Subgroups — #15892891

Order 3328.

664 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718786	C₂	1	1	yes	maximal, characteristic
2	#13123399	C₂	1	1	yes	maximal, characteristic
3	#13116412	C₂	1	8	yes	maximal, complemented
4	C₂×#11718125	C₂	1	2	yes	maximal
5	#13114545	C₂	1	2	yes	maximal
6	C₂×C₁₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₂²	1	2	yes	—
9	C₂×#88180	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	1	yes	characteristic
11	C₂²×#6099	C₂²	1	2	yes	—
12	#11718786	—	2	2	no	—
13	#11718786	C₂²	1	4	yes	—
14	#11718780	—	2	2	no	—
15	#11718779	C₂²	1	4	yes	—
16	C₂×#88059	—	2	2	no	—
17	#11718763	—	2	2	no	—
18	C₁₃×H(3,ℤ/4)	C₂²	1	4	yes	—
19	#11718125	C₂²	1	8	yes	—
20	#11718124	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
25	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₃	D₄	1	2	yes	—
27	C₁₃×#117	—	2	2	no	—
28	C₁₃×#117	—	2	2	no	—
29	C₁₃×#117	—	2	2	no	—
30	C₁₃×#117	C₂³	1	4	yes	—
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₁₃	—	2	2	no	—
34	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
35	#88180	—	2	1	no	—
36	#88180	—	2	1	no	—
37	#88180	Q₈	1	2	yes	—
38	#88180	Q₈	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	—	2	1	no	—
40	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
41	C₂×#6099	—	2	1	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	C₂³	1	2	yes	—
50	C₂×#6099	C₂³	1	4	yes	—
51	C₂²×#1055	—	2	1	no	—
52	C₂²×#1055	—	2	2	no	—
53	#88059	—	2	8	no	—
54	#88054	—	2	8	no	—
55	#23751	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	2	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₂²⋊C₄	—	2	4	no	—
77	C₁₃×C₂²⋊C₄	—	4	2	no	—
78	C₄²×C₁₃	—	2	4	no	—
79	C₄²×C₁₃	—	4	2	no	—
80	C₂²⋊Dic₁₃	—	2	2	no	—
81	C₂²⋊Dic₁₃	—	2	2	no	—
82	C₄⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄×Dic₁₃	—	2	2	no	—
85	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
86	#6099	—	2	4	no	—
87	#6099	—	2	4	no	—
88	#6099	—	2	4	no	—
89	#6099	—	4	1	no	—
90	#6099	—	4	1	no	—
91	#6099	—	4	2	no	—
92	#6099	—	4	2	no	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	4	2	no	—
96	C₂×#1055	—	4	2	no	—
97	C₂×#1055	—	4	2	no	—
98	C₂×#430	—	13	1	no	—
99	#2905	—	13	1	no	—
100	#2102	—	13	8	no	—
101	C₂×#344	—	13	2	no	—
102	#1733	—	13	2	no	—
103	C₂×H(3,ℤ/4)	D₁₃	1	1	yes	characteristic
104	C₂³×C₁₃	—	8	1	no	—
105	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₁₃	#126	1	2	yes	—
108	C₂³×C₁₃	#126	1	2	yes	—
109	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	8	4	no	—
112	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#123	1	2	yes	—
115	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#124	1	2	yes	—
119	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#125	1	2	yes	—
121	C₂×C₄×C₁₃	#125	1	2	yes	—
122	C₂×C₄×C₁₃	#125	1	2	yes	—
123	C₂×C₄×C₁₃	#125	1	2	yes	—
124	C₂×C₄×C₁₃	#126	1	2	yes	—
125	C₂×C₄×C₁₃	#126	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
130	C₂×Dic₁₃	—	2	4	no	—
131	C₂×Dic₁₃	—	8	2	no	—
132	C₈×C₁₃	—	2	4	no	—
133	#1055	—	4	2	no	—
134	#1055	—	4	4	no	—
135	C₂²×M₁₆	—	13	1	no	—
136	C₂²×M₁₆	—	13	2	no	—
137	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
138	C₂×#117	—	13	1	no	—
139	C₂×#117	C₂×D₁₃	1	2	yes	—
140	#430	—	13	4	no	—
141	#430	—	26	2	no	—
142	C₂×#108	—	26	2	no	—
143	#424	—	26	2	no	—
144	#424	—	26	2	no	—
145	#423	—	13	4	no	—
146	#344	—	13	8	no	—
147	#339	—	13	8	no	—
148	H(3,ℤ/4)	C₂×D₁₃	1	4	yes	—
149	C₂²×C₁₃	—	4	1	no	—
150	C₂²×C₁₃	—	4	2	no	—
151	C₂²×C₁₃	—	4	1	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	#400	1	2	yes	—
155	C₂²×C₁₃	#400	1	1	yes	characteristic
156	C₂²×C₁₃	#400	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#126	1	2	yes	—
159	C₄×C₁₃	—	8	4	no	—
160	C₄×C₁₃	#400	1	2	yes	—
161	C₄×C₁₃	#400	1	1	yes	characteristic
162	C₄×C₁₃	#400	1	1	yes	characteristic
163	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
164	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
165	C₄×C₁₃	C₂×#125	1	2	yes	—
166	Dic₁₃	—	4	4	no	—
167	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	13	4	no	—
171	C₂×M₁₆	—	13	2	no	—
172	C₂×M₁₆	—	26	1	no	—
173	C₂×M₁₆	—	26	2	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	—	26	2	no	—
180	C₂×M₁₆	D₅₂	1	2	yes	—
181	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
182	C₂²×C₈	—	26	1	no	—
183	C₂²×C₈	—	26	2	no	—
184	C₂²×C₈	D₅₂	1	2	yes	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	13	2	no	—
189	#117	—	13	2	no	—
190	#117	—	26	1	no	—
191	#117	—	26	1	no	—
192	#117	C₂²×D₁₃	1	4	yes	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	26	1	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	26	1	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	13	2	no	—
199	#108	—	26	8	no	—
200	C₄⋊C₈	—	26	8	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	C₂×#400	1	1	yes	characteristic
207	C₂⁴	#6129	1	1	yes	characteristic
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	26	2	no	—
210	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
212	C₂²×C₄	#6127	1	2	yes	—
213	C₂²×C₄	#6127	1	2	yes	—
214	C₂²×C₄	#6129	1	1	yes	characteristic
215	C₂²×C₄	#6129	1	2	yes	—
216	C₂²×C₄	#6129	1	2	yes	—
217	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	26	4	no	—
223	M₁₆	—	26	4	no	—
224	M₁₆	—	26	4	no	—
225	M₁₆	—	52	2	no	—
226	M₁₆	—	52	2	no	—
227	M₁₆	—	52	1	no	—
228	M₁₆	—	52	1	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	26	2	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	52	2	no	—
234	C₂×C₈	—	52	2	no	—
235	C₂×C₈	—	52	2	no	—
236	C₂×C₈	C₂×D₅₂	1	4	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	26	2	no	—
240	C₄⋊C₄	—	26	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	26	2	no	—
244	C₂²⋊C₄	—	26	2	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	4	2	no	—
247	C₄²	—	13	4	no	—
248	C₄²	—	26	2	no	—
249	C₁₃	#23751	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	8	1	no	—
251	C₂³	#88133	1	2	yes	—
252	C₂³	#88133	1	2	yes	—
253	C₂³	#88151	1	1	yes	characteristic
254	C₂³	#88151	1	1	yes	characteristic
255	C₂³	C₂×#6129	1	1	yes	characteristic
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	26	4	no	—
259	C₂×C₄	—	104	2	no	—
260	C₂×C₄	#88129	1	1	yes	characteristic
261	C₂×C₄	#88129	1	2	yes	—
262	C₂×C₄	#88129	1	1	yes	characteristic
263	C₂×C₄	#88133	1	2	yes	—
264	C₂×C₄	#88133	1	2	yes	—
265	C₂×C₄	#88140	1	1	yes	characteristic
266	C₂×C₄	#88140	1	1	yes	characteristic
267	C₂×C₄	#88140	1	2	yes	—
268	C₂×C₄	#88143	1	2	yes	—
269	C₂×C₄	#88143	1	2	yes	—
270	C₂×C₄	#88143	1	2	yes	—
271	C₂×C₄	#88143	1	2	yes	—
272	C₂×C₄	#88151	1	1	yes	characteristic
273	C₂×C₄	#88151	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
275	C₂×C₄	C₂×#6127	1	2	yes	—
276	C₂×C₄	C₂×#6129	1	2	yes	—
277	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	52	4	no	—
280	C₈	—	52	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	#11718320	1	1	yes	characteristic
287	C₂²	#11718320	1	2	yes	—
288	C₂²	#11718320	1	1	yes	characteristic
289	C₂²	C₂×#88133	1	2	yes	—
290	C₂²	C₂×#88151	1	1	yes	characteristic
291	C₄	—	8	4	no	—
292	C₄	—	52	4	no	—
293	C₄	#11718320	1	1	yes	characteristic
294	C₄	#11718320	1	2	yes	—
295	C₄	#11718320	1	1	yes	characteristic
296	C₄	C₂×#88129	1	1	yes	characteristic
297	C₄	C₂×#88140	1	1	yes	characteristic
298	C₄	C₂×#88143	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	1	no	—
304	C₂	C₂×#11718320	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88129
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×H(3,ℤ/4)	C₂
socle	C₂×C₁₃	C₂×#400
prime radical	C₂⁴×C₁₃	#41
hypercenter	C₂×H(3,ℤ/4)	D₁₃
hypocenter	C₁₃	#23751
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
3. #13116412	C₂	semidirect
249. C₁₃	#23751	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718786	2
#13123399	2
#13116412	2
C₂×#11718125	2
#13114545	2
C₂×C₁₃×H(3,ℤ/4)	2
#23751	13