Subgroups — #15762006

Order 3328.

610 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123194	C₂	1	1	yes	maximal, characteristic
2	#13114987	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114691	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718077	C₂	1	1	yes	maximal, characteristic
6	C₂×#11718077	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#88201	—	2	1	no	—
10	C₂×#88201	—	2	1	no	—
11	C₂×#88128	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#88055	—	2	1	no	—
15	C₂×#88054	—	2	2	no	—
16	#11718649	—	2	2	no	—
17	#11718649	C₄	1	4	yes	—
18	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
21	C₁₃×#332	—	2	2	no	—
22	#11718142	—	2	1	no	—
23	#11718139	—	2	2	no	—
24	#11718077	—	2	2	no	—
25	#11718077	—	2	2	no	—
26	C₂²×Q₈×C₁₃	—	2	1	no	—
27	C₂²×D₅₂	—	2	1	no	—
28	C₂²×D₅₂	—	2	1	no	—
29	C₂²×D₅₂	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	—	2	1	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₁₃×C₄⋊Q₈	—	2	1	no	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×C₄⋊Q₈	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	#88201	—	4	2	no	—
40	#88201	—	4	2	no	—
41	#88201	—	4	2	no	—
42	#88201	—	4	2	no	—
43	C₂×#6103	—	2	4	no	—
44	C₂×#6103	—	2	4	no	—
45	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
48	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
49	C₂²×#1055	—	2	1	no	—
50	C₂²×#1055	—	2	1	no	—
51	#88128	—	2	1	no	—
52	#88128	—	2	1	no	—
53	#88128	—	2	1	no	—
54	#88128	—	2	1	no	—
55	C₁₃×C₄⋊C₈	—	2	1	no	—
56	C₁₃×C₄⋊C₈	—	2	1	no	—
57	C₁₃×C₄⋊C₈	—	2	1	no	—
58	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
59	#88055	—	4	2	no	—
60	#88054	—	4	4	no	—
61	C₄⋊#1055	—	2	1	no	—
62	C₄⋊#1055	—	2	1	no	—
63	C₄⋊#1055	—	2	1	no	—
64	C₄⋊#1055	D₄	1	2	yes	—
65	#9704	—	13	1	no	maximal, Sylow
66	C₂³×D₁₃	—	2	2	no	—
67	C₂×Q₈×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×Q₈×C₁₃	—	4	1	no	—
70	C₂×Q₈×C₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
78	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
79	C₂×D₅₂	—	2	2	no	—
80	C₂×D₅₂	—	2	2	no	—
81	C₂×D₅₂	—	4	4	no	—
82	C₂×D₅₂	—	4	1	no	—
83	C₂×D₅₂	—	4	1	no	—
84	C₂×D₅₂	—	4	1	no	—
85	C₂×D₅₂	—	4	1	no	—
86	C₂×D₅₂	—	4	1	no	—
87	C₂×D₅₂	—	4	1	no	—
88	C₂×D₅₂	—	4	1	no	—
89	C₂×D₅₂	—	4	1	no	—
90	C₂×D₅₂	—	4	2	no	—
91	C₂×D₅₂	—	4	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	—	4	1	no	—
95	C₂×C₈×C₁₃	—	4	1	no	—
96	C₂×C₈×C₁₃	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	#41	1	1	yes	characteristic
104	C₄²×C₁₃	#41	1	1	yes	characteristic
105	#6103	—	4	8	no	—
106	#6103	—	4	8	no	—
107	C₄×Dic₁₃	—	2	1	no	—
108	C₄×Dic₁₃	—	2	1	no	—
109	C₄×Dic₁₃	—	2	2	no	—
110	C₄×Dic₁₃	—	2	2	no	—
111	C₄×Dic₁₃	—	4	2	no	—
112	C₄×Dic₁₃	D₈	1	2	yes	—
113	C₄×Dic₁₃	SD₁₆	1	2	yes	—
114	C₂×#1055	—	2	2	no	—
115	C₂×#1055	—	4	1	no	—
116	C₂×#1055	—	4	1	no	—
117	C₂×#1055	—	4	1	no	—
118	C₂×#1055	—	4	2	no	—
119	#2858	—	13	1	no	—
120	#1842	—	13	1	no	—
121	#1767	—	13	1	no	—
122	C₂×#332	—	13	1	no	—
123	C₂×#332	D₁₃	1	1	yes	characteristic, complemented
124	C₂×#331	—	13	1	no	—
125	C₂×#331	—	13	1	no	—
126	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×D₁₃	—	4	2	no	—
128	C₂²×D₁₃	—	4	4	no	—
129	C₂²×D₁₃	—	4	2	no	—
130	C₂²×D₁₃	—	4	4	no	—
131	C₂²×D₁₃	—	4	2	no	—
132	Q₈×C₁₃	—	8	1	no	—
133	Q₈×C₁₃	—	8	1	no	—
134	Q₈×C₁₃	—	8	1	no	—
135	Q₈×C₁₃	—	8	1	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₁₃	#137	1	2	yes	—
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	—	2	2	no	—
154	C₂×Dic₁₃	—	2	2	no	—
155	C₂×Dic₁₃	—	2	2	no	—
156	C₂×Dic₁₃	—	2	2	no	—
157	C₂×Dic₁₃	#102	1	4	yes	—
158	D₅₂	—	8	2	no	—
159	D₅₂	—	8	2	no	—
160	D₅₂	—	8	1	no	—
161	D₅₂	—	8	1	no	—
162	D₅₂	—	8	1	no	—
163	D₅₂	—	8	1	no	—
164	D₅₂	—	8	2	no	—
165	D₅₂	—	8	1	no	—
166	D₅₂	—	8	1	no	—
167	C₈×C₁₃	—	4	2	no	—
168	C₈×C₁₃	—	8	1	no	—
169	#1055	—	4	2	no	—
170	#1055	—	8	1	no	—
171	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
172	C₂×#127	—	13	1	no	—
173	C₂×#124	—	26	1	no	—
174	C₂×#124	—	26	1	no	—
175	C₂×C₄⋊C₈	—	13	1	no	—
176	C₂×C₄⋊C₈	—	13	1	no	—
177	C₂×C₄⋊C₈	—	26	2	no	—
178	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
180	C₂×#97	—	26	1	no	—
181	#390	—	13	4	no	—
182	#390	—	26	2	no	—
183	C₄³	—	13	1	no	—
184	#336	—	26	1	no	—
185	#332	—	2	2	no	—
186	#332	—	26	2	no	—
187	#331	—	26	2	no	—
188	#331	—	26	2	no	—
189	C₂²×C₁₃	#390	1	1	yes	characteristic
190	C₂²×C₁₃	#413	1	2	yes	—
191	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
192	C₂²×C₁₃	#417	1	2	yes	—
193	C₂²×C₁₃	#418	1	1	yes	characteristic
194	C₂×D₁₃	—	8	4	no	—
195	C₂×D₁₃	—	8	4	no	—
196	C₂×D₁₃	—	8	2	no	—
197	C₂×D₁₃	—	8	2	no	—
198	C₂×D₁₃	—	8	2	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	8	2	no	—
202	C₄×C₁₃	#459	1	1	yes	characteristic
203	C₄×C₁₃	#459	1	1	yes	characteristic
204	C₄×C₁₃	#460	1	1	yes	characteristic
205	C₄×C₁₃	#460	1	1	yes	characteristic
206	C₄×C₁₃	#489	1	2	yes	—
207	C₄×C₁₃	#496	1	1	yes	characteristic
208	C₄×C₁₃	#497	1	1	yes	characteristic
209	Dic₁₃	—	2	4	no	—
210	Dic₁₃	—	4	2	no	—
211	C₂²×Q₈	—	2	1	no	—
212	C₂²×D₄	—	13	2	no	—
213	C₂²×D₄	—	26	1	no	—
214	C₂²×D₄	—	26	1	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	26	1	no	—
218	C₂²×C₈	—	26	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	#127	—	26	1	no	—
224	#127	—	26	1	no	—
225	#127	—	26	1	no	—
226	#127	—	26	1	no	—
227	#124	—	52	2	no	—
228	#124	—	52	2	no	—
229	#124	—	52	2	no	—
230	#124	—	52	2	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	C₄×D₁₃	1	2	yes	—
233	C₂×C₂²⋊C₄	—	26	4	no	—
234	C₂×C₂²⋊C₄	—	26	4	no	—
235	C₂×C₄²	—	13	1	no	—
236	C₂×C₄²	—	13	1	no	—
237	C₂×C₄²	—	13	2	no	—
238	C₂×C₄²	—	13	2	no	—
239	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	13	2	no	—
244	C₄⋊C₈	—	26	1	no	—
245	C₄⋊C₈	—	26	1	no	—
246	C₄⋊C₈	—	26	1	no	—
247	C₄⋊C₈	—	52	4	no	—
248	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
249	#97	—	52	2	no	—
250	C₂×C₁₃	#1547	1	2	yes	—
251	C₂×C₁₃	#1557	1	2	yes	—
252	C₂×C₁₃	#1959	1	1	yes	characteristic
253	C₂×C₁₃	#1967	1	1	yes	characteristic
254	C₂×C₁₃	#1982	1	1	yes	characteristic
255	D₁₃	—	16	2	no	—
256	C₂⁴	—	26	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×D₄	—	26	2	no	—
264	C₂×D₄	—	26	2	no	—
265	C₂×D₄	—	52	4	no	—
266	C₂×D₄	—	52	1	no	—
267	C₂×D₄	—	52	1	no	—
268	C₂×D₄	—	52	1	no	—
269	C₂×D₄	—	52	1	no	—
270	C₂×D₄	—	52	1	no	—
271	C₂×D₄	—	52	1	no	—
272	C₂×D₄	—	52	2	no	—
273	C₂×D₄	—	52	2	no	—
274	C₂×D₄	—	52	1	no	—
275	C₂×D₄	—	52	1	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	13	2	no	—
278	C₂²×C₄	—	13	2	no	—
279	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
281	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	26	2	no	—
288	C₂×C₈	—	52	1	no	—
289	C₂×C₈	—	52	1	no	—
290	C₂×C₈	—	52	2	no	—
291	C₂×C₈	—	52	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₂²⋊C₄	—	52	8	no	—
297	C₂²⋊C₄	—	52	8	no	—
298	C₄²	—	13	2	no	—
299	C₄²	—	13	2	no	—
300	C₄²	—	26	1	no	—
301	C₄²	—	26	2	no	—
302	C₄²	—	26	1	no	—
303	C₄²	—	26	2	no	—
304	C₄²	—	52	2	no	—
305	C₄²	#6127	1	1	yes	characteristic
306	C₄²	D₄×D₁₃	1	1	yes	characteristic
307	C₄²	#6129	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
309	C₁₃	#9704	1	1	yes	characteristic, complemented, Sylow
310	C₂³	—	52	4	no	—
311	C₂³	—	52	2	no	—
312	C₂³	—	52	2	no	—
313	C₂³	—	52	4	no	—
314	C₂³	—	52	2	no	—
315	C₂³	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
316	Q₈	—	8	1	no	—
317	Q₈	—	8	1	no	—
318	Q₈	—	8	1	no	—
319	Q₈	—	8	1	no	—
320	D₄	—	104	2	no	—
321	D₄	—	104	2	no	—
322	D₄	—	104	1	no	—
323	D₄	—	104	1	no	—
324	D₄	—	104	2	no	—
325	D₄	—	104	1	no	—
326	D₄	—	104	1	no	—
327	D₄	—	104	1	no	—
328	D₄	—	104	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	13	4	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	26	2	no	—
339	C₂×C₄	#88135	1	1	yes	characteristic
340	C₂×C₄	#88138	1	1	yes	characteristic
341	C₂×C₄	#88139	1	1	yes	characteristic
342	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
343	C₂×C₄	#88165	1	1	yes	characteristic
344	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
345	C₂×C₄	#88168	1	1	yes	characteristic
346	C₂×C₄	#88168	1	2	yes	—
347	C₂×C₄	#88172	1	2	yes	—
348	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
349	C₂×C₄	#88193	1	1	yes	characteristic
350	C₂×C₄	#88194	1	1	yes	characteristic
351	C₈	—	4	2	no	—
352	C₈	—	8	1	no	—
353	C₈	—	52	2	no	—
354	C₈	—	104	1	no	—
355	C₂²	—	104	2	no	—
356	C₂²	—	104	2	no	—
357	C₂²	—	104	2	no	—
358	C₂²	—	104	4	no	—
359	C₂²	—	104	4	no	—
360	C₂²	#11718412	1	2	yes	—
361	C₂²	D₁₃×#102	1	1	yes	characteristic
362	C₂²	#11718431	1	1	yes	characteristic
363	C₂²	#11718464	1	2	yes	—
364	C₂²	#11718618	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	8	2	no	—
368	C₄	—	26	4	no	—
369	C₄	—	52	2	no	—
370	C₄	#11718423	1	1	yes	characteristic
371	C₄	#11718447	1	1	yes	characteristic
372	C₄	#11718456	1	2	yes	—
373	C₄	#11718811	1	1	yes	characteristic
374	C₄	#11718824	1	1	yes	characteristic
375	C₄	#11718837	1	1	yes	characteristic
376	C₄	#11718851	1	1	yes	characteristic
377	C₂	—	208	2	no	—
378	C₂	#13113512	1	2	yes	—
379	C₂	#13113557	1	2	yes	—
380	C₂	#13115649	1	1	yes	characteristic
381	C₂	#13115684	1	1	yes	characteristic
382	C₂	#13115761	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#332	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#332	D₁₃
hypocenter	C₁₃	#9704
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
2. #13114987	C₂	semidirect
3. #13114691	C₂	semidirect
4. C₂×#11718139	C₂	semidirect
7. C₂×C₁₃×#332	C₂	semidirect
123. C₂×#332	D₁₃	semidirect
309. C₁₃	#9704	semidirect

Maximal subgroups

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

subgroup	index
#13123194	2
#13114987	2
#13114691	2
C₂×#11718139	2
C₂×#11718077	2
C₂×C₁₃×#332	2
#9704	13