Subgroups — #16795897

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	#13131728	C₂	1	2	yes	maximal, complemented
2	#13130485	C₂	1	2	yes	maximal, complemented
3	#13130468	C₂	1	2	yes	maximal, complemented
4	#13130434	C₂	1	2	yes	maximal, complemented
5	#13130417	C₂	1	2	yes	maximal, complemented
6	#13130397	C₂	1	1	yes	maximal, characteristic, complemented
7	#13130197	C₂	1	2	yes	maximal, complemented
8	#13130040	C₂	1	1	yes	maximal, characteristic, complemented
9	#13125416	C₂	1	1	yes	maximal, characteristic
10	#11719430	C₂²	1	2	yes	—
11	#11719159	—	2	4	no	—
12	#11719076	C₂²	1	4	yes	—
13	#11719075	—	2	4	no	—
14	#11719073	C₂²	1	4	yes	—
15	#11719054	C₂²	1	2	yes	—
16	#11719049	C₂²	1	1	yes	characteristic
17	#11719049	C₂²	1	1	yes	characteristic
18	C₁₃×#500	C₄	1	1	yes	characteristic
19	#11718749	C₄	1	2	yes	—
20	#11718544	C₄	1	2	yes	—
21	#11718544	C₄	1	2	yes	—
22	#11718533	C₂²	1	2	yes	—
23	#11718533	C₂²	1	2	yes	—
24	#11718356	C₄	1	1	yes	characteristic
25	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
26	#11718254	C₂²	1	2	yes	—
27	#11718253	C₂²	1	2	yes	—
28	#11718251	C₂²	1	1	yes	characteristic
29	#11718251	C₂²	1	1	yes	characteristic
30	C₈×#1057	C₂²	1	1	yes	characteristic
31	C₈×#1057	C₂²	1	1	yes	characteristic
32	#11718243	C₂²	1	4	yes	—
33	#11718243	C₂²	1	4	yes	—
34	C₂×#6131	—	2	2	no	—
35	C₂×#6118	—	2	4	no	—
36	C₂×D₁₃⋊C₈	—	2	4	no	—
37	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	#88201	—	2	2	no	—
39	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
40	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	#88149	—	2	2	no	—
44	#88149	C₂×C₄	1	4	yes	—
45	#88147	—	2	2	no	—
46	#88147	C₂³	1	4	yes	—
47	#88128	C₂×C₄	1	2	yes	—
48	C₄²×D₁₃	C₂³	1	1	yes	characteristic
49	#88112	—	2	1	no	—
50	#88112	D₄	1	2	yes	—
51	#88111	—	2	2	no	—
52	#88111	—	2	1	no	—
53	#88111	—	2	4	no	—
54	#88111	—	2	4	no	—
55	#88111	D₄	1	2	yes	—
56	#88110	—	2	2	no	—
57	C₄⋊#1057	C₂³	1	4	yes	—
58	C₄×#1057	C₂³	1	1	yes	characteristic
59	C₄×#1057	C₂³	1	1	yes	characteristic
60	C₄×#1057	C₂³	1	1	yes	characteristic
61	C₄×#1057	C₂³	1	1	yes	characteristic
62	C₁₃×#107	C₂×C₄	1	2	yes	—
63	C₁₃×#107	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
66	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
67	#88049	C₂×C₄	1	4	yes	—
68	#88049	C₂×C₄	1	4	yes	—
69	C₄×#1055	C₂×C₄	1	1	yes	characteristic
70	#17970	—	13	1	no	maximal, Sylow
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	#6131	—	8	4	no	—
73	C₂×D₅₂	—	2	4	no	—
74	C₂×D₅₂	—	2	2	no	—
75	C₂×D₅₂	—	2	2	no	—
76	C₂×D₅₂	C₂²×C₄	1	4	yes	—
77	C₂×C₄×D₁₃	—	2	4	no	—
78	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
81	C₂×#1057	—	2	4	no	—
82	C₂×#1057	—	2	1	no	—
83	C₂×#1057	—	2	1	no	—
84	C₂×#1057	C₂×D₄	1	2	yes	—
85	C₂×#1057	C₂×D₄	1	2	yes	—
86	#6118	—	8	4	no	—
87	D₁₃⋊C₈	—	8	4	no	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	#6103	—	2	4	no	—
94	#6103	—	2	4	no	—
95	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄×Dic₁₃	#41	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
98	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
99	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
100	D₁₀₄	—	2	4	no	—
101	D₁₀₄	—	2	4	no	—
102	C₈×D₁₃	—	2	2	no	—
103	C₈×D₁₃	—	2	2	no	—
104	#3169	—	13	1	no	—
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	C₂²×D₁₃	—	2	4	no	—
114	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
115	Q₈×C₁₃	—	8	2	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
123	D₅₂	—	2	4	no	—
124	D₅₂	—	2	4	no	—
125	D₅₂	—	8	2	no	—
126	D₅₂	—	8	2	no	—
127	D₅₂	—	8	4	no	—
128	C₄×D₁₃	—	2	1	no	—
129	C₄×D₁₃	—	2	1	no	—
130	C₄×D₁₃	—	2	2	no	—
131	C₄×D₁₃	—	4	1	no	—
132	C₄×D₁₃	—	4	4	no	—
133	C₄×D₁₃	—	4	4	no	—
134	#1057	—	2	2	no	—
135	#1057	—	2	2	no	—
136	#1057	—	4	1	no	—
137	#1057	—	4	1	no	—
138	#1057	—	4	4	no	—
139	C₈×C₁₃	#131	1	2	yes	—
140	C₈×C₁₃	#131	1	2	yes	—
141	#1055	—	2	1	no	—
142	#1055	—	2	1	no	—
143	#532	—	13	2	no	—
144	#500	#261	1	1	yes	characteristic
145	#493	—	13	1	no	—
146	#486	—	13	2	no	—
147	#466	—	13	2	no	—
148	#466	—	13	2	no	—
149	C₄×D₈	—	13	2	no	—
150	C₄×D₈	—	13	2	no	—
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	—	13	1	no	—
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	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂×D₁₃	—	2	1	no	—
169	C₂×D₁₃	—	2	1	no	—
170	C₂×D₁₃	—	2	1	no	—
171	C₂×D₁₃	—	4	4	no	—
172	C₂×D₁₃	—	4	4	no	—
173	C₂×D₁₃	—	4	4	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	4	4	no	—
177	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
178	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
179	Dic₁₃	—	2	1	no	—
180	Dic₁₃	—	2	1	no	—
181	Dic₁₃	#576	1	2	yes	—
182	C₂×#41	—	26	2	no	—
183	C₂×D₈	—	13	2	no	—
184	C₂×D₈	—	13	2	no	—
185	C₂×M₁₆	—	26	4	no	—
186	C₂²×C₈	—	13	1	no	—
187	C₂²×C₈	—	13	1	no	—
188	C₂²×C₈	—	26	4	no	—
189	C₄⋊Q₈	C₂×#261	1	2	yes	—
190	#127	—	13	2	no	—
191	#124	—	26	2	no	—
192	#121	—	13	4	no	—
193	#121	—	26	2	no	—
194	C₄×D₄	—	13	4	no	—
195	C₄×D₄	—	26	2	no	—
196	C₂×C₄²	—	13	1	no	—
197	#107	C₂×#261	1	2	yes	—
198	#107	C₂×#261	1	2	yes	—
199	C₄⋊C₈	—	13	2	no	—
200	C₄⋊C₈	—	13	4	no	—
201	C₄⋊C₈	—	26	1	no	—
202	#102	—	13	4	no	—
203	#102	—	13	4	no	—
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	—	13	1	no	—
215	C₄×C₈	—	13	1	no	—
216	C₄×C₈	—	13	1	no	—
217	C₄×C₈	C₂×#261	1	1	yes	characteristic
218	C₂×C₁₃	#2931	1	1	yes	characteristic
219	C₂×C₁₃	#2948	1	1	yes	characteristic
220	C₂×C₁₃	#2957	1	1	yes	characteristic
221	D₁₃	—	4	1	no	—
222	D₁₃	—	8	4	no	—
223	#41	—	104	4	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	13	4	no	—
226	C₂×D₄	—	26	4	no	—
227	C₂×D₄	—	26	2	no	—
228	C₂×D₄	—	26	2	no	—
229	C₂²×C₄	—	13	1	no	—
230	C₂²×C₄	—	13	1	no	—
231	C₂²×C₄	—	13	1	no	—
232	C₂²×C₄	—	26	4	no	—
233	D₈	—	26	4	no	—
234	D₈	—	26	4	no	—
235	M₁₆	—	104	4	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	13	1	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	26	2	no	—
242	C₂×C₈	—	26	1	no	—
243	C₂×C₈	—	26	1	no	—
244	C₂×C₈	—	26	4	no	—
245	C₂×C₈	—	104	4	no	—
246	C₂×C₈	C₂²×#261	1	1	yes	characteristic
247	C₂×C₈	C₂²×#261	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	C₂²×#261	1	4	yes	—
250	C₂²⋊C₄	—	26	4	no	—
251	C₂²⋊C₄	—	26	4	no	—
252	C₄²	—	13	1	no	—
253	C₄²	—	13	1	no	—
254	C₄²	—	13	1	no	—
255	C₄²	C₂²×#261	1	1	yes	characteristic
256	C₁₃	#17970	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	13	1	no	—
258	C₂³	—	26	4	no	—
259	Q₈	—	8	2	no	—
260	D₄	—	26	4	no	—
261	D₄	—	26	4	no	—
262	D₄	—	104	2	no	—
263	D₄	—	104	2	no	—
264	D₄	—	104	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	13	1	no	—
267	C₂×C₄	—	13	1	no	—
268	C₂×C₄	—	13	1	no	—
269	C₂×C₄	—	26	2	no	—
270	C₂×C₄	—	26	1	no	—
271	C₂×C₄	—	26	1	no	—
272	C₂×C₄	—	52	1	no	—
273	C₂×C₄	—	52	4	no	—
274	C₂×C₄	—	52	4	no	—
275	C₂×C₄	D₄×#261	1	1	yes	characteristic
276	C₂×C₄	D₄×#261	1	1	yes	characteristic
277	C₂×C₄	C₂³×#261	1	1	yes	characteristic
278	C₈	—	26	2	no	—
279	C₈	—	26	2	no	—
280	C₈	—	26	1	no	—
281	C₈	—	26	1	no	—
282	C₈	—	52	4	no	—
283	C₈	—	52	1	no	—
284	C₈	—	52	1	no	—
285	C₈	#88240	1	2	yes	—
286	C₈	#88240	1	2	yes	—
287	C₂²	—	26	1	no	—
288	C₂²	—	26	1	no	—
289	C₂²	—	26	1	no	—
290	C₂²	—	52	4	no	—
291	C₂²	—	52	4	no	—
292	C₂²	—	52	4	no	—
293	C₂²	C₂×D₄×#261	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	4	4	no	—
297	C₄	—	13	2	no	—
298	C₄	—	26	1	no	—
299	C₄	—	26	1	no	—
300	C₄	C₂×#88240	1	1	yes	characteristic
301	C₄	C₂×#88240	1	1	yes	characteristic
302	C₂	—	52	1	no	—
303	C₂	—	104	4	no	—
304	C₂	#13131561	1	1	yes	characteristic
305	C₂	#13131595	1	1	yes	characteristic
306	C₂	#13131616	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	#17970
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. #13131728	C₂	semidirect
2. #13130485	C₂	semidirect
3. #13130468	C₂	semidirect
4. #13130434	C₂	semidirect
5. #13130417	C₂	semidirect
6. #13130397	C₂	semidirect
7. #13130197	C₂	semidirect
8. #13130040	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
#13131728	2
#13130485	2
#13130468	2
#13130434	2
#13130417	2
#13130397	2
#13130197	2
#13130040	2
#13125416	2
#17970	13