Subgroups — #15781458

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
2	#13116592	C₂	1	1	yes	maximal, characteristic
3	#13112919	C₂	1	4	yes	maximal
4	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88055	C₂²	1	2	yes	—
9	C₂×#88054	C₄	1	2	yes	—
10	C₂×#88052	C₂²	1	2	yes	—
11	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
12	#11718340	—	2	1	no	—
13	#11718340	—	2	1	no	—
14	#11718340	—	2	1	no	—
15	#11718340	—	2	1	no	—
16	C₁₃×#346	—	2	1	no	—
17	C₄×C₁₃×C₁₆	—	2	1	no	—
18	C₄⋊#6090	—	2	1	no	—
19	C₄⋊#6090	—	2	1	no	—
20	C₂²×Dic₂₆	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
25	C₂×#6101	C₈	1	2	yes	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	2	yes	—
28	C₄×Dic₂₆	—	2	1	no	—
29	C₄×Dic₂₆	—	2	1	no	—
30	C₄×Dic₂₆	—	2	1	no	—
31	C₄×Dic₂₆	—	2	1	no	—
32	C₁₃×M₃₂	—	2	2	no	—
33	C₁₃×M₃₂	Q₈	1	4	yes	complemented
34	C₂×C₁₃×C₁₆	—	2	2	no	—
35	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
39	#88055	—	2	2	no	—
40	#88055	—	2	2	no	—
41	#88054	—	2	2	no	—
42	#88054	—	2	2	no	—
43	#88052	—	2	4	no	—
44	C₂×#6090	—	2	2	no	—
45	C₄⋊#1055	—	2	1	no	—
46	C₄⋊#1055	—	2	1	no	—
47	#13268	—	13	1	no	maximal, Sylow
48	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
52	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
53	C₂×Dic₂₆	—	2	1	no	—
54	C₂×Dic₂₆	—	2	1	no	—
55	C₂×Dic₂₆	—	4	2	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₁₃	#41	1	2	yes	—
63	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
66	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
67	C₄⋊Dic₁₃	—	2	1	no	—
68	C₄⋊Dic₁₃	—	2	1	no	—
69	#6101	—	2	2	no	—
70	#6101	—	2	2	no	—
71	C₄×Dic₁₃	—	2	2	no	—
72	C₄×Dic₁₃	—	2	2	no	—
73	C₂×#1055	—	2	2	no	—
74	C₂×#1055	—	2	4	no	—
75	C₁₃×C₁₆	—	2	4	no	—
76	#6090	—	4	2	no	—
77	C₂×#446	—	13	1	no	—
78	#2150	—	13	1	no	—
79	C₄×M₃₂	D₁₃	1	1	yes	characteristic
80	#1377	—	13	4	no	—
81	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	4	1	no	—
84	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#131	1	2	yes	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	—	2	2	no	—
94	C₂×Dic₁₃	—	4	2	no	—
95	C₂×Dic₁₃	—	4	2	no	—
96	Dic₂₆	—	4	2	no	—
97	Dic₂₆	—	4	2	no	—
98	C₈×C₁₃	—	2	1	no	—
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
101	C₈×C₁₃	#124	1	1	yes	characteristic
102	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
103	#1055	—	4	2	no	—
104	C₂×C₄×Q₈	—	13	1	no	—
105	C₂×M₃₂	—	13	2	no	—
106	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
107	#446	—	26	1	no	—
108	#446	—	26	1	no	—
109	#446	—	26	1	no	—
110	#446	—	26	1	no	—
111	C₂×C₄⋊C₈	—	13	2	no	—
112	C₂×C₄⋊C₈	—	13	1	no	—
113	C₂×#97	—	13	2	no	—
114	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
115	C₄⋊C₁₆	—	26	1	no	—
116	C₄⋊C₁₆	—	26	1	no	—
117	#346	—	2	1	no	—
118	C₄×C₁₆	—	2	1	no	—
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
123	C₄×C₁₃	—	2	1	no	—
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	#349	1	2	yes	—
127	C₄×C₁₃	#391	1	1	yes	characteristic
128	C₄×C₁₃	#408	1	1	yes	characteristic
129	C₄×C₁₃	#446	1	2	yes	—
130	Dic₁₃	—	4	2	no	—
131	Dic₁₃	—	4	2	no	—
132	C₂²×Q₈	—	13	1	no	—
133	C₂²×C₈	—	13	2	no	—
134	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
136	C₄×Q₈	—	26	1	no	—
137	C₄×Q₈	—	26	1	no	—
138	C₄×Q₈	—	26	1	no	—
139	C₄×Q₈	—	26	1	no	—
140	C₂×C₄⋊C₄	—	13	2	no	—
141	C₂×C₄⋊C₄	—	13	1	no	—
142	C₂×C₄²	—	13	2	no	—
143	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
144	M₃₂	—	2	2	no	—
145	M₃₂	—	26	4	no	—
146	M₃₂	Dic₂₆	1	4	yes	complemented
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	26	2	no	—
149	C₄⋊C₈	—	26	2	no	—
150	C₄⋊C₈	—	26	2	no	—
151	C₄⋊C₈	—	26	1	no	—
152	C₄⋊C₈	—	26	1	no	—
153	#97	—	26	2	no	—
154	#97	—	26	2	no	—
155	C₄×C₈	D₅₂	1	1	yes	characteristic
156	C₄×C₈	D₅₂	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₂×C₁₃	—	2	2	no	—
160	C₂×C₁₃	#1984	1	1	yes	characteristic
161	C₂×C₁₃	C₂×#349	1	1	yes	characteristic
162	C₂×C₁₃	#2115	1	1	yes	characteristic
163	C₂×Q₈	—	26	1	no	—
164	C₂×Q₈	—	26	1	no	—
165	C₂×Q₈	—	52	2	no	—
166	C₂²×C₄	—	13	2	no	—
167	C₂²×C₄	—	13	2	no	—
168	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	26	4	no	—
173	C₂×C₈	—	26	2	no	—
174	C₂×C₈	#6103	1	1	yes	characteristic
175	C₂×C₈	#6103	1	1	yes	characteristic
176	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
177	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
178	C₂×C₈	#6127	1	2	yes	—
179	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
180	C₄⋊C₄	—	26	2	no	—
181	C₄⋊C₄	—	26	2	no	—
182	C₄⋊C₄	—	26	1	no	—
183	C₄⋊C₄	—	26	1	no	—
184	C₄²	—	26	2	no	—
185	C₄²	—	26	2	no	—
186	C₄²	#6094	1	1	yes	characteristic
187	C₄²	#6094	1	1	yes	characteristic
188	C₄²	#6103	1	1	yes	characteristic
189	C₄²	#6103	1	1	yes	characteristic
190	C₁₆	—	2	4	no	—
191	C₁₆	—	52	2	no	—
192	C₁₃	#13268	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
194	Q₈	—	52	2	no	—
195	Q₈	—	52	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	26	2	no	—
199	C₂×C₄	—	26	2	no	—
200	C₂×C₄	—	52	2	no	—
201	C₂×C₄	—	52	2	no	—
202	C₂×C₄	#88060	1	1	yes	characteristic
203	C₂×C₄	#88060	1	1	yes	characteristic
204	C₂×C₄	#88060	1	1	yes	characteristic
205	C₂×C₄	#88060	1	1	yes	characteristic
206	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
207	C₂×C₄	C₂×#6094	1	1	yes	characteristic
208	C₂×C₄	#88155	1	2	yes	—
209	C₂×C₄	C₂×#6103	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	52	2	no	—
213	C₈	#88123	1	1	yes	characteristic
214	C₈	#88129	1	1	yes	characteristic
215	C₈	#88178	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₈×Dic₂₆	1	2	yes	—
219	C₂²	C₂×#88060	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	52	2	no	—
224	C₄	—	52	2	no	—
225	C₄	#11718106	1	2	yes	—
226	C₄	#11718340	1	2	yes	—
227	C₄	#11718590	1	1	yes	characteristic
228	C₄	#11718771	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	#13115772	1	1	yes	characteristic
231	C₂	C₂×#11718106	1	1	yes	characteristic
232	C₂	#13116456	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#6103
commutator, prime radical	C₂³×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#13268
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
33. C₁₃×M₃₂	Q₈	semidirect
146. M₃₂	Dic₂₆	semidirect
192. C₁₃	#13268	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718340	2
#13116592	2
#13112919	2
C₄×C₁₃×M₃₂	2
#13268	13