Subgroups — #15778965

Order 3328.

446 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718156	C₂	1	2	yes	maximal
3	#13114996	C₂	1	1	yes	maximal, characteristic
4	#13112927	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
8	C₄×C₁₃×Q₁₆	—	2	2	no	—
9	C₄×C₁₃×Q₁₆	C₄	1	4	yes	complemented
10	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#103	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88058	—	2	1	no	—
14	C₂×#88058	C₂²	1	2	yes	—
15	C₂²×#6090	C₂²	1	2	yes	—
16	#11718596	—	2	1	no	—
17	#11718156	—	2	4	no	—
18	C₄⋊#6090	—	2	2	no	—
19	C₂²×Q₈×C₁₃	—	2	1	no	—
20	C₂×C₁₃×Q₁₆	—	2	1	no	—
21	C₂×C₁₃×Q₁₆	—	2	1	no	—
22	C₂×C₁₃×Q₁₆	—	2	1	no	—
23	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×Q₁₆	D₄	1	2	yes	—
25	C₂×C₁₃×Q₁₆	Q₈	1	2	yes	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₁₃	—	2	4	no	—
29	C₄×Q₈×C₁₃	—	4	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₁₃	—	4	1	no	—
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	—	2	2	no	—
35	C₂×C₄⋊Dic₁₃	—	4	1	no	—
36	C₂×C₄⋊Dic₁₃	—	4	1	no	—
37	C₁₃×#107	—	2	1	no	—
38	C₁₃×#107	C₂×C₄	1	2	yes	—
39	C₁₃×#103	—	2	4	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	#88058	—	2	2	no	—
43	#88058	—	2	2	no	—
44	#88058	—	4	2	no	—
45	C₂×#6090	—	2	2	no	—
46	C₂×#6090	—	2	2	no	—
47	C₂×#6090	—	2	2	no	—
48	#12812	—	13	1	no	maximal, Sylow
49	C₂×Q₈×C₁₃	—	2	2	no	—
50	C₂×Q₈×C₁₃	—	2	2	no	—
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	1	no	—
54	C₂×Q₈×C₁₃	—	4	1	no	—
55	C₂²×C₄×C₁₃	—	4	1	no	—
56	C₂²×C₄×C₁₃	—	4	1	no	—
57	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	4	2	no	—
61	C₁₃×Q₁₆	—	2	2	no	—
62	C₁₃×Q₁₆	—	2	2	no	—
63	C₁₃×Q₁₆	—	2	2	no	—
64	C₁₃×Q₁₆	C₄⋊C₄	1	4	yes	complemented
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	4	4	no	—
76	C₁₃×C₄⋊C₄	—	4	1	no	—
77	C₄²×C₁₃	—	4	4	no	—
78	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₁₃	SD₁₆	1	2	yes	—
80	C₄⋊Dic₁₃	—	4	2	no	—
81	C₄⋊Dic₁₃	—	4	2	no	—
82	C₄⋊Dic₁₃	—	8	2	no	—
83	C₄⋊Dic₁₃	—	8	2	no	—
84	#6090	—	4	2	no	—
85	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
86	C₂×C₄⋊C₁₆	—	13	1	no	—
87	C₂×#358	—	13	2	no	—
88	#1845	—	13	1	no	—
89	#1380	—	13	2	no	—
90	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₁₃	—	2	4	no	—
92	Q₈×C₁₃	—	4	2	no	—
93	Q₈×C₁₃	—	4	2	no	—
94	Q₈×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	8	1	no	—
102	C₂×C₄×C₁₃	—	8	1	no	—
103	C₂×C₄×C₁₃	#102	1	2	yes	—
104	C₂×C₄×C₁₃	#102	1	2	yes	—
105	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
106	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
113	C₂×Dic₁₃	—	8	2	no	—
114	C₂×Dic₁₃	—	8	2	no	—
115	C₂×Dic₁₃	—	8	2	no	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₈×C₁₃	#117	1	1	yes	characteristic
119	C₈×C₁₃	#121	1	1	yes	characteristic
120	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
122	C₂×C₄×Q₈	—	2	1	no	—
123	C₂²×C₁₆	—	13	2	no	—
124	C₄×Q₁₆	—	2	2	no	—
125	C₄×Q₁₆	Dic₁₃	1	4	yes	complemented
126	C₂×#107	—	13	2	no	—
127	C₂×#107	—	26	1	no	—
128	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
129	C₂×#103	—	2	1	no	—
130	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
131	#384	—	26	1	no	—
132	C₄⋊C₁₆	—	26	2	no	—
133	#358	—	26	4	no	—
134	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
135	C₂²×C₁₃	#459	1	1	yes	characteristic
136	C₂²×C₁₃	#474	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×Q₃₂	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	—	4	4	no	—
143	C₄×C₁₃	—	8	2	no	—
144	C₄×C₁₃	#358	1	4	yes	complemented
145	C₄×C₁₃	#380	1	1	yes	characteristic
146	C₄×C₁₃	#417	1	1	yes	characteristic
147	C₄×C₁₃	#461	1	1	yes	characteristic
148	C₄×C₁₃	#476	1	1	yes	characteristic
149	Dic₁₃	—	16	2	no	—
150	C₂²×Q₈	—	2	1	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	—	2	1	no	—
153	C₂×Q₁₆	—	2	1	no	—
154	C₂×Q₁₆	Dic₂₆	1	2	yes	—
155	C₂×Q₁₆	D₅₂	1	2	yes	—
156	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
157	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
159	C₄×Q₈	—	2	4	no	—
160	C₄×Q₈	—	4	2	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	4	1	no	—
163	C₂×C₄⋊C₄	—	26	2	no	—
164	C₂×C₄⋊C₄	—	52	1	no	—
165	C₂×C₄⋊C₄	—	52	1	no	—
166	C₂×C₄²	—	4	1	no	—
167	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
168	C₂×C₁₆	—	26	2	no	—
169	C₂×C₁₆	—	26	2	no	—
170	C₂×C₁₆	—	26	2	no	—
171	#107	—	2	1	no	—
172	#107	—	26	2	no	—
173	#107	—	26	2	no	—
174	#107	—	52	2	no	—
175	#107	C₂×Dic₁₃	1	2	yes	—
176	#103	—	2	4	no	—
177	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
178	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
179	C₂×C₁₃	#1802	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
181	C₂×C₁₃	#2139	1	1	yes	characteristic
182	C₂×C₁₃	#2207	1	1	yes	characteristic
183	C₂×C₁₃	#2209	1	1	yes	characteristic
184	C₂×C₁₃	#2225	1	1	yes	characteristic
185	C₂×C₁₃	#2227	1	1	yes	characteristic
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×Q₈	—	4	1	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	52	2	no	—
195	C₂²×C₄	#6104	1	1	yes	characteristic
196	C₂²×C₄	#6104	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
198	Q₁₆	—	2	2	no	—
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	2	2	no	—
201	Q₁₆	C₄⋊Dic₁₃	1	4	yes	complemented
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
205	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
206	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
207	C₂×C₈	#6127	1	1	yes	characteristic
208	C₂×C₈	#6127	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	52	2	no	—
215	C₄⋊C₄	—	52	2	no	—
216	C₄⋊C₄	—	104	2	no	—
217	C₄⋊C₄	—	104	2	no	—
218	C₄²	—	4	4	no	—
219	C₄²	#6105	1	2	yes	—
220	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
221	C₁₆	—	52	2	no	—
222	C₁₃	#12812	1	1	yes	characteristic, complemented, Sylow
223	C₂³	C₂×#6104	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	4	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	—	104	2	no	—
237	C₂×C₄	—	104	2	no	—
238	C₂×C₄	—	104	2	no	—
239	C₂×C₄	#88068	1	2	yes	—
240	C₂×C₄	#88069	1	2	yes	—
241	C₂×C₄	#88072	1	2	yes	—
242	C₂×C₄	#88072	1	2	yes	—
243	C₂×C₄	#88178	1	1	yes	characteristic
244	C₂×C₄	#88184	1	1	yes	characteristic
245	C₂×C₄	C₂×#6105	1	1	yes	characteristic
246	C₂×C₄	#88203	1	1	yes	characteristic
247	C₂×C₄	#88205	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	#88178	1	1	yes	characteristic
251	C₈	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
252	C₈	#88180	1	1	yes	characteristic
253	C₈	#88184	1	1	yes	characteristic
254	C₂²	#11718668	1	1	yes	characteristic
255	C₂²	#11718676	1	1	yes	characteristic
256	C₂²	C₂×#88068	1	1	yes	characteristic
257	C₂²	C₂×#88069	1	1	yes	characteristic
258	C₂²	#11718854	1	1	yes	characteristic
259	C₂²	#11718856	1	1	yes	characteristic
260	C₂²	C₂×#88072	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₄	—	208	2	no	—
265	C₄	#11718156	1	4	yes	complemented
266	C₄	#11718669	1	1	yes	characteristic
267	C₄	#11718680	1	1	yes	characteristic
268	C₄	#11718873	1	1	yes	characteristic
269	C₄	#11718893	1	1	yes	characteristic
270	C₂	#13114836	1	1	yes	characteristic
271	C₂	C₂×#11718156	1	1	yes	characteristic
272	C₂	#13116548	1	1	yes	characteristic
273	C₂	#13116876	1	1	yes	characteristic
274	C₂	#13116886	1	1	yes	characteristic
275	C₂	#13116978	1	1	yes	characteristic
276	C₂	#13116988	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#12812
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×C₁₃×Q₁₆	C₄	semidirect
64. C₁₃×Q₁₆	C₄⋊C₄	semidirect
125. C₄×Q₁₆	Dic₁₃	semidirect
144. C₄×C₁₃	#358	semidirect
201. Q₁₆	C₄⋊Dic₁₃	semidirect
222. C₁₃	#12812	semidirect
265. C₄	#11718156	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6090	2
C₂×#11718156	2
#13114996	2
#13112927	2
C₂×C₄×C₁₃×Q₁₆	2
#12812	13