Subgroups — #15779425

Order 3328.

486 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718097	C₂	1	2	yes	maximal, complemented
3	#13114865	C₂	1	1	yes	maximal, characteristic
4	#13112928	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	#11718873	—	2	1	no	—
10	C₂×#88058	—	2	4	no	—
11	C₂×#88058	—	2	1	no	—
12	C₂×#88058	C₂²	1	1	yes	characteristic
13	C₂×#88058	C₂²	1	1	yes	characteristic
14	C₂×#88058	C₂²	1	2	yes	—
15	C₁₃×C₂²⋊C₁₆	C₄	1	4	yes	complemented
16	#11718143	—	2	1	no	—
17	#11718097	—	2	4	no	—
18	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
19	C₂³×Dic₁₃	—	4	1	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
27	C₂×C₄⋊Dic₁₃	—	2	4	no	—
28	C₂×C₄⋊Dic₁₃	—	2	2	no	—
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	—	2	1	no	—
31	C₂×C₄⋊Dic₁₃	—	4	2	no	—
32	C₂×C₄⋊Dic₁₃	—	4	1	no	—
33	C₂×C₁₃×C₁₆	—	2	2	no	—
34	C₂×C₁₃×C₁₆	—	2	2	no	—
35	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
36	#88071	—	4	1	no	—
37	#88058	—	2	4	no	—
38	#88058	—	2	2	no	—
39	#88058	—	2	2	no	—
40	#88058	—	2	2	no	—
41	#88058	D₄	1	4	yes	—
42	#12920	—	13	1	no	maximal, Sylow
43	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	4	4	no	—
53	C₂²×Dic₁₃	—	4	2	no	—
54	C₂²×Dic₁₃	—	4	1	no	—
55	C₂²×Dic₁₃	—	4	1	no	—
56	C₂²×Dic₁₃	—	8	2	no	—
57	C₂²×Dic₁₃	—	8	2	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₁₃	#41	1	2	yes	—
67	C₂²⋊Dic₁₃	—	8	2	no	—
68	C₄⋊Dic₁₃	—	2	4	no	—
69	C₄⋊Dic₁₃	—	4	4	no	—
70	C₄⋊Dic₁₃	—	4	2	no	—
71	C₄⋊Dic₁₃	—	4	1	no	—
72	C₄⋊Dic₁₃	—	4	1	no	—
73	C₄⋊Dic₁₃	—	4	1	no	—
74	C₄⋊Dic₁₃	—	4	1	no	—
75	C₁₃×C₁₆	—	2	4	no	—
76	C₂²×#107	—	13	1	no	—
77	C₂×#367	—	13	2	no	—
78	C₂×C₂²⋊C₁₆	D₁₃	1	1	yes	characteristic
79	#1808	—	13	1	no	—
80	#1380	—	13	2	no	—
81	C₂³×C₁₃	—	2	1	no	—
82	C₂³×C₁₃	—	2	1	no	—
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	—	2	1	no	—
85	C₂³×C₁₃	#107	1	2	yes	—
86	C₂³×C₁₃	SD₃₂	1	2	yes	—
87	C₂³×C₁₃	SD₃₂	1	2	yes	—
88	C₂³×C₁₃	C₂×D₈	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	#107	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
102	C₂×Dic₁₃	—	4	4	no	—
103	C₂×Dic₁₃	—	8	8	no	—
104	C₂×Dic₁₃	—	8	2	no	—
105	C₂×Dic₁₃	—	8	2	no	—
106	C₂×Dic₁₃	—	8	1	no	—
107	C₂×Dic₁₃	—	8	1	no	—
108	C₂×Dic₁₃	—	8	1	no	—
109	C₂×Dic₁₃	—	8	1	no	—
110	C₂×Dic₁₃	—	8	1	no	—
111	C₂×Dic₁₃	—	8	1	no	—
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	C₄×D₄	1	2	yes	—
114	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
115	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
116	C₂²×C₄⋊C₄	—	26	1	no	—
117	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
118	C₂×#107	—	13	2	no	—
119	C₂×#107	—	13	1	no	—
120	C₂×#107	—	13	1	no	—
121	C₂×#107	—	26	4	no	—
122	C₂×#107	—	26	1	no	—
123	#380	—	26	1	no	—
124	#367	—	26	4	no	—
125	C₂²⋊C₁₆	Dic₁₃	1	4	yes	complemented
126	#340	—	26	1	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	#367	1	4	yes	complemented
134	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
135	C₂²×C₁₃	#447	1	1	yes	characteristic
136	C₂²×C₁₃	#480	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
139	C₂²×C₁₃	#509	1	1	yes	characteristic
140	C₂²×C₁₃	#510	1	1	yes	characteristic
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	#385	1	1	yes	characteristic
143	C₄×C₁₃	#428	1	1	yes	characteristic
144	C₄×C₁₃	#451	1	1	yes	characteristic
145	C₄×C₁₃	#484	1	1	yes	characteristic
146	Dic₁₃	—	8	4	no	—
147	Dic₁₃	—	16	1	no	—
148	Dic₁₃	—	16	1	no	—
149	C₂³×C₄	—	52	1	no	—
150	C₂³×C₄	D₅₂	1	1	yes	characteristic
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	Dic₂₆	1	2	yes	—
154	C₂²×C₈	D₅₂	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
156	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	26	4	no	—
158	C₂×C₄⋊C₄	—	26	2	no	—
159	C₂×C₄⋊C₄	—	26	1	no	—
160	C₂×C₄⋊C₄	—	26	1	no	—
161	C₂×C₄⋊C₄	—	52	2	no	—
162	C₂×C₄⋊C₄	—	52	1	no	—
163	C₂×C₂²⋊C₄	—	52	1	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	complemented
167	#107	—	13	4	no	—
168	#107	—	26	4	no	—
169	#107	—	26	2	no	—
170	#107	—	26	2	no	—
171	#107	—	26	2	no	—
172	#95	—	52	1	no	—
173	C₂×C₁₃	—	2	4	no	—
174	C₂×C₁₃	#1852	1	1	yes	characteristic
175	C₂×C₁₃	C₂×#367	1	1	yes	characteristic
176	C₂×C₁₃	#2159	1	1	yes	characteristic
177	C₂×C₁₃	#2185	1	1	yes	characteristic
178	C₂×C₁₃	#2187	1	1	yes	characteristic
179	C₂×C₁₃	#2233	1	1	yes	characteristic
180	C₂×C₁₃	#2235	1	1	yes	characteristic
181	C₂⁴	D₁₀₄	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	52	4	no	—
187	C₂²×C₄	—	52	2	no	—
188	C₂²×C₄	—	52	1	no	—
189	C₂²×C₄	—	52	1	no	—
190	C₂²×C₄	—	104	2	no	—
191	C₂²×C₄	—	104	2	no	—
192	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
193	C₂²×C₄	Dic₅₂	1	2	yes	—
194	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
195	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
201	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
202	C₂×C₈	D₄×D₁₃	1	2	yes	—
203	C₂×C₈	#6129	1	2	yes	—
204	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
205	C₄⋊C₄	—	26	4	no	—
206	C₄⋊C₄	—	52	4	no	—
207	C₄⋊C₄	—	52	2	no	—
208	C₄⋊C₄	—	52	1	no	—
209	C₄⋊C₄	—	52	1	no	—
210	C₄⋊C₄	—	52	1	no	—
211	C₄⋊C₄	—	52	1	no	—
212	C₂²⋊C₄	—	104	2	no	—
213	C₁₆	—	2	4	no	—
214	C₁₃	#12920	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	#88040	1	2	yes	—
220	C₂³	#88040	1	2	yes	—
221	C₂³	#88058	1	2	yes	—
222	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	52	4	no	—
230	C₂×C₄	—	104	8	no	—
231	C₂×C₄	—	104	2	no	—
232	C₂×C₄	—	104	2	no	—
233	C₂×C₄	—	104	1	no	—
234	C₂×C₄	—	104	1	no	—
235	C₂×C₄	—	104	1	no	—
236	C₂×C₄	—	104	1	no	—
237	C₂×C₄	—	104	1	no	—
238	C₂×C₄	—	104	1	no	—
239	C₂×C₄	#88058	1	2	yes	—
240	C₂×C₄	#88136	1	1	yes	characteristic
241	C₂×C₄	#88140	1	1	yes	characteristic
242	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
243	C₂×C₄	#88162	1	1	yes	characteristic
244	C₂×C₄	#88163	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	#88132	1	2	yes	—
248	C₈	D₄×Dic₁₃	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	#11718097	1	4	yes	complemented
256	C₂²	#11718393	1	1	yes	characteristic
257	C₂²	#11718397	1	1	yes	characteristic
258	C₂²	C₂×#88040	1	1	yes	characteristic
259	C₂²	C₂×#88040	1	1	yes	characteristic
260	C₂²	#11718569	1	1	yes	characteristic
261	C₂²	#11718570	1	1	yes	characteristic
262	C₂²	C₂×#88058	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	104	4	no	—
265	C₄	—	208	1	no	—
266	C₄	—	208	1	no	—
267	C₄	#11718383	1	1	yes	characteristic
268	C₄	#11718400	1	1	yes	characteristic
269	C₄	#11718611	1	1	yes	characteristic
270	C₄	#11718782	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#13115026	1	1	yes	characteristic
273	C₂	C₂×#11718097	1	1	yes	characteristic
274	C₂	#13116624	1	1	yes	characteristic
275	C₂	#13116755	1	1	yes	characteristic
276	C₂	#13116767	1	1	yes	characteristic
277	C₂	#13117022	1	1	yes	characteristic
278	C₂	#13117034	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₁₃
hypocenter	C₁₃	#12920
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#11718097	C₂	semidirect
4. #13112928	C₂	semidirect
15. C₁₃×C₂²⋊C₁₆	C₄	semidirect
35. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
125. C₂²⋊C₁₆	Dic₁₃	semidirect
133. C₂²×C₁₃	#367	semidirect
166. C₂×C₁₆	C₂×Dic₁₃	semidirect
214. C₁₃	#12920	semidirect
255. C₂²	#11718097	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88058	2
C₂×#11718097	2
#13114865	2
#13112928	2
C₂×C₁₃×C₂²⋊C₁₆	2
#12920	13