Subgroups — #16017214

Order 3328.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718840	C₂	1	2	yes	maximal
2	C₄×#88158	C₂	1	1	yes	maximal, characteristic
3	#13116883	C₂	1	4	yes	maximal, complemented
4	#13116878	C₂	1	4	yes	maximal, complemented
5	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116556	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×Q₁₆	—	2	1	no	—
9	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
11	C₂×#88158	C₂²	1	1	yes	characteristic
12	C₄×#6127	—	2	1	no	—
13	C₁₃×#494	—	2	2	no	—
14	C₁₃×#494	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88069	C₂²	1	4	yes	—
17	#11718840	—	2	8	no	—
18	C₂×#88068	C₂²	1	4	yes	—
19	#11718774	—	2	1	no	—
20	#11718767	C₂²	1	1	yes	characteristic
21	C₂×#88052	C₂²	1	2	yes	—
22	C₄×Dic₅₂	C₂²	1	2	yes	—
23	C₄×D₁₀₄	C₂²	1	2	yes	—
24	C₄×#6095	—	2	2	no	—
25	#11718156	C₂²	1	8	yes	—
26	C₄⋊#6090	C₂²	1	4	yes	—
27	C₂²×Q₈×C₁₃	—	2	2	no	—
28	C₂×#6127	—	2	1	no	—
29	C₂×C₁₃×Q₁₆	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	—	2	2	no	—
31	C₂×C₁₃×Q₁₆	—	2	4	no	—
32	C₂×C₁₃×Q₁₆	—	2	4	no	—
33	C₂×C₁₃×Q₁₆	—	4	4	no	—
34	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	—	2	4	no	—
38	C₁₃×C₄⋊Q₈	—	4	1	no	—
39	C₁₃×C₄⋊Q₈	—	4	1	no	—
40	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₄×C₂²⋊D₁₃	—	4	2	no	—
43	#88180	—	2	1	no	—
44	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
45	#88158	—	2	2	no	—
46	#88158	D₄	1	4	yes	—
47	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
48	C₂×#6095	—	2	1	no	—
49	C₄×D₅₂	—	2	2	no	—
50	#88126	—	4	1	no	—
51	C₄²×D₁₃	—	4	1	no	—
52	C₄×Dic₂₆	—	2	2	no	—
53	C₄×C₈×C₁₃	D₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	2	yes	—
55	#88069	—	2	8	no	—
56	#88068	—	2	8	no	—
57	#88061	—	2	2	no	—
58	#88058	C₂³	1	2	yes	—
59	#88057	—	2	1	no	—
60	#88056	—	2	2	no	—
61	#88052	—	2	4	no	—
62	C₂×#6090	C₂³	1	4	yes	—
63	#33583	—	13	1	no	maximal, Sylow
64	C₂×Q₈×C₁₃	—	2	4	no	—
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂×Q₈×C₁₃	—	4	4	no	—
67	C₂×Q₈×C₁₃	—	4	4	no	—
68	C₂²×C₄×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	D₈	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₁₃	—	4	1	no	—
72	#6127	—	2	4	no	—
73	#6127	—	4	2	no	—
74	C₂×D₅₂	—	2	1	no	—
75	C₂×C₄×D₁₃	—	4	1	no	—
76	C₂×C₄×D₁₃	—	4	2	no	—
77	C₂×Dic₂₆	—	2	1	no	—
78	C₁₃×Q₁₆	—	2	4	no	—
79	C₁₃×Q₁₆	—	2	4	no	—
80	C₁₃×Q₁₆	—	4	8	no	—
81	C₁₃×Q₁₆	—	4	4	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	4	4	no	—
89	C₁₃×C₄⋊C₄	—	4	4	no	—
90	C₄²×C₁₃	D₈	1	1	yes	characteristic
91	C₄²×C₁₃	D₈	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₂²⋊Dic₁₃	—	4	1	no	—
94	#6103	—	4	2	no	—
95	C₄⋊Dic₁₃	—	2	2	no	—
96	#6101	—	4	2	no	—
97	C₄×Dic₁₃	—	4	2	no	—
98	C₄×Dic₁₃	—	4	1	no	—
99	Dic₅₂	—	2	1	no	—
100	Dic₅₂	C₂×D₄	1	2	yes	—
101	D₁₀₄	—	2	1	no	—
102	D₁₀₄	C₂×D₄	1	2	yes	—
103	#6095	—	2	4	no	—
104	#6090	—	2	4	no	—
105	C₂×#510	—	13	2	no	—
106	C₂×#494	D₁₃	1	1	yes	characteristic, complemented
107	C₄×#135	—	13	1	no	—
108	#2209	—	13	4	no	—
109	#2208	—	13	4	no	—
110	#2150	—	13	1	no	—
111	#2141	—	13	2	no	—
112	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₁₃	—	4	1	no	—
114	Q₈×C₁₃	—	4	4	no	—
115	Q₈×C₁₃	—	4	4	no	—
116	Q₈×C₁₃	—	8	4	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	8	4	no	—
120	C₂×C₄×C₁₃	#121	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₁₃	—	4	4	no	—
127	C₂×Dic₁₃	—	4	1	no	—
128	C₂×Dic₁₃	—	4	2	no	—
129	D₅₂	—	2	2	no	—
130	D₅₂	—	4	1	no	—
131	C₄×D₁₃	—	4	4	no	—
132	C₄×D₁₃	—	8	4	no	—
133	Dic₂₆	—	2	2	no	—
134	Dic₂₆	—	4	1	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	#121	1	2	yes	—
137	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×#135	—	13	1	no	—
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×Q₁₆	C₂×D₁₃	1	2	yes	—
142	C₂×C₄⋊Q₈	—	2	1	no	—
143	C₄×#41	—	26	1	no	—
144	#510	—	26	8	no	—
145	C₂×Q₃₂	—	13	4	no	—
146	C₂×SD₃₂	—	13	4	no	—
147	C₂×M₃₂	—	13	2	no	—
148	#494	—	2	2	no	—
149	#494	C₂×D₁₃	1	4	yes	—
150	C₄×Q₁₆	—	13	2	no	—
151	C₄×SD₁₆	—	26	2	no	—
152	C₄×D₈	—	13	2	no	—
153	#427	—	13	1	no	—
154	#416	—	26	1	no	—
155	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
156	C₄⋊C₁₆	—	13	4	no	—
157	#358	—	13	8	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	#459	1	2	yes	—
160	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
161	C₂×D₁₃	—	4	2	no	—
162	C₂×D₁₃	—	8	2	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	8	4	no	—
165	C₄×C₁₃	#459	1	2	yes	—
166	C₄×C₁₃	#510	1	4	yes	complemented
167	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
168	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	Dic₁₃	—	4	2	no	—
170	Dic₁₃	—	8	2	no	—
171	C₂×#41	—	26	1	no	—
172	C₂²×Q₈	—	2	2	no	—
173	#135	—	13	4	no	—
174	#135	—	26	2	no	—
175	C₂×Q₁₆	—	2	4	no	—
176	C₂×Q₁₆	—	2	4	no	—
177	C₂×Q₁₆	—	2	4	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	4	4	no	—
180	C₂×Q₁₆	—	13	1	no	—
181	C₂×Q₁₆	C₂²×D₁₃	1	4	yes	—
182	C₂×SD₁₆	—	26	1	no	—
183	C₂×D₈	—	13	1	no	—
184	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	4	no	—
187	C₄⋊Q₈	—	4	1	no	—
188	C₄⋊Q₈	—	4	1	no	—
189	C₄×Q₈	—	26	2	no	—
190	C₄×D₄	—	26	2	no	—
191	C₄×D₄	—	52	2	no	—
192	#117	—	26	1	no	—
193	#117	—	52	1	no	—
194	C₂×C₄⋊C₄	—	4	2	no	—
195	C₂×C₄²	—	52	1	no	—
196	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
197	Q₃₂	—	26	8	no	—
198	SD₃₂	—	26	8	no	—
199	M₃₂	—	26	4	no	—
200	C₂×C₁₆	—	13	4	no	—
201	#107	—	13	2	no	—
202	#106	—	26	1	no	—
203	#103	—	26	2	no	—
204	#102	—	26	2	no	—
205	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
206	C₄×C₈	C₂²×D₁₃	1	2	yes	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
209	C₂×C₁₃	C₂×#510	1	2	yes	—
210	D₁₃	—	8	2	no	—
211	#41	—	26	4	no	—
212	#41	—	52	2	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	4	4	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	26	1	no	—
218	C₂×D₄	—	26	1	no	—
219	C₂×D₄	—	52	1	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	52	1	no	—
222	C₂²×C₄	—	52	2	no	—
223	C₂²×C₄	#6104	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
225	Q₁₆	—	2	4	no	—
226	Q₁₆	—	2	4	no	—
227	Q₁₆	—	4	8	no	—
228	Q₁₆	—	4	4	no	—
229	Q₁₆	—	13	2	no	—
230	Q₁₆	—	26	1	no	—
231	SD₁₆	—	26	4	no	—
232	D₈	—	13	2	no	—
233	D₈	—	26	1	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	D₄×D₁₃	1	2	yes	—
236	C₂×C₈	#6129	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	26	2	no	—
243	C₄⋊C₄	—	52	2	no	—
244	C₂²⋊C₄	—	52	1	no	—
245	C₂²⋊C₄	—	52	2	no	—
246	C₄²	—	52	1	no	—
247	C₄²	—	52	2	no	—
248	C₄²	#6104	1	1	yes	characteristic
249	C₄²	#6104	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
251	C₁₆	—	26	4	no	—
252	C₁₃	#33583	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	52	1	no	—
254	C₂³	C₂×#6104	1	1	yes	characteristic
255	Q₈	—	4	4	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	8	4	no	—
258	Q₈	—	26	2	no	—
259	Q₈	—	52	1	no	—
260	D₄	—	26	2	no	—
261	D₄	—	52	4	no	—
262	D₄	—	52	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	—	52	1	no	—
267	C₂×C₄	—	52	4	no	—
268	C₂×C₄	—	52	2	no	—
269	C₂×C₄	—	104	4	no	—
270	C₂×C₄	C₂×#6104	1	2	yes	—
271	C₂×C₄	C₂×#6104	1	1	yes	characteristic
272	C₂×C₄	C₂×#6104	1	2	yes	—
273	C₂×C₄	#88186	1	2	yes	—
274	C₂×C₄	#88192	1	2	yes	—
275	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	#88192	1	2	yes	—
278	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
279	C₈	C₂×#6129	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	52	2	no	—
282	C₂²	—	104	2	no	—
283	C₂²	#11718733	1	2	yes	—
284	C₂²	C₂²×#6104	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	8	4	no	—
287	C₄	—	52	2	no	—
288	C₄	—	104	2	no	—
289	C₄	#11718733	1	2	yes	—
290	C₄	#11718840	1	4	yes	complemented
291	C₄	C₂×#88186	1	1	yes	characteristic
292	C₄	C₂×#88192	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	104	2	no	—
295	C₂	C₂×#11718733	1	1	yes	characteristic
296	C₂	C₂×#11718840	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88158	C₂²
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	#33583
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #13116883	C₂	semidirect
4. #13116878	C₂	semidirect
5. #13116592	C₂	semidirect
6. #13116556	C₂	semidirect
7. C₂×C₁₃×#494	C₂	semidirect
106. C₂×#494	D₁₃	semidirect
166. C₄×C₁₃	#510	semidirect
252. C₁₃	#33583	semidirect
290. C₄	#11718840	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718840	2
C₄×#88158	2
#13116883	2
#13116878	2
#13116592	2
#13116556	2
C₂×C₁₃×#494	2
#33583	13