Subgroups — #15832068

Order 3328.

730 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	2	yes	maximal
2	#13113611	C₂	1	8	yes	maximal, complemented
3	#13113595	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×#6099	—	2	2	no	—
7	C₁₃×#431	C₄	1	4	yes	—
8	C₁₃×#431	C₂²	1	4	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
12	#11718585	—	2	2	no	—
13	#11718585	—	2	2	no	—
14	#11718585	C₂²	1	4	yes	—
15	#11718585	C₂²	1	4	yes	—
16	C₂×C₄⋊#1055	—	2	2	no	—
17	C₂×C₄⋊#1055	C₂²	1	4	yes	—
18	#11718050	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₃	D₄	1	2	yes	—
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
31	C₂×#6099	—	2	4	no	—
32	C₂×#6099	—	2	4	no	—
33	C₂×#6099	—	4	8	no	—
34	C₂²×#1055	—	2	4	no	—
35	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
36	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₁₃	C₂³	1	4	yes	—
39	C₄⋊#1055	—	2	4	no	—
40	C₄⋊#1055	—	2	4	no	—
41	C₄⋊#1055	—	2	4	no	—
42	C₄⋊#1055	C₂³	1	8	yes	—
43	#17814	—	13	1	no	maximal, Sylow
44	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	D₈	1	4	yes	—
54	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₁₃	—	4	4	no	—
62	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
63	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₁₃	C₂×D₄	1	2	yes	—
69	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
72	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
73	#6099	—	4	8	no	—
74	#6099	—	4	8	no	—
75	C₂×#1055	—	2	8	no	—
76	C₂×#1055	—	4	8	no	—
77	C₂×#431	D₁₃	1	1	yes	characteristic
78	C₂×#423	—	13	2	no	—
79	#1571	—	13	8	no	—
80	#1567	—	13	4	no	—
81	C₂³×C₁₃	—	4	1	no	—
82	C₂³×C₁₃	—	4	1	no	—
83	C₂³×C₁₃	#107	1	4	yes	—
84	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	4	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	—	4	1	no	—
92	C₂×C₄×C₁₃	—	4	1	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	#107	1	4	yes	—
95	C₂×C₄×C₁₃	#107	1	8	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
105	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₈×C₁₃	—	2	8	no	—
110	#1055	—	4	8	no	—
111	C₂²×M₁₆	—	26	2	no	—
112	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
113	#431	Dic₁₃	1	4	yes	—
114	#431	C₂×D₁₃	1	4	yes	—
115	#423	—	13	4	no	—
116	#423	—	13	4	no	—
117	#423	—	26	2	no	—
118	#423	—	26	2	no	—
119	C₂×C₄⋊C₈	—	13	4	no	—
120	C₂×C₄⋊C₈	—	26	2	no	—
121	C₂×C₄⋊C₈	Dic₁₃	1	2	yes	—
122	C₂×C₂²⋊C₈	Dic₁₃	1	2	yes	—
123	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
124	#335	—	13	16	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	1	no	—
129	C₂²×C₁₃	—	4	4	no	—
130	C₂²×C₁₃	C₂×#107	1	4	yes	—
131	C₂²×C₁₃	C₂×#107	1	2	yes	—
132	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	C₂×#107	1	4	yes	—
138	C₄×C₁₃	C₂×#107	1	2	yes	—
139	C₄×C₁₃	#430	1	2	yes	—
140	C₄×C₁₃	#568	1	2	yes	—
141	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
143	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
144	C₂³×C₄	D₅₂	1	1	yes	characteristic
145	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂×M₁₆	—	26	4	no	—
147	C₂×M₁₆	—	26	4	no	—
148	C₂×M₁₆	—	52	8	no	—
149	C₂²×C₈	—	26	4	no	—
150	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
151	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
152	C₂×C₄²	Dic₂₆	1	2	yes	—
153	C₂×C₄²	D₅₂	1	2	yes	—
154	C₂×C₄²	D₅₂	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
156	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
157	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
159	C₄⋊C₈	—	13	8	no	—
160	C₄⋊C₈	—	26	4	no	—
161	C₄⋊C₈	—	26	4	no	—
162	C₄⋊C₈	—	26	4	no	—
163	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
164	C₂²⋊C₈	C₂×Dic₁₃	1	8	yes	—
165	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
166	C₄×C₈	C₂²×D₁₃	1	4	yes	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	2	4	no	—
169	C₂×C₁₃	#2905	1	1	yes	characteristic
170	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
171	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
172	C₂⁴	C₄⋊Dic₁₃	1	1	yes	characteristic
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	D₁₀₄	1	4	yes	—
176	C₂²×C₄	Dic₅₂	1	4	yes	—
177	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
178	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
179	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
183	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
185	C₂²×C₄	C₂×D₅₂	1	2	yes	—
186	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
187	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
188	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
189	M₁₆	—	52	8	no	—
190	M₁₆	—	52	8	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	26	8	no	—
193	C₂×C₈	—	52	8	no	—
194	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
195	C₄²	C₄⋊Dic₁₃	1	2	yes	—
196	C₄²	C₄⋊Dic₁₃	1	2	yes	—
197	C₄²	C₄⋊Dic₁₃	1	2	yes	—
198	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
199	C₄²	C₂×Dic₂₆	1	2	yes	—
200	C₄²	C₂×D₅₂	1	2	yes	—
201	C₄²	C₂×D₅₂	1	1	yes	characteristic
202	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
203	C₄²	C₂²×Dic₁₃	1	2	yes	—
204	C₄²	C₂³×D₁₃	1	1	yes	characteristic
205	C₁₃	#17814	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	4	1	no	—
207	C₂³	—	4	1	no	—
208	C₂³	#88058	1	4	yes	—
209	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
210	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	#88058	1	4	yes	—
220	C₂×C₄	#88058	1	8	yes	—
221	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
222	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
223	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
224	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
225	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
231	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
233	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
234	C₈	—	2	8	no	—
235	C₈	—	52	8	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	4	4	no	—
241	C₂²	C₂×#88058	1	4	yes	—
242	C₂²	C₂×#88058	1	2	yes	—
243	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	C₂×#88058	1	4	yes	—
249	C₄	C₂×#88058	1	2	yes	—
250	C₄	#11718786	1	2	yes	—
251	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
252	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
253	C₄	#11719524	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	2	4	no	—
256	C₂	#13123400	1	1	yes	characteristic
257	C₂	C₂²×#88058	1	1	yes	characteristic
258	C₂	C₂×#11718786	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#431	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	#17814
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #13113611	C₂	semidirect
3. #13113595	C₂	semidirect
18. #11718050	C₂²	semidirect
205. C₁₃	#17814	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
#13113611	2
#13113595	2
C₂×C₁₃×#431	2
#17814	13