Subgroups — #16785592

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718268	C₂	1	2	yes	maximal
2	C₂×#11718268	C₂	1	2	yes	maximal
3	#13130500	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#6118	C₂²	1	4	yes	—
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₄	1	1	yes	characteristic
13	C₄×C₈×D₁₃	—	2	4	no	—
14	#11718290	—	2	2	no	—
15	#11718290	—	2	2	no	—
16	#11718268	—	2	4	no	—
17	#11718268	C₄	1	8	yes	complemented
18	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
19	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
21	C₂²×#1057	—	2	4	no	—
22	C₂×#6118	—	2	4	no	—
23	C₂×#6118	—	2	4	no	—
24	C₂×#6118	—	2	4	no	—
25	C₂×#6118	—	2	4	no	—
26	C₂×#6118	—	2	4	no	—
27	C₂×#6118	C₂×C₄	1	8	yes	—
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	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	D₄	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	2	no	—
37	C₂×C₈×D₁₃	—	2	2	no	—
38	C₂×C₈×D₁₃	—	2	2	no	—
39	C₂×C₈×D₁₃	—	2	2	no	—
40	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₈×D₁₃	D₄	1	4	yes	—
42	C₄²×D₁₃	—	2	2	no	—
43	C₄²×D₁₃	—	2	2	no	—
44	C₄×C₈×C₁₃	—	2	2	no	—
45	C₈×Dic₁₃	—	2	2	no	—
46	C₈×Dic₁₃	—	2	2	no	—
47	C₄×#1055	—	2	2	no	—
48	#10822	—	13	1	no	maximal, Sylow
49	C₂³×D₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×D₁₃	—	2	2	no	—
57	C₂×C₄×D₁₃	—	2	2	no	—
58	C₂×C₄×D₁₃	—	2	2	no	—
59	C₂×C₄×D₁₃	—	2	2	no	—
60	C₂×C₄×D₁₃	—	2	2	no	—
61	C₂×C₄×D₁₃	—	2	2	no	—
62	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
64	C₂×C₄×D₁₃	#41	1	2	yes	—
65	C₂×C₄×D₁₃	#41	1	2	yes	—
66	C₂×C₄×D₁₃	#41	1	2	yes	—
67	C₂×C₄×D₁₃	#41	1	2	yes	—
68	C₂×#1057	—	2	8	no	—
69	C₂×#1057	—	4	4	no	—
70	C₂×#1057	—	4	4	no	—
71	#6118	—	2	8	no	—
72	#6118	—	2	8	no	—
73	#6118	—	4	4	no	—
74	#6118	—	4	4	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	C₄²	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₃	—	2	1	no	—
82	C₄²×C₁₃	—	2	1	no	—
83	C₄×Dic₁₃	—	2	1	no	—
84	C₄×Dic₁₃	—	2	1	no	—
85	C₄×Dic₁₃	—	2	2	no	—
86	C₄×Dic₁₃	—	2	2	no	—
87	C₂×#1055	—	2	1	no	—
88	C₂×#1055	—	2	1	no	—
89	C₂×#1055	—	2	1	no	—
90	C₂×#1055	—	2	1	no	—
91	C₂×#1055	C₄²	1	2	yes	—
92	C₂×#1055	C₂²⋊C₄	1	2	yes	—
93	C₈×D₁₃	—	2	8	no	—
94	C₈×D₁₃	—	2	8	no	—
95	C₂²×C₄×C₈	—	13	1	no	—
96	C₂×#341	—	13	2	no	—
97	C₂×#341	—	13	2	no	—
98	#1728	—	13	2	no	—
99	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×D₁₃	—	2	1	no	—
101	C₂²×D₁₃	—	2	1	no	—
102	C₂²×D₁₃	—	2	1	no	—
103	C₂²×D₁₃	—	2	1	no	—
104	C₂²×D₁₃	C₄×Q₈	1	2	yes	—
105	C₂²×D₁₃	C₂²⋊Q₈	1	2	yes	—
106	C₂²×D₁₃	#125	1	2	yes	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	#108	1	2	yes	—
112	C₂×C₄×C₁₃	#108	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	—	2	1	no	—
121	C₂×Dic₁₃	—	2	1	no	—
122	C₂×Dic₁₃	—	2	1	no	—
123	C₂×Dic₁₃	#108	1	2	yes	—
124	C₂×Dic₁₃	#108	1	2	yes	—
125	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
126	C₂×Dic₁₃	#121	1	2	yes	—
127	C₂×Dic₁₃	#124	1	2	yes	—
128	C₄×D₁₃	—	2	8	no	—
129	C₄×D₁₃	—	2	8	no	—
130	C₄×D₁₃	#117	1	4	yes	—
131	C₄×D₁₃	C₄×D₄	1	4	yes	—
132	C₄×D₁₃	#123	1	4	yes	—
133	C₄×D₁₃	#126	1	4	yes	—
134	#1057	—	4	8	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	2	2	no	—
137	#1055	—	2	2	no	—
138	#1055	—	2	2	no	—
139	C₂²×M₁₆	—	13	4	no	—
140	C₂³×C₈	—	13	1	no	—
141	C₂³×C₈	—	13	1	no	—
142	C₂²×C₄²	—	13	1	no	—
143	C₂×C₄×C₈	—	13	1	no	—
144	C₂×C₄×C₈	—	13	1	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	26	4	no	—
147	C₂×C₄×C₈	#261	1	1	yes	characteristic
148	#341	—	13	8	no	—
149	#341	—	26	4	no	—
150	#336	—	26	2	no	—
151	#336	—	26	2	no	—
152	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₁₃	#380	1	1	yes	characteristic
154	C₂²×C₁₃	#382	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
159	C₂×D₁₃	—	2	2	no	—
160	C₂×D₁₃	—	2	2	no	—
161	C₂×D₁₃	—	2	2	no	—
162	C₂×D₁₃	—	2	2	no	—
163	C₂×D₁₃	—	2	2	no	—
164	C₂×D₁₃	—	2	2	no	—
165	C₂×D₁₃	#387	1	4	yes	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	#376	1	1	yes	characteristic
169	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
170	C₄×C₁₃	#381	1	1	yes	characteristic
171	C₄×C₁₃	#383	1	1	yes	characteristic
172	Dic₁₃	—	2	2	no	—
173	Dic₁₃	—	2	2	no	—
174	Dic₁₃	#388	1	4	yes	—
175	C₂³×C₄	—	13	1	no	—
176	C₂³×C₄	—	13	1	no	—
177	C₂³×C₄	—	13	1	no	—
178	C₂×M₁₆	—	13	8	no	—
179	C₂×M₁₆	—	26	4	no	—
180	C₂×M₁₆	—	26	4	no	—
181	C₂×M₁₆	—	26	4	no	—
182	C₂×M₁₆	—	26	4	no	—
183	C₂×M₁₆	—	26	4	no	—
184	C₂²×C₈	—	13	1	no	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	13	4	no	—
187	C₂²×C₈	—	13	4	no	—
188	C₂²×C₈	—	26	2	no	—
189	C₂²×C₈	—	26	2	no	—
190	C₂²×C₈	—	26	2	no	—
191	C₂²×C₈	—	26	2	no	—
192	C₂²×C₈	—	26	4	no	—
193	C₂²×C₈	C₂×#261	1	1	yes	characteristic
194	C₂²×C₈	C₂×#261	1	1	yes	characteristic
195	C₂×C₄²	—	13	1	no	—
196	C₂×C₄²	—	13	1	no	—
197	C₂×C₄²	—	13	1	no	—
198	C₂×C₄²	—	26	2	no	—
199	C₂×C₄²	—	26	2	no	—
200	C₂×C₄²	C₂×#261	1	1	yes	characteristic
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	—	26	2	no	—
203	C₄×C₈	—	26	2	no	—
204	C₄×C₈	—	26	2	no	—
205	C₂×C₁₃	#1434	1	1	yes	characteristic
206	C₂×C₁₃	C₄×#108	1	1	yes	characteristic
207	C₂×C₁₃	C₄×#108	1	1	yes	characteristic
208	C₂×C₁₃	#1810	1	1	yes	characteristic
209	C₂×C₁₃	#1810	1	1	yes	characteristic
210	C₂×C₁₃	#1837	1	1	yes	characteristic
211	C₂×C₁₃	#1837	1	1	yes	characteristic
212	D₁₃	—	2	4	no	—
213	C₂⁴	—	13	1	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	2	no	—
216	C₂²×C₄	—	13	2	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	13	2	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	1	no	—
222	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	26	2	no	—
224	C₂²×C₄	—	26	2	no	—
225	C₂²×C₄	—	26	2	no	—
226	C₂²×C₄	—	26	2	no	—
227	C₂²×C₄	—	26	2	no	—
228	C₂²×C₄	—	26	2	no	—
229	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
230	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
231	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
232	M₁₆	—	26	8	no	—
233	M₁₆	—	26	8	no	—
234	M₁₆	—	52	4	no	—
235	M₁₆	—	52	4	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	13	2	no	—
241	C₂×C₈	—	13	2	no	—
242	C₂×C₈	—	26	8	no	—
243	C₂×C₈	—	26	8	no	—
244	C₂×C₈	—	26	1	no	—
245	C₂×C₈	—	26	1	no	—
246	C₂×C₈	—	26	1	no	—
247	C₂×C₈	—	26	1	no	—
248	C₂×C₈	—	26	8	no	—
249	C₂×C₈	—	52	4	no	—
250	C₂×C₈	—	52	4	no	—
251	C₂×C₈	C₄×#261	1	2	yes	—
252	C₂×C₈	C₂²⋊#261	1	2	yes	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	26	2	no	—
256	C₄²	—	26	2	no	—
257	C₄²	—	26	1	no	—
258	C₄²	—	26	1	no	—
259	C₁₃	#10822	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	13	2	no	—
261	C₂³	—	13	2	no	—
262	C₂³	—	13	2	no	—
263	C₂³	—	26	1	no	—
264	C₂³	—	26	1	no	—
265	C₂³	—	26	1	no	—
266	C₂³	—	26	1	no	—
267	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	13	2	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	4	no	—
276	C₂×C₄	—	13	4	no	—
277	C₂×C₄	—	13	4	no	—
278	C₂×C₄	—	13	4	no	—
279	C₂×C₄	—	13	2	no	—
280	C₂×C₄	—	13	2	no	—
281	C₂×C₄	—	26	8	no	—
282	C₂×C₄	—	26	8	no	—
283	C₂×C₄	—	26	1	no	—
284	C₂×C₄	—	26	1	no	—
285	C₂×C₄	—	26	1	no	—
286	C₂×C₄	—	26	1	no	—
287	C₂×C₄	#88103	1	2	yes	—
288	C₂×C₄	#88104	1	2	yes	—
289	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
290	C₂×C₄	#88237	1	1	yes	characteristic
291	C₂×C₄	#88237	1	1	yes	characteristic
292	C₂×C₄	#88237	1	1	yes	characteristic
293	C₂×C₄	#88237	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	26	2	no	—
298	C₈	—	26	2	no	—
299	C₈	—	52	8	no	—
300	C₂²	—	13	4	no	—
301	C₂²	—	26	2	no	—
302	C₂²	—	26	2	no	—
303	C₂²	—	26	2	no	—
304	C₂²	—	26	2	no	—
305	C₂²	—	26	2	no	—
306	C₂²	—	26	2	no	—
307	C₂²	C₄×C₄⋊#261	1	1	yes	characteristic
308	C₂²	#11719060	1	1	yes	characteristic
309	C₂²	C₂×#88103	1	1	yes	characteristic
310	C₂²	C₂×#88104	1	1	yes	characteristic
311	C₂²	D₁₃⋊#108	1	1	yes	characteristic
312	C₂²	D₁₃⋊#108	1	1	yes	characteristic
313	C₂²	#11719136	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	13	4	no	—
317	C₄	—	26	2	no	—
318	C₄	—	26	2	no	—
319	C₄	#11719057	1	1	yes	characteristic
320	C₄	#11719061	1	1	yes	characteristic
321	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
322	C₄	#11719135	1	1	yes	characteristic
323	C₂	—	26	4	no	—
324	C₂	#13130012	1	1	yes	characteristic
325	C₂	C₄×#88103	1	1	yes	characteristic
326	C₂	C₄×#88104	1	1	yes	characteristic
327	C₂	#13130715	1	1	yes	characteristic
328	C₂	#13130716	1	1	yes	characteristic
329	C₂	#13130760	1	1	yes	characteristic
330	C₂	#13130761	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×Q₈
cosocle	C₂²×C₄×D₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#10822
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
17. #11718268	C₄	semidirect
259. C₁₃	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718268	2
#13130500	2
C₂×C₄×C₈×D₁₃	2
#10822	13