Subgroups — #16726598

Order 3328.

2666 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	#13128039	C₂	1	2	yes	maximal
2	#13123436	C₂	1	8	yes	maximal, complemented
3	#13123431	C₂	1	8	yes	maximal, complemented
4	#13123426	C₂	1	6	yes	maximal, complemented
5	C₂×#11718767	C₂	1	6	yes	maximal, complemented
6	C₂²×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#131	C₄	1	16	yes	complemented
9	C₂×C₁₃×#131	C₂²	1	4	yes	—
10	C₂×C₁₃×#131	C₂²	1	4	yes	—
11	C₂²×C₁₃×M₁₆	C₂²	1	3	yes	—
12	C₂³×C₈×C₁₃	C₂²	1	3	yes	—
13	#11719532	C₂²	1	16	yes	—
14	#41×Dic₁₃	—	2	8	no	—
15	#11719517	—	2	2	no	—
16	#11719503	—	2	6	no	—
17	C₂×#88180	C₂²	1	12	yes	—
18	#11718782	C₂²	1	48	yes	complemented
19	#11718767	C₂²	1	4	yes	—
20	#11718767	C₂²	1	12	yes	—
21	#11718767	C₂²	1	12	yes	complemented
22	#11718767	C₂²	1	12	yes	complemented
23	C₂×#88058	C₂²	1	12	yes	complemented
24	C₂×#88057	C₂²	1	12	yes	complemented
25	C₂×C₁₃×#41	C₂×C₄	1	8	yes	—
26	C₂×C₁₃×#41	D₄	1	4	yes	—
27	C₂×C₁₃×#41	Q₈	1	8	yes	—
28	C₂×C₁₃×#41	C₂³	1	4	yes	—
29	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×D₄×C₁₃	D₄	1	3	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
32	C₁₃×#131	C₂×C₄	1	32	yes	complemented
33	C₁₃×#131	C₂×C₄	1	32	yes	complemented
34	C₂×C₁₃×M₁₆	C₂×C₄	1	24	yes	complemented
35	C₂×C₁₃×M₁₆	C₂³	1	12	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	24	yes	complemented
37	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
41	C₂×C₂²⋊Dic₁₃	—	2	12	no	—
42	Q₈×Dic₁₃	—	2	16	no	—
43	D₄×Dic₁₃	—	2	48	no	—
44	#88180	—	2	12	no	—
45	#88180	—	2	12	no	—
46	#88180	C₂³	1	8	yes	—
47	#88180	C₂³	1	24	yes	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
49	C₂×C₄×Dic₁₃	—	2	12	no	—
50	#88058	C₂³	1	8	yes	—
51	#88058	C₂³	1	24	yes	—
52	#88057	C₂³	1	8	yes	—
53	#88057	C₂³	1	24	yes	—
54	#60108	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×D₄	1	3	yes	—
56	C₁₃×#41	C₄⋊C₄	1	32	yes	—
57	C₁₃×#41	C₂²×C₄	1	16	yes	—
58	C₁₃×#41	C₂×Q₈	1	16	yes	—
59	C₂×Q₈×C₁₃	C₄⋊C₄	1	8	yes	—
60	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
61	C₂×D₄×C₁₃	C₄⋊C₄	1	24	yes	—
62	C₂×D₄×C₁₃	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	24	no	—
70	C₁₃×M₁₆	C₂²×C₄	1	48	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	—
75	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
78	C₂²⋊Dic₁₃	—	2	8	no	—
79	C₂²⋊Dic₁₃	—	2	24	no	—
80	C₄⋊Dic₁₃	—	2	24	no	—
81	C₄⋊Dic₁₃	C₂⁴	1	16	yes	—
82	C₄×Dic₁₃	—	2	8	no	—
83	C₄×Dic₁₃	—	2	24	no	—
84	C₂²×#131	D₁₃	1	1	yes	characteristic
85	#3428	—	13	2	no	—
86	#2913	—	13	8	no	—
87	#2912	—	13	8	no	—
88	#2911	—	13	6	no	—
89	C₂×#427	—	13	6	no	—
90	C₂³×C₁₃	—	4	6	no	—
91	C₂³×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
92	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
94	Q₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
95	D₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
98	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
100	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
101	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
104	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
105	C₂×Dic₁₃	—	2	16	no	—
106	C₂×Dic₁₃	—	8	24	no	—
107	C₈×C₁₃	C₂³×C₄	1	2	yes	—
108	C₈×C₁₃	C₂³×C₄	1	2	yes	—
109	C₈×C₁₃	C₂³×C₄	1	12	yes	—
110	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
111	C₂×#131	Dic₁₃	1	16	yes	complemented
112	C₂×#131	C₂×D₁₃	1	4	yes	—
113	C₂×#131	C₂×D₁₃	1	4	yes	—
114	C₂²×M₁₆	C₂×D₁₃	1	3	yes	—
115	C₂³×C₈	C₂×D₁₃	1	3	yes	—
116	#520	—	13	16	no	—
117	#519	—	26	2	no	—
118	#518	—	26	6	no	—
119	C₄×#41	—	26	8	no	—
120	C₂×#117	—	13	12	no	—
121	#428	—	13	48	no	—
122	#427	—	13	4	no	—
123	#427	—	13	12	no	—
124	#427	—	13	12	no	—
125	#427	—	13	12	no	—
126	C₂×#107	—	13	12	no	—
127	C₂×#106	—	13	12	no	—
128	C₂²×C₁₃	—	2	12	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	4	12	no	—
131	C₂²×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
132	C₂²×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
133	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
134	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
135	C₄×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
136	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
137	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
138	Dic₁₃	—	4	16	no	—
139	C₂×#41	Dic₂₆	1	8	yes	—
140	C₂×#41	D₅₂	1	4	yes	—
141	C₂×#41	C₂×Dic₁₃	1	8	yes	—
142	C₂×#41	C₂²×D₁₃	1	4	yes	—
143	C₂²×Q₈	D₅₂	1	1	yes	characteristic
144	C₂²×D₄	D₅₂	1	3	yes	—
145	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
146	#131	C₂×Dic₁₃	1	32	yes	complemented
147	#131	C₂×Dic₁₃	1	32	yes	complemented
148	C₂×M₁₆	C₂×Dic₁₃	1	24	yes	complemented
149	C₂×M₁₆	C₂²×D₁₃	1	12	yes	—
150	C₂²×C₈	C₂×Dic₁₃	1	24	yes	complemented
151	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
153	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
154	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
155	C₄×Q₈	—	26	16	no	—
156	C₄×D₄	—	26	48	no	—
157	#117	—	13	8	no	—
158	#117	—	13	24	no	—
159	#117	—	26	12	no	—
160	#117	—	26	12	no	—
161	C₂×C₄⋊C₄	—	13	24	no	—
162	C₂×C₂²⋊C₄	—	26	12	no	—
163	C₂×C₄²	—	26	12	no	—
164	#107	—	13	8	no	—
165	#107	—	13	24	no	—
166	#106	—	13	8	no	—
167	#106	—	13	24	no	—
168	C₂×C₁₃	—	2	2	no	—
169	C₂×C₁₃	—	2	12	no	—
170	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₁₃	#3584	1	2	yes	—
172	C₂⁴	C₂×D₅₂	1	3	yes	—
173	#41	C₄⋊Dic₁₃	1	32	yes	—
174	#41	C₂×Dic₂₆	1	16	yes	—
175	#41	C₂²×Dic₁₃	1	16	yes	—
176	C₂×Q₈	C₄⋊Dic₁₃	1	8	yes	—
177	C₂×Q₈	C₂×D₅₂	1	4	yes	—
178	C₂×D₄	C₄⋊Dic₁₃	1	24	yes	—
179	C₂×D₄	C₂×D₅₂	1	12	yes	—
180	C₂²×C₄	—	26	24	no	—
181	C₂²×C₄	C₂×Dic₂₆	1	12	yes	—
182	C₂²×C₄	C₂×D₅₂	1	3	yes	—
183	C₂²×C₄	C₂×D₅₂	1	6	yes	—
184	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
185	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
186	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
187	M₁₆	C₂²×Dic₁₃	1	48	yes	—
188	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
189	C₂×C₈	C₂²×Dic₁₃	1	12	yes	—
190	C₂×C₈	C₂²×Dic₁₃	1	24	yes	—
191	C₂×C₈	C₂²×Dic₁₃	1	24	yes	—
192	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
193	C₂×C₈	C₂³×D₁₃	1	6	yes	—
194	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
195	C₄⋊C₄	—	13	16	no	—
196	C₄⋊C₄	—	26	24	no	—
197	C₂²⋊C₄	—	26	8	no	—
198	C₂²⋊C₄	—	26	24	no	—
199	C₄²	—	26	24	no	—
200	C₄²	—	26	8	no	—
201	C₁₃	#60108	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	4	6	no	—
203	C₂³	C₂×C₄⋊Dic₁₃	1	12	yes	—
204	C₂³	C₂²×D₅₂	1	1	yes	characteristic
205	C₂³	C₂²×D₅₂	1	6	yes	—
206	Q₈	C₂×C₄⋊Dic₁₃	1	16	yes	—
207	D₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
208	C₂×C₄	—	26	16	no	—
209	C₂×C₄	—	104	24	no	—
210	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
211	C₂×C₄	C₂×C₄⋊Dic₁₃	1	24	yes	—
212	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
213	C₂×C₄	C₂²×Dic₂₆	1	12	yes	—
214	C₂×C₄	C₂²×D₅₂	1	6	yes	—
215	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
216	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
217	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
218	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
219	C₈	C₂³×Dic₁₃	1	2	yes	—
220	C₈	C₂³×Dic₁₃	1	2	yes	—
221	C₈	C₂³×Dic₁₃	1	12	yes	—
222	C₂²	—	2	12	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	12	no	—
225	C₂²	C₂²×C₄⋊Dic₁₃	1	2	yes	—
226	C₂²	C₂²×C₄⋊Dic₁₃	1	12	yes	—
227	C₂²	C₂³×D₅₂	1	1	yes	characteristic
228	C₄	—	52	16	no	—
229	C₄	C₂²×C₄⋊Dic₁₃	1	2	yes	—
230	C₄	C₂²×C₄⋊Dic₁₃	1	12	yes	—
231	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
232	C₄	C₂⁴×Dic₁₃	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	—	2	12	no	—
235	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic
236	C₂	#13129610	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂³×D₄
prime radical	C₂²×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×#131	D₁₃
hypocenter	C₁₃	#60108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #13123436	C₂	semidirect
3. #13123431	C₂	semidirect
4. #13123426	C₂	semidirect
5. C₂×#11718767	C₂	semidirect
8. C₂×C₁₃×#131	C₄	semidirect
18. #11718782	C₂²	semidirect
21. #11718767	C₂²	semidirect
22. #11718767	C₂²	semidirect
23. C₂×#88058	C₂²	semidirect
24. C₂×#88057	C₂²	semidirect
32. C₁₃×#131	C₂×C₄	semidirect
33. C₁₃×#131	C₂×C₄	semidirect
34. C₂×C₁₃×M₁₆	C₂×C₄	semidirect
36. C₂²×C₈×C₁₃	C₂×C₄	semidirect
111. C₂×#131	Dic₁₃	semidirect
146. #131	C₂×Dic₁₃	semidirect
147. #131	C₂×Dic₁₃	semidirect
148. C₂×M₁₆	C₂×Dic₁₃	semidirect
150. C₂²×C₈	C₂×Dic₁₃	semidirect
201. C₁₃	#60108	semidirect

Maximal subgroups

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

subgroup	index
#13128039	2
#13123436	2
#13123431	2
#13123426	2
C₂×#11718767	2
C₂²×C₁₃×#131	2
#60108	13