Subgroups — C₂²×#11718659 #16726567

Order 3328.

3006 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#1055	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88180	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718659	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#88047	C₂	1	2	yes	maximal, complemented
5	C₂²×#88048	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	—	2	6	no	—
9	C₂×C₁₃×#117	C₂²	1	12	yes	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₁₃×C₂²⋊C₄	—	2	1	no	—
12	C₂²×C₄²×C₁₃	—	2	1	no	—
13	C₂²×C₂²⋊Dic₁₃	—	2	1	no	—
14	C₂×#88180	—	2	6	no	—
15	C₂×#88180	C₂²	1	12	yes	—
16	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂²×C₄×Dic₁₃	—	2	1	no	—
18	C₂³×#1055	C₄	1	16	yes	complemented
19	C₂³×#1055	C₂²	1	1	yes	characteristic
20	C₂³×#1055	C₂²	1	1	yes	characteristic
21	C₂³×#1055	C₂²	1	12	yes	—
22	#11718659	C₂²	1	64	yes	direct product factor
23	C₂×#88048	C₂²	1	24	yes	complemented
24	C₂×#88047	C₂²	1	24	yes	complemented
25	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	—	2	2	no	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₁₃	D₄	1	6	yes	—
29	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
33	C₂³×Dic₁₃	—	2	2	no	—
34	C₁₃×#117	—	2	24	no	—
35	C₁₃×#117	C₂³	1	16	yes	—
36	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
38	C₂×C₁₃×C₂²⋊C₄	—	2	12	no	—
39	C₂×C₄²×C₁₃	—	2	12	no	—
40	C₂×C₂²⋊Dic₁₃	—	2	12	no	—
41	#88180	—	2	24	no	—
42	#88180	C₂³	1	16	yes	—
43	C₂×C₄⋊Dic₁₃	—	2	6	no	—
44	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
45	C₂×C₄×Dic₁₃	—	2	12	no	—
46	C₂²×#1055	C₂×C₄	1	8	yes	—
47	C₂²×#1055	C₂×C₄	1	8	yes	—
48	C₂²×#1055	C₂×C₄	1	96	yes	complemented
49	C₂²×#1055	C₂³	1	6	yes	—
50	C₂²×#1055	C₂³	1	6	yes	—
51	C₂²×#1055	C₂³	1	16	yes	—
52	#88048	C₂³	1	32	yes	complemented
53	#88047	C₂³	1	32	yes	complemented
54	C₂²×#427	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	—	2	8	no	—
56	C₂⁴×C₁₃	C₄⋊C₄	1	8	yes	—
57	C₂⁴×C₁₃	C₂×D₄	1	6	yes	—
58	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	12	no	—
60	C₂²×C₄×C₁₃	—	4	8	no	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	48	yes	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
69	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
72	C₂²×Dic₁₃	—	2	12	no	—
73	C₂²×Dic₁₃	—	4	8	no	—
74	C₁₃×C₄⋊C₄	—	2	12	no	—
75	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
76	C₁₃×C₂²⋊C₄	—	2	16	no	—
77	C₄²×C₁₃	—	2	16	no	—
78	C₂²⋊Dic₁₃	—	2	16	no	—
79	C₄⋊Dic₁₃	—	2	12	no	—
80	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
81	C₄×Dic₁₃	—	2	16	no	—
82	C₂×#1055	C₂²×C₄	1	24	yes	—
83	C₂×#1055	C₂²×C₄	1	24	yes	—
84	C₂×#1055	C₂²×C₄	1	64	yes	complemented
85	C₂×#1055	C₂⁴	1	4	yes	—
86	C₂×#1055	C₂⁴	1	4	yes	—
87	C₂⁴×C₈	—	13	1	no	—
88	C₂²×#117	—	13	1	no	—
89	C₂²×#117	D₁₃	1	1	yes	characteristic
90	C₂×#427	—	13	24	no	—
91	C₂²×#107	—	13	2	no	—
92	C₂²×#106	—	13	2	no	—
93	C₂³×C₁₃	—	2	24	no	—
94	C₂³×C₁₃	—	2	32	no	—
95	C₂³×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
96	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₂³×C₁₃	#135	1	8	yes	—
98	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
99	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
100	C₂×C₄×C₁₃	—	2	8	no	—
101	C₂×C₄×C₁₃	—	4	24	no	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
109	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
110	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
111	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
112	C₂×Dic₁₃	—	2	8	no	—
113	C₂×Dic₁₃	—	4	24	no	—
114	#1055	C₂³×C₄	1	8	yes	—
115	#1055	C₂³×C₄	1	8	yes	—
116	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
117	C₂³×C₈	—	13	16	no	—
118	C₂³×C₈	—	13	1	no	—
119	C₂³×C₈	—	13	1	no	—
120	C₂³×C₈	—	13	12	no	—
121	C₂×#117	—	2	6	no	—
122	C₂×#117	—	13	12	no	—
123	C₂×#117	—	26	6	no	—
124	C₂×#117	C₂×D₁₃	1	12	yes	—
125	C₂²×C₄⋊C₄	—	13	2	no	—
126	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
127	C₂²×C₂²⋊C₄	—	2	1	no	—
128	C₂²×C₂²⋊C₄	—	26	1	no	—
129	C₂²×C₄²	—	2	1	no	—
130	C₂²×C₄²	—	26	1	no	—
131	#427	—	13	64	no	—
132	C₂×#107	—	13	24	no	—
133	C₂×#106	—	13	24	no	—
134	C₂²×C₁₃	—	2	48	no	—
135	C₂²×C₁₃	—	2	8	no	—
136	C₂²×C₁₃	#427	1	16	yes	complemented
137	C₂²×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
138	C₂²×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
139	C₂²×C₁₃	C₂×#135	1	12	yes	—
140	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
141	C₄×C₁₃	—	4	8	no	—
142	C₄×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
143	C₄×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
144	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
145	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
146	Dic₁₃	—	4	8	no	—
147	C₂⁵	C₂²⋊D₁₃	1	1	yes	characteristic
148	C₂³×C₄	—	2	2	no	—
149	C₂³×C₄	—	26	2	no	—
150	C₂³×C₄	Dic₂₆	1	8	yes	—
151	C₂³×C₄	C₄×D₁₃	1	8	yes	—
152	C₂³×C₄	C₂²⋊D₁₃	1	6	yes	—
153	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
154	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
155	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
156	C₂²×C₈	—	13	8	no	—
157	C₂²×C₈	—	13	8	no	—
158	C₂²×C₈	—	13	96	no	—
159	C₂²×C₈	—	13	6	no	—
160	C₂²×C₈	—	13	6	no	—
161	C₂²×C₈	—	13	16	no	—
162	#117	—	2	24	no	—
163	#117	—	13	16	no	—
164	#117	—	26	24	no	—
165	#117	C₂²×D₁₃	1	16	yes	—
166	C₂×C₄⋊C₄	—	2	6	no	—
167	C₂×C₄⋊C₄	—	13	12	no	—
168	C₂×C₄⋊C₄	—	26	6	no	—
169	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
170	C₂×C₂²⋊C₄	—	2	12	no	—
171	C₂×C₂²⋊C₄	—	26	12	no	—
172	C₂×C₄²	—	2	12	no	—
173	C₂×C₄²	—	26	12	no	—
174	#107	—	13	32	no	—
175	#106	—	13	32	no	—
176	C₂×C₁₃	—	2	8	no	—
177	C₂×C₁₃	C₂×#427	1	12	yes	complemented
178	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₁₃	C₂²×#135	1	2	yes	—
180	C₂⁴	—	2	8	no	—
181	C₂⁴	#6101	1	8	yes	—
182	C₂⁴	C₂×C₂²⋊D₁₃	1	6	yes	—
183	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
184	C₂²×C₄	—	2	12	no	—
185	C₂²×C₄	—	4	8	no	—
186	C₂²×C₄	—	26	12	no	—
187	C₂²×C₄	—	52	8	no	—
188	C₂²×C₄	#6101	1	48	yes	—
189	C₂²×C₄	#6101	1	8	yes	—
190	C₂²×C₄	C₂×Dic₂₆	1	24	yes	—
191	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
192	C₂²×C₄	C₂×C₄×D₁₃	1	24	yes	—
193	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₁₃	1	12	yes	—
197	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
198	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
199	C₂×C₈	—	13	24	no	—
200	C₂×C₈	—	13	24	no	—
201	C₂×C₈	—	13	64	no	—
202	C₂×C₈	—	13	4	no	—
203	C₂×C₈	—	13	4	no	—
204	C₄⋊C₄	—	2	12	no	—
205	C₄⋊C₄	—	13	8	no	—
206	C₄⋊C₄	—	26	12	no	—
207	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
208	C₂²⋊C₄	—	2	16	no	—
209	C₂²⋊C₄	—	26	16	no	—
210	C₄²	—	2	16	no	—
211	C₄²	—	26	16	no	—
212	C₁₃	C₂²×#427	1	1	yes	characteristic, complemented, Sylow
213	C₂³	—	2	32	no	—
214	C₂³	—	2	24	no	—
215	C₂³	C₂×#6101	1	24	yes	—
216	C₂³	C₂×#6101	1	4	yes	—
217	C₂³	#88204	1	8	yes	—
218	C₂³	C₂²×C₂²⋊D₁₃	1	4	yes	—
219	C₂³	C₂²×C₂²⋊D₁₃	1	3	yes	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	4	24	no	—
222	C₂×C₄	—	26	8	no	—
223	C₂×C₄	—	52	24	no	—
224	C₂×C₄	C₂×#6101	1	48	yes	—
225	C₂×C₄	C₂×#6101	1	12	yes	—
226	C₂×C₄	C₂×#6101	1	24	yes	—
227	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
228	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
229	C₂×C₄	C₂²×C₄×D₁₃	1	8	yes	—
230	C₂×C₄	C₂²×C₄×D₁₃	1	6	yes	—
231	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
232	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
233	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
234	C₈	—	13	8	no	—
235	C₈	—	13	8	no	—
236	C₂²	—	2	48	no	—
237	C₂²	—	2	8	no	—
238	C₂²	#11718659	1	16	yes	direct product factor
239	C₂²	C₂²×#6101	1	8	yes	—
240	C₂²	C₂²×#6101	1	6	yes	—
241	C₂²	C₂×#88204	1	12	yes	—
242	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
243	C₄	—	4	8	no	—
244	C₄	—	52	8	no	—
245	C₄	C₂²×#6101	1	8	yes	—
246	C₄	C₂²×#6101	1	6	yes	—
247	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
248	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
249	C₂	—	2	8	no	—
250	C₂	C₂×#11718659	1	12	yes	direct product factor
251	C₂	C₂³×#6101	1	1	yes	characteristic
252	C₂	C₂²×#88204	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#117	C₂
socle	C₂⁴×C₁₃	C₂×D₄
prime radical	C₂⁵×C₁₃	D₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×#117	D₁₃
hypocenter	C₁₃	C₂²×#427
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
3. C₂×#11718659	C₂	direct (N × H)
4. C₂²×#88047	C₂	semidirect
5. C₂²×#88048	C₂	semidirect
18. C₂³×#1055	C₄	semidirect
22. #11718659	C₂²	direct (N × H)
23. C₂×#88048	C₂²	semidirect
24. C₂×#88047	C₂²	semidirect
48. C₂²×#1055	C₂×C₄	semidirect
52. #88048	C₂³	semidirect
53. #88047	C₂³	semidirect
84. C₂×#1055	C₂²×C₄	semidirect
136. C₂²×C₁₃	#427	semidirect
177. C₂×C₁₃	C₂×#427	semidirect
212. C₁₃	C₂²×#427	semidirect
238. C₂²	#11718659	direct (N × H)
250. C₂	C₂×#11718659	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#1055	2
C₂²×#88180	2
C₂×#11718659	2
C₂²×#88047	2
C₂²×#88048	2
C₂²×C₁₃×#117	2
C₂²×#427	13

Subgroups — C22×#11718659 #16726567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718659 #16726567