Subgroups — #15778134

Order 3328.

526 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718101	C₂	1	2	yes	maximal
3	C₄×#88058	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112928	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅₂	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88061	—	2	1	no	—
11	C₂×#88058	C₂²	1	1	yes	characteristic
12	C₂×#88058	C₂²	1	2	yes	—
13	C₄×C₄⋊Dic₁₃	—	2	1	no	—
14	C₄×D₁₀₄	—	2	2	no	—
15	C₄×D₁₀₄	C₄	1	4	yes	complemented
16	C₁₃×#346	—	2	2	no	—
17	#11718143	—	2	1	no	—
18	#11718101	C₄	1	8	yes	complemented
19	C₂²×D₅₂	—	2	1	no	—
20	C₂²×C₄×D₁₃	—	4	1	no	—
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×#6103	—	4	1	no	—
25	C₂×C₄⋊Dic₁₃	—	2	1	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×C₄×Dic₁₃	—	4	1	no	—
28	C₂×D₁₀₄	—	2	1	no	—
29	C₂×D₁₀₄	—	2	2	no	—
30	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
31	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
32	C₄×D₅₂	—	2	4	no	—
33	C₄×D₅₂	—	4	2	no	—
34	C₂×C₁₃×C₁₆	—	2	2	no	—
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
37	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃	D₄	1	2	yes	—
39	#88071	—	4	1	no	—
40	#88061	—	2	4	no	—
41	#88058	—	2	1	no	—
42	#88058	—	2	2	no	—
43	#88058	C₂×C₄	1	2	yes	—
44	#88058	C₂×C₄	1	4	yes	complemented
45	#12637	—	13	1	no	maximal, Sylow
46	C₂³×D₁₃	—	4	1	no	—
47	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	4	2	no	—
51	C₂²×Dic₁₃	—	4	1	no	—
52	C₂×D₅₂	—	2	2	no	—
53	C₂×D₅₂	—	2	4	no	—
54	C₂×D₅₂	—	4	1	no	—
55	C₂×D₅₂	—	4	2	no	—
56	C₂×C₄×D₁₃	—	4	4	no	—
57	C₂×C₄×D₁₃	—	8	4	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	#41	1	2	yes	—
64	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₁₃	SD₁₆	1	2	yes	—
66	#6103	—	4	4	no	—
67	C₄⋊Dic₁₃	—	2	2	no	—
68	C₄⋊Dic₁₃	—	2	4	no	—
69	C₄⋊Dic₁₃	—	4	1	no	—
70	C₄⋊Dic₁₃	—	4	2	no	—
71	C₄×Dic₁₃	—	8	2	no	—
72	D₁₀₄	—	2	2	no	—
73	D₁₀₄	—	2	4	no	—
74	D₁₀₄	C₄²	1	4	yes	complemented
75	C₁₃×C₁₆	—	2	4	no	—
76	C₂×C₄×D₈	—	13	1	no	—
77	C₂×#357	—	13	2	no	—
78	C₂×#346	D₁₃	1	1	yes	characteristic, complemented
79	C₄×#107	—	13	1	no	—
80	#1380	—	13	2	no	—
81	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
82	C₂²×D₁₃	—	4	2	no	—
83	C₂²×D₁₃	—	4	4	no	—
84	C₂²×D₁₃	—	8	2	no	—
85	C₂²×D₁₃	—	8	1	no	—
86	C₂²×D₁₃	—	8	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	#102	1	2	yes	—
91	C₂×C₄×C₁₃	#102	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#135	1	2	yes	—
96	C₂×Dic₁₃	—	4	2	no	—
97	C₂×Dic₁₃	—	4	4	no	—
98	C₂×Dic₁₃	—	8	2	no	—
99	C₂×Dic₁₃	—	8	2	no	—
100	C₂×Dic₁₃	—	8	2	no	—
101	D₅₂	—	2	4	no	—
102	D₅₂	—	4	2	no	—
103	D₅₂	—	4	4	no	—
104	C₄×D₁₃	—	8	8	no	—
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
107	C₈×C₁₃	#117	1	1	yes	characteristic
108	C₈×C₁₃	C₄×D₄	1	2	yes	—
109	C₂²×D₈	—	13	1	no	—
110	C₂×C₄×D₄	—	26	1	no	—
111	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
112	C₄×D₈	—	13	4	no	—
113	C₄×D₈	—	26	2	no	—
114	C₂×#107	—	13	2	no	—
115	C₂×#107	—	13	1	no	—
116	C₂×#102	—	26	1	no	—
117	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
118	C₄×C₄⋊C₄	—	26	1	no	—
119	#357	—	13	8	no	—
120	#346	—	2	2	no	—
121	#340	—	26	1	no	—
122	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
123	C₂²×C₁₃	C₄×D₈	1	2	yes	—
124	C₂²×C₁₃	#509	1	1	yes	characteristic
125	C₂²×C₁₃	#509	1	2	yes	—
126	C₂²×C₁₃	#510	1	1	yes	characteristic
127	C₂×D₁₃	—	4	4	no	—
128	C₂×D₁₃	—	8	2	no	—
129	C₂×D₁₃	—	8	2	no	—
130	C₂×D₁₃	—	8	4	no	—
131	C₂×D₁₃	—	8	4	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₁₃	#416	1	1	yes	characteristic
136	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
137	Dic₁₃	—	8	2	no	—
138	Dic₁₃	—	8	4	no	—
139	C₂²×D₄	—	26	1	no	—
140	C₂³×C₄	—	52	1	no	—
141	C₂×D₈	—	13	4	no	—
142	C₂×D₈	—	13	2	no	—
143	C₂×D₈	—	26	2	no	—
144	C₂×D₈	—	26	1	no	—
145	C₂²×C₈	D₅₂	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
147	C₄×D₄	—	26	4	no	—
148	C₄×D₄	—	52	2	no	—
149	C₂×C₄⋊C₄	—	26	1	no	—
150	C₂×C₄⋊C₄	—	26	2	no	—
151	C₂×C₂²⋊C₄	—	52	1	no	—
152	C₂×C₄²	—	52	1	no	—
153	C₂×C₄²	D₅₂	1	1	yes	characteristic
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
157	#107	—	13	4	no	—
158	#107	—	13	2	no	—
159	#107	—	26	2	no	—
160	#107	—	26	1	no	—
161	#102	—	26	4	no	—
162	C₄×C₈	C₄×D₁₃	1	2	yes	—
163	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
164	#95	—	52	1	no	—
165	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
166	C₂×C₁₃	#2139	1	1	yes	characteristic
167	C₂×C₁₃	#2140	1	1	yes	characteristic
168	C₂×C₁₃	#2175	1	2	yes	—
169	C₂×C₁₃	#2177	1	2	yes	—
170	D₁₃	—	8	4	no	—
171	C₂⁴	—	52	1	no	—
172	C₂×D₄	—	26	4	no	—
173	C₂×D₄	—	26	2	no	—
174	C₂×D₄	—	52	1	no	—
175	C₂×D₄	—	52	2	no	—
176	C₂²×C₄	—	52	2	no	—
177	C₂²×C₄	—	52	1	no	—
178	C₂²×C₄	—	52	4	no	—
179	C₂²×C₄	—	104	4	no	—
180	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
181	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
183	D₈	—	13	4	no	—
184	D₈	—	26	2	no	—
185	D₈	—	26	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	#6103	1	2	yes	—
188	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
189	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
190	C₂×C₈	#6127	1	2	yes	—
191	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
192	C₄⋊C₄	—	26	4	no	—
193	C₄⋊C₄	—	26	2	no	—
194	C₄⋊C₄	—	52	1	no	—
195	C₄⋊C₄	—	52	2	no	—
196	C₂²⋊C₄	—	52	4	no	—
197	C₄²	—	104	2	no	—
198	C₄²	#6095	1	2	yes	—
199	C₄²	#6103	1	2	yes	—
200	C₁₆	—	2	4	no	—
201	C₁₃	#12637	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	52	2	no	—
203	C₂³	—	52	4	no	—
204	C₂³	—	104	2	no	—
205	C₂³	—	104	1	no	—
206	C₂³	—	104	1	no	—
207	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
208	D₄	—	26	4	no	—
209	D₄	—	52	2	no	—
210	D₄	—	52	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	52	2	no	—
215	C₂×C₄	—	52	4	no	—
216	C₂×C₄	—	104	8	no	—
217	C₂×C₄	—	104	2	no	—
218	C₂×C₄	—	104	2	no	—
219	C₂×C₄	—	104	2	no	—
220	C₂×C₄	#88061	1	2	yes	—
221	C₂×C₄	#88061	1	2	yes	—
222	C₂×C₄	C₄×D₅₂	1	2	yes	—
223	C₂×C₄	C₂×#6095	1	1	yes	characteristic
224	C₂×C₄	#88158	1	2	yes	—
225	C₂×C₄	C₂×#6103	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	C₄²×D₁₃	1	1	yes	characteristic
228	C₈	#88126	1	1	yes	characteristic
229	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
230	C₂²	—	52	4	no	—
231	C₂²	—	104	4	no	—
232	C₂²	—	104	4	no	—
233	C₂²	—	104	2	no	—
234	C₂²	—	104	2	no	—
235	C₂²	C₄×D₁₀₄	1	2	yes	—
236	C₂²	#11718569	1	2	yes	—
237	C₂²	#11718569	1	1	yes	characteristic
238	C₂²	#11718570	1	1	yes	characteristic
239	C₂²	C₂×#88061	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	104	4	no	—
243	C₄	—	104	2	no	—
244	C₄	C₄×#6095	1	2	yes	—
245	C₄	C₄×#6103	1	1	yes	characteristic
246	C₄	#11718774	1	1	yes	characteristic
247	C₂	—	104	4	no	—
248	C₂	C₄×#88061	1	1	yes	characteristic
249	C₂	#13116549	1	1	yes	characteristic
250	C₂	#13116553	1	1	yes	characteristic
251	C₂	#13116702	1	2	yes	—
252	C₂	#13116713	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#12637
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. C₄×#88058	C₂	semidirect
4. #13112928	C₂	semidirect
5. C₂×C₁₃×#346	C₂	semidirect
15. C₄×D₁₀₄	C₄	semidirect
18. #11718101	C₄	semidirect
36. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
44. #88058	C₂×C₄	semidirect
74. D₁₀₄	C₄²	semidirect
78. C₂×#346	D₁₃	semidirect
156. C₂×C₁₆	C₄×D₁₃	semidirect
201. C₁₃	#12637	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₀₄	2
C₂×#11718101	2
C₄×#88058	2
#13112928	2
C₂×C₁₃×#346	2
#12637	13