Subgroups — #15753885

Order 3328.

696 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114689	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718136	C₂	1	2	yes	maximal
4	#13112719	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	—	2	1	no	—
8	C₂×C₄×D₄×C₁₃	—	2	1	no	—
9	C₂×C₄×D₄×C₁₃	C₄	1	2	yes	—
10	C₁₃×#390	—	2	1	no	—
11	C₁₃×#390	—	2	1	no	—
12	C₁₃×#390	C₄	1	2	yes	—
13	C₁₃×#384	C₂²	1	1	yes	characteristic
14	C₄³×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	C₂×C₄⋊#1055	C₂²	1	2	yes	—
17	C₂×#88043	—	2	1	no	—
18	C₂×C₄×#1055	—	2	1	no	—
19	#11718136	C₄	1	8	yes	complemented
20	#11718116	—	2	1	no	—
21	#11718116	—	2	1	no	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂²×D₄×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	—	2	2	no	—
26	C₁₃×#127	—	2	2	no	—
27	C₁₃×#127	C₂×C₄	1	4	yes	—
28	C₄×D₄×C₁₃	—	4	4	no	—
29	C₄×D₄×C₁₃	—	4	4	no	—
30	C₄×D₄×C₁₃	—	4	2	no	—
31	C₄×D₄×C₁₃	—	4	2	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×#1055	—	2	2	no	—
46	C₂²×#1055	—	2	2	no	—
47	C₄⋊#1055	—	2	2	no	—
48	C₄⋊#1055	—	2	2	no	—
49	C₄⋊#1055	C₂×C₄	1	4	yes	complemented
50	C₄⋊#1055	C₂×C₄	1	4	yes	complemented
51	#88043	—	4	2	no	—
52	C₄×#1055	—	4	2	no	—
53	#8234	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	—	2	2	no	—
56	C₂×D₄×C₁₃	—	2	4	no	—
57	C₂×D₄×C₁₃	—	2	4	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	4	2	no	—
60	C₂×D₄×C₁₃	—	4	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂×D₄×C₁₃	—	4	2	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	4	1	no	—
65	C₂×D₄×C₁₃	—	4	1	no	—
66	C₂×D₄×C₁₃	C₄²	1	4	yes	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	4	4	no	—
69	C₂²×C₄×C₁₃	—	4	4	no	—
70	C₂²×C₄×C₁₃	—	4	4	no	—
71	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₄⋊C₄	—	4	1	no	—
80	C₁₃×C₄⋊C₄	—	4	1	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₁₃×C₂²⋊C₄	—	4	4	no	—
83	C₁₃×C₂²⋊C₄	—	4	4	no	—
84	C₁₃×C₂²⋊C₄	—	4	4	no	—
85	C₄²×C₁₃	—	2	2	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	1	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₄²×C₁₃	—	4	1	no	—
91	C₄²×C₁₃	D₈	1	2	yes	—
92	C₄²×C₁₃	SD₁₆	1	2	yes	—
93	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₃	#41	1	2	yes	—
95	C₂×#1055	—	2	4	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	4	2	no	—
98	C₂×#1055	—	4	2	no	—
99	C₂×#1055	—	4	2	no	—
100	C₂×#1055	—	4	2	no	—
101	C₄×#127	D₁₃	1	1	yes	characteristic
102	#1767	—	13	1	no	—
103	C₄×C₄⋊C₈	—	13	1	no	—
104	C₂×#331	—	13	2	no	—
105	#1296	—	13	2	no	—
106	C₂³×C₁₃	—	2	4	no	—
107	C₂³×C₁₃	—	4	2	no	—
108	C₂³×C₁₃	—	4	2	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	—	4	2	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	—	4	2	no	—
113	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	D₄×C₁₃	—	4	4	no	—
115	D₄×C₁₃	—	4	4	no	—
116	D₄×C₁₃	—	4	4	no	—
117	D₄×C₁₃	—	4	4	no	—
118	D₄×C₁₃	—	4	4	no	—
119	D₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	8	8	no	—
135	C₂×C₄×C₁₃	#100	1	2	yes	—
136	C₂×C₄×C₁₃	#102	1	2	yes	—
137	C₂×C₄×C₁₃	#102	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#135	1	2	yes	—
145	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
146	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
148	#1055	—	4	4	no	—
149	#1055	—	4	4	no	—
150	C₂×#127	C₂×D₁₃	1	1	yes	characteristic
151	C₂×C₄×D₄	—	2	1	no	—
152	C₂×C₄×D₄	—	2	1	no	—
153	C₂×C₄×D₄	Dic₁₃	1	2	yes	—
154	C₂×C₄⋊C₈	—	13	2	no	—
155	C₂×C₄⋊C₈	—	13	2	no	—
156	C₂×#97	—	26	1	no	—
157	C₂×C₄×C₈	—	26	1	no	—
158	#390	—	2	1	no	—
159	#390	—	2	1	no	—
160	#390	Dic₁₃	1	2	yes	—
161	#384	C₂×D₁₃	1	1	yes	characteristic
162	C₄³	C₂×D₁₃	1	1	yes	characteristic
163	#336	—	26	1	no	—
164	#336	—	26	1	no	—
165	#331	—	13	8	no	—
166	C₂²×C₁₃	—	4	4	no	—
167	C₂²×C₁₃	—	4	4	no	—
168	C₂²×C₁₃	—	4	4	no	—
169	C₂²×C₁₃	—	8	2	no	—
170	C₂²×C₁₃	—	8	2	no	—
171	C₂²×C₁₃	—	8	2	no	—
172	C₂²×C₁₃	—	8	2	no	—
173	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
175	C₂²×C₁₃	#413	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
177	C₂²×C₁₃	#416	1	1	yes	characteristic
178	C₂²×C₁₃	#417	1	1	yes	characteristic
179	C₂²×C₁₃	#418	1	1	yes	characteristic
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	4	1	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	C₄×D₈	1	2	yes	—
188	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
189	C₄×C₁₃	#440	1	2	yes	—
190	C₄×C₁₃	#442	1	2	yes	—
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	—	2	1	no	—
193	C₂²×D₄	C₂×Dic₁₃	1	2	yes	—
194	C₂³×C₄	—	2	2	no	—
195	C₂²×C₈	—	26	2	no	—
196	C₂²×C₈	—	26	2	no	—
197	#127	—	2	2	no	—
198	#127	C₄×D₁₃	1	4	yes	—
199	C₄×D₄	—	4	4	no	—
200	C₄×D₄	—	4	4	no	—
201	C₄×D₄	—	4	2	no	—
202	C₄×D₄	—	4	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
216	C₄⋊C₈	—	13	4	no	—
217	C₄⋊C₈	—	13	4	no	—
218	C₄⋊C₈	—	26	2	no	—
219	C₄⋊C₈	—	26	2	no	—
220	#97	—	52	2	no	—
221	C₄×C₈	—	52	2	no	—
222	C₂×C₁₃	—	8	4	no	—
223	C₂×C₁₃	#1545	1	1	yes	characteristic
224	C₂×C₁₃	#1547	1	1	yes	characteristic
225	C₂×C₁₃	#1551	1	1	yes	characteristic
226	C₂×C₁₃	#1553	1	1	yes	characteristic
227	C₂×C₁₃	C₄×#100	1	1	yes	characteristic
228	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
229	C₂×C₁₃	#1762	1	1	yes	characteristic
230	C₂⁴	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	C₄×Dic₁₃	1	4	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
254	C₂×C₈	—	26	4	no	—
255	C₂×C₈	—	26	4	no	—
256	C₂×C₈	—	52	2	no	—
257	C₂×C₈	—	52	2	no	—
258	C₂×C₈	—	52	2	no	—
259	C₂×C₈	—	52	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	4	4	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	#6104	1	2	yes	—
274	C₄²	#6105	1	2	yes	—
275	C₄²	C₂×C₄×D₁₃	1	2	yes	—
276	C₄²	#6127	1	2	yes	—
277	C₁₃	#8234	1	1	yes	characteristic, complemented, Sylow
278	C₂³	—	2	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	2	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
286	D₄	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	8	no	—
307	C₂×C₄	#88072	1	2	yes	—
308	C₂×C₄	#88072	1	2	yes	—
309	C₂×C₄	#88073	1	2	yes	—
310	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
311	C₂×C₄	#88180	1	1	yes	characteristic
312	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
313	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
314	C₂×C₄	C₂×#6104	1	1	yes	characteristic
315	C₂×C₄	#88186	1	2	yes	—
316	C₂×C₄	C₂×#6105	1	1	yes	characteristic
317	C₂×C₄	#88203	1	1	yes	characteristic
318	C₂×C₄	#88204	1	2	yes	—
319	C₂×C₄	#88205	1	1	yes	characteristic
320	C₈	—	52	4	no	—
321	C₈	—	52	4	no	—
322	C₂²	—	4	4	no	—
323	C₂²	—	4	4	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
330	C₂²	C₂×#88072	1	1	yes	characteristic
331	C₂²	#11718876	1	1	yes	characteristic
332	C₂²	C₂×#88073	1	1	yes	characteristic
333	C₂²	#11718893	1	1	yes	characteristic
334	C₂²	#11718894	1	1	yes	characteristic
335	C₂²	#11718898	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	1	no	—
342	C₄	—	8	2	no	—
343	C₄	C₄×#6104	1	2	yes	—
344	C₄	#11718675	1	2	yes	—
345	C₄	C₄×#6105	1	2	yes	—
346	C₄	#11718679	1	2	yes	—
347	C₂	—	8	4	no	—
348	C₂	#13113504	1	1	yes	characteristic
349	C₂	#13113513	1	1	yes	characteristic
350	C₂	#13113531	1	1	yes	characteristic
351	C₂	#13113541	1	1	yes	characteristic
352	C₂	C₄×#88073	1	1	yes	characteristic
353	C₂	C₄×#88072	1	1	yes	characteristic
354	C₂	#13114661	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#127	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×C₁₃×#127	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#127	D₁₃
hypocenter	C₁₃	#8234
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #13114689	C₂	semidirect
2. C₄×C₄⋊#1055	C₂	semidirect
4. #13112719	C₂	semidirect
19. #11718136	C₄	semidirect
49. C₄⋊#1055	C₂×C₄	semidirect
50. C₄⋊#1055	C₂×C₄	semidirect
277. C₁₃	#8234	semidirect

Maximal subgroups

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

subgroup	index
#13114689	2
C₄×C₄⋊#1055	2
C₂×#11718136	2
#13112719	2
C₄×C₁₃×#127	2
#8234	13