Subgroups — #15833628

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125415	C₂	1	1	yes	maximal, characteristic
2	#13123549	C₂	1	2	yes	maximal, complemented
3	#13113667	C₂	1	8	yes	maximal, complemented
4	#13113595	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113288	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88155	—	2	2	no	—
8	#11719182	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718776	C₄	1	4	yes	—
13	#11718771	—	2	2	no	—
14	#11718585	C₂²	1	2	yes	—
15	#11718356	C₄	1	1	yes	characteristic
16	#11718355	C₄	1	2	yes	—
17	C₄×D₁₀₄	C₂²	1	4	yes	—
18	#11718349	—	2	2	no	—
19	#11718349	C₂²	1	4	yes	—
20	C₈×D₅₂	—	2	2	no	—
21	C₈×D₅₂	C₂²	1	4	yes	—
22	#11718343	C₄	1	1	yes	characteristic
23	C₁₃×#322	C₂²	1	4	yes	—
24	#11718051	C₂²	1	8	yes	—
25	#11718050	C₂²	1	4	yes	—
26	C₂×#6127	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	1	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	#88184	—	2	2	no	—
33	#88184	C₂×C₄	1	4	yes	—
34	C₂×#6099	—	2	2	no	—
35	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
36	#88155	—	4	8	no	—
37	C₂×#6094	—	2	4	no	—
38	C₂×C₈×D₁₃	—	2	4	no	—
39	#88129	—	2	2	no	—
40	#88128	C₂×C₄	1	2	yes	—
41	C₄×D₅₂	—	2	2	no	—
42	C₄×D₅₂	C₂³	1	4	yes	—
43	#88123	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₃	—	2	1	no	—
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	C₄×C₈×C₁₃	C₂³	1	2	yes	—
51	#88061	C₂×C₄	1	8	yes	—
52	#88060	—	2	4	no	—
53	#88060	—	2	4	no	—
54	#88058	C₂×C₄	1	4	yes	—
55	C₄⋊#1055	—	2	2	no	—
56	C₄⋊#1055	C₂³	1	4	yes	—
57	#17975	—	13	1	no	maximal, Sylow
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	—	2	4	no	—
62	#6127	—	4	4	no	—
63	#6127	—	4	4	no	—
64	C₂×D₅₂	—	2	2	no	—
65	C₂×D₅₂	—	2	2	no	—
66	C₂×D₅₂	C₂²×C₄	1	4	yes	—
67	C₂×C₄×D₁₃	—	2	4	no	—
68	C₂×Dic₂₆	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	2	no	—
73	C₂×C₈×C₁₃	D₈	1	4	yes	complemented
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	#41	1	1	yes	characteristic
81	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
82	#6103	—	2	4	no	—
83	#6103	—	2	4	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
86	#6099	—	4	4	no	—
87	C₂×#1055	—	2	4	no	—
88	D₁₀₄	—	4	4	no	—
89	#6094	—	4	8	no	—
90	C₈×D₁₃	—	4	8	no	—
91	#3169	—	13	1	no	—
92	#2931	—	13	2	no	—
93	#1581	—	13	8	no	—
94	#1567	—	13	1	no	—
95	#1492	—	13	2	no	—
96	C₂×#322	D₁₃	1	1	yes	characteristic, complemented
97	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂²×D₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₁₃	#135	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	C₂×#41	1	1	yes	characteristic
109	C₂²⋊D₁₃	—	4	8	no	—
110	C₂×Dic₁₃	—	2	4	no	—
111	D₅₂	—	2	8	no	—
112	D₅₂	—	4	4	no	—
113	D₅₂	—	4	4	no	—
114	C₄×D₁₃	—	4	4	no	—
115	C₄×D₁₃	—	4	4	no	—
116	Dic₂₆	—	4	4	no	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	—	4	2	no	—
121	C₈×C₁₃	C₂×D₈	1	4	yes	—
122	#1055	—	4	4	no	—
123	C₂×#131	—	26	2	no	—
124	#532	—	13	2	no	—
125	#500	—	13	1	no	—
126	#493	—	13	1	no	—
127	#486	—	13	2	no	—
128	#466	—	13	4	no	—
129	C₄×D₈	—	13	4	no	—
130	#436	—	13	4	no	—
131	#436	—	26	2	no	—
132	C₈×D₄	—	13	4	no	—
133	C₈×D₄	—	26	2	no	—
134	#423	—	13	2	no	—
135	#408	—	26	2	no	—
136	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#335	—	13	4	no	—
140	D₄⋊C₈	—	13	8	no	—
141	#322	C₂×D₁₃	1	4	yes	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	C₄×D₈	1	2	yes	—
145	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂×D₁₃	—	4	4	no	—
147	C₂×D₁₃	—	4	4	no	—
148	C₂×D₁₃	—	4	4	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	C₄×D₈	1	2	yes	—
152	C₄×C₁₃	#444	1	2	yes	—
153	C₄×C₁₃	#568	1	2	yes	—
154	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
156	Dic₁₃	—	4	4	no	—
157	C₂×#41	—	26	2	no	—
158	C₂×D₈	—	13	4	no	—
159	#131	—	52	8	no	—
160	C₂×M₁₆	—	26	2	no	—
161	C₂×M₁₆	—	26	4	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	26	4	no	—
164	C₂²×C₈	C₄×D₁₃	1	2	yes	—
165	C₂²×C₈	D₅₂	1	1	yes	characteristic
166	C₂²×C₈	D₅₂	1	1	yes	characteristic
167	C₄⋊Q₈	—	13	2	no	—
168	#127	—	13	2	no	—
169	#124	—	26	2	no	—
170	#121	—	13	4	no	—
171	#121	—	26	2	no	—
172	C₄×D₄	—	13	4	no	—
173	C₄×D₄	—	26	2	no	—
174	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
175	#107	—	13	4	no	—
176	C₄⋊C₈	—	13	4	no	—
177	C₄⋊C₈	—	26	2	no	—
178	#102	—	13	8	no	—
179	C₂²⋊C₈	—	26	4	no	—
180	C₂²⋊C₈	—	26	4	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
183	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
184	C₄×C₈	D₅₂	1	2	yes	—
185	C₄×C₈	C₂²×D₁₃	1	2	yes	—
186	C₄×C₈	C₂²×D₁₃	1	2	yes	—
187	C₄×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂×C₁₃	—	2	2	no	—
189	C₂×C₁₃	#2932	1	1	yes	characteristic
190	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
191	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
192	D₁₃	—	8	4	no	—
193	#41	—	52	4	no	—
194	#41	—	52	4	no	—
195	C₂×Q₈	—	26	2	no	—
196	C₂×D₄	—	13	4	no	—
197	C₂×D₄	—	26	4	no	—
198	C₂×D₄	—	26	2	no	—
199	C₂×D₄	—	26	2	no	—
200	C₂²×C₄	—	26	4	no	—
201	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
203	C₂²×C₄	#6127	1	1	yes	characteristic
204	D₈	—	52	4	no	—
205	M₁₆	—	52	4	no	—
206	M₁₆	—	52	8	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	26	4	no	—
212	C₂×C₈	—	52	8	no	—
213	C₂×C₈	D₁₀₄	1	4	yes	complemented
214	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
216	C₂×C₈	C₂×D₅₂	1	2	yes	—
217	C₂×C₈	C₂×D₅₂	1	2	yes	—
218	C₄⋊C₄	—	13	4	no	—
219	C₄⋊C₄	—	26	2	no	—
220	C₂²⋊C₄	—	26	4	no	—
221	C₂²⋊C₄	—	26	4	no	—
222	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
223	C₄²	C₂×D₅₂	1	1	yes	characteristic
224	C₄²	#6127	1	1	yes	characteristic
225	C₄²	C₂³×D₁₃	1	1	yes	characteristic
226	C₁₃	#17975	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	26	4	no	—
228	C₂³	C₄×D₅₂	1	1	yes	characteristic
229	Q₈	—	52	4	no	—
230	D₄	—	26	8	no	—
231	D₄	—	52	8	no	—
232	D₄	—	52	4	no	—
233	D₄	—	52	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	26	4	no	—
237	C₂×C₄	—	52	4	no	—
238	C₂×C₄	—	52	4	no	—
239	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
240	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
241	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
242	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
243	C₂×C₄	#88158	1	2	yes	—
244	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
246	C₂×C₄	C₂×#6127	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	—	52	4	no	—
252	C₈	C₂×D₁₀₄	1	4	yes	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	52	4	no	—
256	C₂²	—	52	4	no	—
257	C₂²	—	52	4	no	—
258	C₂²	C₄×D₁₀₄	1	2	yes	—
259	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	52	4	no	—
263	C₄	C₄×D₁₀₄	1	2	yes	—
264	C₄	#11718378	1	2	yes	—
265	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
266	C₄	C₂×#88158	1	1	yes	characteristic
267	C₄	#11719450	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	104	4	no	—
270	C₂	#13123555	1	1	yes	characteristic
271	C₂	C₂×C₄×D₁₀₄	1	1	yes	characteristic
272	C₂	C₂×#11718378	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×#322	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125415	C₂
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#17975
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #13123549	C₂	semidirect
3. #13113667	C₂	semidirect
4. #13113595	C₂	semidirect
5. #13113288	C₂	semidirect
6. C₂×C₁₃×#322	C₂	semidirect
73. C₂×C₈×C₁₃	D₈	semidirect
96. C₂×#322	D₁₃	semidirect
213. C₂×C₈	D₁₀₄	semidirect
226. C₁₃	#17975	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13123549	2
#13113667	2
#13113595	2
#13113288	2
C₂×C₁₃×#322	2
#17975	13