Subgroups — #15762165

Order 3328.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718356	C₂	1	1	yes	maximal, characteristic
2	#13115763	C₂	1	2	yes	maximal, complemented
3	C₂×#11718050	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718077	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₄	—	2	1	no	—
7	C₂²×D₁₀₄	—	2	1	no	—
8	C₂×#88128	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	—	2	1	no	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×#88049	—	2	4	no	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	#11718356	—	2	2	no	—
16	#11718356	C₄	1	4	yes	complemented
17	C₁₃×#336	—	2	1	no	—
18	#11718077	—	2	4	no	—
19	#11718050	—	2	2	no	—
20	C₂²×D₅₂	—	2	2	no	—
21	C₂²×D₅₂	—	2	2	no	—
22	C₂²×D₅₂	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
27	C₂×C₄²×C₁₃	—	2	1	no	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×#1055	—	2	2	no	—
30	C₂×D₁₀₄	—	2	4	no	—
31	C₂×D₁₀₄	—	2	4	no	—
32	C₂×D₁₀₄	—	4	2	no	—
33	C₂×D₁₀₄	—	4	2	no	—
34	C₂×D₁₀₄	—	4	2	no	—
35	C₂×D₁₀₄	—	4	2	no	—
36	#88128	—	2	2	no	—
37	#88128	—	2	2	no	—
38	#88128	—	2	2	no	—
39	#88128	C₂×C₄	1	4	yes	—
40	C₁₃×C₄⋊C₈	—	2	1	no	—
41	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
42	C₄×C₈×C₁₃	—	4	2	no	—
43	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₃	D₄	1	2	yes	—
45	C₁₃×#95	—	2	1	no	—
46	#88049	—	4	8	no	—
47	C₄⋊#1055	—	2	2	no	—
48	C₄⋊#1055	—	2	2	no	—
49	#9734	—	13	1	no	maximal, Sylow
50	C₂³×D₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
55	C₂×D₅₂	—	2	4	no	—
56	C₂×D₅₂	—	2	4	no	—
57	C₂×D₅₂	—	2	4	no	—
58	C₂×D₅₂	—	4	2	no	—
59	C₂×D₅₂	—	4	4	no	—
60	C₂×D₅₂	—	4	2	no	—
61	C₂×D₅₂	—	4	4	no	—
62	C₂×D₅₂	—	4	2	no	—
63	C₂×D₅₂	—	4	2	no	—
64	C₂×D₅₂	—	4	2	no	—
65	C₂×D₅₂	—	4	2	no	—
66	C₂×D₅₂	—	4	2	no	—
67	C₂×D₅₂	—	8	4	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	4	1	no	—
72	C₂×C₈×C₁₃	—	4	1	no	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃	D₈	1	2	yes	—
76	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	D₈	1	4	yes	complemented
79	C₄²×C₁₃	—	4	1	no	—
80	C₄²×C₁₃	—	4	1	no	—
81	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	#41	1	1	yes	characteristic
85	C₂×#1055	—	4	2	no	—
86	C₂×#1055	—	4	4	no	—
87	D₁₀₄	—	4	8	no	—
88	D₁₀₄	—	4	8	no	—
89	C₂×#493	—	13	1	no	—
90	#1982	—	13	2	no	—
91	C₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
92	C₂×#335	—	13	1	no	—
93	C₂×#331	—	13	2	no	—
94	C₂³×C₁₃	#121	1	1	yes	characteristic
95	C₂²×D₁₃	—	4	4	no	—
96	C₂²×D₁₃	—	8	2	no	—
97	C₂²×D₁₃	—	8	2	no	—
98	C₂²×D₁₃	—	8	4	no	—
99	C₂²×D₁₃	—	8	4	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#135	1	2	yes	—
114	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	D₅₂	—	4	4	no	—
117	D₅₂	—	4	4	no	—
118	D₅₂	—	4	8	no	—
119	D₅₂	—	8	4	no	—
120	D₅₂	—	8	4	no	—
121	D₅₂	—	8	2	no	—
122	D₅₂	—	8	2	no	—
123	D₅₂	—	8	2	no	—
124	D₅₂	—	8	2	no	—
125	C₈×C₁₃	—	2	2	no	—
126	C₈×C₁₃	—	4	2	no	—
127	C₈×C₁₃	#102	1	4	yes	complemented
128	#1055	—	8	2	no	—
129	C₂²×D₈	—	26	1	no	—
130	C₂²×D₈	—	26	1	no	—
131	C₂×#127	—	13	2	no	—
132	#493	—	13	4	no	—
133	#493	—	26	2	no	—
134	C₂×C₄⋊C₈	—	13	2	no	—
135	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
136	C₂×#102	—	26	4	no	—
137	C₂×C₄×C₈	—	2	1	no	—
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
140	#336	—	2	1	no	—
141	#335	—	26	2	no	—
142	#331	—	26	4	no	—
143	C₂²×C₁₃	#390	1	1	yes	characteristic
144	C₂²×C₁₃	#459	1	1	yes	characteristic
145	C₂²×C₁₃	#460	1	1	yes	characteristic
146	C₂²×C₁₃	#466	1	2	yes	—
147	C₂²×C₁₃	#472	1	2	yes	—
148	C₂×D₁₃	—	8	4	no	—
149	C₂×D₁₃	—	8	4	no	—
150	C₂×D₁₃	—	8	4	no	—
151	C₂×D₁₃	—	16	4	no	—
152	C₂×D₁₃	—	16	4	no	—
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	4	4	no	—
156	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
157	C₄×C₁₃	#418	1	1	yes	characteristic
158	C₄×C₁₃	C₄×D₈	1	2	yes	—
159	C₄×C₁₃	#459	1	1	yes	characteristic
160	C₄×C₁₃	#460	1	1	yes	characteristic
161	C₄×C₁₃	#493	1	1	yes	characteristic
162	C₄×C₁₃	#495	1	1	yes	characteristic
163	C₂²×D₄	—	26	2	no	—
164	C₂²×D₄	—	26	2	no	—
165	C₂²×D₄	—	26	2	no	—
166	C₂×D₈	—	26	4	no	—
167	C₂×D₈	—	26	4	no	—
168	C₂×D₈	—	52	2	no	—
169	C₂×D₈	—	52	2	no	—
170	C₂×D₈	—	52	2	no	—
171	C₂×D₈	—	52	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	26	2	no	—
174	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
176	#127	—	13	4	no	—
177	#127	—	26	2	no	—
178	#127	—	26	2	no	—
179	#127	—	26	2	no	—
180	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
181	C₂×C₄²	—	2	1	no	—
182	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	26	2	no	—
185	C₄⋊C₈	—	26	2	no	—
186	C₄⋊C₈	D₅₂	1	2	yes	—
187	#102	—	52	8	no	—
188	C₄×C₈	—	4	2	no	—
189	C₄×C₈	C₄×D₁₃	1	2	yes	—
190	C₄×C₈	D₅₂	1	2	yes	—
191	#95	—	2	1	no	—
192	C₂×C₁₃	#1667	1	2	yes	—
193	C₂×C₁₃	#1668	1	2	yes	—
194	C₂×C₁₃	#1970	1	1	yes	characteristic
195	C₂×C₁₃	#1978	1	1	yes	characteristic
196	C₂×C₁₃	#1982	1	1	yes	characteristic
197	D₁₃	—	16	4	no	—
198	C₂⁴	—	52	2	no	—
199	C₂×D₄	—	26	4	no	—
200	C₂×D₄	—	26	4	no	—
201	C₂×D₄	—	26	4	no	—
202	C₂×D₄	—	52	2	no	—
203	C₂×D₄	—	52	2	no	—
204	C₂×D₄	—	52	4	no	—
205	C₂×D₄	—	52	4	no	—
206	C₂×D₄	—	52	2	no	—
207	C₂×D₄	—	52	2	no	—
208	C₂×D₄	—	52	2	no	—
209	C₂×D₄	—	52	2	no	—
210	C₂×D₄	—	52	2	no	—
211	C₂×D₄	—	104	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
214	C₂²×C₄	#6127	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
216	D₈	—	52	8	no	—
217	D₈	—	52	8	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	52	2	no	—
224	C₂×C₈	—	52	4	no	—
225	C₂×C₈	#6103	1	2	yes	—
226	C₂×C₈	#6103	1	2	yes	—
227	C₂×C₈	#6104	1	2	yes	—
228	C₂×C₈	#6106	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	D₁₀₄	1	4	yes	complemented
231	C₄²	—	4	1	no	—
232	C₄²	—	4	1	no	—
233	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
234	C₄²	C₂×D₅₂	1	1	yes	characteristic
235	C₄²	C₂×D₅₂	1	1	yes	characteristic
236	C₄²	#6127	1	1	yes	characteristic
237	C₁₃	#9734	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	52	4	no	—
239	C₂³	—	104	2	no	—
240	C₂³	—	104	2	no	—
241	C₂³	—	104	4	no	—
242	C₂³	—	104	4	no	—
243	C₂³	#88184	1	1	yes	characteristic
244	D₄	—	52	4	no	—
245	D₄	—	52	8	no	—
246	D₄	—	52	4	no	—
247	D₄	—	104	4	no	—
248	D₄	—	104	4	no	—
249	D₄	—	104	2	no	—
250	D₄	—	104	2	no	—
251	D₄	—	104	2	no	—
252	D₄	—	104	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
258	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
259	C₂×C₄	#88128	1	1	yes	characteristic
260	C₂×C₄	#88129	1	1	yes	characteristic
261	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
262	C₂×C₄	#88158	1	2	yes	—
263	C₂×C₄	C₂×#6103	1	1	yes	characteristic
264	C₂×C₄	#88184	1	1	yes	characteristic
265	C₂×C₄	C₂×#6104	1	1	yes	characteristic
266	C₂×C₄	C₂×#6106	1	1	yes	characteristic
267	C₂×C₄	#88203	1	1	yes	characteristic
268	C₂×C₄	#88203	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	104	2	no	—
272	C₈	#88049	1	4	yes	complemented
273	C₂²	—	104	4	no	—
274	C₂²	—	104	4	no	—
275	C₂²	—	104	4	no	—
276	C₂²	—	208	4	no	—
277	C₂²	—	208	4	no	—
278	C₂²	#11718600	1	1	yes	characteristic
279	C₂²	#11718676	1	1	yes	characteristic
280	C₂²	#11718688	1	1	yes	characteristic
281	C₂²	#11718776	1	2	yes	—
282	C₂²	#11718803	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	4	no	—
286	C₄	C₄×D₁₀₄	1	2	yes	—
287	C₄	#11718356	1	1	yes	characteristic
288	C₄	#11718357	1	1	yes	characteristic
289	C₄	C₂×#88049	1	1	yes	characteristic
290	C₄	#11718662	1	1	yes	characteristic
291	C₄	#11718676	1	1	yes	characteristic
292	C₄	#11718688	1	1	yes	characteristic
293	C₂	—	208	4	no	—
294	C₂	#13114191	1	2	yes	—
295	C₂	#13114196	1	2	yes	—
296	C₂	#13115702	1	1	yes	characteristic
297	C₂	#13115746	1	1	yes	characteristic
298	C₂	#13115763	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₈×C₁₃×C₄⋊C₄	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂×#11718356	C₂
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#9734
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #13115763	C₂	semidirect
3. C₂×#11718050	C₂	semidirect
4. C₂×#11718077	C₂	semidirect
5. C₈×C₁₃×C₄⋊C₄	C₂	semidirect
16. #11718356	C₄	semidirect
78. C₁₃×C₄⋊C₄	D₈	semidirect
91. C₈×C₄⋊C₄	D₁₃	semidirect
127. C₈×C₁₃	#102	semidirect
230. C₄⋊C₄	D₁₀₄	semidirect
237. C₁₃	#9734	semidirect
272. C₈	#88049	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718356	2
#13115763	2
C₂×#11718050	2
C₂×#11718077	2
C₈×C₁₃×C₄⋊C₄	2
#9734	13