Subgroups — C₂²×#11718762 #16726310

Order 3328.

3518 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	C₂³×#6099	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88180	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718762	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#88055	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₁₃	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₄	1	16	yes	complemented
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	12	yes	—
11	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊Dic₁₃	C₄	1	2	yes	—
13	C₂×#88180	C₄	1	12	yes	—
14	C₂×#88180	C₂²	1	12	yes	—
15	C₂²×C₄⋊Dic₁₃	C₄	1	2	yes	—
16	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
17	C₂²×#6099	C₄	1	16	yes	complemented
18	C₂²×#6099	C₂²	1	12	yes	—
19	C₂³×#1055	C₂²	1	2	yes	—
20	#11718762	C₂²	1	64	yes	direct product factor
21	C₂×#88055	C₂²	1	24	yes	complemented
22	C₂×C₈×Dic₁₃	C₂²	1	24	yes	complemented
23	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
30	C₂³×Dic₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	96	yes	complemented
34	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	16	yes	—
37	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	24	yes	—
38	#88180	C₂×C₄	1	48	yes	—
39	#88180	C₂³	1	16	yes	—
40	C₂×C₄⋊Dic₁₃	C₂×C₄	1	24	yes	—
41	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
42	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
43	C₂×#6099	C₂×C₄	1	96	yes	complemented
44	C₂×#6099	C₂³	1	16	yes	—
45	C₂²×#1055	C₂×C₄	1	16	yes	—
46	C₂²×#1055	C₂³	1	12	yes	—
47	#88055	C₂³	1	32	yes	complemented
48	C₈×Dic₁₃	C₂³	1	32	yes	complemented
49	C₂²×#405	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	—	2	8	no	—
51	C₂⁴×C₁₃	C₄²	1	8	yes	—
52	C₂⁴×C₁₃	C₂²×C₄	1	6	yes	—
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₄²	1	48	yes	—
55	C₂²×C₄×C₁₃	C₄²	1	8	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
65	C₂²×Dic₁₃	—	2	16	no	—
66	C₂²×Dic₁₃	C₂²×C₄	1	24	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	64	yes	complemented
70	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
71	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
72	C₂²⋊Dic₁₃	C₂²×C₄	1	32	yes	—
73	C₄⋊Dic₁₃	C₂²×C₄	1	32	yes	—
74	C₄×Dic₁₃	C₂²×C₄	1	24	yes	—
75	C₄×Dic₁₃	C₂⁴	1	8	yes	—
76	#6099	C₂²×C₄	1	64	yes	complemented
77	C₂×#1055	C₂²×C₄	1	48	yes	—
78	C₂×#1055	C₂⁴	1	8	yes	—
79	C₂³×M₁₆	—	13	1	no	—
80	C₂⁴×C₈	D₁₃	1	1	yes	characteristic
81	C₂²×#117	—	13	1	no	—
82	C₂×#405	—	13	24	no	—
83	C₂²×#97	—	13	2	no	—
84	C₂²×C₄×C₈	—	13	2	no	—
85	C₂³×C₁₃	—	2	24	no	—
86	C₂³×C₁₃	—	2	32	no	—
87	C₂³×C₁₃	C₂×C₄²	1	24	yes	—
88	C₂³×C₁₃	C₂×C₄²	1	4	yes	—
89	C₂³×C₁₃	#131	1	8	yes	—
90	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
91	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
92	C₂×C₄×C₁₃	C₂×C₄²	1	48	yes	—
93	C₂×C₄×C₁₃	C₂×C₄²	1	12	yes	—
94	C₂×C₄×C₁₃	C₂×C₄²	1	24	yes	—
95	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
96	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
97	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
98	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
99	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
100	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
101	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
102	C₂×Dic₁₃	—	2	48	no	—
103	C₂×Dic₁₃	C₂³×C₄	1	16	yes	—
104	C₈×C₁₃	C₂³×C₄	1	8	yes	—
105	C₈×C₁₃	C₂³×C₄	1	8	yes	—
106	#1055	C₂³×C₄	1	16	yes	—
107	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
108	C₂²×M₁₆	—	13	16	no	—
109	C₂²×M₁₆	—	13	12	no	—
110	C₂³×C₈	—	13	2	no	—
111	C₂³×C₈	Dic₁₃	1	16	yes	complemented
112	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
113	C₂³×C₈	C₂×D₁₃	1	12	yes	—
114	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
115	C₂×#117	—	13	12	no	—
116	C₂×#117	—	13	12	no	—
117	C₂²×C₄⋊C₄	—	13	2	no	—
118	C₂²×C₂²⋊C₄	—	13	2	no	—
119	C₂²×C₄²	—	13	2	no	—
120	#405	—	13	64	no	—
121	C₂×#97	—	13	24	no	—
122	C₂×C₄×C₈	—	13	24	no	—
123	C₂²×C₁₃	—	2	48	no	—
124	C₂²×C₁₃	—	2	8	no	—
125	C₂²×C₁₃	#405	1	16	yes	complemented
126	C₂²×C₁₃	C₂²×C₄²	1	8	yes	—
127	C₂²×C₁₃	C₂²×C₄²	1	6	yes	—
128	C₂²×C₁₃	C₂×#131	1	12	yes	—
129	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
130	C₄×C₁₃	C₂²×C₄²	1	6	yes	—
131	C₄×C₁₃	C₂²×C₄²	1	8	yes	—
132	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
133	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
134	Dic₁₃	—	2	16	no	—
135	C₂⁵	C₄×D₁₃	1	1	yes	characteristic
136	C₂³×C₄	—	13	4	no	—
137	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
138	C₂³×C₄	C₄×D₁₃	1	6	yes	—
139	C₂³×C₄	C₄×D₁₃	1	8	yes	—
140	C₂³×C₄	C₂×Dic₁₃	1	8	yes	—
141	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
142	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
143	C₂×M₁₆	—	13	96	no	—
144	C₂×M₁₆	—	13	16	no	—
145	C₂²×C₈	—	13	16	no	—
146	C₂²×C₈	—	13	12	no	—
147	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
148	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
149	C₂²×C₈	C₂×Dic₁₃	1	96	yes	complemented
150	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
151	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
152	C₂²×C₈	C₂²×D₁₃	1	16	yes	—
153	#117	—	13	48	no	—
154	#117	—	13	16	no	—
155	C₂×C₄⋊C₄	—	13	24	no	—
156	C₂×C₂²⋊C₄	—	13	24	no	—
157	C₂×C₄²	—	13	12	no	—
158	C₂×C₄²	—	13	12	no	—
159	#97	—	13	32	no	—
160	C₄×C₈	—	13	32	no	—
161	C₂×C₁₃	—	2	8	no	—
162	C₂×C₁₃	C₂×#405	1	12	yes	complemented
163	C₂×C₁₃	C₂³×C₄²	1	1	yes	characteristic
164	C₂×C₁₃	C₂²×#131	1	2	yes	—
165	C₂⁴	—	2	8	no	—
166	C₂⁴	C₄×Dic₁₃	1	8	yes	—
167	C₂⁴	C₂×C₄×D₁₃	1	6	yes	—
168	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	—	13	24	no	—
170	C₂²×C₄	—	26	16	no	—
171	C₂²×C₄	C₄×Dic₁₃	1	48	yes	—
172	C₂²×C₄	C₄×Dic₁₃	1	8	yes	—
173	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
174	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
175	C₂²×C₄	C₂×C₄×D₁₃	1	12	yes	—
176	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
177	C₂²×C₄	C₂×C₄×D₁₃	1	24	yes	—
178	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
179	C₂²×C₄	C₂²×Dic₁₃	1	24	yes	—
180	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
181	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
182	M₁₆	—	13	64	no	—
183	C₂×C₈	—	13	48	no	—
184	C₂×C₈	—	13	8	no	—
185	C₂×C₈	C₂²×Dic₁₃	1	24	yes	—
186	C₂×C₈	C₂²×Dic₁₃	1	24	yes	—
187	C₂×C₈	C₂²×Dic₁₃	1	64	yes	complemented
188	C₂×C₈	C₂³×D₁₃	1	4	yes	—
189	C₂×C₈	C₂³×D₁₃	1	4	yes	—
190	C₄⋊C₄	—	13	32	no	—
191	C₂²⋊C₄	—	13	32	no	—
192	C₄²	—	13	24	no	—
193	C₄²	—	13	8	no	—
194	C₁₃	C₂²×#405	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	2	24	no	—
196	C₂³	—	2	32	no	—
197	C₂³	#88155	1	8	yes	—
198	C₂³	C₂×C₄×Dic₁₃	1	24	yes	—
199	C₂³	C₂×C₄×Dic₁₃	1	4	yes	—
200	C₂³	C₂²×C₄×D₁₃	1	4	yes	—
201	C₂³	C₂²×C₄×D₁₃	1	3	yes	—
202	C₂×C₄	—	13	16	no	—
203	C₂×C₄	—	26	48	no	—
204	C₂×C₄	C₂×C₄×Dic₁₃	1	48	yes	—
205	C₂×C₄	C₂×C₄×Dic₁₃	1	12	yes	—
206	C₂×C₄	C₂×C₄×Dic₁₃	1	24	yes	—
207	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
208	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
209	C₂×C₄	C₂²×C₄×D₁₃	1	6	yes	—
210	C₂×C₄	C₂²×C₄×D₁₃	1	8	yes	—
211	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
212	C₂×C₄	C₂³×Dic₁₃	1	8	yes	—
213	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
214	C₈	—	13	16	no	—
215	C₈	C₂³×Dic₁₃	1	8	yes	—
216	C₈	C₂³×Dic₁₃	1	8	yes	—
217	C₂²	—	2	8	no	—
218	C₂²	—	2	48	no	—
219	C₂²	#11718762	1	16	yes	direct product factor
220	C₂²	C₂×#88155	1	12	yes	—
221	C₂²	C₂²×C₄×Dic₁₃	1	8	yes	—
222	C₂²	C₂²×C₄×Dic₁₃	1	6	yes	—
223	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
224	C₄	—	26	16	no	—
225	C₄	C₂²×C₄×Dic₁₃	1	6	yes	—
226	C₄	C₂²×C₄×Dic₁₃	1	8	yes	—
227	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
228	C₄	C₂⁴×Dic₁₃	1	1	yes	characteristic
229	C₂	—	2	8	no	—
230	C₂	C₂×#11718762	1	12	yes	direct product factor
231	C₂	C₂³×C₄×Dic₁₃	1	1	yes	characteristic
232	C₂	C₂²×#88155	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂⁴×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂⁵×C₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	C₂²×#405
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#11718762	C₂	direct (N × H)
4. C₂²×#88055	C₂	semidirect
5. C₂²×C₈×Dic₁₃	C₂	semidirect
8. C₂³×C₈×C₁₃	C₄	semidirect
17. C₂²×#6099	C₄	semidirect
20. #11718762	C₂²	direct (N × H)
21. C₂×#88055	C₂²	semidirect
22. C₂×C₈×Dic₁₃	C₂²	semidirect
33. C₂²×C₈×C₁₃	C₂×C₄	semidirect
43. C₂×#6099	C₂×C₄	semidirect
47. #88055	C₂³	semidirect
48. C₈×Dic₁₃	C₂³	semidirect
69. C₂×C₈×C₁₃	C₂²×C₄	semidirect
76. #6099	C₂²×C₄	semidirect
111. C₂³×C₈	Dic₁₃	semidirect
125. C₂²×C₁₃	#405	semidirect
149. C₂²×C₈	C₂×Dic₁₃	semidirect
162. C₂×C₁₃	C₂×#405	semidirect
187. C₂×C₈	C₂²×Dic₁₃	semidirect
194. C₁₃	C₂²×#405	semidirect
219. C₂²	#11718762	direct (N × H)
230. C₂	C₂×#11718762	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6099	2
C₂²×#88180	2
C₂×#11718762	2
C₂²×#88055	2
C₂²×C₈×Dic₁₃	2
C₂⁴×C₈×C₁₃	2
C₂²×#405	13

Subgroups — C22×#11718762 #16726310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718762 #16726310