Subgroups — #16018915

Order 3328.

688 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125406	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117100	C₂	1	2	yes	maximal, complemented
3	#13117077	C₂	1	2	yes	maximal, complemented
4	#13116873	C₂	1	2	yes	maximal, complemented
5	#13116868	C₂	1	2	yes	maximal, complemented
6	#13116765	C₂	1	2	yes	maximal, complemented
7	#13116748	C₂	1	2	yes	maximal, complemented
8	#13116639	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×D₈	—	2	1	no	—
11	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
12	C₂×C₁₃×#127	—	2	1	no	—
13	#11719184	—	2	1	no	—
14	C₁₃×#493	—	2	1	no	—
15	C₁₃×#493	—	2	1	no	—
16	C₁₃×#493	C₂²	1	2	yes	—
17	C₁₃×#493	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88067	C₂²	1	4	yes	complemented
20	C₂×#88066	C₂²	1	4	yes	complemented
21	#11718777	C₂²	1	1	yes	characteristic
22	#11718776	C₂²	1	1	yes	characteristic
23	#11718775	—	2	1	no	—
24	C₄×Dic₅₂	C₂²	1	1	yes	characteristic, complemented
25	#11718358	—	2	1	no	—
26	#11718357	C₂²	1	1	yes	characteristic, complemented
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic, complemented
28	C₄×#6095	—	2	1	no	—
29	#11718344	C₂²	1	1	yes	characteristic, complemented
30	#11718154	C₂²	1	4	yes	complemented
31	#11718154	C₂²	1	4	yes	complemented
32	C₂²⋊#6090	C₂²	1	2	yes	—
33	C₄⋊#6090	C₂²	1	2	yes	complemented
34	#11718053	C₂²	1	2	yes	complemented
35	C₂²×D₄×C₁₃	—	2	2	no	—
36	C₂²×D₄×C₁₃	—	4	1	no	—
37	C₂²×D₄×C₁₃	—	4	1	no	—
38	C₂×C₁₃×D₈	—	2	2	no	—
39	C₂×C₁₃×D₈	—	2	2	no	—
40	C₂×C₁₃×D₈	—	2	2	no	—
41	C₂×C₁₃×D₈	—	4	4	no	—
42	C₂×C₁₃×D₈	—	4	4	no	—
43	C₂×C₁₃×D₈	C₂³	1	4	yes	—
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#127	—	2	2	no	—
47	C₁₃×#127	—	2	2	no	—
48	C₁₃×#127	—	4	1	no	—
49	C₁₃×#127	—	4	1	no	—
50	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
51	#88184	—	2	1	no	—
52	#88183	—	4	1	no	—
53	C₄×C₂²⋊D₁₃	—	4	1	no	—
54	#88178	—	2	1	no	—
55	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
56	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
57	C₂×#6095	—	2	1	no	—
58	#88130	—	4	1	no	—
59	#88129	—	2	1	no	—
60	C₄×D₅₂	—	2	1	no	—
61	#88126	—	4	1	no	—
62	#88124	—	2	1	no	—
63	C₄×Dic₂₆	—	2	1	no	—
64	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	#88067	—	4	4	no	—
69	#88066	—	4	4	no	—
70	#88061	—	2	1	no	—
71	#88061	—	2	1	no	—
72	#88058	C₂³	1	1	yes	characteristic
73	#88058	C₂³	1	1	yes	characteristic
74	#88057	—	2	1	no	—
75	#88056	—	2	1	no	—
76	#88056	—	2	1	no	—
77	C₂×#6090	C₂³	1	4	yes	—
78	#33783	—	13	1	no	maximal, Sylow
79	C₂⁴×C₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	2	4	no	—
81	C₂×D₄×C₁₃	—	4	4	no	—
82	C₂×D₄×C₁₃	—	4	4	no	—
83	C₂×D₄×C₁₃	—	4	2	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	4	2	no	—
86	C₂×D₄×C₁₃	—	4	2	no	—
87	C₂×D₄×C₁₃	—	4	2	no	—
88	C₂×D₄×C₁₃	—	8	4	no	—
89	C₂×D₄×C₁₃	—	8	4	no	—
90	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₃	—	4	1	no	—
94	C₂×D₅₂	—	2	1	no	—
95	C₂×C₄×D₁₃	—	4	1	no	—
96	C₂×Dic₂₆	—	2	1	no	—
97	C₁₃×D₈	—	2	4	no	—
98	C₁₃×D₈	—	2	4	no	—
99	C₁₃×D₈	—	4	4	no	—
100	C₁₃×D₈	—	4	4	no	—
101	C₁₃×D₈	—	4	4	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₄²×C₁₃	D₈	1	1	yes	characteristic
109	C₄²×C₁₃	D₈	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂²⋊Dic₁₃	—	4	1	no	—
113	#6103	—	4	1	no	—
114	#6103	—	4	1	no	—
115	#6103	—	4	1	no	—
116	#6103	—	4	1	no	—
117	C₄⋊Dic₁₃	—	2	1	no	—
118	C₄⋊Dic₁₃	—	2	1	no	—
119	#6101	—	4	1	no	—
120	#6101	—	4	1	no	—
121	#6101	—	4	1	no	—
122	#6101	—	4	1	no	—
123	C₄×Dic₁₃	—	4	1	no	—
124	Dic₅₂	—	2	2	no	—
125	D₁₀₄	—	2	2	no	—
126	#6095	—	4	2	no	—
127	#6090	—	2	4	no	—
128	#3168	—	13	1	no	—
129	C₂×#493	D₁₃	1	1	yes	characteristic, complemented
130	#2250	—	13	2	no	—
131	#2244	—	13	2	no	—
132	#2207	—	13	2	no	—
133	#2206	—	13	2	no	—
134	#2187	—	13	2	no	—
135	#2184	—	13	2	no	—
136	#2163	—	13	1	no	—
137	C₂³×C₁₃	—	4	4	no	—
138	C₂³×C₁₃	—	8	4	no	—
139	C₂³×C₁₃	—	8	4	no	—
140	C₂³×C₁₃	—	8	4	no	—
141	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂²×D₁₃	—	4	1	no	—
143	D₄×C₁₃	—	4	4	no	—
144	D₄×C₁₃	—	4	4	no	—
145	D₄×C₁₃	—	8	4	no	—
146	D₄×C₁₃	—	8	4	no	—
147	D₄×C₁₃	—	8	4	no	—
148	D₄×C₁₃	—	8	4	no	—
149	D₄×C₁₃	—	8	4	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	1	no	—
152	C₂×C₄×C₁₃	—	2	1	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂²⋊D₁₃	—	8	2	no	—
161	C₂×Dic₁₃	—	4	1	no	—
162	C₂×Dic₁₃	—	4	1	no	—
163	C₂×Dic₁₃	—	4	1	no	—
164	D₅₂	—	4	1	no	—
165	D₅₂	—	4	1	no	—
166	C₄×D₁₃	—	8	2	no	—
167	Dic₂₆	—	4	1	no	—
168	Dic₂₆	—	4	1	no	—
169	C₈×C₁₃	—	2	1	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
172	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
173	C₂²×D₈	—	2	1	no	—
174	C₂²×D₈	C₂×D₁₃	1	2	yes	—
175	C₂×#127	—	2	1	no	—
176	#529	—	26	1	no	—
177	C₂×SD₃₂	—	13	4	no	—
178	C₂×D₁₆	—	13	4	no	—
179	#499	—	13	1	no	—
180	#495	—	13	1	no	—
181	#493	—	2	1	no	—
182	#493	—	2	1	no	—
183	#493	C₂×D₁₃	1	2	yes	—
184	#493	C₂×D₁₃	1	2	yes	—
185	#488	—	26	1	no	—
186	#467	—	13	1	no	—
187	#466	—	13	1	no	—
188	#465	—	26	1	no	—
189	C₄×Q₁₆	—	13	1	no	—
190	C₄×SD₁₆	—	26	1	no	—
191	C₄×D₈	—	13	1	no	—
192	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
193	C₄⋊C₁₆	—	13	2	no	—
194	#357	—	13	4	no	—
195	#357	—	13	4	no	—
196	C₂²⋊C₁₆	—	13	2	no	—
197	#346	—	13	2	no	—
198	C₂²×C₁₃	—	2	1	no	—
199	C₂²×C₁₃	—	2	1	no	—
200	C₂²×C₁₃	—	8	4	no	—
201	C₂²×C₁₃	—	8	4	no	—
202	C₂²×C₁₃	—	8	4	no	—
203	C₂²×C₁₃	—	16	8	no	—
204	C₂²×C₁₃	#509	1	2	yes	—
205	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
206	C₂×D₁₃	—	8	1	no	—
207	C₂×D₁₃	—	8	1	no	—
208	C₂×D₁₃	—	8	1	no	—
209	C₄×C₁₃	—	2	1	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	#509	1	2	yes	—
214	C₄×C₁₃	#535	1	1	yes	characteristic
215	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	Dic₁₃	—	8	1	no	—
217	Dic₁₃	—	8	1	no	—
218	Dic₁₃	—	8	1	no	—
219	C₂²×D₄	—	2	2	no	—
220	C₂²×D₄	—	4	1	no	—
221	C₂²×D₄	—	4	1	no	—
222	C₂×Q₁₆	—	13	1	no	—
223	C₂×SD₁₆	—	26	1	no	—
224	C₂×D₈	—	2	2	no	—
225	C₂×D₈	—	2	2	no	—
226	C₂×D₈	—	2	2	no	—
227	C₂×D₈	—	4	4	no	—
228	C₂×D₈	—	4	4	no	—
229	C₂×D₈	—	13	1	no	—
230	C₂×D₈	C₂²×D₁₃	1	4	yes	—
231	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	#127	—	2	2	no	—
234	#127	—	2	2	no	—
235	#127	—	4	1	no	—
236	#127	—	4	1	no	—
237	#126	—	52	1	no	—
238	#125	—	26	1	no	—
239	#124	—	26	1	no	—
240	#123	—	52	1	no	—
241	C₂²⋊Q₈	—	26	1	no	—
242	#121	—	26	1	no	—
243	C₄×Q₈	—	26	1	no	—
244	C₄×D₄	—	26	1	no	—
245	C₄×D₄	—	52	1	no	—
246	#117	—	52	1	no	—
247	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
248	SD₃₂	—	52	4	no	—
249	D₁₆	—	52	4	no	—
250	C₂×C₁₆	—	13	4	no	—
251	#107	—	13	1	no	—
252	#107	—	13	1	no	—
253	#106	—	26	1	no	—
254	#103	—	26	1	no	—
255	#103	—	26	1	no	—
256	#102	—	26	1	no	—
257	#102	—	26	1	no	—
258	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
259	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
262	C₂×C₁₃	—	2	2	no	—
263	C₂×C₁₃	—	16	4	no	—
264	C₂×C₁₃	#3205	1	1	yes	characteristic
265	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
266	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
267	D₁₃	—	16	1	no	—
268	C₂⁴	—	4	2	no	—
269	C₂×Q₈	—	26	1	no	—
270	C₂×D₄	—	2	4	no	—
271	C₂×D₄	—	4	4	no	—
272	C₂×D₄	—	4	4	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	8	4	no	—
279	C₂×D₄	—	8	4	no	—
280	C₂×D₄	—	26	1	no	—
281	C₂×D₄	—	52	1	no	—
282	C₂²×C₄	—	52	1	no	—
283	C₂²×C₄	#6104	1	1	yes	characteristic
284	C₂²×C₄	#6104	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
286	Q₁₆	—	26	2	no	—
287	SD₁₆	—	52	2	no	—
288	D₈	—	2	4	no	—
289	D₈	—	2	4	no	—
290	D₈	—	4	4	no	—
291	D₈	—	4	4	no	—
292	D₈	—	4	4	no	—
293	D₈	—	26	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
300	C₄⋊C₄	—	26	1	no	—
301	C₄⋊C₄	—	26	1	no	—
302	C₄⋊C₄	—	52	1	no	—
303	C₄⋊C₄	—	52	1	no	—
304	C₄⋊C₄	—	52	1	no	—
305	C₄⋊C₄	—	52	1	no	—
306	C₂²⋊C₄	—	52	1	no	—
307	C₂²⋊C₄	—	52	1	no	—
308	C₂²⋊C₄	—	52	1	no	—
309	C₂²⋊C₄	—	52	1	no	—
310	C₂²⋊C₄	—	52	1	no	—
311	C₄²	—	52	1	no	—
312	C₄²	#6104	1	1	yes	characteristic
313	C₄²	#6104	1	1	yes	characteristic
314	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
315	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
316	C₁₆	—	26	4	no	—
317	C₁₃	#33783	1	1	yes	characteristic, complemented, Sylow
318	C₂³	—	4	4	no	—
319	C₂³	—	8	4	no	—
320	C₂³	—	8	4	no	—
321	C₂³	—	8	4	no	—
322	C₂³	—	52	1	no	—
323	C₂³	C₂×#6104	1	1	yes	characteristic
324	Q₈	—	52	1	no	—
325	Q₈	—	52	1	no	—
326	D₄	—	4	4	no	—
327	D₄	—	4	4	no	—
328	D₄	—	8	4	no	—
329	D₄	—	8	4	no	—
330	D₄	—	8	4	no	—
331	D₄	—	8	4	no	—
332	D₄	—	8	4	no	—
333	D₄	—	52	1	no	—
334	D₄	—	52	1	no	—
335	D₄	—	104	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	52	1	no	—
341	C₂×C₄	—	52	1	no	—
342	C₂×C₄	—	52	1	no	—
343	C₂×C₄	—	104	2	no	—
344	C₂×C₄	C₂×#6104	1	1	yes	characteristic
345	C₂×C₄	C₂×#6104	1	1	yes	characteristic
346	C₂×C₄	C₂×#6104	1	1	yes	characteristic
347	C₂×C₄	C₂×#6104	1	1	yes	characteristic
348	C₂×C₄	C₂×#6104	1	1	yes	characteristic
349	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	#88186	1	2	yes	—
353	C₈	#88251	1	2	yes	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	8	4	no	—
357	C₂²	—	8	4	no	—
358	C₂²	—	8	4	no	—
359	C₂²	—	16	8	no	—
360	C₂²	—	104	1	no	—
361	C₂²	—	104	1	no	—
362	C₂²	—	104	1	no	—
363	C₂²	#11718808	1	2	yes	—
364	C₂²	C₂²×#6104	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	104	1	no	—
370	C₄	—	104	1	no	—
371	C₄	—	104	1	no	—
372	C₄	#11718808	1	2	yes	—
373	C₄	C₂×#88186	1	1	yes	characteristic
374	C₄	#11719510	1	1	yes	characteristic
375	C₂	—	2	2	no	—
376	C₂	—	16	4	no	—
377	C₂	—	208	1	no	—
378	C₂	#13125754	1	1	yes	characteristic
379	C₂	C₂×#11718808	1	1	yes	characteristic
380	C₂	C₂×#11718808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#493	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₂×#493	D₁₃
hypocenter	C₁₃	#33783
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #13125406	C₂	semidirect
2. #13117100	C₂	semidirect
3. #13117077	C₂	semidirect
4. #13116873	C₂	semidirect
5. #13116868	C₂	semidirect
6. #13116765	C₂	semidirect
7. #13116748	C₂	semidirect
8. #13116639	C₂	semidirect
9. C₂×C₁₃×#493	C₂	semidirect
19. C₂×#88067	C₂²	semidirect
20. C₂×#88066	C₂²	semidirect
24. C₄×Dic₅₂	C₂²	semidirect
26. #11718357	C₂²	semidirect
27. C₄×D₁₀₄	C₂²	semidirect
29. #11718344	C₂²	semidirect
30. #11718154	C₂²	semidirect
31. #11718154	C₂²	semidirect
33. C₄⋊#6090	C₂²	semidirect
34. #11718053	C₂²	semidirect
129. C₂×#493	D₁₃	semidirect
317. C₁₃	#33783	semidirect

Maximal subgroups

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

subgroup	index
#13125406	2
#13117100	2
#13117077	2
#13116873	2
#13116868	2
#13116765	2
#13116748	2
#13116639	2
C₂×C₁₃×#493	2
#33783	13