Subgroups — #15756662

Order 3328.

422 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114953	C₂	1	1	yes	maximal, characteristic
2	C₄²×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718071	C₂	1	2	yes	maximal
4	C₂×#11718072	C₂	1	2	yes	maximal
5	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄³×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#1055	C₂²	1	2	yes	—
9	C₂×C₄×#1055	C₄	1	1	yes	characteristic
10	C₂×C₄×#1055	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₄	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	C₁₃×#336	—	2	2	no	—
16	#11718116	—	2	2	no	—
17	#11718072	—	2	4	no	—
18	#11718071	—	2	4	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	2	yes	—
23	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×#1055	—	2	2	no	—
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₁₃×C₄⋊C₈	—	2	2	no	—
31	C₁₃×C₄⋊C₈	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
33	C₄⋊#1055	—	2	2	no	—
34	C₄×#1055	—	2	2	no	—
35	C₄×#1055	—	2	2	no	—
36	C₄×#1055	—	2	2	no	—
37	C₄×#1055	C₈	1	4	yes	complemented
38	C₄×#1055	D₄	1	2	yes	—
39	C₄×#1055	D₄	1	2	yes	—
40	C₄×#1055	Q₈	1	2	yes	—
41	C₄×#1055	Q₈	1	2	yes	—
42	#8807	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	—	4	2	no	—
49	C₂×C₈×C₁₃	—	4	4	no	—
50	C₄²×C₁₃	—	2	4	no	—
51	C₄²×C₁₃	—	2	1	no	—
52	C₄²×C₁₃	—	2	1	no	—
53	C₄²×C₁₃	—	2	4	no	—
54	C₄²×C₁₃	C₂×C₈	1	2	yes	—
55	C₄²×C₁₃	M₁₆	1	2	yes	—
56	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	—	2	1	no	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	—	4	2	no	—
65	C₂×#1055	—	4	4	no	—
66	C₂×#1055	C₄⋊C₄	1	2	yes	—
67	C₂×#1055	C₄⋊C₄	1	2	yes	—
68	C₂×#1055	C₄⋊C₄	1	2	yes	—
69	C₂×#1055	C₄⋊C₄	1	2	yes	—
70	C₂×#1055	C₂×C₈	1	2	yes	—
71	C₂×#1055	C₂×C₈	1	2	yes	—
72	C₂×#1055	M₁₆	1	2	yes	—
73	C₂×#1055	M₁₆	1	2	yes	—
74	#1831	—	13	1	no	—
75	#1831	D₁₃	1	1	yes	characteristic
76	C₄²×C₈	—	13	1	no	—
77	C₂×#335	—	13	2	no	—
78	C₂×#334	—	13	2	no	—
79	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₁₃	#108	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#135	1	4	yes	—
98	C₈×C₁₃	—	8	2	no	—
99	#1055	—	2	2	no	—
100	#1055	—	2	2	no	—
101	#1055	—	8	2	no	—
102	#1055	C₄⋊C₈	1	4	yes	—
103	#1055	C₄⋊C₈	1	4	yes	—
104	C₂×C₄⋊C₈	—	13	2	no	—
105	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₂×C₄×C₈	—	13	1	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₂×C₄×C₈	—	13	1	no	—
110	C₂×C₄×C₈	—	13	1	no	—
111	C₂×C₄×C₈	—	13	1	no	—
112	C₄³	C₂×D₁₃	1	1	yes	characteristic
113	#336	—	2	2	no	—
114	#336	—	26	2	no	—
115	#335	—	26	4	no	—
116	#334	—	26	4	no	—
117	C₂²×C₁₃	#384	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₁₃	#423	1	1	yes	characteristic
120	C₂²×C₁₃	#427	1	2	yes	—
121	C₂²×C₁₃	C₂×#108	1	2	yes	—
122	C₄×C₁₃	—	2	4	no	—
123	C₄×C₁₃	—	2	4	no	—
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
127	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₄×C₁₃	#423	1	1	yes	characteristic
129	C₄×C₁₃	#423	1	1	yes	characteristic
130	C₄×C₁₃	#495	1	2	yes	—
131	C₄×C₁₃	#499	1	2	yes	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	13	1	no	—
134	C₂²×C₈	—	13	1	no	—
135	C₂²×C₈	—	13	1	no	—
136	C₂²×C₈	—	13	1	no	—
137	C₂²×C₈	—	26	2	no	—
138	C₂×C₄²	Dic₂₆	1	2	yes	—
139	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
140	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
141	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
142	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	26	2	no	—
146	C₄⋊C₈	—	26	2	no	—
147	C₄×C₈	—	13	4	no	—
148	C₄×C₈	—	13	2	no	—
149	C₄×C₈	—	13	2	no	—
150	C₄×C₈	—	13	2	no	—
151	C₄×C₈	—	13	2	no	—
152	C₄×C₈	—	26	2	no	—
153	C₄×C₈	—	26	2	no	—
154	C₄×C₈	—	26	2	no	—
155	C₂×C₁₃	#1564	1	2	yes	—
156	C₂×C₁₃	#1568	1	2	yes	—
157	C₂×C₁₃	#1842	1	1	yes	characteristic
158	C₂×C₁₃	#1846	1	1	yes	characteristic
159	C₂×C₁₃	#1848	1	1	yes	characteristic
160	C₂²×C₄	#6101	1	2	yes	—
161	C₂²×C₄	#6101	1	2	yes	—
162	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	4	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	13	2	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	13	2	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	13	2	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	26	1	no	—
176	C₂×C₈	—	26	1	no	—
177	C₂×C₈	—	26	1	no	—
178	C₂×C₈	—	26	1	no	—
179	C₂×C₈	—	52	2	no	—
180	C₂×C₈	—	52	4	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	4	no	—
184	C₄²	—	2	4	no	—
185	C₄²	C₈×D₁₃	1	2	yes	—
186	C₄²	#6094	1	2	yes	—
187	C₄²	D₄×D₁₃	1	1	yes	characteristic
188	C₄²	D₄×D₁₃	1	1	yes	characteristic
189	C₄²	Q₈×D₁₃	1	1	yes	characteristic
190	C₄²	Q₈×D₁₃	1	1	yes	characteristic
191	C₁₃	#8807	1	1	yes	characteristic, complemented, Sylow
192	C₂³	C₂×#6101	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	#88054	1	4	yes	—
199	C₂×C₄	#88062	1	4	yes	—
200	C₂×C₄	#88142	1	1	yes	characteristic
201	C₂×C₄	#88142	1	1	yes	characteristic
202	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
203	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
205	C₂×C₄	C₂×#6094	1	1	yes	characteristic
206	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
208	C₂×C₄	#88194	1	1	yes	characteristic
209	C₂×C₄	#88198	1	1	yes	characteristic
210	C₂×C₄	#88204	1	4	yes	—
211	C₈	—	8	2	no	—
212	C₈	—	13	4	no	—
213	C₈	—	13	4	no	—
214	C₈	—	26	2	no	—
215	C₈	—	26	2	no	—
216	C₈	—	104	2	no	—
217	C₂²	#11718637	1	1	yes	characteristic
218	C₂²	#11718659	1	2	yes	—
219	C₂²	C₂×#88054	1	1	yes	characteristic
220	C₂²	#11718779	1	1	yes	characteristic
221	C₂²	C₂×#88062	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
227	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
228	C₄	#11718498	1	1	yes	characteristic
229	C₄	#11718498	1	1	yes	characteristic
230	C₄	#11718720	1	2	yes	—
231	C₄	#11718725	1	2	yes	—
232	C₂	#13113582	1	2	yes	—
233	C₂	#13113597	1	2	yes	—
234	C₂	#13114987	1	1	yes	characteristic
235	C₂	#13114998	1	1	yes	characteristic
236	C₂	#13115007	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×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#8807
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
37. C₄×#1055	C₈	semidirect
191. C₁₃	#8807	semidirect

Maximal subgroups

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

subgroup	index
#13114953	2
C₄²×#1055	2
C₂×#11718071	2
C₂×#11718072	2
C₁₃×#1831	2
#8807	13