Subgroups — #15781172

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718158	C₂	1	4	yes	maximal, complemented
2	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115000	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#135	—	2	1	no	—
6	C₂×C₁₃×#135	C₂²	1	2	yes	—
7	C₁₃×#532	—	2	1	no	—
8	C₁₃×#495	—	2	1	no	—
9	C₁₃×#495	C₄	1	2	yes	complemented
10	C₁₃×#494	C₄	1	1	yes	characteristic, complemented
11	C₁₃×#493	C₄	1	1	yes	characteristic, complemented
12	C₁₃×#492	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88058	—	2	1	no	—
15	C₂×#88057	C₂²	1	2	yes	—
16	C₂×#88052	C₂²	1	2	yes	—
17	#11718597	—	2	1	no	—
18	#11718158	—	2	8	no	—
19	#11718053	—	2	1	no	—
20	C₄×#6090	—	2	1	no	—
21	C₂×C₁₃×#41	—	2	2	no	—
22	C₁₃×#135	—	2	4	no	—
23	C₁₃×#135	—	2	4	no	—
24	C₁₃×#135	—	4	2	no	—
25	C₁₃×#135	—	4	2	no	—
26	C₂×C₁₃×Q₁₆	—	2	1	no	—
27	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×SD₁₆	—	2	2	no	—
29	C₂×C₁₃×SD₁₆	—	2	2	no	—
30	C₂×C₁₃×D₈	—	2	1	no	—
31	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×#127	—	2	1	no	—
36	C₁₃×#124	—	2	2	no	—
37	C₁₃×#121	—	4	2	no	—
38	C₄×D₄×C₁₃	—	4	2	no	—
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	#88071	—	4	2	no	—
46	#88058	—	4	2	no	—
47	#88057	—	2	4	no	—
48	#88052	—	2	2	no	—
49	#88052	D₄	1	4	yes	complemented
50	C₂×#6090	—	2	2	no	—
51	#13211	—	13	1	no	maximal, Sylow
52	C₁₃×#41	—	4	2	no	—
53	C₁₃×#41	—	4	2	no	—
54	C₁₃×#41	—	4	2	no	—
55	C₁₃×#41	—	4	2	no	—
56	C₂×Q₈×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	D₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	4	2	no	—
64	C₁₃×Q₁₆	—	2	4	no	—
65	C₁₃×Q₁₆	—	4	2	no	—
66	C₁₃×SD₁₆	—	4	4	no	—
67	C₁₃×SD₁₆	—	4	4	no	—
68	C₁₃×D₈	—	2	4	no	—
69	C₁₃×D₈	—	4	2	no	—
70	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	D₈	1	2	yes	—
73	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	2	yes	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₂²⋊C₄	—	4	4	no	—
80	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₁₃	SD₁₆	1	2	yes	—
83	C₄⋊Dic₁₃	—	4	2	no	—
84	C₄⋊Dic₁₃	—	4	2	no	—
85	#6090	—	2	4	no	—
86	#3147	D₁₃	1	1	yes	characteristic
87	C₂×#360	—	13	4	no	—
88	C₄×M₃₂	—	13	1	no	—
89	#1846	—	13	1	no	—
90	C₂³×C₁₃	—	4	2	no	—
91	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₁₃	—	4	2	no	—
93	Q₈×C₁₃	—	4	2	no	—
94	D₄×C₁₃	—	4	2	no	—
95	D₄×C₁₃	—	4	2	no	—
96	D₄×C₁₃	—	8	4	no	—
97	D₄×C₁₃	—	8	2	no	—
98	D₄×C₁₃	—	8	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	8	2	no	—
102	C₂×C₄×C₁₃	—	8	2	no	—
103	C₂×C₄×C₁₃	#102	1	2	yes	—
104	C₂×C₄×C₁₃	#102	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×Dic₁₃	—	8	4	no	—
112	C₂×Dic₁₃	—	8	2	no	—
113	C₈×C₁₃	#102	1	2	yes	—
114	C₈×C₁₃	#102	1	2	yes	—
115	C₈×C₁₃	C₄×D₄	1	2	yes	—
116	C₈×C₁₃	#121	1	2	yes	—
117	C₂×#135	—	2	1	no	—
118	C₂×#135	C₂×D₁₃	1	2	yes	—
119	#532	—	2	1	no	—
120	C₂×M₃₂	—	13	2	no	—
121	#495	—	2	1	no	—
122	#495	Dic₁₃	1	2	yes	complemented
123	#494	Dic₁₃	1	1	yes	characteristic, complemented
124	#493	Dic₁₃	1	1	yes	characteristic, complemented
125	#492	—	2	1	no	—
126	C₂×#107	—	26	1	no	—
127	C₂×#106	—	13	2	no	—
128	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
129	#382	—	26	1	no	—
130	#360	—	26	8	no	—
131	#346	—	26	1	no	—
132	C₄×C₁₆	—	26	1	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	8	4	no	—
135	C₂²×C₁₃	—	8	2	no	—
136	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
137	C₂²×C₁₃	#486	1	1	yes	characteristic
138	C₂²×C₁₃	#493	1	1	yes	characteristic
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	8	2	no	—
142	C₄×C₁₃	#390	1	1	yes	characteristic
143	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
144	C₄×C₁₃	#486	1	1	yes	characteristic
145	C₄×C₁₃	#492	1	1	yes	characteristic
146	Dic₁₃	—	16	2	no	—
147	C₂×#41	—	2	2	no	—
148	#135	—	2	4	no	—
149	#135	—	2	4	no	—
150	#135	—	4	2	no	—
151	#135	—	4	2	no	—
152	C₂×Q₁₆	—	2	1	no	—
153	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
154	C₂×SD₁₆	—	2	2	no	—
155	C₂×SD₁₆	—	2	2	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
158	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	1	no	—
161	#127	—	2	1	no	—
162	#124	—	2	2	no	—
163	#121	—	4	2	no	—
164	C₄×D₄	—	4	2	no	—
165	C₂×C₄⋊C₄	—	26	2	no	—
166	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
167	M₃₂	—	13	4	no	—
168	M₃₂	—	26	2	no	—
169	C₂×C₁₆	—	26	2	no	—
170	#107	—	52	2	no	—
171	#106	—	26	4	no	—
172	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
173	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
176	#95	—	52	2	no	—
177	C₂×C₁₃	—	2	1	no	—
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	16	2	no	—
180	C₂×C₁₃	#1957	1	1	yes	characteristic
181	C₂×C₁₃	#2145	1	2	yes	—
182	#41	—	4	2	no	—
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	52	2	no	—
192	C₂²×C₄	#6104	1	2	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
194	Q₁₆	—	2	4	no	—
195	Q₁₆	—	4	2	no	—
196	SD₁₆	—	4	4	no	—
197	SD₁₆	—	4	4	no	—
198	D₈	—	2	4	no	—
199	D₈	—	4	2	no	—
200	C₂×C₈	#6104	1	2	yes	—
201	C₂×C₈	#6105	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
204	C₂×C₈	D₄×D₁₃	1	2	yes	—
205	C₂×C₈	#6129	1	2	yes	—
206	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	52	2	no	—
210	C₄⋊C₄	—	52	2	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	#6105	1	2	yes	—
213	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
214	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
215	C₁₆	—	26	4	no	—
216	C₁₃	#13211	1	1	yes	characteristic, complemented, Sylow
217	C₂³	—	4	2	no	—
218	C₂³	C₂×#6104	1	1	yes	characteristic
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	104	4	no	—
231	C₂×C₄	—	104	2	no	—
232	C₂×C₄	#88072	1	2	yes	—
233	C₂×C₄	#88072	1	2	yes	—
234	C₂×C₄	C₂×#6104	1	1	yes	characteristic
235	C₂×C₄	C₂×#6105	1	1	yes	characteristic
236	C₂×C₄	C₂×#6105	1	1	yes	characteristic
237	C₂×C₄	#88190	1	1	yes	characteristic
238	C₂×C₄	#88194	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
240	C₈	#88072	1	2	yes	—
241	C₈	#88072	1	2	yes	—
242	C₈	D₄×Dic₁₃	1	2	yes	—
243	C₈	#88192	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	8	2	no	—
247	C₂²	#11718731	1	1	yes	characteristic
248	C₂²	#11718734	1	1	yes	characteristic
249	C₂²	C₂×#88072	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	208	2	no	—
254	C₄	#11718731	1	1	yes	characteristic
255	C₄	#11718737	1	1	yes	characteristic
256	C₄	C₂×#88072	1	1	yes	characteristic
257	C₄	#11718882	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	16	2	no	—
261	C₂	#13115643	1	1	yes	characteristic
262	C₂	#13116572	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting, prime radical	C₁₃×#3147	C₂
socle	C₂²×C₁₃	C₂×#102
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#3147	D₁₃
hypocenter	C₁₃	#13211
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#11718158	C₂	semidirect
2. C₄×#88052	C₂	semidirect
3. #13115000	C₂	semidirect
9. C₁₃×#495	C₄	semidirect
10. C₁₃×#494	C₄	semidirect
11. C₁₃×#493	C₄	semidirect
49. #88052	D₄	semidirect
122. #495	Dic₁₃	semidirect
123. #494	Dic₁₃	semidirect
124. #493	Dic₁₃	semidirect
216. C₁₃	#13211	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718158	2
C₄×#88052	2
#13115000	2
C₁₃×#3147	2
#13211	13