄	—	52	4	no	—
257	C₂×C₄	—	52	4	no	—
258	C₂×C₄	D₄×#261	1	1	yes	characteristic
259	C₂×C₄	D₄×#261	1	1	yes	characteristic
260	C₂×C₄	C₂³×#261	1	1	yes	characteristic
261	C₈	—	13	2	no	—
262	C₈	—	13	2	no	—
263	C₈	—	26	1	no	—
264	C₈	—	26	1	no	—
265	C₈	—	26	4	no	—
266	C₈	—	26	2	no	—
267	C₈	—	52	4	no	—
268	C₈	#88240	1	4	yes	complemented
269	C₂²	—	13	2	no	—
270	C₂²	—	26	1	no	—
271	C₂²	—	26	1	no	—
272	C₂²	—	52	4	no	—
273	C₂²	—	52	4	no	—
274	C₂²	—	52	4	no	—
275	C₂²	C₂×D₄×#261	1	1	yes	characteristic
276	C₄	—	4	4	no	—
277	C₄	—	13	1	no	—
278	C₄	—	13	1	no	—
279	C₄	—	26	1	no	—
280	C₄	—	26	1	no	—
281	C₄	D₈×#261	1	2	yes	—
282	C₄	#11719111	1	2	yes	—
283	C₄	C₂×#88240	1	2	yes	—
284	C₂	—	26	2	no	—
285	C₂	—	104	4	no	—
286	C₂	#13131561	1	1	yes	characteristic
287	C₂	C₂×D₈×#261	1	1	yes	characteristic
288	C₂	C₂×#11719111	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#500	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125416	C₂
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#500	#261
hypocenter	C₁₃	#17956
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
1. #13131561	C₂	semidirect
2. #13130468	C₂	semidirect
3. #13130417	C₂	semidirect
4. #13130394	C₂	semidirect
5. #13130183	C₂	semidirect
6. C₈×D₁₃⋊C₈	C₂	semidirect
84. D₁₃⋊C₈	D₈	semidirect
130. C₈×C₁₃	#131	semidirect
240. C₁₃	#17956	semidirect
268. C₈	#88240	semidirect

Maximal subgroups

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

subgroup	index
#13131561	2
#13130468	2
#13130417	2
#13130394	2
#13130183	2
C₈×D₁₃⋊C₈	2
#13125416	2
#17956	13