Subgroups — #16017595

Order 3328.

542 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88158	C₂	1	1	yes	maximal, characteristic
2	#13116997	C₂	1	4	yes	maximal, complemented
3	#13116990	C₂	1	4	yes	maximal, complemented
4	#13116622	C₂	1	2	yes	maximal, complemented
5	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116546	C₂	1	2	yes	maximal
7	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#125	—	2	1	no	—
9	C₂×#88158	C₂²	1	1	yes	characteristic
10	C₄×#6127	—	2	1	no	—
11	C₁₃×#499	—	2	2	no	—
12	C₁₃×#499	C₂²	1	4	yes	—
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₁₃×#106	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718774	—	2	1	no	—
17	#11718767	C₂²	1	1	yes	characteristic
18	C₂×#88052	C₂²	1	2	yes	—
19	C₄×Dic₅₂	C₂²	1	2	yes	—
20	C₄×D₁₀₄	C₂²	1	2	yes	—
21	C₄×#6095	—	2	2	no	—
22	#11718084	C₂²	1	4	yes	—
23	#11718083	C₂²	1	4	yes	—
24	#11718082	C₂²	1	4	yes	—
25	#11718081	C₂²	1	4	yes	—
26	C₄⋊#6090	C₂²	1	4	yes	—
27	C₂×#6127	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×#125	—	2	4	no	—
31	C₁₃×#125	—	4	1	no	—
32	C₁₃×#125	—	4	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₃	—	4	2	no	—
37	#88180	—	2	1	no	—
38	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
39	#88158	—	2	2	no	—
40	#88158	Q₈	1	4	yes	—
41	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
42	C₂×#6095	—	2	1	no	—
43	C₄×D₅₂	—	2	2	no	—
44	#88126	—	4	1	no	—
45	C₄²×D₁₃	—	4	1	no	—
46	C₄×Dic₂₆	—	2	2	no	—
47	C₁₃×#107	—	2	2	no	—
48	C₁₃×#107	C₂³	1	4	yes	—
49	C₁₃×#106	—	2	4	no	—
50	C₄×C₈×C₁₃	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	2	yes	—
52	#88061	—	2	2	no	—
53	#88058	C₂³	1	2	yes	—
54	#88057	—	2	1	no	—
55	#88056	—	2	2	no	—
56	#88052	—	2	4	no	—
57	C₂×#6090	C₂³	1	4	yes	—
58	#33630	—	13	1	no	maximal, Sylow
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	D₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₃	—	4	1	no	—
63	#6127	—	2	4	no	—
64	#6127	—	4	2	no	—
65	C₂×D₅₂	—	2	1	no	—
66	C₂×C₄×D₁₃	—	4	1	no	—
67	C₂×C₄×D₁₃	—	4	2	no	—
68	C₂×Dic₂₆	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₁₃	#41	1	2	yes	—
74	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	—	4	2	no	—
77	C₁₃×C₄⋊C₄	—	4	4	no	—
78	C₁₃×C₄⋊C₄	—	4	4	no	—
79	C₄²×C₁₃	D₈	1	1	yes	characteristic
80	C₄²×C₁₃	D₈	1	1	yes	characteristic
81	C₄²×C₁₃	C₂×D₄	1	2	yes	—
82	C₂²⋊Dic₁₃	—	4	1	no	—
83	#6103	—	4	2	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	#6101	—	4	2	no	—
86	C₄×Dic₁₃	—	4	2	no	—
87	C₄×Dic₁₃	—	4	1	no	—
88	Dic₅₂	—	2	1	no	—
89	Dic₅₂	C₂×Q₈	1	2	yes	—
90	D₁₀₄	—	2	1	no	—
91	D₁₀₄	C₂×Q₈	1	2	yes	—
92	#6095	—	2	4	no	—
93	#6090	—	2	4	no	—
94	C₂×#499	D₁₃	1	1	yes	characteristic, complemented
95	C₄×#135	—	13	1	no	—
96	#2229	—	13	4	no	—
97	#2228	—	13	4	no	—
98	#2159	—	13	2	no	—
99	#2150	—	13	1	no	—
100	#2139	—	13	2	no	—
101	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₁₃	—	4	1	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	8	4	no	—
106	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#137	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₂²⋊D₁₃	—	4	4	no	—
113	C₂×Dic₁₃	—	4	1	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	D₅₂	—	2	2	no	—
116	D₅₂	—	4	1	no	—
117	C₄×D₁₃	—	4	4	no	—
118	C₄×D₁₃	—	8	4	no	—
119	Dic₂₆	—	2	2	no	—
120	Dic₂₆	—	4	1	no	—
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
123	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×#135	—	13	1	no	—
126	C₂×#125	—	2	1	no	—
127	C₄×#41	—	26	1	no	—
128	C₂×M₃₂	—	13	2	no	—
129	#499	—	2	2	no	—
130	#499	C₂×D₁₃	1	4	yes	—
131	C₄×Q₁₆	—	13	2	no	—
132	C₄×SD₁₆	—	26	2	no	—
133	C₄×D₈	—	13	2	no	—
134	#427	—	13	1	no	—
135	C₂×#107	C₂×D₁₃	1	2	yes	—
136	C₂×#106	—	2	1	no	—
137	#416	—	26	1	no	—
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#367	—	13	4	no	—
140	#366	—	13	4	no	—
141	C₄⋊C₁₆	—	13	4	no	—
142	#358	—	13	4	no	—
143	#357	—	13	4	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	#474	1	2	yes	—
146	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₁₃	—	4	2	no	—
148	C₂×D₁₃	—	8	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	8	4	no	—
152	C₄×C₁₃	#474	1	2	yes	—
153	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
154	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
155	Dic₁₃	—	4	2	no	—
156	Dic₁₃	—	8	2	no	—
157	C₂×#41	—	26	1	no	—
158	#135	—	13	4	no	—
159	#135	—	26	2	no	—
160	C₂×Q₁₆	—	13	1	no	—
161	C₂×SD₁₆	—	26	1	no	—
162	C₂×D₈	—	13	1	no	—
163	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
165	#125	—	2	4	no	—
166	#125	—	4	1	no	—
167	#125	—	4	1	no	—
168	C₄×Q₈	—	26	2	no	—
169	C₄×D₄	—	26	2	no	—
170	C₄×D₄	—	52	2	no	—
171	#117	—	26	1	no	—
172	#117	—	52	1	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	4	2	no	—
175	C₂×C₄²	—	52	1	no	—
176	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
177	M₃₂	—	26	4	no	—
178	C₂×C₁₆	—	13	4	no	—
179	#107	—	2	2	no	—
180	#107	—	13	2	no	—
181	#107	C₂²×D₁₃	1	4	yes	—
182	#106	—	2	4	no	—
183	#106	—	26	1	no	—
184	#103	—	26	2	no	—
185	#102	—	26	2	no	—
186	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
187	C₄×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂×C₁₃	—	2	2	no	—
189	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
190	C₂×C₁₃	#3414	1	2	yes	—
191	D₁₃	—	8	2	no	—
192	#41	—	26	4	no	—
193	#41	—	52	2	no	—
194	C₂×Q₈	—	26	1	no	—
195	C₂×D₄	—	26	1	no	—
196	C₂×D₄	—	52	1	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	52	1	no	—
199	C₂²×C₄	—	52	2	no	—
200	C₂²×C₄	#6104	1	2	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
202	Q₁₆	—	13	2	no	—
203	Q₁₆	—	26	1	no	—
204	SD₁₆	—	26	4	no	—
205	D₈	—	13	2	no	—
206	D₈	—	26	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	Q₈×D₁₃	1	2	yes	—
209	C₂×C₈	#6131	1	2	yes	—
210	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
211	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	26	2	no	—
218	C₄⋊C₄	—	52	2	no	—
219	C₂²⋊C₄	—	52	1	no	—
220	C₂²⋊C₄	—	52	2	no	—
221	C₄²	—	52	1	no	—
222	C₄²	—	52	2	no	—
223	C₄²	#6104	1	1	yes	characteristic
224	C₄²	#6104	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
226	C₁₆	—	26	4	no	—
227	C₁₃	#33630	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	52	1	no	—
229	C₂³	C₂×#6104	1	1	yes	characteristic
230	Q₈	—	26	2	no	—
231	Q₈	—	52	1	no	—
232	D₄	—	26	2	no	—
233	D₄	—	52	4	no	—
234	D₄	—	52	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	—	52	1	no	—
239	C₂×C₄	—	52	4	no	—
240	C₂×C₄	—	52	2	no	—
241	C₂×C₄	—	104	4	no	—
242	C₂×C₄	C₂×#6104	1	2	yes	—
243	C₂×C₄	C₂×#6104	1	2	yes	—
244	C₂×C₄	C₂×#6104	1	1	yes	characteristic
245	C₂×C₄	#88196	1	2	yes	—
246	C₂×C₄	#88200	1	2	yes	—
247	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	#88200	1	2	yes	—
250	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
251	C₈	C₂×#6131	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	52	2	no	—
254	C₂²	—	104	2	no	—
255	C₂²	#11718748	1	2	yes	—
256	C₂²	C₂²×#6104	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	8	4	no	—
260	C₄	—	52	2	no	—
261	C₄	—	104	2	no	—
262	C₄	#11718748	1	2	yes	—
263	C₄	C₂×#88196	1	1	yes	characteristic
264	C₄	C₂×#88200	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	104	2	no	—
267	C₂	C₂×#11718748	1	1	yes	characteristic
268	C₂	#13127970	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#499	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88158	C₂²
hypercenter	C₂×#499	D₁₃
hypocenter	C₁₃	#33630
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #13116997	C₂	semidirect
3. #13116990	C₂	semidirect
4. #13116622	C₂	semidirect
5. #13116592	C₂	semidirect
7. C₂×C₁₃×#499	C₂	semidirect
94. C₂×#499	D₁₃	semidirect
227. C₁₃	#33630	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88158	2
#13116997	2
#13116990	2
#13116622	2
#13116592	2
#13116546	2
C₂×C₁₃×#499	2
#33630	13