Subgroups — #15778251

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116592	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718104	C₂	1	2	yes	maximal
3	C₄×#88055	C₂	1	1	yes	maximal, characteristic
4	#13112915	C₂	1	2	yes	maximal
5	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88055	C₄	1	1	yes	characteristic
9	C₂×#88055	C₄	1	1	yes	characteristic
10	C₂×#88055	C₂²	1	1	yes	characteristic
11	C₂×#88055	C₂²	1	1	yes	characteristic
12	C₂×#88052	C₂²	1	2	yes	—
13	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	C₁₃×C₄⋊C₁₆	C₄	1	4	yes	complemented
16	#11718104	—	2	4	no	—
17	C₄⋊#6090	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	—	2	2	no	—
22	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₁₃×M₃₂	—	2	2	no	—
27	C₁₃×M₃₂	—	2	2	no	—
28	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	#88055	—	2	1	no	—
32	#88055	—	2	1	no	—
33	#88055	—	2	2	no	—
34	#88055	—	2	2	no	—
35	#88055	D₄	1	2	yes	—
36	#88055	Q₈	1	2	yes	—
37	#88052	—	2	2	no	—
38	#88052	—	2	2	no	—
39	C₂×#6090	C₂×C₄	1	4	yes	—
40	C₄×#1055	—	2	1	no	—
41	C₄×#1055	—	2	1	no	—
42	#12658	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	2	1	no	—
46	C₂²×Dic₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	C₄²	1	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
56	C₄²×C₁₃	C₄²	1	2	yes	—
57	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
58	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
59	C₄×Dic₁₃	—	2	1	no	—
60	C₄×Dic₁₃	—	2	1	no	—
61	C₄×Dic₁₃	—	4	4	no	—
62	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
63	C₄×Dic₁₃	M₁₆	1	2	yes	—
64	C₂×#1055	—	2	1	no	—
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	—	2	1	no	—
67	C₂×#1055	—	2	1	no	—
68	C₂×#1055	C₄⋊C₄	1	2	yes	—
69	C₂×#1055	M₁₆	1	2	yes	—
70	C₁₃×C₁₆	—	2	4	no	—
71	#6090	—	2	4	no	—
72	#2150	—	13	1	no	—
73	#2150	D₁₃	1	1	yes	characteristic
74	C₂×#347	—	13	2	no	—
75	C₄×#97	—	13	1	no	—
76	#1375	—	13	2	no	—
77	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	#97	1	2	yes	—
80	C₂×C₄×C₁₃	#97	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	4	2	no	—
90	C₂×Dic₁₃	—	4	2	no	—
91	C₈×C₁₃	—	2	2	no	—
92	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
93	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
94	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
95	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
96	#1055	—	2	2	no	—
97	#1055	—	2	2	no	—
98	C₂×M₃₂	—	13	2	no	—
99	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
100	C₂×#97	—	13	1	no	—
101	C₂×#97	—	13	1	no	—
102	C₂×#97	—	13	1	no	—
103	C₂×#97	—	13	1	no	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
106	C₄³	—	13	1	no	—
107	C₄⋊C₁₆	—	13	4	no	—
108	C₄⋊C₁₆	Dic₁₃	1	4	yes	complemented
109	#347	—	26	4	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
112	C₂²×C₁₃	#423	1	1	yes	characteristic
113	C₂²×C₁₃	#432	1	1	yes	characteristic
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	#347	1	4	yes	complemented
116	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
118	C₄×C₁₃	#423	1	1	yes	characteristic
119	C₄×C₁₃	#432	1	1	yes	characteristic
120	Dic₁₃	—	4	2	no	—
121	Dic₁₃	—	4	2	no	—
122	C₂²×C₈	—	13	1	no	—
123	C₂²×C₈	—	13	1	no	—
124	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
126	C₂×C₄²	—	13	1	no	—
127	C₂×C₄²	—	13	1	no	—
128	C₂×C₄²	—	26	2	no	—
129	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	—	2	2	no	—
132	M₃₂	—	26	2	no	—
133	M₃₂	—	26	2	no	—
134	C₂×C₁₆	—	13	4	no	—
135	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
136	#97	—	13	2	no	—
137	#97	—	13	2	no	—
138	#97	—	26	2	no	—
139	#97	—	26	2	no	—
140	#97	—	26	1	no	—
141	#97	—	26	1	no	—
142	C₄×C₈	—	26	1	no	—
143	C₄×C₈	—	26	1	no	—
144	C₄×C₈	C₄×D₁₃	1	2	yes	—
145	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
146	C₂×C₁₃	—	2	1	no	—
147	C₂×C₁₃	—	2	1	no	—
148	C₂×C₁₃	#1767	1	1	yes	characteristic
149	C₂×C₁₃	C₂×#347	1	2	yes	—
150	C₂²×C₄	—	26	1	no	—
151	C₂²×C₄	—	26	1	no	—
152	C₂²×C₄	#6094	1	2	yes	—
153	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	13	2	no	—
157	C₂×C₈	—	13	2	no	—
158	C₂×C₈	—	26	1	no	—
159	C₂×C₈	—	26	1	no	—
160	C₂×C₈	—	26	1	no	—
161	C₂×C₈	—	26	1	no	—
162	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
164	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
165	C₂×C₈	#6129	1	1	yes	characteristic
166	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
167	C₂×C₈	#6131	1	1	yes	characteristic
168	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
169	C₄²	—	13	2	no	—
170	C₄²	—	13	2	no	—
171	C₄²	—	26	1	no	—
172	C₄²	—	26	1	no	—
173	C₄²	—	52	4	no	—
174	C₄²	#6094	1	1	yes	characteristic
175	C₄²	#6094	1	1	yes	characteristic
176	C₄²	C₄×Dic₁₃	1	2	yes	—
177	C₁₆	—	2	4	no	—
178	C₁₆	—	26	4	no	—
179	C₁₃	#12658	1	1	yes	characteristic, complemented, Sylow
180	C₂³	C₂×#6094	1	1	yes	characteristic
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	26	2	no	—
183	C₂×C₄	—	26	2	no	—
184	C₂×C₄	—	52	2	no	—
185	C₂×C₄	—	52	2	no	—
186	C₂×C₄	#88055	1	2	yes	—
187	C₂×C₄	#88055	1	2	yes	—
188	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
189	C₂×C₄	#88146	1	1	yes	characteristic
190	C₂×C₄	C₂×#6094	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
192	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	26	2	no	—
196	C₈	—	26	2	no	—
197	C₈	#88141	1	1	yes	characteristic
198	C₈	#88147	1	1	yes	characteristic
199	C₈	D₄×Dic₁₃	1	1	yes	characteristic
200	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
201	C₂²	—	2	2	no	—
202	C₂²	#11718496	1	1	yes	characteristic
203	C₂²	#11718498	1	1	yes	characteristic
204	C₂²	C₂×#88055	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	52	2	no	—
207	C₄	—	52	2	no	—
208	C₄	#11718104	1	4	yes	complemented
209	C₄	#11718496	1	1	yes	characteristic
210	C₄	#11718498	1	1	yes	characteristic
211	C₄	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
212	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	#13114691	1	1	yes	characteristic
216	C₂	C₂×#11718104	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2150	C₂
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2150	D₁₃
hypocenter	C₁₃	#12658
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #2150

Complements

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

N	H ≅ G/N	type
15. C₁₃×C₄⋊C₁₆	C₄	semidirect
17. C₄⋊#6090	C₄	semidirect
108. C₄⋊C₁₆	Dic₁₃	semidirect
115. C₄×C₁₃	#347	semidirect
179. C₁₃	#12658	semidirect
208. C₄	#11718104	semidirect

Maximal subgroups

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

subgroup	index
#13116592	2
C₂×#11718104	2
C₄×#88055	2
#13112915	2
C₁₃×#2150	2
#12658	13