Subgroups — #16789117

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊#108	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88101	C₂	1	1	yes	maximal, characteristic
3	#13129901	C₂	1	4	yes	maximal, complemented
4	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#261	—	2	1	no	—
6	C₂×D₁₃⋊M₁₆	—	2	1	no	—
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	D₁₃⋊#108	—	2	2	no	—
9	D₁₃⋊#108	C₄	1	4	yes	complemented
10	C₂×#88104	—	2	1	no	—
11	C₂×#88103	C₂²	1	2	yes	—
12	C₂×#88101	—	2	1	no	—
13	C₂×#88101	—	2	2	no	—
14	C₂×#88101	C₄	1	4	yes	—
15	C₂×#88101	C₂²	1	2	yes	—
16	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
17	C₂×#88052	C₄	1	1	yes	characteristic, complemented
18	D₁₃×M₃₂	—	2	2	no	—
19	D₁₃×M₃₂	—	2	2	no	—
20	C₂×#88038	C₂²	1	2	yes	—
21	C₂×C₁₆×D₁₃	C₄	1	2	yes	complemented
22	C₂³×#261	—	4	1	no	—
23	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
24	C₂×#6122	—	4	1	no	—
25	C₂×C₄⋊#261	—	2	4	no	—
26	C₂×C₄⋊#261	—	2	2	no	—
27	C₂×C₄⋊#261	—	4	2	no	—
28	C₂×C₄⋊#261	—	4	1	no	—
29	D₁₃⋊M₁₆	—	2	4	no	—
30	D₁₃⋊M₁₆	—	4	2	no	—
31	C₂×#6118	—	2	2	no	—
32	C₂×D₁₃⋊C₈	—	4	1	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	Q₈	1	1	yes	characteristic
35	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
39	C₂×C₈×D₁₃	D₄	1	2	yes	—
40	C₂×C₈×D₁₃	D₄	1	2	yes	—
41	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
43	#88104	—	2	4	no	—
44	#88103	—	2	2	no	—
45	#88103	C₂×C₄	1	4	yes	complemented
46	#88101	—	2	4	no	—
47	#88101	—	2	2	no	—
48	#88101	C₂×C₄	1	4	yes	—
49	C₁₃×M₃₂	—	2	2	no	—
50	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
51	#88052	—	2	2	no	—
52	C₂×#6090	C₂×C₄	1	2	yes	complemented
53	#88038	—	2	2	no	—
54	#88038	—	2	2	no	—
55	C₁₆×D₁₃	—	2	4	no	—
56	C₂×#1383	—	13	1	no	maximal, Sylow
57	C₂³×D₁₃	SD₁₆	1	1	yes	characteristic
58	C₂²×#261	—	4	4	no	—
59	C₂²×#261	—	4	2	no	—
60	C₂²×#261	—	8	4	no	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	SD₁₆	1	1	yes	characteristic
63	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×D₁₃	D₈	1	2	yes	—
67	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
68	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
69	C₂×C₄×D₁₃	Q₁₆	1	2	yes	—
70	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
71	#6122	—	4	4	no	—
72	C₂×#1057	—	4	2	no	—
73	C₄⋊#261	—	2	4	no	—
74	C₄⋊#261	—	4	4	no	—
75	C₄⋊#261	—	4	2	no	—
76	#6118	—	2	4	no	—
77	#6118	—	4	2	no	—
78	D₁₃⋊C₈	—	4	4	no	—
79	C₂×C₈×C₁₃	C₄²	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
84	C₂×#1055	C₄⋊C₄	1	2	yes	—
85	C₂×#1055	C₄⋊C₄	1	2	yes	—
86	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
87	C₈×D₁₃	C₄²	1	2	yes	—
88	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
89	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
90	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
91	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
92	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
93	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
96	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
97	C₁₃×C₁₆	—	2	2	no	—
98	#6090	—	2	2	no	—
99	C₂²×M₃₂	—	13	1	no	—
100	C₂²×#108	—	13	1	no	—
101	C₂²×#106	—	13	1	no	—
102	#1383	—	13	4	no	—
103	C₂³×C₁₃	#106	1	1	yes	characteristic
104	C₂²×D₁₃	—	2	4	no	—
105	C₂²×D₁₃	#102	1	2	yes	—
106	C₂²×D₁₃	#103	1	2	yes	—
107	C₂²×D₁₃	#106	1	1	yes	characteristic
108	C₂²×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×#261	—	4	4	no	—
110	C₂×#261	—	8	4	no	—
111	C₂×#261	—	8	8	no	—
112	C₂×C₄×C₁₃	#95	1	2	yes	—
113	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#107	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×Dic₁₃	#102	1	2	yes	—
117	C₂×Dic₁₃	#103	1	2	yes	—
118	C₂×Dic₁₃	#106	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₄×D₁₃	#95	1	2	yes	—
121	C₄×D₁₃	#102	1	2	yes	—
122	C₄×D₁₃	#102	1	2	yes	—
123	C₄×D₁₃	#103	1	2	yes	—
124	C₄×D₁₃	#103	1	2	yes	—
125	C₄×D₁₃	#107	1	2	yes	—
126	C₄×D₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×D₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
129	C₄×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
130	#1057	—	4	4	no	—
131	C₈×C₁₃	#95	1	2	yes	—
132	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
133	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	#1055	#95	1	2	yes	—
135	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
136	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×M₁₆	—	26	1	no	—
138	C₂³×C₈	—	13	1	no	—
139	C₂²×C₄⋊C₄	—	26	1	no	—
140	C₂×M₃₂	—	13	1	no	—
141	C₂×M₃₂	—	13	2	no	—
142	C₂×M₃₂	—	26	2	no	—
143	C₂×M₃₂	—	26	2	no	—
144	C₂×M₃₂	#261	1	1	yes	characteristic, complemented
145	C₂²×C₁₆	—	13	2	no	—
146	C₂×#108	—	13	2	no	—
147	C₂×#108	—	13	4	no	—
148	C₂×#108	—	26	1	no	—
149	C₂×#108	—	26	2	no	—
150	C₂×#106	—	13	2	no	—
151	C₂×#106	—	13	4	no	—
152	C₂×#106	—	26	1	no	—
153	C₂×#106	—	26	2	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	#340	1	2	yes	—
156	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
157	C₂×D₁₃	—	2	2	no	—
158	C₂×D₁₃	—	2	4	no	—
159	C₂×D₁₃	—	2	4	no	—
160	C₂×D₁₃	#340	1	2	yes	—
161	C₂×D₁₃	#365	1	4	yes	—
162	C₂×D₁₃	C₂×#102	1	1	yes	characteristic
163	C₂×D₁₃	C₂×#103	1	1	yes	characteristic
164	#261	—	8	4	no	—
165	C₄×C₁₃	#340	1	2	yes	—
166	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
167	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
168	Dic₁₃	#340	1	2	yes	—
169	Dic₁₃	C₂×#102	1	1	yes	characteristic
170	Dic₁₃	C₂×#103	1	1	yes	characteristic
171	C₂³×C₄	—	13	1	no	—
172	C₂³×C₄	—	52	1	no	—
173	C₂×M₁₆	—	26	2	no	—
174	C₂×M₁₆	—	26	4	no	—
175	C₂×M₁₆	—	52	2	no	—
176	C₂×M₁₆	—	52	1	no	—
177	C₂²×C₈	—	13	1	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	—	13	2	no	—
180	C₂²×C₈	—	13	1	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₂²×C₈	—	13	2	no	—
183	C₂²×C₈	—	13	2	no	—
184	C₂²×C₈	—	13	1	no	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	52	1	no	—
187	C₂²×C₈	C₂×#261	1	1	yes	characteristic
188	C₂×C₄⋊C₄	—	26	4	no	—
189	C₂×C₄⋊C₄	—	26	2	no	—
190	C₂×C₄⋊C₄	—	52	1	no	—
191	C₂×C₄⋊C₄	—	52	2	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	—	26	2	no	—
194	M₃₂	—	26	2	no	—
195	M₃₂	—	26	2	no	—
196	C₂×C₁₆	—	13	2	no	—
197	C₂×C₁₆	—	26	4	no	—
198	C₂×C₁₆	C₂×#261	1	2	yes	complemented
199	#108	—	13	4	no	—
200	#108	—	26	2	no	—
201	#108	—	26	4	no	—
202	#106	—	13	4	no	—
203	#106	—	26	4	no	—
204	#106	—	26	2	no	—
205	C₂×C₁₃	—	2	2	no	—
206	C₂×C₁₃	C₂×#340	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#365	1	2	yes	—
208	D₁₃	—	2	1	no	—
209	D₁₃	—	2	1	no	—
210	D₁₃	#1383	1	4	yes	complemented
211	C₂⁴	—	13	1	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	2	no	—
216	C₂²×C₄	—	13	2	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	13	2	no	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	1	no	—
221	C₂²×C₄	—	52	4	no	—
222	C₂²×C₄	—	52	2	no	—
223	C₂²×C₄	—	104	4	no	—
224	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
225	M₁₆	—	26	4	no	—
226	M₁₆	—	52	2	no	—
227	M₁₆	—	52	4	no	—
228	C₂×C₈	—	13	2	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	—	13	1	no	—
232	C₂×C₈	—	13	2	no	—
233	C₂×C₈	—	13	2	no	—
234	C₂×C₈	—	13	2	no	—
235	C₂×C₈	—	13	2	no	—
236	C₂×C₈	—	13	1	no	—
237	C₂×C₈	—	13	1	no	—
238	C₂×C₈	—	13	2	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	—	13	1	no	—
241	C₂×C₈	—	13	2	no	—
242	C₂×C₈	—	52	4	no	—
243	C₂×C₈	—	52	2	no	—
244	C₂×C₈	C₄×#261	1	2	yes	—
245	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
246	C₂×C₈	C₂²⋊#261	1	2	yes	—
247	C₂×C₈	C₂²×#261	1	1	yes	characteristic
248	C₄⋊C₄	—	26	4	no	—
249	C₄⋊C₄	—	52	4	no	—
250	C₄⋊C₄	—	52	2	no	—
251	C₁₆	—	2	2	no	—
252	C₁₆	—	26	2	no	—
253	C₁₃	C₂×#1383	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	13	1	no	—
255	C₂³	—	13	2	no	—
256	C₂³	—	13	2	no	—
257	C₂³	—	13	1	no	—
258	C₂³	—	26	4	no	—
259	C₂³	#88101	1	1	yes	characteristic
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	13	1	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	13	2	no	—
265	C₂×C₄	—	13	2	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	13	2	no	—
268	C₂×C₄	—	13	1	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	13	1	no	—
271	C₂×C₄	—	13	1	no	—
272	C₂×C₄	—	13	1	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	52	4	no	—
275	C₂×C₄	—	104	8	no	—
276	C₂×C₄	—	104	4	no	—
277	C₂×C₄	#88101	1	1	yes	characteristic
278	C₂×C₄	#88102	1	2	yes	—
279	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
281	C₈	—	13	2	no	—
282	C₈	—	13	1	no	—
283	C₈	—	13	1	no	—
284	C₈	—	52	4	no	—
285	C₈	(C₂×#261)⋊C₄	1	2	yes	—
286	C₈	C₂×C₄×#261	1	1	yes	characteristic
287	C₈	C₂×C₂²⋊#261	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	13	2	no	—
290	C₂²	—	13	4	no	—
291	C₂²	—	13	1	no	—
292	C₂²	—	13	1	no	—
293	C₂²	—	26	4	no	—
294	C₂²	—	26	4	no	—
295	C₂²	—	26	2	no	—
296	C₂²	#11718265	1	2	yes	—
297	C₂²	C₂×#88101	1	1	yes	characteristic
298	C₄	—	13	2	no	—
299	C₄	—	13	1	no	—
300	C₄	—	13	1	no	—
301	C₄	—	104	4	no	—
302	C₄	#11718265	1	2	yes	—
303	C₄	C₂×#88102	1	1	yes	characteristic
304	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
305	C₂	—	2	2	no	—
306	C₂	—	13	4	no	—
307	C₂	—	26	1	no	—
308	C₂	—	26	1	no	—
309	C₂	C₂×#11718265	1	1	yes	characteristic
310	C₂	D₁₃⋊#365	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#88101
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂³×D₁₃	SD₁₆
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	C₂×#1383
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13129901	C₂	semidirect
9. D₁₃⋊#108	C₄	semidirect
16. C₂×C₁₃×M₃₂	C₄	semidirect
17. C₂×#88052	C₄	semidirect
21. C₂×C₁₆×D₁₃	C₄	semidirect
45. #88103	C₂×C₄	semidirect
50. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
52. C₂×#6090	C₂×C₄	semidirect
144. C₂×M₃₂	#261	semidirect
198. C₂×C₁₆	C₂×#261	semidirect
210. D₁₃	#1383	semidirect
253. C₁₃	C₂×#1383	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃⋊#108	2
C₂²×#88101	2
#13129901	2
C₂×D₁₃×M₃₂	2
C₂×#1383	13