Subgroups — #16018298

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718344	C₂	1	1	yes	maximal, characteristic, complemented
2	#13125264	C₂	1	1	yes	maximal, characteristic
3	#13117082	C₂	1	2	yes	maximal, complemented
4	#13117078	C₂	1	2	yes	maximal, complemented
5	#13117068	C₂	1	4	yes	maximal, complemented
6	#13116549	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88158	—	2	1	no	—
9	C₂×#88158	C₂²	1	2	yes	—
10	#11719182	—	2	1	no	—
11	C₂×#88124	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88058	C₂²	1	2	yes	—
15	C₂×#88057	—	2	1	no	—
16	#11718360	C₂²	1	1	yes	characteristic
17	#11718357	—	2	1	no	—
18	#11718357	C₂²	1	2	yes	—
19	#11718356	C₂²	1	1	yes	characteristic
20	#11718354	—	2	1	no	—
21	#11718344	—	2	1	no	—
22	#11718344	—	2	1	no	—
23	#11718344	C₂²	1	2	yes	—
24	#11718344	C₂²	1	2	yes	—
25	C₁₃×#346	C₂²	1	2	yes	—
26	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
27	#11718101	C₂²	1	8	yes	—
28	#11718095	C₂²	1	8	yes	—
29	C₂×#6127	—	2	2	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	#88184	—	4	2	no	—
34	C₂×C₄⋊Dic₁₃	—	2	2	no	—
35	C₂×#6101	—	4	2	no	—
36	C₂×Dic₅₂	—	2	1	no	—
37	C₂×Dic₅₂	C₂³	1	2	yes	—
38	#88158	—	2	8	no	—
39	#88158	—	4	4	no	—
40	C₂×D₁₀₄	—	2	1	no	—
41	C₂×D₁₀₄	C₂³	1	2	yes	—
42	C₂×#6095	—	2	2	no	—
43	C₂×#6095	—	2	2	no	—
44	#88129	—	2	2	no	—
45	#88128	—	2	1	no	—
46	C₄×D₅₂	—	4	2	no	—
47	#88124	—	2	2	no	—
48	#88124	—	2	2	no	—
49	#88124	—	4	2	no	—
50	#88123	—	2	1	no	—
51	C₁₃×M₃₂	—	2	4	no	—
52	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	#88058	—	2	2	no	—
58	#88058	C₂³	1	4	yes	—
59	#88057	—	2	2	no	—
60	#88057	—	2	2	no	—
61	#33708	—	13	1	no	maximal, Sylow
62	C₂²×C₄×C₁₃	D₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃	—	4	2	no	—
65	C₂²×Dic₁₃	—	4	2	no	—
66	#6127	—	4	4	no	—
67	#6127	—	4	4	no	—
68	C₂×D₅₂	—	2	2	no	—
69	C₂×D₅₂	—	4	2	no	—
70	C₂×C₄×D₁₃	—	4	2	no	—
71	C₂×Dic₂₆	—	2	2	no	—
72	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	#41	1	4	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₄²×C₁₃	D₈	1	2	yes	—
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	#6103	—	4	4	no	—
82	C₄⋊Dic₁₃	—	2	4	no	—
83	C₄⋊Dic₁₃	—	4	2	no	—
84	C₄⋊Dic₁₃	—	4	2	no	—
85	#6101	—	4	4	no	—
86	#6101	—	4	4	no	—
87	Dic₅₂	—	2	4	no	—
88	Dic₅₂	—	4	2	no	—
89	D₁₀₄	—	2	4	no	—
90	D₁₀₄	—	4	2	no	—
91	#6095	—	4	4	no	—
92	#6095	—	4	4	no	—
93	C₁₃×C₁₆	—	2	4	no	—
94	C₂×#499	—	13	1	no	—
95	#3147	—	13	1	no	—
96	#2245	—	13	2	no	—
97	#2244	—	13	2	no	—
98	#2242	—	13	4	no	—
99	#2139	—	13	4	no	—
100	C₄×M₃₂	D₁₃	1	1	yes	characteristic, complemented
101	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	#124	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₂²⋊D₁₃	—	8	4	no	—
111	C₂×Dic₁₃	—	4	4	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₂×Dic₁₃	—	8	4	no	—
114	D₅₂	—	4	2	no	—
115	D₅₂	—	4	2	no	—
116	D₅₂	—	8	4	no	—
117	C₄×D₁₃	—	8	4	no	—
118	Dic₂₆	—	4	2	no	—
119	Dic₂₆	—	4	2	no	—
120	C₈×C₁₃	#124	1	2	yes	—
121	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
122	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
123	C₈×C₁₃	C₂×#41	1	2	yes	—
124	C₂×#135	—	13	2	no	—
125	C₂×#135	—	26	1	no	—
126	#532	—	26	1	no	—
127	C₂×#125	—	26	1	no	—
128	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
129	#499	—	13	2	no	—
130	#499	—	13	2	no	—
131	#499	—	26	1	no	—
132	#499	—	26	1	no	—
133	#495	—	13	2	no	—
134	#495	—	26	1	no	—
135	#494	—	13	1	no	—
136	#493	—	13	1	no	—
137	#492	—	26	1	no	—
138	C₂×#107	—	13	2	no	—
139	C₂×#106	—	26	1	no	—
140	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
141	#358	—	13	8	no	—
142	#357	—	13	8	no	—
143	#346	C₂×D₁₃	1	2	yes	—
144	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	#486	1	2	yes	—
147	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₁₃	—	8	2	no	—
149	C₂×D₁₃	—	8	4	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	#486	1	2	yes	—
153	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
154	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
155	Dic₁₃	—	8	2	no	—
156	Dic₁₃	—	8	4	no	—
157	C₂×#41	—	26	2	no	—
158	#135	—	26	8	no	—
159	#135	—	52	4	no	—
160	C₂×Q₁₆	—	13	2	no	—
161	C₂×Q₁₆	—	26	1	no	—
162	C₂×SD₁₆	—	26	2	no	—
163	C₂×SD₁₆	—	26	2	no	—
164	C₂×D₈	—	13	2	no	—
165	C₂×D₈	—	26	1	no	—
166	C₂²×C₈	D₅₂	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	26	1	no	—
169	#127	—	26	1	no	—
170	#125	—	26	2	no	—
171	#125	—	26	2	no	—
172	#125	—	52	2	no	—
173	#124	—	26	2	no	—
174	#121	—	52	2	no	—
175	C₄×D₄	—	52	2	no	—
176	C₂×C₄⋊C₄	—	26	2	no	—
177	C₂×C₄⋊C₄	—	52	2	no	—
178	C₂×C₄²	D₅₂	1	1	yes	characteristic
179	M₃₂	—	2	4	no	—
180	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
181	#107	—	13	4	no	—
182	#107	—	26	2	no	—
183	#106	—	26	2	no	—
184	#106	—	26	2	no	—
185	C₄×C₈	D₅₂	1	1	yes	characteristic
186	C₄×C₈	D₅₂	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
191	C₂×C₁₃	#3414	1	2	yes	—
192	D₁₃	—	16	2	no	—
193	#41	—	52	4	no	—
194	#41	—	52	4	no	—
195	C₂×Q₈	—	26	2	no	—
196	C₂×D₄	—	26	2	no	—
197	C₂×D₄	—	52	2	no	—
198	C₂×D₄	—	52	2	no	—
199	C₂²×C₄	—	52	2	no	—
200	C₂²×C₄	—	52	2	no	—
201	C₂²×C₄	D₁₀₄	1	2	yes	—
202	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
203	Q₁₆	—	26	4	no	—
204	Q₁₆	—	52	2	no	—
205	SD₁₆	—	52	4	no	—
206	SD₁₆	—	52	4	no	—
207	D₈	—	26	4	no	—
208	D₈	—	52	2	no	—
209	C₂×C₈	#6095	1	4	yes	—
210	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
211	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
212	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
213	C₂×C₈	#6127	1	4	yes	—
214	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
215	C₄⋊C₄	—	26	4	no	—
216	C₄⋊C₄	—	52	2	no	—
217	C₄⋊C₄	—	52	4	no	—
218	C₄⋊C₄	—	52	4	no	—
219	C₄⋊C₄	—	52	2	no	—
220	C₂²⋊C₄	—	52	4	no	—
221	C₄²	D₁₀₄	1	2	yes	—
222	C₄²	C₂×D₅₂	1	1	yes	characteristic
223	C₄²	C₂×D₅₂	1	1	yes	characteristic
224	C₁₆	—	2	4	no	—
225	C₁₃	#33708	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	52	2	no	—
227	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
228	Q₈	—	52	2	no	—
229	Q₈	—	52	2	no	—
230	D₄	—	52	2	no	—
231	D₄	—	52	2	no	—
232	D₄	—	104	4	no	—
233	D₄	—	104	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	52	2	no	—
236	C₂×C₄	—	52	4	no	—
237	C₂×C₄	—	104	4	no	—
238	C₂×C₄	—	104	4	no	—
239	C₂×C₄	#88129	1	2	yes	—
240	C₂×C₄	C₂×#6095	1	2	yes	—
241	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
242	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
243	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
245	C₈	#88129	1	2	yes	—
246	C₈	C₂×#6095	1	2	yes	—
247	C₈	C₂×#6095	1	2	yes	—
248	C₈	C₂×#6127	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	104	2	no	—
251	C₂²	—	104	4	no	—
252	C₂²	#11718355	1	2	yes	—
253	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	104	2	no	—
257	C₄	—	104	4	no	—
258	C₄	#11718355	1	2	yes	—
259	C₄	C₂×#88129	1	1	yes	characteristic
260	C₄	C₂²×#6095	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	208	2	no	—
263	C₂	C₂×#11718355	1	1	yes	characteristic
264	C₂	#13127972	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13125264	C₂
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#33708
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#11718344	C₂	semidirect
3. #13117082	C₂	semidirect
4. #13117078	C₂	semidirect
5. #13117068	C₂	semidirect
6. #13116549	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
100. C₄×M₃₂	D₁₃	semidirect
225. C₁₃	#33708	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718344	2
#13125264	2
#13117082	2
#13117078	2
#13117068	2
#13116549	2
C₄×C₁₃×M₃₂	2
#33708	13