Subgroups — #16807177

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719096	C₂	1	2	yes	maximal, complemented
2	#13131228	C₂	1	4	yes	maximal, complemented
3	#13131225	C₂	1	4	yes	maximal, complemented
4	#13131177	C₂	1	1	yes	maximal, characteristic, complemented
5	#13131080	C₂	1	2	yes	maximal, complemented
6	#13131047	C₂	1	1	yes	maximal, characteristic, complemented
7	#13123738	C₂	1	1	yes	maximal, characteristic
8	#11719453	C₄	1	1	yes	characteristic
9	C₂×#88161	C₂²	1	2	yes	—
10	C₂×#88106	C₂²	1	2	yes	—
11	C₂×#88106	C₂²	1	1	yes	characteristic
12	C₂²×#6092	C₂²	1	1	yes	characteristic
13	#11719096	—	2	2	no	—
14	#11719096	—	2	2	no	—
15	#11719096	C₂²	1	4	yes	complemented
16	#11719096	C₂²	1	4	yes	complemented
17	C₂×#88098	—	2	2	no	—
18	C₂×D₁₃⋊C₁₆	—	2	2	no	—
19	C₁₃×#430	C₄	1	1	yes	characteristic
20	#11718800	C₄	1	2	yes	—
21	#11718783	C₂²	1	1	yes	characteristic
22	#11718556	C₂²	1	4	yes	—
23	#11718555	C₄	1	4	yes	—
24	#11718270	C₂²	1	8	yes	complemented
25	#11718261	C₂²	1	4	yes	complemented
26	#11718258	C₂²	1	2	yes	complemented
27	#11718257	C₂²	1	2	yes	complemented
28	C₂×#6127	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
31	#88180	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂³	1	1	yes	characteristic
33	#88163	C₈	1	4	yes	—
34	#88162	C₈	1	4	yes	—
35	#88161	—	2	2	no	—
36	#88161	—	2	2	no	—
37	#88161	C₂×C₄	1	4	yes	—
38	#88161	C₂³	1	4	yes	—
39	#88158	C₂×C₄	1	4	yes	—
40	C₂×#6094	—	2	2	no	—
41	C₂×C₈×D₁₃	—	2	2	no	—
42	#88106	—	2	2	no	—
43	#88106	—	2	2	no	—
44	#88106	C₂³	1	4	yes	—
45	C₂×#6092	—	2	2	no	—
46	C₂×#6092	D₄	1	2	yes	—
47	C₂×#6092	D₄	1	2	yes	—
48	C₂×#6092	C₂³	1	2	yes	—
49	C₂×#6092	C₂³	1	1	yes	characteristic
50	C₂×#6092	C₂³	1	1	yes	characteristic
51	#88098	—	2	4	no	—
52	#88098	—	2	4	no	—
53	D₁₃⋊C₁₆	—	2	4	no	—
54	D₁₃⋊C₁₆	—	2	4	no	—
55	C₁₃×#108	C₂×C₄	1	4	yes	—
56	#88065	C₂×C₄	1	8	yes	—
57	#88055	C₂×C₄	1	2	yes	—
58	C₈×Dic₁₃	C₂³	1	2	yes	—
59	#33441	—	13	1	no	maximal, Sylow
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	—	2	2	no	—
62	#6127	—	2	2	no	—
63	#6127	—	2	2	no	—
64	#6127	C₂×C₈	1	4	yes	—
65	#6127	C₂²×C₄	1	4	yes	—
66	C₂×D₅₂	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₁₃	—	2	2	no	—
68	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
69	C₁₃×M₁₆	—	2	2	no	—
70	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
71	C₁₃×M₁₆	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂²⋊Dic₁₃	—	2	1	no	—
75	C₄⋊Dic₁₃	—	2	1	no	—
76	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
77	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
78	C₂×#1055	C₂×D₄	1	2	yes	—
79	C₂×#1055	#41	1	2	yes	—
80	C₂×#1055	C₂⁴	1	1	yes	characteristic
81	Dic₅₂	C₂×C₈	1	4	yes	—
82	D₁₀₄	C₂×C₈	1	4	yes	—
83	#6095	C₂×C₈	1	8	yes	—
84	#6094	—	2	4	no	—
85	#6094	—	2	4	no	—
86	C₈×D₁₃	—	2	4	no	—
87	C₈×D₁₃	—	2	4	no	—
88	#6092	—	2	4	no	—
89	#6092	—	2	2	no	—
90	#6092	C₂×D₄	1	1	yes	characteristic
91	#6092	C₂×D₄	1	2	yes	—
92	#6092	C₂×D₄	1	1	yes	characteristic
93	C₂×#504	—	13	2	no	—
94	#2955	—	13	1	no	—
95	#2171	—	13	4	no	—
96	#2170	—	13	4	no	—
97	#2153	—	13	1	no	—
98	#2117	—	13	2	no	—
99	#2108	—	13	1	no	—
100	C₂³×C₁₃	#131	1	1	yes	characteristic
101	C₂²×D₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂²⋊D₁₃	—	2	4	no	—
107	C₂²⋊D₁₃	—	2	4	no	—
108	C₂×Dic₁₃	—	2	2	no	—
109	C₂×Dic₁₃	—	2	2	no	—
110	D₅₂	—	2	2	no	—
111	D₅₂	C₂²×C₈	1	4	yes	—
112	C₄×D₁₃	—	2	4	no	—
113	C₄×D₁₃	—	2	4	no	—
114	Dic₂₆	—	2	2	no	—
115	Dic₂₆	C₂²×C₈	1	4	yes	—
116	C₈×C₁₃	—	2	4	no	—
117	C₈×C₁₃	C₂²×C₈	1	4	yes	—
118	#1055	C₄×D₄	1	2	yes	—
119	#1055	C₂²×D₄	1	1	yes	characteristic
120	#1055	C₂×#41	1	1	yes	characteristic
121	#575	—	13	1	no	—
122	C₂×#131	—	13	2	no	—
123	#504	—	13	4	no	—
124	#504	—	13	4	no	—
125	#504	—	26	2	no	—
126	#504	—	26	2	no	—
127	C₂×M₃₂	—	13	1	no	—
128	C₂×M₃₂	—	13	2	no	—
129	C₂×M₃₂	—	26	2	no	—
130	C₂²×C₁₆	—	13	1	no	—
131	C₂²×C₁₆	—	26	2	no	—
132	#444	—	13	4	no	—
133	#443	—	13	4	no	—
134	#430	#261	1	1	yes	characteristic
135	#421	—	13	2	no	—
136	#405	—	13	1	no	—
137	#364	—	13	4	no	—
138	#350	—	13	8	no	—
139	#346	—	13	2	no	—
140	C₄×C₁₆	—	13	2	no	—
141	C₂²×C₁₃	—	4	1	no	—
142	C₂²×C₁₃	C₈×D₄	1	2	yes	—
143	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
144	C₂×D₁₃	—	2	4	no	—
145	C₂×D₁₃	—	4	4	no	—
146	C₄×C₁₃	C₈×D₄	1	2	yes	—
147	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
149	Dic₁₃	—	2	4	no	—
150	Dic₁₃	—	4	2	no	—
151	C₂×#41	—	13	2	no	—
152	#137	—	13	4	no	—
153	AGL(1,ℤ/8)	—	13	4	no	—
154	#135	—	13	4	no	—
155	#131	—	13	4	no	—
156	#131	—	13	4	no	—
157	#131	—	26	2	no	—
158	#131	—	26	2	no	—
159	C₂×M₁₆	—	26	2	no	—
160	C₂×M₁₆	C₂×#261	1	2	yes	—
161	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	26	2	no	—
164	#117	—	13	1	no	—
165	M₃₂	—	13	4	no	—
166	M₃₂	—	26	4	no	—
167	M₃₂	—	26	4	no	—
168	M₃₂	—	26	2	no	—
169	M₃₂	—	26	2	no	—
170	C₂×C₁₆	—	13	2	no	—
171	C₂×C₁₆	—	13	2	no	—
172	C₂×C₁₆	—	13	1	no	—
173	C₂×C₁₆	—	13	1	no	—
174	C₂×C₁₆	—	13	2	no	—
175	C₂×C₁₆	—	26	4	no	—
176	C₂×C₁₆	—	26	4	no	—
177	C₂×C₁₆	—	26	2	no	—
178	#108	C₂×#261	1	4	yes	—
179	#104	—	13	8	no	—
180	#97	—	13	2	no	—
181	C₄×C₈	—	13	2	no	—
182	C₂×C₁₃	—	2	1	no	—
183	C₂×C₁₃	—	2	2	no	—
184	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
185	D₁₃	—	4	4	no	—
186	#41	—	13	4	no	—
187	#41	—	13	4	no	—
188	#41	—	26	2	no	—
189	#41	—	26	2	no	—
190	C₂×Q₈	—	13	2	no	—
191	C₂×D₄	—	13	2	no	—
192	C₂×D₄	—	26	2	no	—
193	C₂²×C₄	—	26	2	no	—
194	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
195	Q₁₆	—	13	4	no	—
196	SD₁₆	—	13	8	no	—
197	D₈	—	13	4	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	26	4	no	—
200	M₁₆	—	26	4	no	—
201	M₁₆	D₁₃⋊C₈	1	4	yes	—
202	M₁₆	C₂²×#261	1	4	yes	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	1	no	—
207	C₂×C₈	—	13	1	no	—
208	C₂×C₈	—	26	4	no	—
209	C₂×C₈	—	26	4	no	—
210	C₂×C₈	C₂²×#261	1	2	yes	—
211	C₄⋊C₄	—	26	1	no	—
212	C₂²⋊C₄	—	26	1	no	—
213	C₄²	—	13	2	no	—
214	C₁₆	—	13	1	no	—
215	C₁₆	—	13	2	no	—
216	C₁₆	—	13	1	no	—
217	C₁₆	—	26	2	no	—
218	C₁₆	—	26	4	no	—
219	C₁₃	#33441	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	26	2	no	—
221	C₂³	#88240	1	1	yes	characteristic
222	Q₈	—	13	4	no	—
223	Q₈	—	26	2	no	—
224	D₄	—	13	4	no	—
225	D₄	—	26	4	no	—
226	D₄	—	26	4	no	—
227	D₄	—	26	2	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	26	4	no	—
230	C₂×C₄	—	26	4	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
233	C₂×C₄	D₄×#261	1	2	yes	—
234	C₂×C₄	#88240	1	1	yes	characteristic
235	C₂×C₄	C₂³×#261	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	—	13	1	no	—
238	C₈	—	13	1	no	—
239	C₈	—	13	2	no	—
240	C₈	C₂×D₁₃⋊C₈	1	4	yes	—
241	C₂²	—	4	1	no	—
242	C₂²	—	26	4	no	—
243	C₂²	—	52	4	no	—
244	C₂²	#11719073	1	2	yes	—
245	C₂²	C₂×#88240	1	1	yes	characteristic
246	C₄	—	26	4	no	—
247	C₄	—	52	2	no	—
248	C₄	#11719073	1	2	yes	—
249	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
250	C₄	C₂×D₄×#261	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	52	4	no	—
254	C₂	C₂×#11719073	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#430	C₄
socle	C₂×C₁₃	C₂×C₈×D₄
prime radical	#11719453	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#430	#261
hypocenter	C₁₃	#33441
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	semidirect
2. #13131228	C₂	semidirect
3. #13131225	C₂	semidirect
4. #13131177	C₂	semidirect
5. #13131080	C₂	semidirect
6. #13131047	C₂	semidirect
15. #11719096	C₂²	semidirect
16. #11719096	C₂²	semidirect
24. #11718270	C₂²	semidirect
25. #11718261	C₂²	semidirect
26. #11718258	C₂²	semidirect
27. #11718257	C₂²	semidirect
219. C₁₃	#33441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719096	2
#13131228	2
#13131225	2
#13131177	2
#13131080	2
#13131047	2
#13123738	2
#33441	13