Subgroups — #16018444

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
2	#13117123	C₂	1	4	yes	maximal, complemented
3	#13117116	C₂	1	2	yes	maximal, complemented
4	#13117109	C₂	1	2	yes	maximal, complemented
5	#13116625	C₂	1	4	yes	maximal
6	C₄×#88052	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	—	2	1	no	—
9	C₂×#88123	—	2	1	no	—
10	C₁₃×#500	C₂²	1	1	yes	characteristic
11	C₁₃×#500	C₂²	1	1	yes	characteristic
12	C₁₃×#499	—	2	1	no	—
13	C₁₃×#499	C₂²	1	2	yes	—
14	C₁₃×#498	—	2	1	no	—
15	C₁₃×#427	—	2	1	no	—
16	C₁₃×#427	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88058	—	2	1	no	—
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88052	C₂²	1	2	yes	—
21	#11718343	—	2	1	no	—
22	#11718343	—	2	1	no	—
23	#11718343	C₂²	1	2	yes	—
24	#11718343	C₂²	1	2	yes	—
25	#11718082	C₂²	1	8	yes	—
26	#11718081	C₂²	1	8	yes	—
27	#11718053	C₂²	1	2	yes	—
28	C₄×#6090	C₂²	1	2	yes	—
29	C₂²×Dic₂₆	—	4	1	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×C₄⋊Q₈	—	2	1	no	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×#125	—	2	2	no	—
35	C₁₃×C₂²⋊Q₈	—	4	1	no	—
36	C₁₃×C₂²⋊Q₈	—	4	1	no	—
37	C₄×Q₈×C₁₃	—	4	1	no	—
38	C₄×Q₈×C₁₃	—	4	1	no	—
39	C₁₃×#117	—	2	2	no	—
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₃	—	2	2	no	—
42	C₂×C₄⋊Dic₁₃	—	4	1	no	—
43	#88123	—	2	2	no	—
44	#88123	—	2	2	no	—
45	#88123	—	4	1	no	—
46	#88123	—	4	1	no	—
47	C₁₃×#107	—	2	1	no	—
48	C₁₃×#107	—	2	1	no	—
49	C₁₃×#107	C₂³	1	2	yes	—
50	C₁₃×#107	C₂³	1	2	yes	—
51	C₁₃×#106	—	2	2	no	—
52	C₁₃×#106	—	2	2	no	—
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	—	2	2	no	—
59	#88058	—	2	2	no	—
60	#88058	C₂³	1	4	yes	—
61	#88052	Q₈	1	8	yes	complemented
62	C₂×#6090	C₂³	1	4	yes	—
63	#33731	—	13	1	no	maximal, Sylow
64	C₂×Q₈×C₁₃	—	4	1	no	—
65	C₂×Q₈×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	4	2	no	—
70	C₂×Dic₂₆	—	4	2	no	—
71	C₂×Dic₂₆	—	4	2	no	—
72	C₂×Dic₂₆	—	8	4	no	—
73	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	—	4	1	no	—
84	C₁₃×C₄⋊C₄	—	4	1	no	—
85	C₁₃×C₂²⋊C₄	—	4	2	no	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	D₈	1	1	yes	characteristic
88	C₄²×C₁₃	D₈	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	—	4	2	no	—
93	C₄⋊Dic₁₃	—	4	2	no	—
94	C₄⋊Dic₁₃	—	4	2	no	—
95	#6090	C₂×Q₈	1	8	yes	complemented
96	#3160	D₁₃	1	1	yes	characteristic
97	C₂×#500	—	13	1	no	—
98	#2255	—	13	4	no	—
99	#2254	—	13	2	no	—
100	#2252	—	13	2	no	—
101	#2159	—	13	4	no	—
102	C₄×M₃₂	—	13	1	no	—
103	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
104	Q₈×C₁₃	—	8	2	no	—
105	Q₈×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃	—	4	4	no	—
119	C₂×Dic₁₃	—	8	4	no	—
120	Dic₂₆	—	8	4	no	—
121	Dic₂₆	—	8	4	no	—
122	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
123	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
124	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
125	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
126	#533	—	2	1	no	—
127	C₂×C₄⋊Q₈	—	26	1	no	—
128	C₂×M₃₂	—	13	2	no	—
129	#500	—	13	2	no	—
130	#500	—	13	2	no	—
131	#500	—	26	1	no	—
132	#500	—	26	1	no	—
133	#500	C₂×D₁₃	1	1	yes	characteristic
134	#500	C₂×D₁₃	1	1	yes	characteristic
135	#499	—	2	1	no	—
136	#499	C₂×D₁₃	1	2	yes	—
137	#498	—	2	1	no	—
138	#427	—	2	1	no	—
139	#427	C₂×D₁₃	1	2	yes	—
140	C₂×#107	—	13	2	no	—
141	C₂×#107	—	26	1	no	—
142	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
143	#367	—	13	8	no	—
144	#366	—	13	8	no	—
145	#346	—	13	2	no	—
146	C₄×C₁₆	—	13	2	no	—
147	C₂²×C₁₃	—	2	1	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	#500	1	2	yes	—
150	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	8	2	no	—
154	C₄×C₁₃	—	8	2	no	—
155	C₄×C₁₃	#500	1	2	yes	—
156	C₄×C₁₃	#509	1	2	yes	—
157	C₄×C₁₃	#510	1	2	yes	—
158	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
160	Dic₁₃	—	8	4	no	—
161	C₂²×Q₈	—	52	1	no	—
162	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	1	no	—
165	C₄⋊Q₈	—	2	1	no	—
166	C₄⋊Q₈	—	26	2	no	—
167	C₄⋊Q₈	—	26	2	no	—
168	C₄⋊Q₈	—	52	1	no	—
169	C₄⋊Q₈	—	52	1	no	—
170	#125	—	2	2	no	—
171	C₂²⋊Q₈	—	4	1	no	—
172	C₂²⋊Q₈	—	4	1	no	—
173	C₄×Q₈	—	4	1	no	—
174	C₄×Q₈	—	4	1	no	—
175	#117	—	2	2	no	—
176	C₂×C₄⋊C₄	—	26	2	no	—
177	C₂×C₄⋊C₄	—	52	1	no	—
178	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
179	M₃₂	—	13	8	no	—
180	C₂×C₁₆	—	13	4	no	—
181	#107	—	2	1	no	—
182	#107	—	2	1	no	—
183	#107	—	13	4	no	—
184	#107	—	26	2	no	—
185	#107	—	26	2	no	—
186	#107	—	26	2	no	—
187	#107	C₂²×D₁₃	1	2	yes	—
188	#107	C₂²×D₁₃	1	2	yes	—
189	#106	—	2	2	no	—
190	#106	—	2	2	no	—
191	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₂×C₁₃	—	2	2	no	—
196	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
197	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
198	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
199	C₂×Q₈	—	4	1	no	—
200	C₂×Q₈	—	4	1	no	—
201	C₂×Q₈	—	52	2	no	—
202	C₂×Q₈	—	52	2	no	—
203	C₂×Q₈	—	104	4	no	—
204	C₂²×C₄	—	52	2	no	—
205	C₂²×C₄	#6104	1	1	yes	characteristic
206	C₂²×C₄	#6104	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	C₂×C₈	#6107	1	4	yes	—
209	C₂×C₈	Q₈×D₁₃	1	4	yes	—
210	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	26	4	no	—
221	C₄⋊C₄	—	52	2	no	—
222	C₄⋊C₄	—	52	2	no	—
223	C₄⋊C₄	—	52	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	#6104	1	1	yes	characteristic
227	C₄²	#6104	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
230	C₁₆	—	13	8	no	—
231	C₁₃	#33731	1	1	yes	characteristic, complemented, Sylow
232	C₂³	C₂×#6104	1	1	yes	characteristic
233	Q₈	—	8	2	no	—
234	Q₈	—	8	2	no	—
235	Q₈	—	104	4	no	—
236	Q₈	—	104	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	52	4	no	—
242	C₂×C₄	—	104	4	no	—
243	C₂×C₄	C₂×#6104	1	1	yes	characteristic
244	C₂×C₄	C₂×#6104	1	1	yes	characteristic
245	C₂×C₄	C₂×#6104	1	1	yes	characteristic
246	C₂×C₄	C₂×#6104	1	1	yes	characteristic
247	C₂×C₄	C₂×#6104	1	1	yes	characteristic
248	C₂×C₄	C₂×#6107	1	2	yes	—
249	C₂×C₄	#88198	1	2	yes	—
250	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₈	C₂×#6107	1	2	yes	—
252	C₈	C₂×#6107	1	2	yes	—
253	C₈	#88198	1	2	yes	—
254	C₈	C₂×Q₈×D₁₃	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	#11718739	1	2	yes	—
258	C₂²	C₂²×#6104	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	8	2	no	—
262	C₄	—	8	2	no	—
263	C₄	—	104	4	no	—
264	C₄	#11718739	1	2	yes	—
265	C₄	#11718854	1	2	yes	—
266	C₄	#11718856	1	2	yes	—
267	C₄	C₂²×#6107	1	1	yes	characteristic
268	C₄	C₂×#88198	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	C₂×#11718739	1	1	yes	characteristic
271	C₂	C₂×#11718854	1	1	yes	characteristic
272	C₂	C₂×#11718856	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#33731
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #13117123	C₂	semidirect
3. #13117116	C₂	semidirect
4. #13117109	C₂	semidirect
61. #88052	Q₈	semidirect
95. #6090	C₂×Q₈	semidirect
231. C₁₃	#33731	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718343	2
#13117123	2
#13117116	2
#13117109	2
#13116625	2
C₄×#88052	2
C₁₃×#3160	2
#33731	13