Subgroups — #16017659

Order 3328.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125348	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718857	C₂	1	1	yes	maximal, characteristic
3	#13117010	C₂	1	4	yes	maximal, complemented
4	#13117005	C₂	1	4	yes	maximal, complemented
5	#13116609	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116608	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718103	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116572	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2145	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
11	C₂×#88202	—	2	1	no	—
12	#11719222	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₁₃×#427	C₂²	1	1	yes	characteristic
15	C₂×#88077	—	2	1	no	—
16	#11718857	—	2	1	no	—
17	#11718857	—	2	1	no	—
18	#11718857	C₂²	1	2	yes	—
19	#11718857	C₂²	1	2	yes	—
20	C₂×#88059	C₂²	1	1	yes	characteristic
21	#11718767	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×#88052	C₂²	1	1	yes	characteristic
24	#11718659	—	2	1	no	—
25	#11718539	—	2	1	no	—
26	#11718535	—	2	1	no	—
27	#11718519	C₂²	1	2	yes	complemented
28	#11718516	C₂²	1	2	yes	complemented
29	C₁₃×#360	C₂²	1	4	yes	—
30	#11718158	C₂²	1	4	yes	—
31	#11718105	C₂²	1	4	yes	complemented
32	#11718103	C₂²	1	2	yes	complemented
33	#11718103	C₂²	1	2	yes	complemented
34	#11718079	C₂²	1	4	yes	complemented
35	C₂×C₁₃×#41	—	2	1	no	—
36	C₁₃×#135	—	2	1	no	—
37	C₁₃×#135	—	2	1	no	—
38	C₁₃×#135	Q₈	1	2	yes	complemented
39	C₁₃×#135	C₂³	1	2	yes	—
40	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
41	C₂×C₁₃×SD₁₆	—	2	1	no	—
42	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic, complemented
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#117	—	2	1	no	—
45	#88202	—	2	2	no	—
46	#88202	—	2	2	no	—
47	#88202	—	4	1	no	—
48	#88202	—	4	1	no	—
49	#88180	—	2	1	no	—
50	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
51	C₂×#6099	—	2	1	no	—
52	C₂×#6099	—	4	1	no	—
53	C₂²×#1055	—	4	1	no	—
54	C₂²×#1055	D₄	1	1	yes	characteristic
55	#88144	—	2	2	no	—
56	#88142	—	2	2	no	—
57	#88132	—	4	2	no	—
58	C₄×Dic₂₆	—	4	2	no	—
59	C₁₃×M₃₂	—	2	1	no	—
60	C₁₃×M₃₂	C₂³	1	2	yes	—
61	C₂×C₁₃×C₁₆	—	2	1	no	—
62	C₁₃×#107	—	2	1	no	—
63	C₁₃×#106	C₂³	1	2	yes	—
64	#88077	—	2	2	no	—
65	#88077	—	2	2	no	—
66	#88059	—	2	1	no	—
67	#88059	C₂³	1	2	yes	—
68	#88058	—	2	1	no	—
69	#88057	C₂³	1	2	yes	—
70	C₈×Dic₁₃	D₄	1	1	yes	characteristic
71	C₈×Dic₁₃	D₄	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
73	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
74	#88052	—	2	1	no	—
75	#88052	C₂³	1	2	yes	—
76	C₂×#6090	—	2	1	no	—
77	#88048	—	2	2	no	—
78	#88047	—	2	2	no	—
79	#33637	—	13	1	no	maximal, Sylow
80	C₁₃×#41	—	2	2	no	—
81	C₁₃×#41	—	2	2	no	—
82	C₁₃×#41	—	4	1	no	—
83	C₁₃×#41	—	4	1	no	—
84	C₂×Q₈×C₁₃	—	2	1	no	—
85	C₂×D₄×C₁₃	—	2	1	no	—
86	C₂×D₄×C₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	—	4	1	no	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₁₃	—	2	1	no	—
90	C₂×Dic₂₆	—	4	2	no	—
91	C₁₃×Q₁₆	—	2	1	no	—
92	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	—
93	C₁₃×SD₁₆	—	2	2	no	—
94	C₁₃×SD₁₆	—	2	2	no	—
95	C₁₃×D₈	—	2	1	no	—
96	C₁₃×D₈	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₄²×C₁₃	—	4	1	no	—
106	C₂²⋊Dic₁₃	—	4	1	no	—
107	C₄⋊Dic₁₃	—	2	2	no	—
108	C₄⋊Dic₁₃	—	4	2	no	—
109	#6101	—	4	2	no	—
110	#6101	—	4	2	no	—
111	C₄×Dic₁₃	—	2	1	no	—
112	C₄×Dic₁₃	—	4	1	no	—
113	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
115	#6099	—	2	2	no	—
116	#6099	—	4	1	no	—
117	#6099	—	4	2	no	—
118	#6099	—	4	2	no	—
119	C₂×#1055	—	4	1	no	—
120	C₂×#1055	—	4	2	no	—
121	C₂×#1055	—	4	2	no	—
122	C₂×#1055	SD₁₆	1	1	yes	characteristic
123	C₂×#1055	SD₁₆	1	1	yes	characteristic
124	C₂×#1055	SD₁₆	1	1	yes	characteristic
125	C₂×#1055	SD₁₆	1	1	yes	characteristic
126	C₂×#1055	C₂×D₄	1	1	yes	characteristic
127	C₂×#1055	C₂×D₄	1	1	yes	characteristic
128	C₁₃×C₁₆	—	2	2	no	—
129	#6090	—	2	2	no	—
130	#3160	—	13	1	no	—
131	C₂×#443	—	13	1	no	—
132	#2231	—	13	4	no	—
133	#2230	—	13	4	no	—
134	#2155	—	13	1	no	—
135	#2155	—	13	1	no	—
136	C₂×#364	—	13	1	no	—
137	#2145	—	13	1	no	—
138	#2145	D₁₃	1	1	yes	characteristic
139	C₂³×C₁₃	—	4	1	no	—
140	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	Q₈×C₁₃	—	2	2	no	—
142	Q₈×C₁₃	—	4	1	no	—
143	D₄×C₁₃	—	2	2	no	—
144	D₄×C₁₃	—	4	2	no	—
145	D₄×C₁₃	—	4	2	no	—
146	D₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	—	4	1	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×Dic₁₃	—	2	1	no	—
158	C₂×Dic₁₃	—	2	1	no	—
159	C₂×Dic₁₃	—	4	2	no	—
160	C₂×Dic₁₃	—	4	2	no	—
161	Dic₂₆	—	8	4	no	—
162	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
163	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
164	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
165	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
166	#1055	—	4	2	no	—
167	#1055	C₂×SD₁₆	1	1	yes	characteristic
168	#1055	C₂×SD₁₆	1	1	yes	characteristic
169	#1055	C₂×SD₁₆	1	1	yes	characteristic
170	#1055	C₂×SD₁₆	1	1	yes	characteristic
171	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
172	C₂×#131	—	26	1	no	—
173	#533	—	26	1	no	—
174	C₂×M₃₂	—	13	1	no	—
175	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
176	#500	—	26	1	no	—
177	#499	—	13	2	no	—
178	#499	—	26	1	no	—
179	#498	—	13	2	no	—
180	#443	—	13	2	no	—
181	#443	—	13	2	no	—
182	#443	—	26	1	no	—
183	#443	—	26	1	no	—
184	C₂×#108	—	13	1	no	—
185	#427	—	13	1	no	—
186	#427	—	26	1	no	—
187	#427	C₂×D₁₃	1	1	yes	characteristic
188	C₂×#104	—	26	1	no	—
189	C₂×C₄×C₈	—	13	1	no	—
190	#365	—	13	4	no	—
191	#365	—	13	4	no	—
192	#364	—	13	2	no	—
193	#364	—	13	2	no	—
194	#360	—	13	4	no	—
195	#360	C₂×D₁₃	1	4	yes	—
196	C₂²×C₁₃	—	4	1	no	—
197	C₂²×C₁₃	—	4	2	no	—
198	C₂²×C₁₃	—	8	2	no	—
199	C₂²×C₁₃	#476	1	1	yes	characteristic
200	C₂²×C₁₃	#476	1	1	yes	characteristic
201	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₄×C₁₃	—	4	2	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	#476	1	1	yes	characteristic
205	C₄×C₁₃	#476	1	1	yes	characteristic
206	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
207	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
208	Dic₁₃	—	2	2	no	—
209	Dic₁₃	—	4	1	no	—
210	Dic₁₃	—	8	2	no	—
211	C₂×#41	—	2	1	no	—
212	#135	—	2	1	no	—
213	#135	—	2	1	no	—
214	#135	Dic₂₆	1	2	yes	complemented
215	#135	C₂²×D₁₃	1	2	yes	—
216	C₂×Q₁₆	Dic₂₆	1	1	yes	characteristic, complemented
217	C₂×SD₁₆	—	2	1	no	—
218	C₂×D₈	Dic₂₆	1	1	yes	characteristic, complemented
219	#131	—	26	2	no	—
220	#131	—	26	2	no	—
221	#131	—	52	1	no	—
222	#131	—	52	1	no	—
223	C₂×M₁₆	—	26	1	no	—
224	C₂×M₁₆	—	52	1	no	—
225	C₂²×C₈	—	13	1	no	—
226	C₂²×C₈	—	52	1	no	—
227	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
228	C₄⋊Q₈	—	26	2	no	—
229	#125	—	26	2	no	—
230	C₂²⋊Q₈	—	52	2	no	—
231	C₄×Q₈	—	52	2	no	—
232	#117	—	2	1	no	—
233	#117	—	26	1	no	—
234	C₂×C₄²	—	13	1	no	—
235	M₃₂	—	2	1	no	—
236	M₃₂	—	13	2	no	—
237	M₃₂	—	26	1	no	—
238	M₃₂	C₂²×D₁₃	1	2	yes	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	26	1	no	—
241	#108	—	13	2	no	—
242	#108	—	26	1	no	—
243	#107	—	2	1	no	—
244	#107	—	26	2	no	—
245	#107	—	26	1	no	—
246	#106	—	13	2	no	—
247	#106	—	26	2	no	—
248	#106	C₂²×D₁₃	1	2	yes	—
249	#104	—	26	2	no	—
250	#104	—	26	2	no	—
251	C₄×C₈	—	13	1	no	—
252	C₄×C₈	—	13	1	no	—
253	C₄×C₈	—	13	1	no	—
254	C₄×C₈	—	13	1	no	—
255	C₂×C₁₃	—	2	1	no	—
256	C₂×C₁₃	—	2	1	no	—
257	C₂×C₁₃	—	2	1	no	—
258	C₂×C₁₃	—	8	2	no	—
259	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	4	1	no	—
263	#41	—	4	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	52	2	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	26	1	no	—
270	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	C₂×Dic₂₆	1	2	yes	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	C₂×Dic₂₆	1	2	yes	—
277	M₁₆	—	26	2	no	—
278	M₁₆	—	52	2	no	—
279	M₁₆	—	52	2	no	—
280	M₁₆	—	52	1	no	—
281	C₂×C₈	—	13	1	no	—
282	C₂×C₈	—	13	1	no	—
283	C₂×C₈	—	13	1	no	—
284	C₂×C₈	—	13	1	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	13	1	no	—
287	C₂×C₈	—	52	2	no	—
288	C₂×C₈	—	52	2	no	—
289	C₂×C₈	—	52	1	no	—
290	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
291	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
292	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
293	C₂×C₈	#6129	1	1	yes	characteristic
294	C₂×C₈	#6129	1	1	yes	characteristic
295	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₄⋊C₄	—	52	2	no	—
299	C₄⋊C₄	—	52	2	no	—
300	C₄⋊C₄	—	52	2	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₂²⋊C₄	—	52	1	no	—
303	C₄²	—	4	1	no	—
304	C₄²	—	13	1	no	—
305	C₄²	—	13	1	no	—
306	C₄²	—	26	1	no	—
307	C₄²	—	52	1	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	26	2	no	—
310	C₁₃	#33637	1	1	yes	characteristic, complemented, Sylow
311	C₂³	—	4	1	no	—
312	C₂³	#88165	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	4	1	no	—
315	Q₈	—	104	4	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	26	1	no	—
325	C₂×C₄	—	26	1	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	#88132	1	1	yes	characteristic
329	C₂×C₄	#88132	1	1	yes	characteristic
330	C₂×C₄	#88165	1	1	yes	characteristic
331	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
332	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
334	C₈	—	13	1	no	—
335	C₈	—	13	1	no	—
336	C₈	—	13	1	no	—
337	C₈	—	13	1	no	—
338	C₈	—	52	2	no	—
339	C₈	#88132	1	1	yes	characteristic
340	C₈	#88132	1	1	yes	characteristic
341	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
342	C₈	C₂×#6129	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#11718422	1	1	yes	characteristic
347	C₂²	#11718422	1	1	yes	characteristic
348	C₂²	C₂×#88165	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	26	2	no	—
352	C₄	—	52	1	no	—
353	C₄	—	104	2	no	—
354	C₄	#11718422	1	1	yes	characteristic
355	C₄	#11718422	1	1	yes	characteristic
356	C₄	C₂×#88132	1	1	yes	characteristic
357	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×#11718422	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718422
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2145	C₂
socle	C₂×C₁₃	C₂×#476
prime radical	C₂×C₁₃×#135	C₂²
hypercenter	#2145	D₁₃
hypocenter	C₁₃	#33637
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2145

Complements

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

N	H ≅ G/N	type
1. #13125348	C₂	semidirect
3. #13117010	C₂	semidirect
4. #13117005	C₂	semidirect
5. #13116609	C₂	semidirect
6. #13116608	C₂	semidirect
7. C₂×#11718103	C₂	semidirect
27. #11718519	C₂²	semidirect
28. #11718516	C₂²	semidirect
31. #11718105	C₂²	semidirect
32. #11718103	C₂²	semidirect
33. #11718103	C₂²	semidirect
34. #11718079	C₂²	semidirect
38. C₁₃×#135	Q₈	semidirect
40. C₂×C₁₃×Q₁₆	Q₈	semidirect
42. C₂×C₁₃×D₈	Q₈	semidirect
214. #135	Dic₂₆	semidirect
216. C₂×Q₁₆	Dic₂₆	semidirect
218. C₂×D₈	Dic₂₆	semidirect
310. C₁₃	#33637	semidirect

Maximal subgroups

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

subgroup	index
#13125348	2
C₂×#11718857	2
#13117010	2
#13117005	2
#13116609	2
#13116608	2
C₂×#11718103	2
#13116572	2
C₁₃×#2145	2
#33637	13