Subgroups — #16015934

Order 3328.

636 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127938	C₂	1	1	yes	maximal, characteristic
2	#13123444	C₂	1	1	yes	maximal, characteristic
3	#13123428	C₂	1	1	yes	maximal, characteristic
4	#13116608	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116607	C₂	1	4	yes	maximal, complemented
6	C₂×#11718086	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718105	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×#365	C₂	1	4	yes	maximal, complemented
9	C₁₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719450	—	2	1	no	—
11	C₂×#88155	C₂²	1	1	yes	characteristic
12	#11719228	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
15	C₁₃×#427	C₂²	1	1	yes	characteristic
16	#11718785	—	2	1	no	—
17	C₂×#88059	C₂²	1	1	yes	characteristic
18	C₂×#88057	C₂²	1	1	yes	characteristic
19	C₂×#88052	C₂²	1	1	yes	characteristic
20	#11718656	—	2	1	no	—
21	#11718568	C₄	1	4	yes	complemented
22	D₁₃×M₃₂	C₂²	1	4	yes	—
23	#11718562	C₄	1	4	yes	complemented
24	#11718554	—	2	2	no	—
25	D₁₃×#108	C₂²	1	4	yes	—
26	#11718542	—	2	2	no	—
27	#11718521	C₂²	1	2	yes	—
28	D₁₃×#106	C₂²	1	2	yes	—
29	C₁₃×#365	C₂²	1	2	yes	—
30	C₁₃×#365	C₂²	1	2	yes	—
31	#11718105	C₂²	1	4	yes	—
32	#11718086	C₂²	1	4	yes	—
33	#11718079	C₂²	1	2	yes	—
34	#11718079	C₂²	1	2	yes	—
35	C₂×#6127	D₄	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	—	2	1	no	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#117	—	2	1	no	—
39	C₂×C₄⋊Dic₁₃	—	2	1	no	—
40	C₂×#6099	—	2	1	no	—
41	C₂×#6099	D₄	1	1	yes	characteristic
42	#88161	—	4	2	no	—
43	M₁₆×D₁₃	—	2	4	no	—
44	#88155	—	4	2	no	—
45	#88155	C₂×C₄	1	2	yes	—
46	#88155	C₂×C₄	1	2	yes	—
47	#88155	Q₈	1	2	yes	—
48	#88155	Q₈	1	2	yes	—
49	C₂×#6094	D₄	1	2	yes	—
50	C₂×C₈×D₁₃	C₂³	1	2	yes	—
51	#88146	—	2	2	no	—
52	C₄⋊C₄×D₁₃	—	2	2	no	—
53	#88135	—	4	2	no	—
54	C₄×D₅₂	—	4	1	no	—
55	C₄×Dic₂₆	—	4	1	no	—
56	C₁₃×M₃₂	C₂×C₄	1	2	yes	complemented
57	C₁₃×M₃₂	C₂³	1	2	yes	—
58	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	C₂³	1	2	yes	—
61	C₁₃×#107	—	2	1	no	—
62	C₁₃×#106	C₂³	1	1	yes	characteristic
63	C₁₃×#106	C₂³	1	1	yes	characteristic
64	#88062	—	2	4	no	—
65	#88059	—	2	1	no	—
66	#88059	C₂³	1	2	yes	—
67	#88057	—	2	1	no	—
68	#88057	C₂³	1	2	yes	—
69	#88052	C₂×C₄	1	2	yes	complemented
70	#88052	C₂³	1	2	yes	—
71	C₂×#6090	C₂×C₄	1	2	yes	complemented
72	#88048	—	2	2	no	—
73	#88047	—	2	2	no	—
74	#88038	C₂×C₄	1	8	yes	complemented
75	C₁₆×D₁₃	C₂×C₄	1	8	yes	complemented
76	#33404	—	13	1	no	maximal, Sylow
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₃	SD₁₆	1	2	yes	—
79	C₂²×Dic₁₃	—	4	1	no	—
80	#6127	C₄⋊C₄	1	2	yes	—
81	#6127	C₄⋊C₄	1	2	yes	—
82	#6127	SD₁₆	1	2	yes	—
83	#6127	SD₁₆	1	2	yes	—
84	C₂×D₅₂	SD₁₆	1	1	yes	characteristic
85	C₂×C₄×D₁₃	—	4	2	no	—
86	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
87	C₂×Dic₂₆	SD₁₆	1	1	yes	characteristic
88	C₁₃×M₁₆	—	2	2	no	—
89	C₁₃×M₁₆	—	4	1	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₂²⋊C₄	—	4	1	no	—
100	C₄²×C₁₃	—	4	1	no	—
101	#6103	—	4	2	no	—
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	—	4	1	no	—
104	#6101	—	4	2	no	—
105	C₄×Dic₁₃	—	4	2	no	—
106	#6099	—	2	2	no	—
107	#6099	—	4	1	no	—
108	#6099	C₄⋊C₄	1	2	yes	—
109	#6099	C₄⋊C₄	1	2	yes	—
110	C₂×#1055	—	4	1	no	—
111	C₂×#1055	C₂×D₄	1	2	yes	—
112	#6094	—	4	4	no	—
113	#6094	C₄⋊C₄	1	4	yes	—
114	#6094	C₄⋊C₄	1	4	yes	—
115	C₈×D₁₃	—	4	4	no	—
116	C₈×D₁₃	C₂²×C₄	1	4	yes	—
117	C₈×D₁₃	C₂×Q₈	1	4	yes	—
118	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
119	#6090	C₂²×C₄	1	4	yes	—
120	#3407	—	13	1	no	—
121	#2916	—	13	1	no	—
122	#2912	—	13	1	no	—
123	#2155	—	13	1	no	—
124	#2155	—	13	4	no	—
125	#2155	D₁₃	1	1	yes	characteristic, complemented
126	C₂×#365	—	13	1	no	—
127	C₂×#365	—	13	1	no	—
128	C₂×#365	—	13	4	no	—
129	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂²×D₁₃	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₁₃	#106	1	4	yes	—
140	C₂²⋊D₁₃	#106	1	4	yes	—
141	C₂×Dic₁₃	—	4	2	no	—
142	C₂×Dic₁₃	—	16	1	no	—
143	C₂×Dic₁₃	C₂×SD₁₆	1	2	yes	—
144	D₅₂	#106	1	2	yes	—
145	D₅₂	#106	1	2	yes	—
146	C₄×D₁₃	—	16	2	no	—
147	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
148	C₄×D₁₃	C₂×SD₁₆	1	4	yes	—
149	Dic₂₆	#106	1	2	yes	—
150	Dic₂₆	#106	1	2	yes	—
151	C₈×C₁₃	—	4	2	no	—
152	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
156	#1055	—	4	2	no	—
157	#1055	C₂×C₄⋊C₄	1	4	yes	—
158	#568	—	26	1	no	—
159	C₂×#131	—	13	1	no	—
160	#520	—	26	1	no	—
161	#504	—	13	4	no	—
162	#504	—	13	4	no	—
163	C₂×M₃₂	—	13	1	no	—
164	C₂×M₃₂	—	13	4	no	—
165	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
166	#430	—	26	2	no	—
167	#430	—	26	1	no	—
168	C₂×#108	—	13	1	no	—
169	C₂×#108	—	13	4	no	—
170	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
171	#428	—	26	2	no	—
172	#428	—	26	1	no	—
173	#427	—	13	2	no	—
174	#427	C₂×D₁₃	1	1	yes	characteristic
175	C₂×#106	—	13	1	no	—
176	C₂×#106	—	13	2	no	—
177	#365	—	13	4	no	—
178	#365	—	13	4	no	—
179	#365	—	13	2	no	—
180	#365	—	13	2	no	—
181	#365	C₂×D₁₃	1	2	yes	—
182	#365	C₂×D₁₃	1	2	yes	—
183	C₂²×C₁₃	—	4	1	no	—
184	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
186	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
187	C₂×D₁₃	—	4	2	no	—
188	C₂×D₁₃	C₂×#106	1	4	yes	—
189	C₄×C₁₃	—	8	1	no	—
190	C₄×C₁₃	—	8	1	no	—
191	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
192	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
193	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
194	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
195	Dic₁₃	—	8	2	no	—
196	Dic₁₃	C₂×#106	1	4	yes	—
197	C₂×#41	—	13	1	no	—
198	#131	—	13	2	no	—
199	#131	—	13	2	no	—
200	#131	—	13	2	no	—
201	#131	—	13	2	no	—
202	#131	—	52	2	no	—
203	#131	—	52	2	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	13	1	no	—
206	C₂×M₁₆	—	13	2	no	—
207	C₂×M₁₆	—	26	4	no	—
208	C₂×M₁₆	—	26	1	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
211	C₄×Q₈	—	52	1	no	—
212	C₄×D₄	—	52	1	no	—
213	C₄×D₄	—	52	2	no	—
214	#117	—	2	1	no	—
215	#117	—	26	2	no	—
216	C₂×C₄⋊C₄	—	26	2	no	—
217	C₂×C₄⋊C₄	—	26	1	no	—
218	M₃₂	—	13	2	no	—
219	M₃₂	—	13	2	no	—
220	M₃₂	—	13	8	no	—
221	M₃₂	C₄×D₁₃	1	2	yes	complemented
222	M₃₂	C₂²×D₁₃	1	2	yes	—
223	C₂×C₁₆	—	13	2	no	—
224	C₂×C₁₆	—	13	8	no	—
225	C₂×C₁₆	C₄×D₁₃	1	2	yes	complemented
226	#108	—	2	1	no	—
227	#108	—	13	2	no	—
228	#108	—	26	4	no	—
229	#108	—	26	1	no	—
230	#108	C₂²×D₁₃	1	2	yes	—
231	#107	—	2	1	no	—
232	#107	—	26	2	no	—
233	#106	—	13	2	no	—
234	#106	—	26	2	no	—
235	#106	—	26	1	no	—
236	#106	C₂²×D₁₃	1	1	yes	characteristic
237	#106	C₂²×D₁₃	1	1	yes	characteristic
238	C₂×C₁₃	—	2	1	no	—
239	C₂×C₁₃	—	2	1	no	—
240	C₂×C₁₃	—	2	1	no	—
241	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
242	D₁₃	—	2	4	no	—
243	#41	—	13	2	no	—
244	#41	—	13	2	no	—
245	#41	—	13	2	no	—
246	#41	—	13	2	no	—
247	C₂×Q₈	—	13	1	no	—
248	C₂×D₄	—	13	2	no	—
249	C₂×D₄	—	13	1	no	—
250	C₂²×C₄	—	13	2	no	—
251	C₂²×C₄	—	52	2	no	—
252	C₂²×C₄	—	52	1	no	—
253	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
254	M₁₆	—	2	2	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	13	2	no	—
257	M₁₆	—	13	2	no	—
258	M₁₆	—	13	4	no	—
259	M₁₆	—	13	4	no	—
260	M₁₆	—	26	2	no	—
261	M₁₆	—	52	4	no	—
262	M₁₆	—	52	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	—	13	4	no	—
267	C₂×C₈	—	52	4	no	—
268	C₂×C₈	—	52	1	no	—
269	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
271	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
272	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
273	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	26	2	no	—
278	C₄⋊C₄	—	52	2	no	—
279	C₄⋊C₄	—	52	1	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	52	2	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	52	2	no	—
284	C₁₆	—	13	4	no	—
285	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
286	C₁₃	#33404	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	13	2	no	—
288	C₂³	SD₁₆×D₁₃	1	1	yes	characteristic
289	Q₈	—	13	2	no	—
290	Q₈	—	13	2	no	—
291	D₄	—	13	4	no	—
292	D₄	—	13	4	no	—
293	D₄	—	13	2	no	—
294	D₄	—	13	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	4	no	—
299	C₂×C₄	—	13	4	no	—
300	C₂×C₄	—	52	2	no	—
301	C₂×C₄	—	208	2	no	—
302	C₂×C₄	—	208	1	no	—
303	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
307	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	13	4	no	—
311	C₈	—	52	2	no	—
312	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
313	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
314	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
315	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
316	C₂²	—	4	1	no	—
317	C₂²	—	13	4	no	—
318	C₂²	—	52	2	no	—
319	C₂²	D₁₃×#106	1	1	yes	characteristic
320	C₂²	D₁₃×#106	1	1	yes	characteristic
321	C₂²	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	13	4	no	—
325	C₄	—	104	2	no	—
326	C₄	D₁₃×#106	1	1	yes	characteristic
327	C₄	D₁₃×#106	1	1	yes	characteristic
328	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	26	4	no	—
334	C₂	C₂×D₁₃×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₃×#106
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2155	C₂
socle	C₂×C₁₃	C₂²×#106
prime radical	C₂×#6127	D₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2155	D₁₃
hypocenter	C₁₃	#33404
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
4. #13116608	C₂	semidirect
5. #13116607	C₂	semidirect
6. C₂×#11718086	C₂	semidirect
7. C₂×#11718105	C₂	semidirect
8. D₁₃×#365	C₂	semidirect
9. C₁₃×#2155	C₂	semidirect
21. #11718568	C₄	semidirect
23. #11718562	C₄	semidirect
56. C₁₃×M₃₂	C₂×C₄	semidirect
58. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
69. #88052	C₂×C₄	semidirect
71. C₂×#6090	C₂×C₄	semidirect
74. #88038	C₂×C₄	semidirect
75. C₁₆×D₁₃	C₂×C₄	semidirect
125. #2155	D₁₃	semidirect
221. M₃₂	C₄×D₁₃	semidirect
225. C₂×C₁₆	C₄×D₁₃	semidirect
286. C₁₃	#33404	semidirect

Maximal subgroups

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

subgroup	index
#13127938	2
#13123444	2
#13123428	2
#13116608	2
#13116607	2
C₂×#11718086	2
C₂×#11718105	2
D₁₃×#365	2
C₁₃×#2155	2
#33404	13