Subgroups — #15778138

Order 3328.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718156	C₂	1	2	yes	maximal
2	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
3	C₄×#88058	C₂	1	1	yes	maximal, characteristic
4	#13112927	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
8	C₄×C₁₃×Q₁₆	—	2	2	no	—
9	C₄×C₁₃×Q₁₆	C₄	1	4	yes	complemented
10	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#103	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88058	C₂²	1	2	yes	—
14	C₂²×#6090	C₂²	1	2	yes	—
15	C₄×C₄⋊Dic₁₃	—	2	1	no	—
16	#11718156	C₄	1	8	yes	complemented
17	#11718143	—	2	1	no	—
18	#11718053	—	2	2	no	—
19	C₂²×Q₈×C₁₃	—	2	1	no	—
20	C₂×C₁₃×Q₁₆	—	2	1	no	—
21	C₂×C₁₃×Q₁₆	—	2	2	no	—
22	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₄×Q₈×C₁₃	—	2	4	no	—
27	C₄×Q₈×C₁₃	—	4	2	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
30	C₂×C₄²×C₁₃	—	4	1	no	—
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	C₂×C₄×Dic₁₃	—	4	1	no	—
34	C₁₃×#107	—	2	1	no	—
35	C₁₃×#107	C₂×C₄	1	2	yes	—
36	C₁₃×#103	—	2	4	no	—
37	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃	D₄	1	2	yes	—
39	#88071	—	4	1	no	—
40	#88058	—	2	2	no	—
41	#88058	C₂×C₄	1	4	yes	—
42	C₂×#6090	—	2	2	no	—
43	C₂×#6090	—	2	2	no	—
44	C₂×#6090	C₂×C₄	1	4	yes	—
45	#12638	—	13	1	no	maximal, Sylow
46	C₂×Q₈×C₁₃	—	2	2	no	—
47	C₂×Q₈×C₁₃	—	2	4	no	—
48	C₂×Q₈×C₁₃	—	4	1	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₃	—	4	1	no	—
51	C₂²×C₄×C₁₃	—	4	1	no	—
52	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	—	4	2	no	—
56	C₁₃×Q₁₆	—	2	2	no	—
57	C₁₃×Q₁₆	—	2	4	no	—
58	C₁₃×Q₁₆	C₄²	1	4	yes	complemented
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	#41	1	2	yes	—
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	4	4	no	—
67	C₁₃×C₄⋊C₄	—	4	1	no	—
68	C₄²×C₁₃	—	4	4	no	—
69	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₁₃	SD₁₆	1	2	yes	—
71	C₄⋊Dic₁₃	—	2	4	no	—
72	C₄⋊Dic₁₃	—	4	2	no	—
73	C₄×Dic₁₃	—	8	2	no	—
74	#6090	—	2	4	no	—
75	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
76	C₂×#358	—	13	2	no	—
77	C₂×#346	—	13	1	no	—
78	C₄×#107	—	13	1	no	—
79	#1380	—	13	2	no	—
80	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
81	Q₈×C₁₃	—	2	4	no	—
82	Q₈×C₁₃	—	4	2	no	—
83	Q₈×C₁₃	—	4	4	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	4	4	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	8	1	no	—
91	C₂×C₄×C₁₃	—	8	1	no	—
92	C₂×C₄×C₁₃	#102	1	2	yes	—
93	C₂×C₄×C₁₃	#102	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#135	1	2	yes	—
98	C₂×Dic₁₃	—	4	4	no	—
99	C₂×Dic₁₃	—	8	2	no	—
100	C₂×Dic₁₃	—	8	2	no	—
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
103	C₈×C₁₃	#117	1	1	yes	characteristic
104	C₈×C₁₃	C₄×D₄	1	2	yes	—
105	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
106	C₂×C₄×Q₈	—	2	1	no	—
107	C₂²×C₁₆	—	13	2	no	—
108	C₄×Q₁₆	—	2	2	no	—
109	C₄×Q₁₆	Dic₁₃	1	4	yes	complemented
110	C₂×#107	—	13	2	no	—
111	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
112	C₂×#103	—	2	1	no	—
113	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
114	C₄×C₄⋊C₄	—	26	1	no	—
115	#358	—	13	8	no	—
116	#346	—	26	2	no	—
117	#340	—	26	1	no	—
118	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
119	C₂²×C₁₃	C₄×D₈	1	2	yes	—
120	C₂²×C₁₃	#509	1	1	yes	characteristic
121	C₂²×C₁₃	#510	1	2	yes	—
122	C₂²×C₁₃	#510	1	1	yes	characteristic
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	4	4	no	—
126	C₄×C₁₃	—	8	2	no	—
127	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₁₃	#416	1	1	yes	characteristic
129	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
130	Dic₁₃	—	8	4	no	—
131	C₂²×Q₈	—	2	1	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	1	no	—
134	C₂×Q₁₆	C₄×D₁₃	1	4	yes	—
135	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
136	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	4	2	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	26	2	no	—
143	C₂×C₄²	—	4	1	no	—
144	C₂×C₄²	—	52	1	no	—
145	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
146	C₂×C₁₆	—	13	4	no	—
147	C₂×C₁₆	—	26	2	no	—
148	C₂×C₁₆	—	26	2	no	—
149	#107	—	2	1	no	—
150	#107	—	13	4	no	—
151	#107	—	26	2	no	—
152	#107	C₂×Dic₁₃	1	2	yes	—
153	#103	—	2	4	no	—
154	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
155	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
156	#95	—	52	1	no	—
157	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
158	C₂×C₁₃	#2139	1	1	yes	characteristic
159	C₂×C₁₃	#2141	1	1	yes	characteristic
160	C₂×C₁₃	#2175	1	2	yes	—
161	C₂×C₁₃	#2176	1	2	yes	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	52	2	no	—
169	C₂²×C₄	#6104	1	1	yes	characteristic
170	C₂²×C₄	#6104	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
172	Q₁₆	—	2	2	no	—
173	Q₁₆	—	2	4	no	—
174	Q₁₆	C₄×Dic₁₃	1	4	yes	complemented
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
178	C₂×C₈	#6127	1	2	yes	—
179	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	26	4	no	—
185	C₄⋊C₄	—	52	2	no	—
186	C₄²	—	4	4	no	—
187	C₄²	—	104	2	no	—
188	C₄²	#6105	1	2	yes	—
189	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
190	C₁₆	—	26	4	no	—
191	C₁₃	#12638	1	1	yes	characteristic, complemented, Sylow
192	C₂³	C₂×#6104	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	4	4	no	—
195	Q₈	—	4	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	52	4	no	—
205	C₂×C₄	—	104	2	no	—
206	C₂×C₄	—	104	2	no	—
207	C₂×C₄	#88072	1	2	yes	—
208	C₂×C₄	#88072	1	2	yes	—
209	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
210	C₂×C₄	C₂×#6105	1	1	yes	characteristic
211	C₂×C₄	#88204	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
215	C₈	#88180	1	1	yes	characteristic
216	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
217	C₂²	C₄×#6104	1	2	yes	—
218	C₂²	#11718840	1	2	yes	—
219	C₂²	#11718854	1	1	yes	characteristic
220	C₂²	#11718856	1	1	yes	characteristic
221	C₂²	C₂×#88072	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	2	no	—
226	C₄	—	104	4	no	—
227	C₄	C₄×#6105	1	2	yes	—
228	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
229	C₄	#11718894	1	1	yes	characteristic
230	C₂	C₄×#88072	1	1	yes	characteristic
231	C₂	#13116548	1	1	yes	characteristic
232	C₂	#13116556	1	1	yes	characteristic
233	C₂	#13116698	1	2	yes	—
234	C₂	#13116705	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#12638
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×C₁₃×Q₁₆	C₄	semidirect
16. #11718156	C₄	semidirect
58. C₁₃×Q₁₆	C₄²	semidirect
109. C₄×Q₁₆	Dic₁₃	semidirect
174. Q₁₆	C₄×Dic₁₃	semidirect
191. C₁₃	#12638	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718156	2
C₂×#11718053	2
C₄×#88058	2
#13112927	2
C₂×C₄×C₁₃×Q₁₆	2
#12638	13