Subgroups — #15761461

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115554	C₂	1	2	yes	maximal
2	C₄×#88058	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
4	#13112676	C₂	1	2	yes	maximal
5	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	C₄	1	1	yes	characteristic
10	C₂×#88058	C₄	1	1	yes	characteristic
11	C₂×#88054	—	2	2	no	—
12	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₈²×C₁₃	—	2	2	no	—
15	#11718143	C₄	1	2	yes	—
16	#11718142	—	2	2	no	—
17	#11718050	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	1	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₁₃	—	2	2	no	—
27	C₂²×#1055	—	2	2	no	—
28	C₄×C₈×C₁₃	—	2	1	no	—
29	C₄×C₈×C₁₃	—	2	1	no	—
30	C₄×C₈×C₁₃	—	2	1	no	—
31	C₄×C₈×C₁₃	D₄	1	2	yes	—
32	C₄×C₈×C₁₃	Q₈	1	2	yes	—
33	C₄×C₈×C₁₃	Q₈	1	2	yes	—
34	#88071	—	2	2	no	—
35	#88058	—	2	2	no	—
36	#88058	C₈	1	4	yes	complemented
37	#88054	—	4	4	no	—
38	C₄⋊#1055	—	2	2	no	—
39	C₄⋊#1055	—	2	2	no	—
40	#9594	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	2	no	—
45	C₂²×Dic₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	2	1	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	—	2	1	no	—
52	C₂×C₈×C₁₃	—	2	1	no	—
53	C₂×C₈×C₁₃	—	2	1	no	—
54	C₂×C₈×C₁₃	—	2	1	no	—
55	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
59	C₂×C₈×C₁₃	D₈	1	2	yes	—
60	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
61	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₁₃	#41	1	1	yes	characteristic
65	C₄⋊Dic₁₃	—	2	2	no	—
66	C₄⋊Dic₁₃	—	2	2	no	—
67	C₄⋊Dic₁₃	—	2	2	no	—
68	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
69	C₄×Dic₁₃	—	4	2	no	—
70	C₄×Dic₁₃	—	4	2	no	—
71	C₂×#1055	—	4	2	no	—
72	C₂×#1055	—	4	2	no	—
73	C₂×#1055	—	4	2	no	—
74	#1939	—	13	2	no	—
75	C₄×#107	—	13	1	no	—
76	C₂×#335	—	13	1	no	—
77	C₂×C₈²	D₁₃	1	1	yes	characteristic
78	#1279	—	13	2	no	—
79	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
96	C₂×Dic₁₃	—	2	4	no	—
97	C₂×Dic₁₃	—	4	2	no	—
98	C₂×Dic₁₃	—	4	2	no	—
99	C₂×Dic₁₃	—	4	2	no	—
100	C₂×Dic₁₃	—	4	2	no	—
101	C₂×Dic₁₃	—	4	2	no	—
102	C₈×C₁₃	—	2	4	no	—
103	C₈×C₁₃	—	2	2	no	—
104	C₈×C₁₃	—	2	2	no	—
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	C₄⋊C₈	1	4	yes	complemented
107	#1055	—	8	2	no	—
108	C₂×#107	—	13	1	no	—
109	C₂×#107	—	13	1	no	—
110	C₂×C₄⋊C₈	—	13	2	no	—
111	C₂×C₄⋊C₈	—	26	2	no	—
112	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
115	C₄×C₄⋊C₄	—	13	2	no	—
116	#340	—	13	2	no	—
117	#336	—	26	2	no	—
118	#335	—	26	2	no	—
119	C₈²	—	2	2	no	—
120	C₂²×C₁₃	#389	1	1	yes	characteristic
121	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₁₃	#436	1	1	yes	characteristic
123	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
124	C₂²×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
125	C₂²×C₁₃	#443	1	1	yes	characteristic
126	C₂²×C₁₃	#443	1	1	yes	characteristic
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₄×C₁₃	#424	1	1	yes	characteristic
131	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
132	C₄×C₁₃	#446	1	1	yes	characteristic
133	C₄×C₁₃	#466	1	1	yes	characteristic
134	C₄×C₁₃	#467	1	1	yes	characteristic
135	C₄×C₁₃	#499	1	1	yes	characteristic
136	C₄×C₁₃	#500	1	1	yes	characteristic
137	Dic₁₃	—	4	4	no	—
138	Dic₁₃	—	8	2	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₂²×C₈	—	26	2	no	—
141	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
142	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
143	C₂²×C₈	D₅₂	1	1	yes	characteristic
144	C₂²×C₈	D₅₂	1	1	yes	characteristic
145	C₂×C₄⋊C₄	—	13	2	no	—
146	C₂×C₄⋊C₄	—	13	2	no	—
147	C₂×C₄²	—	26	2	no	—
148	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
149	#107	—	13	4	no	—
150	#107	—	26	2	no	—
151	C₄⋊C₈	—	26	2	no	—
152	C₄⋊C₈	—	26	2	no	—
153	C₄⋊C₈	—	52	4	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	Dic₂₆	1	2	yes	—
158	C₄×C₈	Dic₂₆	1	2	yes	—
159	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
160	#95	—	26	2	no	—
161	C₂×C₁₃	C₈×D₈	1	1	yes	characteristic
162	C₂×C₁₃	C₈×Q₁₆	1	1	yes	characteristic
163	C₂×C₁₃	#1589	1	1	yes	characteristic
164	C₂×C₁₃	#1591	1	1	yes	characteristic
165	C₂×C₁₃	#1939	1	1	yes	characteristic
166	C₂×C₁₃	#1942	1	1	yes	characteristic
167	C₂×C₁₃	#1945	1	1	yes	characteristic
168	C₂²×C₄	—	26	2	no	—
169	C₂²×C₄	—	26	2	no	—
170	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
172	C₂²×C₄	#6127	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	52	2	no	—
183	C₂×C₈	—	52	2	no	—
184	C₂×C₈	—	52	2	no	—
185	C₂×C₈	C₈×D₁₃	1	2	yes	—
186	C₂×C₈	#6094	1	2	yes	—
187	C₂×C₈	D₁₀₄	1	2	yes	—
188	C₂×C₈	Dic₅₂	1	2	yes	—
189	C₂×C₈	#6101	1	2	yes	—
190	C₂×C₈	#6101	1	2	yes	—
191	C₄⋊C₄	—	13	4	no	—
192	C₄⋊C₄	—	26	2	no	—
193	C₄⋊C₄	—	26	2	no	—
194	C₄⋊C₄	—	26	2	no	—
195	C₄²	—	52	2	no	—
196	C₄²	—	52	2	no	—
197	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
198	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
199	C₄²	#6127	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
201	C₁₃	#9594	1	1	yes	characteristic, complemented, Sylow
202	C₂³	C₄×D₅₂	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	26	4	no	—
206	C₂×C₄	—	52	2	no	—
207	C₂×C₄	—	52	2	no	—
208	C₂×C₄	—	52	2	no	—
209	C₂×C₄	—	52	2	no	—
210	C₂×C₄	—	52	2	no	—
211	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
212	C₂×C₄	#88123	1	1	yes	characteristic
213	C₂×C₄	#88124	1	1	yes	characteristic
214	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
215	C₂×C₄	C₂×#6094	1	1	yes	characteristic
216	C₂×C₄	#88155	1	1	yes	characteristic
217	C₂×C₄	#88155	1	1	yes	characteristic
218	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
219	C₂×C₄	#88158	1	1	yes	characteristic
220	C₂×C₄	#88158	1	1	yes	characteristic
221	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
222	C₂×C₄	C₂×#6101	1	1	yes	characteristic
223	C₂×C₄	#88178	1	1	yes	characteristic
224	C₂×C₄	#88184	1	1	yes	characteristic
225	C₈	—	2	4	no	—
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	104	2	no	—
230	C₈	#88054	1	4	yes	complemented
231	C₂²	C₈×D₅₂	1	1	yes	characteristic
232	C₂²	#11718349	1	1	yes	characteristic
233	C₂²	C₄×D₁₀₄	1	1	yes	characteristic
234	C₂²	C₄×Dic₅₂	1	1	yes	characteristic
235	C₂²	#11718361	1	1	yes	characteristic
236	C₂²	#11718361	1	1	yes	characteristic
237	C₂²	#11718589	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	52	4	no	—
241	C₄	—	104	2	no	—
242	C₄	C₈×Dic₂₆	1	1	yes	characteristic
243	C₄	#11718340	1	1	yes	characteristic
244	C₄	#11718343	1	1	yes	characteristic
245	C₄	#11718344	1	1	yes	characteristic
246	C₄	C₂×#88054	1	1	yes	characteristic
247	C₄	#11718763	1	1	yes	characteristic
248	C₄	#11718776	1	1	yes	characteristic
249	C₄	#11718777	1	1	yes	characteristic
250	C₂	C₈×D₁₀₄	1	1	yes	characteristic
251	C₂	C₈×Dic₅₂	1	1	yes	characteristic
252	C₂	#13113722	1	1	yes	characteristic
253	C₂	#13113734	1	1	yes	characteristic
254	C₂	#13115554	1	1	yes	characteristic
255	C₂	#13115569	1	1	yes	characteristic
256	C₂	#13115584	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#9594
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
36. #88058	C₈	semidirect
106. C₈×C₁₃	C₄⋊C₈	semidirect
201. C₁₃	#9594	semidirect
230. C₈	#88054	semidirect

Maximal subgroups

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

subgroup	index
#13115554	2
C₄×#88058	2
C₂×#11718050	2
#13112676	2
C₂×C₈²×C₁₃	2
#9594	13