Subgroups — #16016312

Order 3328.

674 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	C₂×#11718565	C₂	1	1	yes	maximal, characteristic
2	#13123710	C₂	1	2	yes	maximal, complemented
3	#13116711	C₂	1	2	yes	maximal, complemented
4	#13116706	C₂	1	2	yes	maximal, complemented
5	#13116687	C₂	1	4	yes	maximal, complemented
6	#13116546	C₂	1	2	yes	maximal, complemented
7	#13116432	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88158	C₂²	1	2	yes	—
10	#11719209	—	2	2	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	#11718762	C₂²	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	#11718627	—	2	2	no	—
16	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
17	#11718565	C₄	1	8	yes	complemented
18	C₂×#88040	C₂²	1	2	yes	—
19	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
20	#11718550	—	2	4	no	—
21	#11718534	C₂²	1	4	yes	—
22	#11718533	C₂²	1	4	yes	—
23	C₁₃×#367	C₂²	1	2	yes	—
24	C₁₃×#367	C₂²	1	2	yes	—
25	#11718094	C₂²	1	2	yes	—
26	C₁₆×Dic₁₃	C₂²	1	2	yes	—
27	#11718084	C₂²	1	4	yes	—
28	#11718083	C₂²	1	4	yes	—
29	C₂×#6127	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₄×C₂²⋊D₁₃	—	4	4	no	—
33	#88180	D₄	1	1	yes	characteristic
34	C₂×#6099	D₄	1	1	yes	characteristic
35	C₂×Dic₅₂	C₂³	1	2	yes	—
36	#88158	—	2	4	no	—
37	#88158	C₂×C₄	1	8	yes	complemented
38	C₂×D₁₀₄	C₂³	1	2	yes	—
39	C₂×#6095	—	2	2	no	—
40	#88147	—	2	4	no	—
41	#88146	—	4	2	no	—
42	C₄⋊C₄×D₁₃	—	4	2	no	—
43	#88141	—	2	4	no	—
44	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
45	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
47	C₁₃×#107	—	2	2	no	—
48	C₁₃×#107	C₂³	1	4	yes	—
49	#88055	D₄	1	2	yes	—
50	C₈×Dic₁₃	C₂³	1	2	yes	—
51	#88052	—	2	2	no	—
52	C₂×#6090	C₂³	1	2	yes	—
53	#88050	—	2	4	no	—
54	#88049	—	2	4	no	—
55	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
56	#88040	C₂×C₄	1	8	yes	complemented
57	D₂₀₈	C₂×C₄	1	4	yes	complemented
58	#33460	—	13	1	no	maximal, Sylow
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	—	4	2	no	—
62	#6127	—	2	8	no	—
63	#6127	—	4	4	no	—
64	C₂×D₅₂	—	2	2	no	—
65	C₂×C₄×D₁₃	—	4	2	no	—
66	C₂×C₄×D₁₃	—	4	4	no	—
67	C₂×Dic₂₆	—	2	2	no	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	#41	1	2	yes	—
71	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₂²⋊Dic₁₃	D₈	1	2	yes	—
75	#6103	—	4	4	no	—
76	C₄⋊Dic₁₃	D₈	1	2	yes	—
77	#6101	—	4	4	no	—
78	C₄×Dic₁₃	—	4	4	no	—
79	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
80	#6099	—	2	2	no	—
81	C₂×#1055	C₂×D₄	1	2	yes	—
82	Dic₅₂	—	2	2	no	—
83	Dic₅₂	C₂²×C₄	1	4	yes	—
84	D₁₀₄	—	2	2	no	—
85	D₁₀₄	C₂²×C₄	1	4	yes	—
86	#6095	—	2	8	no	—
87	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
88	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
89	#6090	—	2	2	no	—
90	C₂×#508	—	13	1	no	—
91	#2951	—	13	2	no	—
92	#2177	—	13	2	no	—
93	#2176	—	13	2	no	—
94	C₄×SD₃₂	—	13	4	no	—
95	C₂×#367	D₁₃	1	1	yes	characteristic, complemented
96	#2139	—	13	2	no	—
97	#2108	—	13	1	no	—
98	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
99	C₂²×D₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	—	16	2	no	—
102	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#135	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₂²⋊D₁₃	—	4	8	no	—
107	C₂×Dic₁₃	—	4	2	no	—
108	C₂×Dic₁₃	C₂×D₈	1	4	yes	—
109	D₅₂	—	2	4	no	—
110	D₅₂	—	4	2	no	—
111	C₄×D₁₃	—	4	8	no	—
112	C₄×D₁₃	—	16	4	no	—
113	Dic₂₆	—	2	4	no	—
114	Dic₂₆	—	4	2	no	—
115	C₈×C₁₃	C₄×D₄	1	2	yes	—
116	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
118	#1055	—	2	2	no	—
119	C₂×#135	—	13	2	no	—
120	#520	—	26	2	no	—
121	#508	—	13	8	no	—
122	C₂×Q₃₂	—	13	1	no	—
123	C₂×SD₃₂	—	13	2	no	—
124	C₂×D₁₆	—	13	1	no	—
125	C₂×M₃₂	—	13	1	no	—
126	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
127	#440	—	26	4	no	—
128	C₄×Q₁₆	—	13	4	no	—
129	C₄×D₈	—	13	4	no	—
130	C₂×#107	C₂×D₁₃	1	2	yes	—
131	#417	—	26	2	no	—
132	#405	—	13	1	no	—
133	#367	C₂×D₁₃	1	2	yes	—
134	#367	C₂×D₁₃	1	2	yes	—
135	#358	—	13	4	no	—
136	#357	—	13	4	no	—
137	#346	—	13	2	no	—
138	C₄×C₁₆	—	13	2	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	C₄×D₈	1	2	yes	—
141	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₁₃	—	4	4	no	—
143	C₂×D₁₃	—	8	2	no	—
144	C₂×D₁₃	—	8	2	no	—
145	C₄×C₁₃	—	8	4	no	—
146	C₄×C₁₃	C₄×D₈	1	2	yes	—
147	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
149	Dic₁₃	—	2	4	no	—
150	Dic₁₃	—	4	4	no	—
151	C₂×#41	—	26	2	no	—
152	#135	—	13	8	no	—
153	#135	—	26	4	no	—
154	C₂×Q₁₆	—	13	2	no	—
155	C₂×SD₁₆	—	26	2	no	—
156	C₂×D₈	—	13	2	no	—
157	C₂×M₁₆	—	13	1	no	—
158	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
159	C₄×Q₈	—	26	4	no	—
160	C₄×D₄	—	26	4	no	—
161	C₄×D₄	—	52	4	no	—
162	#117	—	13	1	no	—
163	#117	—	52	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	52	2	no	—
166	Q₃₂	—	13	4	no	—
167	SD₃₂	—	13	8	no	—
168	D₁₆	—	13	4	no	—
169	M₃₂	—	26	2	no	—
170	C₂×C₁₆	—	13	2	no	—
171	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
172	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
174	#107	—	2	2	no	—
175	#107	C₂²×D₁₃	1	4	yes	—
176	#103	—	26	4	no	—
177	#102	—	26	4	no	—
178	#97	—	13	2	no	—
179	C₄×C₈	—	13	2	no	—
180	C₂×C₁₃	—	2	2	no	—
181	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
182	C₂×C₁₃	#3416	1	2	yes	—
183	D₁₃	—	8	4	no	—
184	#41	—	26	8	no	—
185	#41	—	52	4	no	—
186	C₂×Q₈	—	26	2	no	—
187	C₂×D₄	—	26	2	no	—
188	C₂×D₄	—	52	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	52	4	no	—
191	C₂²×C₄	—	52	2	no	—
192	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
193	Q₁₆	—	13	4	no	—
194	Q₁₆	—	26	2	no	—
195	SD₁₆	—	26	8	no	—
196	D₈	—	13	4	no	—
197	D₈	—	26	2	no	—
198	M₁₆	—	26	2	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
201	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
202	C₂×C₈	#6131	1	2	yes	—
203	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	13	2	no	—
207	C₄⋊C₄	—	52	4	no	—
208	C₂²⋊C₄	—	13	2	no	—
209	C₂²⋊C₄	—	52	4	no	—
210	C₄²	—	13	2	no	—
211	C₄²	—	52	4	no	—
212	C₁₆	—	26	2	no	—
213	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
214	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
215	C₁₃	#33460	1	1	yes	characteristic, complemented, Sylow
216	C₂³	—	52	2	no	—
217	C₂³	D₈×D₁₃	1	1	yes	characteristic
218	Q₈	—	26	4	no	—
219	Q₈	—	52	2	no	—
220	D₄	—	26	4	no	—
221	D₄	—	52	8	no	—
222	D₄	—	52	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	13	4	no	—
225	C₂×C₄	—	16	2	no	—
226	C₂×C₄	—	52	2	no	—
227	C₂×C₄	—	52	8	no	—
228	C₂×C₄	—	208	4	no	—
229	C₂×C₄	#88147	1	2	yes	—
230	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
231	C₂×C₄	#88173	1	2	yes	—
232	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
233	C₈	—	26	2	no	—
234	C₈	#88147	1	2	yes	—
235	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
236	C₈	C₂×#6131	1	1	yes	characteristic
237	C₂²	—	2	2	no	—
238	C₂²	—	52	4	no	—
239	C₂²	—	104	2	no	—
240	C₂²	—	104	2	no	—
241	C₂²	#11718533	1	2	yes	—
242	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
243	C₄	—	8	4	no	—
244	C₄	—	26	4	no	—
245	C₄	—	52	4	no	—
246	C₄	#11718533	1	2	yes	—
247	C₄	C₂×#88147	1	1	yes	characteristic
248	C₄	C₂×#88173	1	1	yes	characteristic
249	C₂	—	2	2	no	—
250	C₂	—	104	4	no	—
251	C₂	C₂×#11718533	1	1	yes	characteristic
252	C₂	#13127978	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#367	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718565	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#367	D₁₃
hypocenter	C₁₃	#33460
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
2. #13123710	C₂	semidirect
3. #13116711	C₂	semidirect
4. #13116706	C₂	semidirect
5. #13116687	C₂	semidirect
6. #13116546	C₂	semidirect
7. #13116432	C₂	semidirect
8. C₂×C₁₃×#367	C₂	semidirect
17. #11718565	C₄	semidirect
37. #88158	C₂×C₄	semidirect
44. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
55. Dic₁₀₄	C₂×C₄	semidirect
56. #88040	C₂×C₄	semidirect
57. D₂₀₈	C₂×C₄	semidirect
95. C₂×#367	D₁₃	semidirect
171. C₂×C₁₆	C₄×D₁₃	semidirect
215. C₁₃	#33460	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718565	2
#13123710	2
#13116711	2
#13116706	2
#13116687	2
#13116546	2
#13116432	2
C₂×C₁₃×#367	2
#33460	13