Subgroups — #15756579

Order 3328.

422 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114988	C₂	1	1	yes	maximal, characteristic
2	C₄²×#1055	C₂	1	1	yes	maximal, characteristic
3	#13112670	C₂	1	4	yes	maximal
4	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
6	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	—	2	1	no	—
10	C₂×C₄⋊#1055	—	2	1	no	—
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₄	1	2	yes	—
13	C₂×C₄×#1055	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×#1055	—	2	2	no	—
25	C₂²×#1055	C₂×C₄	1	2	yes	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₁₃×C₄⋊C₈	—	2	4	no	—
28	C₁₃×C₄⋊C₈	—	4	2	no	—
29	C₁₃×C₄⋊C₈	—	4	2	no	—
30	C₄⋊#1055	—	2	4	no	—
31	C₄⋊#1055	—	4	2	no	—
32	C₄⋊#1055	—	4	2	no	—
33	C₄×#1055	—	2	2	no	—
34	C₄×#1055	—	2	2	no	—
35	C₄×#1055	—	2	2	no	—
36	C₄×#1055	—	2	4	no	—
37	C₄×#1055	C₈	1	4	yes	complemented
38	#8784	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₃	—	4	2	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₂×C₈×C₁₃	—	4	2	no	—
49	C₄²×C₁₃	—	2	2	no	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₄²×C₁₃	—	2	2	no	—
52	C₄²×C₁₃	C₂×C₈	1	2	yes	—
53	C₄²×C₁₃	M₁₆	1	2	yes	—
54	C₄²×C₁₃	D₈	1	2	yes	—
55	C₄²×C₁₃	SD₁₆	1	2	yes	—
56	C₄²×C₁₃	SD₁₆	1	2	yes	—
57	C₄²×C₁₃	Q₁₆	1	2	yes	—
58	C₄²×C₁₃	#41	1	2	yes	—
59	C₄²×C₁₃	#41	1	2	yes	—
60	C₂×#1055	—	2	2	no	—
61	C₂×#1055	—	2	2	no	—
62	C₂×#1055	—	2	2	no	—
63	C₂×#1055	—	2	2	no	—
64	C₂×#1055	—	4	2	no	—
65	C₂×#1055	—	4	2	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	C₂×C₈	1	4	yes	—
68	C₂×#1055	M₁₆	1	4	yes	—
69	#1842	—	13	1	no	—
70	#1842	D₁₃	1	1	yes	characteristic
71	C₄²×C₈	—	13	1	no	—
72	#1277	—	13	4	no	—
73	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
82	C₂×C₄×C₁₃	#106	1	2	yes	—
83	C₂×C₄×C₁₃	#106	1	2	yes	—
84	C₂×C₄×C₁₃	#107	1	2	yes	—
85	C₂×C₄×C₁₃	#107	1	2	yes	—
86	C₂×C₄×C₁₃	#108	1	2	yes	—
87	C₂×C₄×C₁₃	#108	1	2	yes	—
88	C₂×C₄×C₁₃	#117	1	2	yes	—
89	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#125	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
99	C₈×C₁₃	—	8	2	no	—
100	#1055	—	2	4	no	—
101	#1055	—	2	4	no	—
102	#1055	—	8	2	no	—
103	C₂×C₄⋊C₈	—	2	1	no	—
104	C₂×C₄⋊C₈	—	2	1	no	—
105	C₂×C₄⋊C₈	—	13	2	no	—
106	C₂×C₄⋊C₈	—	26	1	no	—
107	C₂×C₄⋊C₈	—	26	1	no	—
108	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
109	C₂×C₄×C₈	—	13	1	no	—
110	C₂×C₄×C₈	—	13	2	no	—
111	C₂×C₄×C₈	—	13	1	no	—
112	C₂×C₄×C₈	—	13	2	no	—
113	C₄³	C₂×D₁₃	1	1	yes	characteristic
114	C₂²×C₁₃	#382	1	1	yes	characteristic
115	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₁₃	#423	1	1	yes	characteristic
117	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
119	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
120	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	#334	1	4	yes	—
125	C₄×C₁₃	#335	1	4	yes	—
126	C₄×C₁₃	#431	1	2	yes	—
127	C₄×C₁₃	#432	1	2	yes	—
128	C₄×C₁₃	#486	1	1	yes	characteristic
129	C₄×C₁₃	#486	1	1	yes	characteristic
130	C₄×C₁₃	#487	1	1	yes	characteristic
131	C₄×C₁₃	#487	1	1	yes	characteristic
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	13	2	no	—
134	C₂²×C₈	—	13	2	no	—
135	C₂²×C₈	—	26	2	no	—
136	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
137	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
138	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
139	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
141	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
142	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
143	C₄⋊C₈	—	2	4	no	—
144	C₄⋊C₈	—	4	2	no	—
145	C₄⋊C₈	—	4	2	no	—
146	C₄⋊C₈	—	26	4	no	—
147	C₄⋊C₈	—	52	2	no	—
148	C₄⋊C₈	—	52	2	no	—
149	C₄×C₈	—	13	4	no	—
150	C₄×C₈	—	26	4	no	—
151	C₄×C₈	—	26	2	no	—
152	C₄×C₈	—	26	2	no	—
153	C₄×C₈	—	26	2	no	—
154	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
155	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
156	C₂×C₁₃	#1566	1	1	yes	characteristic
157	C₂×C₁₃	#1567	1	1	yes	characteristic
158	C₂×C₁₃	#1831	1	1	yes	characteristic
159	C₂×C₁₃	#1834	1	1	yes	characteristic
160	C₂×C₁₃	#1837	1	1	yes	characteristic
161	C₂²×C₄	#6101	1	1	yes	characteristic
162	C₂²×C₄	#6101	1	1	yes	characteristic
163	C₂²×C₄	#6101	1	1	yes	characteristic
164	C₂²×C₄	#6101	1	1	yes	characteristic
165	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	13	4	no	—
172	C₂×C₈	—	13	4	no	—
173	C₂×C₈	—	26	2	no	—
174	C₂×C₈	—	26	2	no	—
175	C₂×C₈	—	26	2	no	—
176	C₂×C₈	—	26	2	no	—
177	C₂×C₈	—	52	2	no	—
178	C₂×C₈	—	52	2	no	—
179	C₂×C₈	—	52	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₈×D₁₃	1	2	yes	—
184	C₄²	#6094	1	2	yes	—
185	C₄²	#6104	1	2	yes	—
186	C₄²	#6105	1	2	yes	—
187	C₄²	#6106	1	2	yes	—
188	C₄²	#6107	1	2	yes	—
189	C₄²	#6129	1	2	yes	—
190	C₄²	#6131	1	2	yes	—
191	C₁₃	#8784	1	1	yes	characteristic, complemented, Sylow
192	C₂³	C₂×#6101	1	1	yes	characteristic
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	#88047	1	2	yes	—
200	C₂×C₄	#88047	1	2	yes	—
201	C₂×C₄	#88048	1	2	yes	—
202	C₂×C₄	#88048	1	2	yes	—
203	C₂×C₄	#88054	1	2	yes	—
204	C₂×C₄	#88054	1	2	yes	—
205	C₂×C₄	#88062	1	2	yes	—
206	C₂×C₄	#88062	1	2	yes	—
207	C₂×C₄	#88144	1	2	yes	—
208	C₂×C₄	#88146	1	2	yes	—
209	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×#6094	1	1	yes	characteristic
211	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
212	C₂×C₄	C₂×#6104	1	1	yes	characteristic
213	C₂×C₄	C₂×#6105	1	1	yes	characteristic
214	C₂×C₄	#88190	1	1	yes	characteristic
215	C₂×C₄	C₂×#6106	1	1	yes	characteristic
216	C₂×C₄	C₂×#6107	1	1	yes	characteristic
217	C₂×C₄	#88201	1	1	yes	characteristic
218	C₈	—	8	2	no	—
219	C₈	—	26	4	no	—
220	C₈	—	26	4	no	—
221	C₈	—	104	2	no	—
222	C₂²	C₂×#88047	1	1	yes	characteristic
223	C₂²	C₂×#88048	1	1	yes	characteristic
224	C₂²	#11718638	1	1	yes	characteristic
225	C₂²	C₂×#88054	1	1	yes	characteristic
226	C₂²	#11718779	1	1	yes	characteristic
227	C₂²	C₂×#88062	1	1	yes	characteristic
228	C₂²	C₂×#88062	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	#11718071	1	4	yes	—
233	C₄	#11718072	1	4	yes	—
234	C₄	#11718494	1	2	yes	—
235	C₄	#11718496	1	2	yes	—
236	C₄	#11718731	1	1	yes	characteristic
237	C₄	#11718742	1	1	yes	characteristic
238	C₄	#11718749	1	1	yes	characteristic
239	C₄	#11718754	1	1	yes	characteristic
240	C₂	C₂×#11718072	1	1	yes	characteristic
241	C₂	C₂×#11718071	1	1	yes	characteristic
242	C₂	#13113591	1	1	yes	characteristic
243	C₂	#13113594	1	1	yes	characteristic
244	C₂	#13114952	1	1	yes	characteristic
245	C₂	#13114963	1	1	yes	characteristic
246	C₂	#13114973	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1842	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1842	D₁₃
hypocenter	C₁₃	#8784
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
37. C₄×#1055	C₈	semidirect
191. C₁₃	#8784	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
C₄²×#1055	2
#13112670	2
C₁₃×#1842	2
#8784	13