Subgroups — #16019657

Order 3328.

696 conjugacy classes of subgroups.

The subgroup lattice has 268 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	2	yes	maximal, complemented
2	#13126596	C₂	1	2	yes	maximal, complemented
3	#13117086	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116922	C₂	1	2	yes	maximal, complemented
5	#13116877	C₂	1	2	yes	maximal, complemented
6	#13116761	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88158	C₂²	1	4	yes	—
10	#11719258	—	2	2	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	#11718776	C₂²	1	4	yes	—
14	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
15	#11718565	—	2	8	no	—
16	C₂×#88040	C₂²	1	4	yes	—
17	C₂×D₂₀₈	C₂²	1	2	yes	—
18	C₂×D₂₀₈	C₂²	1	2	yes	—
19	#11718509	—	2	2	no	—
20	#11718395	—	2	4	no	—
21	C₄×Dic₅₂	C₂²	1	2	yes	—
22	#11718356	C₂²	1	2	yes	—
23	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
26	#11718101	C₂²	1	4	yes	—
27	#11718096	C₂²	1	1	yes	characteristic
28	#11718095	C₂²	1	2	yes	—
29	C₂×#6127	—	2	4	no	—
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
32	#88194	—	4	2	no	—
33	#88184	—	2	4	no	—
34	C₂×Dic₅₂	C₂³	1	2	yes	—
35	#88158	—	2	8	no	—
36	#88158	—	4	4	no	—
37	C₂×D₁₀₄	—	2	2	no	—
38	C₂×D₁₀₄	C₂³	1	4	yes	—
39	C₂×#6095	—	2	4	no	—
40	#88147	—	4	2	no	—
41	#88138	—	4	4	no	—
42	#88128	—	2	2	no	—
43	C₄×Dic₂₆	—	2	2	no	—
44	C₂×C₁₃×C₁₆	—	2	4	no	—
45	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	#88061	—	2	4	no	—
52	#88058	C₂³	1	2	yes	—
53	#88056	—	2	2	no	—
54	Dic₁₀₄	D₄	1	4	yes	complemented
55	#88040	—	2	8	no	—
56	D₂₀₈	—	2	4	no	—
57	D₂₀₈	—	4	2	no	—
58	#33853	—	13	1	no	maximal, Sylow
59	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₂²⋊D₁₃	—	4	4	no	—
62	C₂×C₂²⋊D₁₃	—	4	4	no	—
63	#6127	—	4	8	no	—
64	#6127	—	8	4	no	—
65	C₂×D₅₂	—	2	4	no	—
66	C₂×D₅₂	—	4	2	no	—
67	C₂×D₅₂	—	4	2	no	—
68	C₂×C₄×D₁₃	—	4	4	no	—
69	C₂×Dic₂₆	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
76	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
77	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
78	#6103	—	4	4	no	—
79	C₄⋊Dic₁₃	—	2	2	no	—
80	#6101	—	4	2	no	—
81	C₄×Dic₁₃	—	4	2	no	—
82	Dic₅₂	—	2	2	no	—
83	Dic₅₂	C₂×D₄	1	4	yes	—
84	D₁₀₄	—	2	4	no	—
85	D₁₀₄	—	2	4	no	—
86	D₁₀₄	—	8	2	no	—
87	#6095	—	4	8	no	—
88	C₁₃×C₁₆	—	2	2	no	—
89	C₁₃×C₁₆	—	2	1	no	—
90	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
91	C₂×#508	—	13	2	no	—
92	#3287	—	13	2	no	—
93	#2246	—	13	1	no	—
94	#2215	—	13	2	no	—
95	#2207	—	13	2	no	—
96	#2186	—	13	4	no	—
97	C₄×Q₃₂	—	13	1	no	—
98	D₄×C₁₆	D₁₃	1	1	yes	characteristic, complemented
99	C₂³×C₁₃	C₂×D₈	1	2	yes	—
100	C₂²×D₁₃	—	4	4	no	—
101	D₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂²⋊D₁₃	—	8	8	no	—
109	C₂²⋊D₁₃	—	16	4	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	D₅₂	—	4	4	no	—
113	D₅₂	—	4	4	no	—
114	D₅₂	—	8	4	no	—
115	D₅₂	—	16	2	no	—
116	C₄×D₁₃	—	8	8	no	—
117	Dic₂₆	—	2	4	no	—
118	Dic₂₆	—	4	2	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	—	2	1	no	—
121	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×#135	—	13	4	no	—
124	#550	—	26	2	no	—
125	#508	—	26	8	no	—
126	C₂×Q₃₂	—	13	1	no	—
127	C₂×SD₃₂	—	13	4	no	—
128	C₂×D₁₆	—	13	2	no	—
129	C₂×D₁₆	—	13	2	no	—
130	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
131	#493	—	13	2	no	—
132	#466	—	13	4	no	—
133	#460	—	26	2	no	—
134	#449	—	26	4	no	—
135	C₄×Q₁₆	—	13	2	no	—
136	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
137	#366	—	13	1	no	—
138	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
139	#358	—	13	2	no	—
140	#357	—	13	4	no	—
141	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
142	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₁₃	—	8	4	no	—
148	C₂×D₁₃	—	8	4	no	—
149	C₂×D₁₃	—	8	4	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	#508	1	2	yes	complemented
153	C₄×C₁₃	D₄²	1	1	yes	characteristic
154	C₄×C₁₃	#576	1	1	yes	characteristic
155	Dic₁₃	—	4	4	no	—
156	Dic₁₃	—	8	2	no	—
157	C₂×#41	—	26	4	no	—
158	#135	—	26	8	no	—
159	#135	—	52	4	no	—
160	C₂×Q₁₆	—	13	2	no	—
161	C₂×SD₁₆	—	26	4	no	—
162	C₂×D₈	—	13	4	no	—
163	C₂×D₈	—	26	2	no	—
164	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
165	#127	—	26	2	no	—
166	#127	—	52	2	no	—
167	#121	—	26	4	no	—
168	#121	—	52	4	no	—
169	C₄×Q₈	—	26	2	no	—
170	C₄×D₄	—	52	2	no	—
171	C₄×D₄	D₅₂	1	1	yes	characteristic
172	Q₃₂	—	13	4	no	—
173	SD₃₂	—	26	8	no	—
174	D₁₆	—	26	4	no	—
175	D₁₆	—	52	2	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
178	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
179	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
180	#107	—	13	2	no	—
181	C₄⋊C₈	D₅₂	1	1	yes	characteristic
182	#103	—	26	2	no	—
183	#102	—	26	4	no	—
184	C₂²⋊C₈	D₅₂	1	2	yes	—
185	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₂×C₁₃	—	4	2	no	—
187	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
188	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
189	C₂×C₁₃	#3415	1	1	yes	characteristic
190	D₁₃	—	16	4	no	—
191	#41	—	52	8	no	—
192	#41	—	104	4	no	—
193	C₂×Q₈	—	26	2	no	—
194	C₂×D₄	—	26	4	no	—
195	C₂×D₄	—	52	4	no	—
196	C₂×D₄	—	52	4	no	—
197	C₂×D₄	—	52	2	no	—
198	C₂×D₄	—	52	2	no	—
199	C₂×D₄	D₁₀₄	1	1	yes	characteristic
200	C₂²×C₄	—	52	4	no	—
201	C₂²×C₄	C₂×D₅₂	1	2	yes	—
202	Q₁₆	—	13	4	no	—
203	Q₁₆	—	26	2	no	—
204	SD₁₆	—	52	8	no	—
205	D₈	—	26	4	no	—
206	D₈	—	26	4	no	—
207	D₈	—	104	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
211	C₂×C₈	C₂×D₅₂	1	2	yes	—
212	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
213	C₄⋊C₄	—	26	2	no	—
214	C₄⋊C₄	—	52	2	no	—
215	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
216	C₂²⋊C₄	—	52	4	no	—
217	C₂²⋊C₄	D₁₀₄	1	2	yes	—
218	C₄²	—	52	2	no	—
219	C₄²	C₂×D₅₂	1	1	yes	characteristic
220	C₁₆	—	2	2	no	—
221	C₁₆	—	2	1	no	—
222	C₁₆	D₄×D₁₃	1	2	yes	—
223	C₁₃	#33853	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	52	4	no	—
225	C₂³	C₂×D₁₀₄	1	2	yes	—
226	Q₈	—	26	4	no	—
227	Q₈	—	52	2	no	—
228	D₄	—	2	2	no	—
229	D₄	—	52	4	no	—
230	D₄	—	52	4	no	—
231	D₄	—	104	8	no	—
232	D₄	—	104	4	no	—
233	D₄	—	208	2	no	—
234	D₄	—	208	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	52	2	no	—
238	C₂×C₄	—	52	2	no	—
239	C₂×C₄	—	104	8	no	—
240	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
241	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
242	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	2	no	—
246	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
247	C₈	#88256	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	104	4	no	—
252	C₂²	—	104	4	no	—
253	C₂²	—	104	4	no	—
254	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	52	4	no	—
258	C₄	—	104	2	no	—
259	C₄	#11718565	1	2	yes	complemented
260	C₄	D₄×D₅₂	1	1	yes	characteristic
261	C₄	#11719457	1	1	yes	characteristic
262	C₂	—	4	2	no	—
263	C₂	—	208	4	no	—
264	C₂	D₄×D₁₀₄	1	1	yes	characteristic
265	C₂	C₂×#11718565	1	1	yes	characteristic
266	C₂	#13127977	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	D₄×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	D₄×C₁₆	D₁₃
hypocenter	C₁₃	#33853
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11718565	C₂	semidirect
2. #13126596	C₂	semidirect
3. #13117086	C₂	semidirect
4. #13116922	C₂	semidirect
5. #13116877	C₂	semidirect
6. #13116761	C₂	semidirect
7. C₄×Dic₁₀₄	C₂	semidirect
8. D₄×C₁₃×C₁₆	C₂	semidirect
54. Dic₁₀₄	D₄	semidirect
98. D₄×C₁₆	D₁₃	semidirect
152. C₄×C₁₃	#508	semidirect
223. C₁₃	#33853	semidirect
259. C₄	#11718565	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
#13126596	2
#13117086	2
#13116922	2
#13116877	2
#13116761	2
C₄×Dic₁₀₄	2
D₄×C₁₃×C₁₆	2
#33853	13