Subgroups — #16726552

Order 3328.

2738 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719524	C₂	1	2	yes	maximal
2	#13123404	C₂	1	16	yes	maximal, complemented
3	#13123400	C₂	1	6	yes	maximal, complemented
4	C₂×#11718585	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₄×C₁₃×#41	C₄	1	4	yes	—
8	C₄×C₁₃×#41	C₄	1	4	yes	—
9	C₄×C₁₃×#41	C₂²	1	8	yes	—
10	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
11	C₂×C₄×D₄×C₁₃	C₄	1	6	yes	—
12	C₂×C₁₃×#117	C₂²	1	3	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
14	#11719524	—	2	16	no	—
15	C₂×#88202	C₂²	1	16	yes	—
16	C₂²×#6099	C₂²	1	12	yes	—
17	#11718660	C₂²	1	48	yes	complemented
18	#11718585	C₂²	1	4	yes	—
19	#11718585	C₂²	1	12	yes	—
20	#11718585	C₂²	1	24	yes	complemented
21	C₂×C₄⋊#1055	C₂²	1	24	yes	complemented
22	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
23	C₂×C₁₃×#41	D₄	1	8	yes	—
24	C₂×C₁₃×#41	Q₈	1	8	yes	—
25	C₂×C₁₃×#41	C₂³	1	4	yes	—
26	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₃	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
30	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
31	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
32	C₄×D₄×C₁₃	C₂×C₄	1	24	yes	—
33	C₄×D₄×C₁₃	C₂×C₄	1	24	yes	—
34	C₁₃×#117	C₂×C₄	1	12	yes	—
35	C₁₃×#117	C₂³	1	12	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
37	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	6	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
44	#88202	—	2	64	no	—
45	C₂×#6099	—	2	48	no	—
46	C₂×#6099	C₂³	1	24	yes	—
47	C₂×#6099	C₂³	1	8	yes	—
48	C₂²×#1055	C₂³	1	24	yes	—
49	C₄⋊#1055	C₂³	1	48	yes	—
50	C₄⋊#1055	C₂³	1	16	yes	—
51	#60099	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₂²×C₄	1	3	yes	—
53	C₁₃×#41	C₄⋊C₄	1	32	yes	—
54	C₁₃×#41	C₂×D₄	1	16	yes	—
55	C₁₃×#41	C₂×Q₈	1	16	yes	—
56	C₂×Q₈×C₁₃	C₄⋊C₄	1	8	yes	—
57	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
58	C₂×D₄×C₁₃	C₄⋊C₄	1	24	yes	—
59	C₂×D₄×C₁₃	C₂²×C₄	1	12	yes	—
60	C₂²×C₄×C₁₃	—	4	12	no	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	C₂²×C₄	1	24	yes	—
72	C₁₃×C₂²⋊C₄	C₂²×C₄	1	24	yes	—
73	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
77	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
78	C₄²×C₁₃	C₂⁴	1	2	yes	—
79	C₄²×C₁₃	C₂⁴	1	6	yes	—
80	#6099	—	2	16	no	—
81	#6099	—	2	48	no	—
82	C₂×#1055	—	2	48	no	—
83	C₂×#1055	C₂⁴	1	16	yes	—
84	C₂×#568	—	13	2	no	—
85	C₂×C₄×#41	D₁₃	1	1	yes	characteristic
86	#2906	—	13	16	no	—
87	#2905	—	13	6	no	—
88	C₂×#423	—	13	6	no	—
89	C₂³×C₁₃	—	4	6	no	—
90	C₂³×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
91	C₂³×C₁₃	C₂³×C₄	1	6	yes	—
92	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
94	D₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	24	no	—
97	C₂×C₄×C₁₃	—	4	12	no	—
98	C₂×C₄×C₁₃	—	4	6	no	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
101	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
103	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
110	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
111	#1055	—	2	16	no	—
112	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
113	#568	—	26	16	no	—
114	C₂×#131	—	13	16	no	—
115	C₂²×M₁₆	—	13	12	no	—
116	C₄×#41	Dic₁₃	1	4	yes	—
117	C₄×#41	Dic₁₃	1	4	yes	—
118	C₄×#41	C₂×D₁₃	1	8	yes	—
119	C₂×C₄×Q₈	Dic₁₃	1	2	yes	—
120	C₂×C₄×D₄	Dic₁₃	1	6	yes	—
121	C₂×#117	C₂×D₁₃	1	3	yes	—
122	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
123	#424	—	13	48	no	—
124	#423	—	13	4	no	—
125	#423	—	13	12	no	—
126	#423	—	13	24	no	—
127	C₂×C₄⋊C₈	—	13	24	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	12	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	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	12	no	—
136	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
137	C₄×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
138	C₄×C₁₃	#568	1	4	yes	complemented
139	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
140	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
141	C₂×#41	Dic₂₆	1	8	yes	—
142	C₂×#41	D₅₂	1	8	yes	—
143	C₂×#41	C₂×Dic₁₃	1	4	yes	—
144	C₂×#41	C₂²×D₁₃	1	4	yes	—
145	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂²×D₄	C₂×Dic₁₃	1	3	yes	—
147	C₂³×C₄	C₂×Dic₁₃	1	6	yes	—
148	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
149	#131	—	26	64	no	—
150	C₂×M₁₆	—	13	8	no	—
151	C₂×M₁₆	—	13	24	no	—
152	C₂×M₁₆	—	26	48	no	—
153	C₂²×C₈	—	13	24	no	—
154	C₄×Q₈	C₂×Dic₁₃	1	8	yes	—
155	C₄×Q₈	C₂×Dic₁₃	1	8	yes	—
156	C₄×D₄	C₂×Dic₁₃	1	24	yes	—
157	C₄×D₄	C₂×Dic₁₃	1	24	yes	—
158	#117	C₂×Dic₁₃	1	12	yes	—
159	#117	C₂²×D₁₃	1	12	yes	—
160	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
161	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	6	yes	—
162	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
163	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
164	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
165	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
166	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
167	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
168	C₄⋊C₈	—	13	16	no	—
169	C₄⋊C₈	—	13	48	no	—
170	C₂×C₁₃	—	2	2	no	—
171	C₂×C₁₃	—	2	12	no	—
172	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₁₃	C₂×#568	1	2	yes	—
174	C₂⁴	C₂²×Dic₁₃	1	3	yes	—
175	#41	C₄⋊Dic₁₃	1	32	yes	—
176	#41	C₂×Dic₂₆	1	16	yes	—
177	#41	C₂×D₅₂	1	16	yes	—
178	C₂×Q₈	C₄⋊Dic₁₃	1	8	yes	—
179	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
180	C₂×D₄	C₄⋊Dic₁₃	1	24	yes	—
181	C₂×D₄	C₂²×Dic₁₃	1	12	yes	—
182	C₂²×C₄	—	4	12	no	—
183	C₂²×C₄	C₂×Dic₂₆	1	12	yes	—
184	C₂²×C₄	C₂×D₅₂	1	12	yes	—
185	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
186	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
187	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
188	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
189	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
190	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
191	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
193	M₁₆	—	26	16	no	—
194	M₁₆	—	26	48	no	—
195	C₂×C₈	—	13	16	no	—
196	C₂×C₈	—	26	48	no	—
197	C₄⋊C₄	C₂²×Dic₁₃	1	24	yes	—
198	C₂²⋊C₄	C₂²×Dic₁₃	1	24	yes	—
199	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
200	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
201	C₄²	C₂²×Dic₁₃	1	6	yes	—
202	C₄²	C₂²×Dic₁₃	1	12	yes	—
203	C₄²	C₂²×Dic₁₃	1	12	yes	—
204	C₄²	C₂³×D₁₃	1	2	yes	—
205	C₄²	C₂³×D₁₃	1	6	yes	—
206	C₁₃	#60099	1	1	yes	characteristic, complemented, Sylow
207	C₂³	—	4	6	no	—
208	C₂³	C₂×C₄⋊Dic₁₃	1	12	yes	—
209	C₂³	C₂³×Dic₁₃	1	6	yes	—
210	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
211	Q₈	C₂×C₄⋊Dic₁₃	1	16	yes	—
212	D₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	24	no	—
215	C₂×C₄	—	4	12	no	—
216	C₂×C₄	—	4	6	no	—
217	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₁₃	1	24	yes	—
219	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
220	C₂×C₄	C₂²×Dic₂₆	1	12	yes	—
221	C₂×C₄	C₂²×D₅₂	1	12	yes	—
222	C₂×C₄	C₂²×D₅₂	1	2	yes	—
223	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
224	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
225	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
227	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
228	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
229	C₈	—	26	16	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	12	no	—
232	C₂²	—	4	12	no	—
233	C₂²	C₂²×C₄⋊Dic₁₃	1	2	yes	—
234	C₂²	C₂²×C₄⋊Dic₁₃	1	12	yes	—
235	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	12	no	—
238	C₄	C₂²×C₄⋊Dic₁₃	1	2	yes	—
239	C₄	C₂²×C₄⋊Dic₁₃	1	12	yes	—
240	C₄	#11719524	1	4	yes	complemented
241	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
242	C₄	C₂³×D₅₂	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	2	12	no	—
245	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic
246	C₂	C₂×#11719524	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₁₃
commutator, socle	C₂²×C₁₃	C₂⁴×C₄
Fitting	C₂×C₄×C₁₃×#41	C₂
prime radical	C₂²×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂×C₄×#41	D₁₃
hypocenter	C₁₃	#60099
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. #13123404	C₂	semidirect
3. #13123400	C₂	semidirect
4. C₂×#11718585	C₂	semidirect
17. #11718660	C₂²	semidirect
20. #11718585	C₂²	semidirect
21. C₂×C₄⋊#1055	C₂²	semidirect
138. C₄×C₁₃	#568	semidirect
206. C₁₃	#60099	semidirect
240. C₄	#11719524	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719524	2
#13123404	2
#13123400	2
C₂×#11718585	2
C₂×C₄×C₁₃×#41	2
#60099	13