Subgroups — #16789307

Order 3328.

490 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×#11718284	C₂	1	2	yes	maximal
2	#13131042	C₂	1	2	yes	maximal, complemented
3	#13129884	C₂	1	2	yes	maximal, complemented
4	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×#88180	C₄	1	1	yes	characteristic
7	C₂²×#6099	C₂²	1	1	yes	characteristic
8	C₂×#88106	C₂²	1	4	yes	—
9	#11718762	—	2	2	no	—
10	#11718762	—	2	2	no	—
11	C₂×#88055	C₄	1	1	yes	characteristic
12	C₂×#88055	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	#11718284	C₄	1	8	yes	complemented
16	#11718258	—	2	2	no	—
17	#11718257	—	2	2	no	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	—	2	1	no	—
20	C₂²×C₈×C₁₃	—	2	1	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	C₈	1	4	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₁₃	C₈	1	2	yes	—
26	#88180	—	2	2	no	—
27	#88180	—	2	2	no	—
28	C₂×C₄⋊Dic₁₃	C₈	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#6099	—	2	2	no	—
32	C₂×#6099	—	2	2	no	—
33	C₂×#6099	C₂×C₄	1	4	yes	—
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂³	1	1	yes	characteristic
36	#88106	—	2	4	no	—
37	#88106	C₂×C₄	1	8	yes	—
38	C₂×#6092	—	2	4	no	—
39	#88055	—	2	2	no	—
40	#88055	—	2	2	no	—
41	C₈×Dic₁₃	—	2	1	no	—
42	C₈×Dic₁₃	—	2	1	no	—
43	C₈×Dic₁₃	D₄	1	2	yes	—
44	C₈×Dic₁₃	D₄	1	2	yes	—
45	#12602	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
56	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
66	C₂²⋊Dic₁₃	—	2	4	no	—
67	C₄⋊Dic₁₃	—	2	4	no	—
68	C₄×Dic₁₃	—	2	1	no	—
69	C₄×Dic₁₃	—	2	1	no	—
70	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
71	C₄×Dic₁₃	M₁₆	1	2	yes	—
72	#6099	—	2	8	no	—
73	C₂×#1055	—	2	1	no	—
74	C₂×#1055	—	2	1	no	—
75	C₂×#1055	C₄²	1	2	yes	—
76	C₂×#1055	C₂²×C₄	1	2	yes	—
77	C₂×#1055	C₂×D₄	1	2	yes	—
78	C₂×#1055	#41	1	2	yes	—
79	#6092	—	2	8	no	—
80	C₂×#405	—	13	1	no	—
81	C₂×#349	—	13	2	no	—
82	C₄×M₃₂	—	13	2	no	—
83	#1377	—	13	2	no	—
84	C₂³×C₁₃	—	2	1	no	—
85	C₂³×C₁₃	—	2	1	no	—
86	C₂³×C₁₃	—	4	2	no	—
87	C₂³×C₁₃	C₄×C₈	1	2	yes	—
88	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×Dic₁₃	—	2	2	no	—
103	C₂×Dic₁₃	—	4	2	no	—
104	C₂×Dic₁₃	—	4	2	no	—
105	C₂×Dic₁₃	C₂²⋊C₈	1	4	yes	—
106	C₈×C₁₃	—	2	2	no	—
107	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
108	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
109	#1055	—	2	2	no	—
110	#1055	C₄×D₄	1	4	yes	—
111	C₂²×M₁₆	—	13	1	no	—
112	C₂³×C₈	#261	1	1	yes	characteristic
113	C₂×#117	—	13	1	no	—
114	C₂×M₃₂	—	13	4	no	—
115	#405	—	26	2	no	—
116	#405	—	26	2	no	—
117	C₂×#97	—	13	1	no	—
118	C₂×#97	—	13	1	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₂×C₄×C₈	—	13	1	no	—
121	#349	—	13	8	no	—
122	#346	—	26	2	no	—
123	C₄×C₁₆	—	26	2	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	4	2	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	—	4	2	no	—
129	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
130	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₁₃	#431	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
135	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₄×C₁₃	#431	1	1	yes	characteristic
137	Dic₁₃	—	2	4	no	—
138	Dic₁₃	—	4	2	no	—
139	C₂³×C₄	C₂×#261	1	1	yes	characteristic
140	C₂×M₁₆	—	13	4	no	—
141	C₂×M₁₆	—	26	2	no	—
142	C₂×M₁₆	—	26	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	1	no	—
148	C₂²×C₈	#1057	1	4	yes	—
149	C₂²×C₈	C₂×#261	1	1	yes	characteristic
150	C₂²×C₈	C₂×#261	1	1	yes	characteristic
151	#117	—	26	2	no	—
152	#117	—	26	2	no	—
153	C₂×C₄⋊C₄	—	13	2	no	—
154	C₂×C₂²⋊C₄	—	13	2	no	—
155	C₂×C₄²	—	13	1	no	—
156	C₂×C₄²	—	13	1	no	—
157	M₃₂	—	13	8	no	—
158	M₃₂	—	26	4	no	—
159	C₂×C₁₆	—	26	4	no	—
160	#97	—	26	2	no	—
161	#97	—	26	2	no	—
162	C₄×C₈	—	13	2	no	—
163	C₄×C₈	—	13	2	no	—
164	C₄×C₈	—	26	1	no	—
165	C₄×C₈	—	26	1	no	—
166	C₂×C₁₃	—	2	1	no	—
167	C₂×C₁₃	—	2	1	no	—
168	C₂×C₁₃	—	4	2	no	—
169	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
170	C₂×C₁₃	#2115	1	2	yes	—
171	C₂⁴	D₁₃⋊C₈	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	13	2	no	—
177	C₂²×C₄	—	13	2	no	—
178	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
179	C₂²×C₄	C₄×#261	1	2	yes	—
180	C₂²×C₄	C₂×#1057	1	2	yes	—
181	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
182	M₁₆	—	26	8	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	13	2	no	—
188	C₂×C₈	—	13	2	no	—
189	C₂×C₈	—	13	2	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	26	1	no	—
192	C₂×C₈	—	26	1	no	—
193	C₂×C₈	C₂×#1057	1	2	yes	—
194	C₂×C₈	C₂×#1057	1	2	yes	—
195	C₂×C₈	#6122	1	2	yes	—
196	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
197	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
198	C₄⋊C₄	—	26	4	no	—
199	C₂²⋊C₄	—	26	4	no	—
200	C₄²	—	13	2	no	—
201	C₄²	—	13	2	no	—
202	C₄²	—	26	1	no	—
203	C₄²	—	26	1	no	—
204	C₁₆	—	26	8	no	—
205	C₁₃	#12602	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	4	2	no	—
209	C₂³	C₄×#1057	1	2	yes	—
210	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	13	4	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	52	2	no	—
220	C₂×C₄	—	52	2	no	—
221	C₂×C₄	C₄×#1057	1	2	yes	—
222	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
224	C₂×C₄	#88237	1	1	yes	characteristic
225	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
226	C₂×C₄	C₂×#6122	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	13	4	no	—
230	C₈	—	26	2	no	—
231	C₈	C₂²⋊#1057	1	2	yes	—
232	C₈	C₂²⋊#1057	1	2	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
239	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
240	C₂²	#11719125	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	26	4	no	—
243	C₄	—	52	2	no	—
244	C₄	#11719119	1	1	yes	characteristic
245	C₄	C₄×#6122	1	1	yes	characteristic
246	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
247	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	C₄×C₂²⋊#1057	1	1	yes	characteristic
252	C₂	#13131073	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂³×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂³×C₈	#261
hypocenter	C₁₃	#12602
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #13131042	C₂	semidirect
3. #13129884	C₂	semidirect
15. #11718284	C₄	semidirect
205. C₁₃	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718284	2
#13131042	2
#13129884	2
C₂×#11718762	2
#12602	13