Subgroups — #15778976

Order 3328.

430 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718095	C₂	1	2	yes	maximal
3	#13115000	C₂	1	1	yes	maximal, characteristic
4	#13112928	C₂	1	2	yes	maximal
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88058	C₂²	1	2	yes	—
11	C₂×#88058	C₂²	1	1	yes	characteristic
12	C₂×#88057	—	2	1	no	—
13	C₂×#88056	—	2	1	no	—
14	#11718597	—	2	1	no	—
15	C₄×Dic₅₂	—	2	2	no	—
16	C₄×Dic₅₂	C₄	1	4	yes	complemented
17	C₁₃×C₄⋊C₁₆	—	2	2	no	—
18	#11718095	—	2	4	no	—
19	C₂²×Dic₂₆	—	2	1	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	—	2	1	no	—
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×#6101	—	4	1	no	—
26	C₂×C₄×Dic₁₃	—	4	1	no	—
27	C₂×Dic₅₂	—	2	1	no	—
28	C₂×Dic₅₂	—	2	1	no	—
29	C₂×Dic₅₂	—	2	1	no	—
30	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
31	C₂×Dic₅₂	D₄	1	2	yes	—
32	C₂×Dic₅₂	Q₈	1	2	yes	—
33	C₄×Dic₂₆	—	2	4	no	—
34	C₄×Dic₂₆	—	4	2	no	—
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₂×C₁₃×C₁₆	—	2	2	no	—
37	C₂×C₁₃×C₁₆	—	2	2	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	#88071	—	4	1	no	—
41	#88071	—	4	1	no	—
42	#88058	—	2	1	no	—
43	#88058	—	2	2	no	—
44	#88058	—	2	2	no	—
45	#88058	C₂×C₄	1	2	yes	—
46	#88057	—	4	2	no	—
47	#88056	—	2	4	no	—
48	#12814	—	13	1	no	maximal, Sylow
49	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	4	1	no	—
53	C₂²×Dic₁₃	—	4	1	no	—
54	C₂²×Dic₁₃	—	4	2	no	—
55	C₂×Dic₂₆	—	2	2	no	—
56	C₂×Dic₂₆	—	2	2	no	—
57	C₂×Dic₂₆	—	2	2	no	—
58	C₂×Dic₂₆	—	4	1	no	—
59	C₂×Dic₂₆	—	4	1	no	—
60	C₂×Dic₂₆	—	4	1	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
69	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
70	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₁₃	SD₁₆	1	2	yes	—
72	C₄⋊Dic₁₃	—	2	2	no	—
73	C₄⋊Dic₁₃	—	4	1	no	—
74	C₄⋊Dic₁₃	—	4	2	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	#6101	—	4	4	no	—
77	C₄×Dic₁₃	—	4	4	no	—
78	Dic₅₂	—	2	2	no	—
79	Dic₅₂	—	2	2	no	—
80	Dic₅₂	—	2	2	no	—
81	Dic₅₂	C₄⋊C₄	1	4	yes	—
82	C₁₃×C₁₆	—	4	2	no	—
83	C₂×C₄×Q₁₆	—	13	1	no	—
84	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic
85	C₂×#358	—	13	2	no	—
86	#1846	—	13	1	no	—
87	#1380	—	13	2	no	—
88	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	#102	1	2	yes	—
92	C₂×C₄×C₁₃	#102	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
99	C₂×Dic₁₃	—	4	2	no	—
100	C₂×Dic₁₃	—	4	2	no	—
101	C₂×Dic₁₃	—	4	2	no	—
102	C₂×Dic₁₃	—	4	2	no	—
103	C₂×Dic₁₃	—	8	2	no	—
104	C₂×Dic₁₃	—	8	2	no	—
105	C₂×Dic₁₃	—	8	2	no	—
106	C₂×Dic₁₃	—	8	2	no	—
107	Dic₂₆	—	2	4	no	—
108	Dic₂₆	—	4	2	no	—
109	Dic₂₆	—	4	2	no	—
110	Dic₂₆	—	4	2	no	—
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₈×C₁₃	#117	1	1	yes	characteristic
114	C₈×C₁₃	#121	1	1	yes	characteristic
115	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂²×Q₁₆	—	13	1	no	—
117	C₂×C₄×Q₈	—	26	1	no	—
118	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
119	C₄×Q₁₆	—	13	4	no	—
120	C₄×Q₁₆	—	26	2	no	—
121	C₂×#107	—	13	1	no	—
122	C₂×#107	—	13	2	no	—
123	C₂×#106	—	26	1	no	—
124	C₂×#103	—	26	1	no	—
125	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
126	#382	—	26	1	no	—
127	C₄⋊C₁₆	—	2	2	no	—
128	#358	—	26	4	no	—
129	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
130	C₂²×C₁₃	#459	1	1	yes	characteristic
131	C₂²×C₁₃	#474	1	1	yes	characteristic
132	C₂²×C₁₃	#508	1	2	yes	—
133	C₂²×C₁₃	#510	1	2	yes	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	#380	1	1	yes	characteristic
137	C₄×C₁₃	#417	1	1	yes	characteristic
138	C₄×C₁₃	#461	1	1	yes	characteristic
139	C₄×C₁₃	#476	1	1	yes	characteristic
140	Dic₁₃	—	4	4	no	—
141	Dic₁₃	—	8	2	no	—
142	Dic₁₃	—	16	2	no	—
143	C₂²×Q₈	—	26	1	no	—
144	C₂×Q₁₆	—	13	2	no	—
145	C₂×Q₁₆	—	13	2	no	—
146	C₂×Q₁₆	—	13	2	no	—
147	C₂×Q₁₆	—	26	1	no	—
148	C₂×Q₁₆	—	26	1	no	—
149	C₂×Q₁₆	—	26	1	no	—
150	C₂²×C₈	D₅₂	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
152	C₄×Q₈	—	26	4	no	—
153	C₄×Q₈	—	52	2	no	—
154	C₂×C₄⋊C₄	—	26	1	no	—
155	C₂×C₄⋊C₄	—	26	2	no	—
156	C₂×C₄⋊C₄	—	52	1	no	—
157	C₂×C₄²	—	52	1	no	—
158	C₂×C₄²	D₅₂	1	1	yes	characteristic
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	#107	—	13	2	no	—
163	#107	—	26	1	no	—
164	#107	—	26	2	no	—
165	#107	—	26	2	no	—
166	#106	—	52	2	no	—
167	#103	—	26	4	no	—
168	C₄×C₈	C₄×D₁₃	1	2	yes	—
169	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
170	#95	—	52	1	no	—
171	#95	—	52	1	no	—
172	C₂×C₁₃	#1802	1	1	yes	characteristic
173	C₂×C₁₃	#2138	1	1	yes	characteristic
174	C₂×C₁₃	#2141	1	1	yes	characteristic
175	C₂×C₁₃	#2211	1	2	yes	—
176	C₂×C₁₃	#2229	1	2	yes	—
177	C₂×Q₈	—	26	2	no	—
178	C₂×Q₈	—	26	2	no	—
179	C₂×Q₈	—	26	2	no	—
180	C₂×Q₈	—	52	1	no	—
181	C₂×Q₈	—	52	1	no	—
182	C₂×Q₈	—	52	1	no	—
183	C₂²×C₄	—	52	1	no	—
184	C₂²×C₄	—	52	1	no	—
185	C₂²×C₄	—	52	2	no	—
186	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
187	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
189	Q₁₆	—	13	4	no	—
190	Q₁₆	—	26	2	no	—
191	Q₁₆	—	26	2	no	—
192	Q₁₆	—	26	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	#6103	1	2	yes	—
196	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
197	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
198	C₂×C₈	#6129	1	1	yes	characteristic
199	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
200	C₂×C₈	#6131	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
202	C₄⋊C₄	—	26	2	no	—
203	C₄⋊C₄	—	52	1	no	—
204	C₄⋊C₄	—	52	4	no	—
205	C₄⋊C₄	—	52	2	no	—
206	C₄⋊C₄	—	52	2	no	—
207	C₄²	—	52	4	no	—
208	C₄²	#6095	1	2	yes	—
209	C₄²	#6103	1	2	yes	—
210	C₁₆	—	4	2	no	—
211	C₁₃	#12814	1	1	yes	characteristic, complemented, Sylow
212	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
213	Q₈	—	26	4	no	—
214	Q₈	—	52	2	no	—
215	Q₈	—	52	2	no	—
216	Q₈	—	52	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	52	2	no	—
220	C₂×C₄	—	52	2	no	—
221	C₂×C₄	—	52	2	no	—
222	C₂×C₄	—	52	2	no	—
223	C₂×C₄	—	104	2	no	—
224	C₂×C₄	—	104	2	no	—
225	C₂×C₄	—	104	2	no	—
226	C₂×C₄	—	104	2	no	—
227	C₂×C₄	#88061	1	2	yes	—
228	C₂×C₄	#88061	1	2	yes	—
229	C₂×C₄	#88149	1	1	yes	characteristic
230	C₂×C₄	#88151	1	1	yes	characteristic
231	C₂×C₄	C₂×#6095	1	1	yes	characteristic
232	C₂×C₄	#88162	1	1	yes	characteristic
233	C₂×C₄	#88163	1	1	yes	characteristic
234	C₂×C₄	C₂×#6103	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
237	C₈	#88146	1	1	yes	characteristic
238	C₈	#88192	1	1	yes	characteristic
239	C₈	#88200	1	1	yes	characteristic
240	C₂²	#11718504	1	1	yes	characteristic
241	C₂²	#11718507	1	1	yes	characteristic
242	C₂²	#11718565	1	2	yes	—
243	C₂²	#11718570	1	2	yes	—
244	C₂²	C₂×#88061	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	52	4	no	—
248	C₄	—	104	2	no	—
249	C₄	—	208	2	no	—
250	C₄	#11718502	1	1	yes	characteristic
251	C₄	#11718503	1	1	yes	characteristic
252	C₄	#11718648	1	1	yes	characteristic
253	C₄	#11718794	1	1	yes	characteristic
254	C₂	#13114837	1	1	yes	characteristic
255	C₂	#13116544	1	1	yes	characteristic
256	C₂	#13116557	1	1	yes	characteristic
257	C₂	#13116899	1	2	yes	—
258	C₂	#13117001	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	#12814
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
16. C₄×Dic₅₂	C₄	semidirect
211. C₁₃	#12814	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
C₂×#11718095	2
#13115000	2
#13112928	2
C₂×C₁₃×C₄⋊C₁₆	2
#12814	13