Subgroups — #16016303

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#13116699	C₂	1	4	yes	maximal, complemented
4	#13116694	C₂	1	2	yes	maximal, complemented
5	#13116682	C₂	1	2	yes	maximal, complemented
6	#13116546	C₂	1	2	yes	maximal, complemented
7	#13116432	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#366	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×#366	C₂²	1	2	yes	—
24	C₁₃×#366	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	#33459	—	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	1	yes	characteristic
77	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
78	#6101	—	4	4	no	—
79	C₄×Dic₁₃	—	4	4	no	—
80	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
81	#6099	—	2	2	no	—
82	C₂×#1055	C₂×D₄	1	2	yes	—
83	Dic₅₂	—	2	2	no	—
84	Dic₅₂	C₂²×C₄	1	4	yes	—
85	D₁₀₄	—	2	2	no	—
86	D₁₀₄	C₂²×C₄	1	4	yes	—
87	#6095	—	2	8	no	—
88	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
89	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
90	#6090	—	2	2	no	—
91	C₂×#508	—	13	1	no	—
92	#2951	—	13	2	no	—
93	#2175	—	13	4	no	—
94	C₄×Q₃₂	—	13	2	no	—
95	C₄×D₁₆	—	13	2	no	—
96	C₂×#366	D₁₃	1	1	yes	characteristic, complemented
97	#2139	—	13	2	no	—
98	#2108	—	13	1	no	—
99	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	4	no	—
102	C₂×C₄×C₁₃	—	16	2	no	—
103	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#135	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₂²⋊D₁₃	—	4	8	no	—
108	C₂×Dic₁₃	—	4	2	no	—
109	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
110	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
111	D₅₂	—	2	4	no	—
112	D₅₂	—	4	2	no	—
113	C₄×D₁₃	—	4	8	no	—
114	C₄×D₁₃	—	16	4	no	—
115	Dic₂₆	—	2	4	no	—
116	Dic₂₆	—	4	2	no	—
117	C₈×C₁₃	C₄×D₄	1	2	yes	—
118	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
120	#1055	—	2	2	no	—
121	C₂×#135	—	13	2	no	—
122	#520	—	26	2	no	—
123	#508	—	13	8	no	—
124	C₂×Q₃₂	—	13	1	no	—
125	C₂×SD₃₂	—	13	2	no	—
126	C₂×D₁₆	—	13	1	no	—
127	C₂×M₃₂	—	13	1	no	—
128	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
129	#440	—	26	4	no	—
130	C₄×Q₁₆	—	13	4	no	—
131	C₄×D₈	—	13	4	no	—
132	C₂×#107	C₂×D₁₃	1	2	yes	—
133	#417	—	26	2	no	—
134	#405	—	13	1	no	—
135	#366	C₂×D₁₃	1	2	yes	—
136	#366	C₂×D₁₃	1	2	yes	—
137	#358	—	13	4	no	—
138	#357	—	13	4	no	—
139	#346	—	13	2	no	—
140	C₄×C₁₆	—	13	2	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	C₄×D₈	1	2	yes	—
144	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₁₃	—	4	4	no	—
146	C₂×D₁₃	—	8	2	no	—
147	C₂×D₁₃	—	8	2	no	—
148	C₄×C₁₃	—	8	4	no	—
149	C₄×C₁₃	C₄×D₈	1	2	yes	—
150	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
152	Dic₁₃	—	2	2	no	—
153	Dic₁₃	—	2	2	no	—
154	Dic₁₃	—	4	4	no	—
155	C₂×#41	—	26	2	no	—
156	#135	—	13	8	no	—
157	#135	—	26	4	no	—
158	C₂×Q₁₆	—	13	2	no	—
159	C₂×SD₁₆	—	26	2	no	—
160	C₂×D₈	—	13	2	no	—
161	C₂×M₁₆	—	13	1	no	—
162	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
163	C₄×Q₈	—	26	4	no	—
164	C₄×D₄	—	26	4	no	—
165	C₄×D₄	—	52	4	no	—
166	#117	—	13	1	no	—
167	#117	—	52	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	52	2	no	—
170	Q₃₂	—	13	4	no	—
171	SD₃₂	—	13	8	no	—
172	D₁₆	—	13	4	no	—
173	M₃₂	—	26	2	no	—
174	C₂×C₁₆	—	13	2	no	—
175	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
176	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
178	#107	—	2	2	no	—
179	#107	C₂²×D₁₃	1	4	yes	—
180	#103	—	26	4	no	—
181	#102	—	26	4	no	—
182	#97	—	13	2	no	—
183	C₄×C₈	—	13	2	no	—
184	C₂×C₁₃	—	2	2	no	—
185	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
186	C₂×C₁₃	#3415	1	1	yes	characteristic
187	C₂×C₁₃	#3417	1	1	yes	characteristic
188	D₁₃	—	8	4	no	—
189	#41	—	26	8	no	—
190	#41	—	52	4	no	—
191	C₂×Q₈	—	26	2	no	—
192	C₂×D₄	—	26	2	no	—
193	C₂×D₄	—	52	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	52	4	no	—
196	C₂²×C₄	—	52	2	no	—
197	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
198	Q₁₆	—	13	4	no	—
199	Q₁₆	—	26	2	no	—
200	SD₁₆	—	26	8	no	—
201	D₈	—	13	4	no	—
202	D₈	—	26	2	no	—
203	M₁₆	—	26	2	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
206	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
207	C₂×C₈	#6131	1	2	yes	—
208	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	13	1	no	—
212	C₄⋊C₄	—	13	1	no	—
213	C₄⋊C₄	—	52	4	no	—
214	C₂²⋊C₄	—	13	2	no	—
215	C₂²⋊C₄	—	52	4	no	—
216	C₄²	—	13	2	no	—
217	C₄²	—	52	4	no	—
218	C₁₆	—	26	2	no	—
219	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
220	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
221	C₁₃	#33459	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	52	2	no	—
223	C₂³	D₈×D₁₃	1	1	yes	characteristic
224	Q₈	—	26	4	no	—
225	Q₈	—	52	2	no	—
226	D₄	—	26	4	no	—
227	D₄	—	52	8	no	—
228	D₄	—	52	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	13	2	no	—
231	C₂×C₄	—	13	2	no	—
232	C₂×C₄	—	16	2	no	—
233	C₂×C₄	—	52	2	no	—
234	C₂×C₄	—	52	8	no	—
235	C₂×C₄	—	208	4	no	—
236	C₂×C₄	#88147	1	2	yes	—
237	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
238	C₂×C₄	#88173	1	2	yes	—
239	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
240	C₈	—	26	2	no	—
241	C₈	#88147	1	2	yes	—
242	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
243	C₈	C₂×#6131	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	52	4	no	—
247	C₂²	—	104	2	no	—
248	C₂²	—	104	2	no	—
249	C₂²	#11718533	1	2	yes	—
250	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
251	C₄	—	8	4	no	—
252	C₄	—	26	2	no	—
253	C₄	—	26	2	no	—
254	C₄	—	52	4	no	—
255	C₄	#11718533	1	2	yes	—
256	C₄	C₂×#88147	1	1	yes	characteristic
257	C₄	C₂×#88173	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	104	4	no	—
260	C₂	C₂×#11718533	1	1	yes	characteristic
261	C₂	#13127973	1	1	yes	characteristic
262	C₂	#13127985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#366	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718565	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#366	D₁₃
hypocenter	C₁₃	#33459
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #13123710	C₂	semidirect
3. #13116699	C₂	semidirect
4. #13116694	C₂	semidirect
5. #13116682	C₂	semidirect
6. #13116546	C₂	semidirect
7. #13116432	C₂	semidirect
8. C₂×C₁₃×#366	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
96. C₂×#366	D₁₃	semidirect
175. C₂×C₁₆	C₄×D₁₃	semidirect
221. C₁₃	#33459	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
#13116699	2
#13116694	2
#13116682	2
#13116546	2
#13116432	2
C₂×C₁₃×#366	2
#33459	13