Subgroups — #16018884

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125415	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117103	C₂	1	2	yes	maximal, complemented
3	#13117072	C₂	1	2	yes	maximal, complemented
4	#13116888	C₂	1	4	yes	maximal, complemented
5	#13116753	C₂	1	2	yes	maximal, complemented
6	#13116748	C₂	1	2	yes	maximal, complemented
7	#13116639	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×SD₁₆	—	2	1	no	—
11	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#124	—	2	1	no	—
13	#11719182	—	2	1	no	—
14	C₁₃×#495	—	2	1	no	—
15	C₁₃×#495	—	2	1	no	—
16	C₁₃×#495	C₂²	1	2	yes	—
17	C₁₃×#495	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88068	C₂²	1	4	yes	complemented
20	C₂×#88066	C₂²	1	4	yes	—
21	#11718776	—	2	1	no	—
22	#11718776	C₂²	1	2	yes	—
23	#11718356	C₂²	1	1	yes	characteristic, complemented
24	#11718355	—	2	1	no	—
25	C₄×D₁₀₄	—	2	1	no	—
26	C₄×D₁₀₄	C₂²	1	2	yes	complemented
27	#11718343	C₂²	1	1	yes	characteristic, complemented
28	#11718156	C₂²	1	4	yes	complemented
29	#11718154	C₂²	1	4	yes	—
30	C₂²⋊#6090	C₂²	1	2	yes	—
31	C₄⋊#6090	C₂²	1	2	yes	complemented
32	#11718053	C₂²	1	2	yes	complemented
33	C₂²×Q₈×C₁₃	—	2	1	no	—
34	C₂²×D₄×C₁₃	—	2	1	no	—
35	C₂×#6127	—	4	1	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	—	4	2	no	—
38	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
39	C₂×C₁₃×SD₁₆	—	2	2	no	—
40	C₂×C₁₃×SD₁₆	—	2	2	no	—
41	C₂×C₁₃×SD₁₆	—	4	4	no	—
42	C₂×C₁₃×D₈	—	2	1	no	—
43	C₂×C₁₃×D₈	—	4	2	no	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#124	—	2	2	no	—
48	C₁₃×#124	—	2	2	no	—
49	C₁₃×#124	—	4	1	no	—
50	C₁₃×#124	—	4	1	no	—
51	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
52	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
53	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
54	#88184	—	2	2	no	—
55	#88184	—	4	1	no	—
56	C₂×D₁₀₄	—	2	1	no	—
57	C₂×D₁₀₄	C₂³	1	2	yes	—
58	#88129	—	4	1	no	—
59	#88128	—	2	1	no	—
60	C₄×D₅₂	—	2	2	no	—
61	C₄×D₅₂	—	4	1	no	—
62	#88123	—	2	1	no	—
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	#88068	—	4	4	no	—
68	#88066	—	4	4	no	—
69	#88061	—	2	2	no	—
70	#88061	—	2	2	no	—
71	#88058	—	2	1	no	—
72	#88058	C₂³	1	2	yes	—
73	C₂×#6090	C₂³	1	4	yes	—
74	#33779	—	13	1	no	maximal, Sylow
75	C₂⁴×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	4	2	no	—
78	C₂×Q₈×C₁₃	—	4	2	no	—
79	C₂×Q₈×C₁₃	—	4	1	no	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂×D₄×C₁₃	—	4	2	no	—
82	C₂×D₄×C₁₃	—	4	2	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₃	—	4	2	no	—
89	#6127	—	8	4	no	—
90	C₂×D₅₂	—	2	2	no	—
91	C₂×D₅₂	—	4	1	no	—
92	C₂×D₅₂	—	4	1	no	—
93	C₂×C₄×D₁₃	—	4	2	no	—
94	C₂×Dic₂₆	—	4	1	no	—
95	C₁₃×Q₁₆	—	2	4	no	—
96	C₁₃×Q₁₆	—	4	2	no	—
97	C₁₃×SD₁₆	—	4	4	no	—
98	C₁₃×SD₁₆	—	4	4	no	—
99	C₁₃×D₈	—	2	4	no	—
100	C₁₃×D₈	—	4	2	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₄²×C₁₃	D₈	1	1	yes	characteristic
112	C₄²×C₁₃	D₈	1	1	yes	characteristic
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	#6103	—	4	2	no	—
116	#6103	—	4	2	no	—
117	C₄⋊Dic₁₃	—	2	2	no	—
118	C₄⋊Dic₁₃	—	4	1	no	—
119	D₁₀₄	—	2	4	no	—
120	D₁₀₄	—	4	2	no	—
121	#6090	—	2	4	no	—
122	#3169	—	13	1	no	—
123	C₂×#495	D₁₃	1	1	yes	characteristic, complemented
124	#2250	—	13	2	no	—
125	#2243	—	13	2	no	—
126	#2210	—	13	4	no	—
127	#2185	—	13	2	no	—
128	#2184	—	13	2	no	—
129	#2163	—	13	1	no	—
130	C₂³×C₁₃	—	4	2	no	—
131	C₂³×C₁₃	—	8	2	no	—
132	C₂³×C₁₃	—	8	2	no	—
133	C₂³×C₁₃	—	8	2	no	—
134	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₁₃	—	4	2	no	—
136	Q₈×C₁₃	—	4	2	no	—
137	Q₈×C₁₃	—	4	2	no	—
138	Q₈×C₁₃	—	8	2	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	8	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	1	no	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂²⋊D₁₃	—	8	4	no	—
155	C₂×Dic₁₃	—	4	2	no	—
156	D₅₂	—	4	2	no	—
157	D₅₂	—	4	2	no	—
158	D₅₂	—	8	2	no	—
159	D₅₂	—	8	2	no	—
160	C₄×D₁₃	—	8	4	no	—
161	Dic₂₆	—	8	2	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
165	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
166	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
167	C₂²×SD₁₆	—	2	1	no	—
168	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
169	#532	—	26	1	no	—
170	C₂×#124	—	2	1	no	—
171	C₂×SD₃₂	—	13	4	no	—
172	C₂×D₁₆	—	13	4	no	—
173	#500	—	13	1	no	—
174	#495	—	2	1	no	—
175	#495	—	2	1	no	—
176	#495	C₂×D₁₃	1	2	yes	—
177	#495	C₂×D₁₃	1	2	yes	—
178	#493	—	13	1	no	—
179	#486	—	26	1	no	—
180	#466	—	13	2	no	—
181	#466	—	26	1	no	—
182	C₄×D₈	—	13	2	no	—
183	C₄×D₈	—	26	1	no	—
184	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
185	C₄⋊C₁₆	—	13	2	no	—
186	#358	—	13	4	no	—
187	#357	—	13	4	no	—
188	C₂²⋊C₁₆	—	13	2	no	—
189	#346	—	13	2	no	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	8	2	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	16	4	no	—
196	C₂²×C₁₃	#509	1	2	yes	—
197	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
198	C₂×D₁₃	—	8	2	no	—
199	C₂×D₁₃	—	8	2	no	—
200	C₂×D₁₃	—	8	2	no	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	—	2	1	no	—
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	#535	1	1	yes	characteristic
208	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	Dic₁₃	—	8	2	no	—
210	C₂×#41	—	52	1	no	—
211	C₂²×Q₈	—	2	1	no	—
212	C₂²×D₄	—	2	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	4	2	no	—
215	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	4	4	no	—
219	C₂×D₈	—	2	1	no	—
220	C₂×D₈	—	4	2	no	—
221	C₂×D₈	—	13	2	no	—
222	C₂×D₈	—	26	1	no	—
223	C₂×D₈	C₂²×D₁₃	1	2	yes	—
224	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₄⋊Q₈	—	26	1	no	—
227	#127	—	26	1	no	—
228	#124	—	2	2	no	—
229	#124	—	2	2	no	—
230	#124	—	4	1	no	—
231	#124	—	4	1	no	—
232	#124	—	52	1	no	—
233	#121	—	26	2	no	—
234	#121	—	52	1	no	—
235	C₄×D₄	—	26	2	no	—
236	C₄×D₄	—	52	1	no	—
237	C₂×C₂²⋊C₄	—	4	1	no	—
238	C₂×C₂²⋊C₄	—	4	1	no	—
239	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
240	SD₃₂	—	52	4	no	—
241	D₁₆	—	52	4	no	—
242	C₂×C₁₆	—	13	4	no	—
243	#107	—	13	2	no	—
244	#107	—	26	1	no	—
245	#102	—	26	2	no	—
246	#102	—	26	2	no	—
247	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
248	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
251	C₂×C₁₃	—	2	2	no	—
252	C₂×C₁₃	—	16	2	no	—
253	C₂×C₁₃	#3205	1	1	yes	characteristic
254	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
255	C₂×C₁₃	#3414	1	1	yes	characteristic
256	D₁₃	—	16	2	no	—
257	C₂⁴	—	4	1	no	—
258	#41	—	104	4	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	52	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	26	2	no	—
269	C₂×D₄	—	52	2	no	—
270	C₂×D₄	—	52	1	no	—
271	C₂×D₄	—	52	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	52	2	no	—
274	C₂²×C₄	#6104	1	1	yes	characteristic
275	C₂²×C₄	#6104	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
277	Q₁₆	—	2	4	no	—
278	Q₁₆	—	4	2	no	—
279	SD₁₆	—	4	4	no	—
280	SD₁₆	—	4	4	no	—
281	D₈	—	2	4	no	—
282	D₈	—	4	2	no	—
283	D₈	—	26	4	no	—
284	D₈	—	52	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
290	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
291	C₄⋊C₄	—	26	2	no	—
292	C₄⋊C₄	—	52	1	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	52	2	no	—
298	C₂²⋊C₄	—	52	2	no	—
299	C₄²	#6104	1	1	yes	characteristic
300	C₄²	#6104	1	1	yes	characteristic
301	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
302	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
303	C₁₆	—	26	4	no	—
304	C₁₃	#33779	1	1	yes	characteristic, complemented, Sylow
305	C₂³	—	4	2	no	—
306	C₂³	—	8	2	no	—
307	C₂³	—	8	2	no	—
308	C₂³	—	8	2	no	—
309	C₂³	—	52	2	no	—
310	C₂³	C₂×#6104	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	2	no	—
314	Q₈	—	104	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	8	2	no	—
318	D₄	—	52	2	no	—
319	D₄	—	52	2	no	—
320	D₄	—	104	2	no	—
321	D₄	—	104	2	no	—
322	D₄	—	104	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	52	2	no	—
330	C₂×C₄	—	104	4	no	—
331	C₂×C₄	C₂×#6104	1	1	yes	characteristic
332	C₂×C₄	C₂×#6104	1	1	yes	characteristic
333	C₂×C₄	C₂×#6104	1	1	yes	characteristic
334	C₂×C₄	C₂×#6104	1	1	yes	characteristic
335	C₂×C₄	C₂×#6104	1	1	yes	characteristic
336	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	#88186	1	2	yes	—
340	C₈	#88251	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	16	4	no	—
347	C₂²	—	104	2	no	—
348	C₂²	—	104	2	no	—
349	C₂²	—	104	2	no	—
350	C₂²	#11718854	1	2	yes	—
351	C₂²	C₂²×#6104	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	8	2	no	—
358	C₄	—	104	2	no	—
359	C₄	C₂×#88186	1	1	yes	characteristic
360	C₄	#11719510	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	16	2	no	—
363	C₂	—	208	2	no	—
364	C₂	#13125754	1	1	yes	characteristic
365	C₂	C₂×#11718854	1	1	yes	characteristic
366	C₂	#13127970	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×#495	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₂×#495	D₁₃
hypocenter	C₁₃	#33779
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. #13125415	C₂	semidirect
2. #13117103	C₂	semidirect
3. #13117072	C₂	semidirect
4. #13116888	C₂	semidirect
5. #13116753	C₂	semidirect
6. #13116748	C₂	semidirect
7. #13116639	C₂	semidirect
8. C₂×C₁₃×#495	C₂	semidirect
19. C₂×#88068	C₂²	semidirect
23. #11718356	C₂²	semidirect
26. C₄×D₁₀₄	C₂²	semidirect
27. #11718343	C₂²	semidirect
28. #11718156	C₂²	semidirect
31. C₄⋊#6090	C₂²	semidirect
32. #11718053	C₂²	semidirect
123. C₂×#495	D₁₃	semidirect
304. C₁₃	#33779	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13117103	2
#13117072	2
#13116888	2
#13116753	2
#13116748	2
#13116639	2
C₂×C₁₃×#495	2
#33779	13