Subgroups — #15767629

Order 3328.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114996	C₂	1	1	yes	maximal, characteristic
2	#13114841	C₂	1	2	yes	maximal
3	#13112720	C₂	1	2	yes	maximal
4	#13112671	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88075	—	2	2	no	—
10	C₂×#88058	—	2	1	no	—
11	C₂×#88058	—	2	1	no	—
12	#11718596	C₂²	1	2	yes	—
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₁₃×#336	—	2	1	no	—
15	C₁₃×#336	—	2	1	no	—
16	#11718121	—	2	2	no	—
17	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	—	2	1	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₃	—	2	2	no	—
22	C₄×Q₈×C₁₃	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
24	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	—	2	1	no	—
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	C₂×C₄⋊Dic₁₃	—	2	2	no	—
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	C₂²×#1055	—	2	2	no	—
31	C₁₃×C₄⋊C₈	—	2	2	no	—
32	C₄×C₈×C₁₃	—	2	2	no	—
33	#88075	—	4	4	no	—
34	#88058	—	4	2	no	—
35	#88058	—	4	2	no	—
36	C₄⋊#1055	—	2	2	no	—
37	C₄⋊#1055	—	2	2	no	—
38	#10617	—	13	1	no	maximal, Sylow
39	C₂×Q₈×C₁₃	—	2	1	no	—
40	C₂×Q₈×C₁₃	—	2	2	no	—
41	C₂×Q₈×C₁₃	—	2	2	no	—
42	C₂×Q₈×C₁₃	D₈	1	2	yes	—
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
48	C₂²×Dic₁₃	—	4	2	no	—
49	C₂×C₈×C₁₃	—	2	1	no	—
50	C₂×C₈×C₁₃	—	2	1	no	—
51	C₂×C₈×C₁₃	—	2	1	no	—
52	C₂×C₈×C₁₃	—	2	1	no	—
53	C₂×C₈×C₁₃	—	4	1	no	—
54	C₂×C₈×C₁₃	—	4	1	no	—
55	C₂×C₈×C₁₃	—	4	1	no	—
56	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	4	2	no	—
60	C₄²×C₁₃	—	4	1	no	—
61	C₄²×C₁₃	—	4	1	no	—
62	C₄²×C₁₃	#41	1	2	yes	—
63	C₄²×C₁₃	#41	1	2	yes	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	C₄⋊Dic₁₃	—	4	2	no	—
67	C₄⋊Dic₁₃	—	4	2	no	—
68	C₄⋊Dic₁₃	—	8	2	no	—
69	C₂×#1055	—	4	2	no	—
70	C₂×#1055	—	4	2	no	—
71	C₂×#1055	—	4	2	no	—
72	#1984	D₁₃	1	1	yes	characteristic
73	#1845	—	13	1	no	—
74	#1803	—	13	2	no	—
75	#1296	—	13	2	no	—
76	#1277	—	13	1	no	—
77	C₂³×C₁₃	#121	1	1	yes	characteristic
78	Q₈×C₁₃	—	2	4	no	—
79	Q₈×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	4	1	no	—
84	C₂×C₄×C₁₃	—	4	1	no	—
85	C₂×C₄×C₁₃	—	4	1	no	—
86	C₂×C₄×C₁₃	—	4	1	no	—
87	C₂×C₄×C₁₃	—	4	1	no	—
88	C₂×C₄×C₁₃	#123	1	2	yes	—
89	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#126	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
96	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#137	1	2	yes	—
98	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
99	C₂×Dic₁₃	—	8	2	no	—
100	C₂×Dic₁₃	—	8	2	no	—
101	C₂×Dic₁₃	—	8	2	no	—
102	C₈×C₁₃	—	2	2	no	—
103	C₈×C₁₃	—	2	2	no	—
104	C₈×C₁₃	—	8	1	no	—
105	#1055	—	8	2	no	—
106	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
107	C₂×#107	—	26	1	no	—
108	C₂×#107	—	26	1	no	—
109	C₂×C₄⋊C₈	—	13	2	no	—
110	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
111	C₂×#103	—	26	2	no	—
112	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
113	#384	—	13	2	no	—
114	#340	—	26	2	no	—
115	#336	—	2	1	no	—
116	#336	—	2	1	no	—
117	C₂²×C₁₃	#397	1	1	yes	characteristic
118	C₂²×C₁₃	#460	1	1	yes	characteristic
119	C₂²×C₁₃	#462	1	1	yes	characteristic
120	C₂²×C₁₃	#466	1	1	yes	characteristic
121	C₂²×C₁₃	#470	1	1	yes	characteristic
122	C₂²×C₁₃	#472	1	1	yes	characteristic
123	C₂²×C₁₃	#473	1	1	yes	characteristic
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	4	2	no	—
126	C₄×C₁₃	—	8	1	no	—
127	C₄×C₁₃	#480	1	2	yes	—
128	C₄×C₁₃	#483	1	1	yes	characteristic
129	C₄×C₁₃	#484	1	1	yes	characteristic
130	C₄×C₁₃	#485	1	2	yes	—
131	C₄×C₁₃	#487	1	1	yes	characteristic
132	C₄×C₁₃	#489	1	1	yes	characteristic
133	Dic₁₃	—	16	2	no	—
134	C₂²×Q₈	D₅₂	1	1	yes	characteristic
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	—	26	2	no	—
137	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
139	C₄×Q₈	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	26	2	no	—
143	C₂×C₄⋊C₄	—	26	2	no	—
144	C₂×C₄⋊C₄	—	26	2	no	—
145	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
148	#107	—	52	2	no	—
149	#107	—	52	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	26	2	no	—
152	C₄⋊C₈	—	26	2	no	—
153	#103	—	52	4	no	—
154	C₄×C₈	—	2	2	no	—
155	C₂×C₁₃	#1668	1	1	yes	characteristic
156	C₂×C₁₃	#1674	1	1	yes	characteristic
157	C₂×C₁₃	#1690	1	1	yes	characteristic
158	C₂×C₁₃	#1696	1	1	yes	characteristic
159	C₂×C₁₃	#2072	1	1	yes	characteristic
160	C₂×C₁₃	#2075	1	1	yes	characteristic
161	C₂×C₁₃	#2078	1	1	yes	characteristic
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	D₁₀₄	1	2	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	52	2	no	—
169	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
170	C₂²×C₄	#6127	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	52	2	no	—
180	C₂×C₈	—	52	2	no	—
181	C₂×C₈	—	52	2	no	—
182	C₂×C₈	#6106	1	2	yes	—
183	C₂×C₈	#6107	1	2	yes	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	52	2	no	—
187	C₄⋊C₄	—	52	2	no	—
188	C₄⋊C₄	—	52	2	no	—
189	C₄⋊C₄	—	52	2	no	—
190	C₄⋊C₄	—	104	2	no	—
191	C₄²	—	4	1	no	—
192	C₄²	—	4	1	no	—
193	C₄²	#6129	1	2	yes	—
194	C₄²	#6129	1	2	yes	—
195	C₁₃	#10617	1	1	yes	characteristic, complemented, Sylow
196	C₂³	#88184	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	4	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	104	2	no	—
208	C₂×C₄	—	104	2	no	—
209	C₂×C₄	—	104	2	no	—
210	C₂×C₄	#88133	1	2	yes	—
211	C₂×C₄	#88140	1	2	yes	—
212	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
213	C₂×C₄	#88158	1	1	yes	characteristic
214	C₂×C₄	#88163	1	2	yes	—
215	C₂×C₄	#88189	1	1	yes	characteristic
216	C₂×C₄	#88190	1	1	yes	characteristic
217	C₂×C₄	C₂×#6106	1	1	yes	characteristic
218	C₂×C₄	C₂×#6107	1	1	yes	characteristic
219	C₂×C₄	#88203	1	1	yes	characteristic
220	C₂×C₄	#88205	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	8	1	no	—
224	C₈	—	104	2	no	—
225	C₂²	#11718614	1	1	yes	characteristic
226	C₂²	#11718688	1	1	yes	characteristic
227	C₂²	#11718692	1	1	yes	characteristic
228	C₂²	#11718776	1	1	yes	characteristic
229	C₂²	#11718798	1	1	yes	characteristic
230	C₂²	#11718803	1	1	yes	characteristic
231	C₂²	#11718804	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	1	no	—
235	C₄	—	208	2	no	—
236	C₄	#11718384	1	2	yes	—
237	C₄	#11718397	1	2	yes	—
238	C₄	#11718704	1	1	yes	characteristic
239	C₄	#11718705	1	1	yes	characteristic
240	C₄	#11718716	1	1	yes	characteristic
241	C₄	#11718742	1	1	yes	characteristic
242	C₂	#13114196	1	1	yes	characteristic
243	C₂	#13114230	1	1	yes	characteristic
244	C₂	#13114330	1	1	yes	characteristic
245	C₂	#13114364	1	1	yes	characteristic
246	C₂	#13116244	1	1	yes	characteristic
247	C₂	#13116259	1	1	yes	characteristic
248	C₂	#13116276	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88184
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1984	C₂
socle, prime radical	C₂³×C₁₃	#121
hypercenter	#1984	D₁₃
hypocenter	C₁₃	#10617
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

Complements

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

N	H ≅ G/N	type
195. C₁₃	#10617	semidirect

Maximal subgroups

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

subgroup	index
#13114996	2
#13114841	2
#13112720	2
#13112671	2
C₁₃×#1984	2
#10617	13