Subgroups — #16018278

Order 3328.

578 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	#13125350	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718360	C₂	1	1	yes	maximal, characteristic
3	#13117082	C₂	1	4	yes	maximal, complemented
4	#13117064	C₂	1	4	yes	maximal, complemented
5	#13116557	C₂	1	4	yes	maximal
6	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₂	—	2	1	no	—
8	C₂²×Dic₅₂	C₂²	1	2	yes	—
9	#11719175	—	2	1	no	—
10	C₂×#88123	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718767	—	2	1	no	—
14	#11718767	C₂²	1	2	yes	—
15	#11718360	—	2	1	no	—
16	#11718360	—	2	1	no	—
17	#11718360	C₂²	1	2	yes	—
18	#11718360	C₂²	1	2	yes	—
19	#11718344	—	2	1	no	—
20	#11718344	C₂²	1	2	yes	—
21	#11718343	C₂²	1	2	yes	—
22	#11718341	—	2	1	no	—
23	C₁₃×#346	C₂²	1	2	yes	—
24	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
25	#11718095	C₂²	1	16	yes	—
26	C₂²×Dic₂₆	—	2	2	no	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	#88180	—	2	2	no	—
31	C₂×C₄⋊Dic₁₃	—	4	2	no	—
32	#88178	—	4	2	no	—
33	C₂×Dic₅₂	—	2	8	no	—
34	C₂×Dic₅₂	—	2	2	no	—
35	C₂×Dic₅₂	—	2	2	no	—
36	C₂×Dic₅₂	—	2	2	no	—
37	C₂×Dic₅₂	—	4	4	no	—
38	C₂×Dic₅₂	C₂³	1	4	yes	—
39	#88124	—	2	2	no	—
40	#88123	—	2	2	no	—
41	#88123	—	2	2	no	—
42	#88123	—	2	2	no	—
43	#88123	—	4	2	no	—
44	C₄×Dic₂₆	—	4	2	no	—
45	C₁₃×M₃₂	—	2	4	no	—
46	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	#88058	—	2	2	no	—
52	#88058	C₂³	1	4	yes	—
53	#88057	—	2	2	no	—
54	#88057	—	2	2	no	—
55	#33705	—	13	1	no	maximal, Sylow
56	C₂²×C₄×C₁₃	D₈	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	—	4	2	no	—
59	C₂×Dic₂₆	—	2	4	no	—
60	C₂×Dic₂₆	—	4	4	no	—
61	C₂×Dic₂₆	—	4	4	no	—
62	C₂×Dic₂₆	—	4	2	no	—
63	C₂×Dic₂₆	—	4	2	no	—
64	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	#41	1	4	yes	—
69	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₄²×C₁₃	D₈	1	2	yes	—
71	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²⋊Dic₁₃	—	4	2	no	—
74	C₄⋊Dic₁₃	—	2	4	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	C₄⋊Dic₁₃	—	4	4	no	—
77	C₄⋊Dic₁₃	—	4	4	no	—
78	#6101	—	4	4	no	—
79	C₄×Dic₁₃	—	4	2	no	—
80	Dic₅₂	—	2	8	no	—
81	Dic₅₂	—	4	4	no	—
82	Dic₅₂	—	4	4	no	—
83	Dic₅₂	—	4	4	no	—
84	C₁₃×C₁₆	—	2	4	no	—
85	#3160	—	13	1	no	—
86	C₂×#494	—	13	1	no	—
87	#2245	—	13	4	no	—
88	#2241	—	13	4	no	—
89	#2141	—	13	4	no	—
90	C₄×M₃₂	D₁₃	1	1	yes	characteristic
91	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	#124	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₁₃	—	4	4	no	—
100	C₂×Dic₁₃	—	4	4	no	—
101	C₂×Dic₁₃	—	8	4	no	—
102	Dic₂₆	—	4	4	no	—
103	Dic₂₆	—	4	4	no	—
104	Dic₂₆	—	8	4	no	—
105	Dic₂₆	—	8	4	no	—
106	C₈×C₁₃	#124	1	2	yes	—
107	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
108	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
109	C₈×C₁₃	C₂×#41	1	2	yes	—
110	C₂²×Q₁₆	—	13	2	no	—
111	C₂²×Q₁₆	—	26	1	no	—
112	#533	—	26	1	no	—
113	C₂×C₄⋊Q₈	—	26	1	no	—
114	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
115	#500	—	13	2	no	—
116	#499	—	13	2	no	—
117	#499	—	26	1	no	—
118	#498	—	26	1	no	—
119	#494	—	13	2	no	—
120	#494	—	13	2	no	—
121	#494	—	26	1	no	—
122	#494	—	26	1	no	—
123	#427	—	13	2	no	—
124	#427	—	26	1	no	—
125	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
126	#358	—	13	16	no	—
127	#346	C₂×D₁₃	1	2	yes	—
128	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	#486	1	2	yes	—
131	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	#486	1	2	yes	—
134	C₄×C₁₃	#510	1	4	yes	—
135	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
136	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
137	Dic₁₃	—	8	4	no	—
138	Dic₁₃	—	8	4	no	—
139	C₂²×Q₈	—	26	2	no	—
140	C₂×Q₁₆	—	13	4	no	—
141	C₂×Q₁₆	—	26	2	no	—
142	C₂×Q₁₆	—	26	2	no	—
143	C₂×Q₁₆	—	26	2	no	—
144	C₂×Q₁₆	—	26	8	no	—
145	C₂×Q₁₆	—	52	4	no	—
146	C₂²×C₈	D₅₂	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
148	C₄⋊Q₈	—	26	2	no	—
149	C₄⋊Q₈	—	26	2	no	—
150	C₄⋊Q₈	—	26	2	no	—
151	C₄⋊Q₈	—	52	2	no	—
152	#125	—	26	2	no	—
153	C₂²⋊Q₈	—	52	2	no	—
154	C₄×Q₈	—	52	2	no	—
155	#117	—	26	2	no	—
156	C₂×C₄⋊C₄	—	52	2	no	—
157	C₂×C₄²	D₅₂	1	1	yes	characteristic
158	M₃₂	—	2	4	no	—
159	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
160	#107	—	13	4	no	—
161	#107	—	26	2	no	—
162	#106	—	26	2	no	—
163	#106	—	26	2	no	—
164	C₄×C₈	D₅₂	1	1	yes	characteristic
165	C₄×C₈	D₅₂	1	1	yes	characteristic
166	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×C₁₃	—	2	2	no	—
169	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#510	1	2	yes	—
171	C₂×Q₈	—	26	4	no	—
172	C₂×Q₈	—	52	2	no	—
173	C₂×Q₈	—	52	2	no	—
174	C₂×Q₈	—	52	4	no	—
175	C₂×Q₈	—	52	4	no	—
176	C₂²×C₄	—	52	2	no	—
177	C₂²×C₄	D₁₀₄	1	2	yes	—
178	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
179	Q₁₆	—	26	8	no	—
180	Q₁₆	—	52	4	no	—
181	Q₁₆	—	52	4	no	—
182	Q₁₆	—	52	4	no	—
183	C₂×C₈	#6095	1	4	yes	—
184	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
185	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
186	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
187	C₂×C₈	#6127	1	4	yes	—
188	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
189	C₄⋊C₄	—	26	4	no	—
190	C₄⋊C₄	—	52	4	no	—
191	C₄⋊C₄	—	52	4	no	—
192	C₄⋊C₄	—	52	4	no	—
193	C₄⋊C₄	—	52	2	no	—
194	C₂²⋊C₄	—	52	2	no	—
195	C₄²	—	52	2	no	—
196	C₄²	D₁₀₄	1	2	yes	—
197	C₄²	C₂×D₅₂	1	1	yes	characteristic
198	C₄²	C₂×D₅₂	1	1	yes	characteristic
199	C₁₆	—	2	4	no	—
200	C₁₃	#33705	1	1	yes	characteristic, complemented, Sylow
201	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
202	Q₈	—	52	4	no	—
203	Q₈	—	52	4	no	—
204	Q₈	—	104	4	no	—
205	Q₈	—	104	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	52	4	no	—
208	C₂×C₄	—	52	4	no	—
209	C₂×C₄	—	104	4	no	—
210	C₂×C₄	#88129	1	2	yes	—
211	C₂×C₄	C₂×#6095	1	2	yes	—
212	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
213	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
214	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
216	C₈	#88129	1	2	yes	—
217	C₈	C₂×#6095	1	2	yes	—
218	C₈	C₂×#6095	1	2	yes	—
219	C₈	C₂×#6127	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#11718355	1	2	yes	—
222	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	104	4	no	—
225	C₄	—	104	4	no	—
226	C₄	#11718355	1	2	yes	—
227	C₄	#11718570	1	4	yes	—
228	C₄	C₂×#88129	1	1	yes	characteristic
229	C₄	C₂²×#6095	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#11718355	1	1	yes	characteristic
232	C₂	C₂×#11718570	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#33705
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13117082	C₂	semidirect
4. #13117064	C₂	semidirect
200. C₁₃	#33705	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
C₂×#11718360	2
#13117082	2
#13117064	2
#13116557	2
C₄×C₁₃×M₃₂	2
#33705	13