Subgroups — #15761691

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88123	C₂	1	1	yes	maximal, characteristic
2	#13114988	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718048	C₂	1	4	yes	maximal
4	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	—	2	2	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718596	C₄	1	1	yes	characteristic
11	C₄×C₄⋊Dic₁₃	—	2	2	no	—
12	#11718590	C₄	1	2	yes	—
13	#11718589	—	2	2	no	—
14	C₂×C₄⋊#1055	—	2	2	no	—
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	#11718048	—	2	8	no	—
17	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	2	yes	—
22	C₂×C₄²×C₁₃	D₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
27	C₂×#6101	—	2	2	no	—
28	C₂×C₄×Dic₁₃	—	2	2	no	—
29	C₂²×#1055	—	2	2	no	—
30	#88123	—	2	2	no	—
31	#88123	C₈	1	4	yes	complemented
32	C₄×Dic₂₆	—	2	4	no	—
33	C₄×Dic₂₆	—	2	4	no	—
34	C₄×C₈×C₁₃	—	2	8	no	—
35	C₄×C₈×C₁₃	—	2	2	no	—
36	C₄×C₈×C₁₃	D₄	1	4	yes	—
37	#88071	—	2	4	no	—
38	C₄⋊#1055	—	2	4	no	—
39	C₄⋊#1055	—	4	4	no	—
40	#9637	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	—	2	2	no	—
47	C₂²×Dic₁₃	—	2	2	no	—
48	C₂×Dic₂₆	—	2	4	no	—
49	C₂×Dic₂₆	—	2	4	no	—
50	C₂×Dic₂₆	—	2	2	no	—
51	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
52	C₂×C₈×C₁₃	—	2	8	no	—
53	C₂×C₈×C₁₃	—	2	4	no	—
54	C₄²×C₁₃	—	2	4	no	—
55	C₄²×C₁₃	—	2	2	no	—
56	C₄²×C₁₃	C₂×C₈	1	2	yes	—
57	C₄²×C₁₃	M₁₆	1	2	yes	—
58	C₄²×C₁₃	SD₁₆	1	4	yes	—
59	C₄²×C₁₃	Q₁₆	1	4	yes	—
60	C₄²×C₁₃	C₂×D₄	1	2	yes	—
61	C₄²×C₁₃	#41	1	2	yes	—
62	C₄⋊Dic₁₃	—	2	8	no	—
63	C₄⋊Dic₁₃	—	2	4	no	—
64	C₄⋊Dic₁₃	—	4	4	no	—
65	#6101	—	4	4	no	—
66	C₄×Dic₁₃	—	4	2	no	—
67	C₄×Dic₁₃	—	4	2	no	—
68	C₂×#1055	—	4	2	no	—
69	C₂×#1055	—	4	4	no	—
70	C₄×C₄⋊Q₈	—	13	1	no	—
71	#1842	—	13	1	no	—
72	C₄²×C₈	D₁₃	1	1	yes	characteristic
73	C₂×Q₈⋊C₈	—	13	4	no	—
74	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
79	C₂×C₄×C₁₃	#103	1	4	yes	—
80	C₂×C₄×C₁₃	#103	1	4	yes	—
81	C₂×C₄×C₁₃	#104	1	4	yes	—
82	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₁₃	#121	1	2	yes	—
84	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#131	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
91	C₂×Dic₁₃	—	2	4	no	—
92	C₂×Dic₁₃	—	4	2	no	—
93	C₂×Dic₁₃	—	4	2	no	—
94	C₂×Dic₁₃	—	4	2	no	—
95	C₂×Dic₁₃	—	4	4	no	—
96	Dic₂₆	—	2	8	no	—
97	Dic₂₆	—	4	4	no	—
98	C₈×C₁₃	—	2	8	no	—
99	#1055	—	8	2	no	—
100	C₂×C₄⋊Q₈	—	13	1	no	—
101	C₂×C₄×Q₈	—	13	2	no	—
102	C₂×C₄⋊C₈	—	13	2	no	—
103	C₂×C₄⋊C₈	—	26	2	no	—
104	C₂×C₄×C₈	—	2	2	no	—
105	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
106	#391	—	13	2	no	—
107	#389	—	26	2	no	—
108	#384	—	13	1	no	—
109	C₄×C₄⋊C₄	—	26	2	no	—
110	C₄³	C₂×D₁₃	1	1	yes	characteristic
111	Q₈⋊C₈	—	26	8	no	—
112	C₂²×C₁₃	#390	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₁₃	#408	1	1	yes	characteristic
115	C₂²×C₁₃	C₂×#103	1	2	yes	—
116	C₂²×C₁₃	C₂×#104	1	2	yes	—
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	2	4	no	—
119	C₄×C₁₃	Q₈⋊C₈	1	8	yes	complemented
120	C₄×C₁₃	C₈×D₄	1	2	yes	—
121	C₄×C₁₃	#436	1	2	yes	—
122	C₄×C₁₃	#487	1	2	yes	—
123	C₄×C₁₃	#492	1	1	yes	characteristic
124	C₄×C₁₃	#494	1	1	yes	characteristic
125	Dic₁₃	—	4	4	no	—
126	Dic₁₃	—	8	2	no	—
127	C₂²×Q₈	—	13	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	26	2	no	—
130	C₄⋊Q₈	—	13	4	no	—
131	C₄⋊Q₈	—	26	2	no	—
132	C₄×Q₈	—	26	4	no	—
133	C₄×Q₈	—	26	4	no	—
134	C₂×C₄⋊C₄	—	13	2	no	—
135	C₂×C₄⋊C₄	—	26	2	no	—
136	C₂×C₄⋊C₄	—	26	2	no	—
137	C₂×C₄⋊C₄	—	26	2	no	—
138	C₂×C₄²	—	26	2	no	—
139	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
140	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
141	C₂×C₄²	D₅₂	1	2	yes	—
142	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
143	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
144	C₄⋊C₈	—	26	4	no	—
145	C₄⋊C₈	—	52	4	no	—
146	C₄×C₈	—	2	8	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	D₅₂	1	4	yes	—
149	#95	—	26	4	no	—
150	C₂×C₁₃	C₂×Q₈⋊C₈	1	2	yes	—
151	C₂×C₁₃	#1493	1	2	yes	—
152	C₂×C₁₃	#1954	1	1	yes	characteristic
153	C₂×C₁₃	#1956	1	1	yes	characteristic
154	C₂×C₁₃	#1962	1	1	yes	characteristic
155	C₂×Q₈	—	13	4	no	—
156	C₂×Q₈	—	26	2	no	—
157	C₂×Q₈	—	26	4	no	—
158	C₂×Q₈	—	26	4	no	—
159	C₂²×C₄	—	26	2	no	—
160	C₂²×C₄	—	26	2	no	—
161	C₂²×C₄	#6103	1	2	yes	—
162	C₂²×C₄	#6103	1	2	yes	—
163	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	52	2	no	—
169	C₂×C₈	—	52	4	no	—
170	C₄⋊C₄	—	26	8	no	—
171	C₄⋊C₄	—	26	4	no	—
172	C₄⋊C₄	—	52	4	no	—
173	C₄⋊C₄	—	52	4	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	52	2	no	—
177	C₄²	—	52	2	no	—
178	C₄²	C₈×D₁₃	1	2	yes	—
179	C₄²	#6094	1	2	yes	—
180	C₄²	#6095	1	4	yes	—
181	C₄²	Dic₅₂	1	4	yes	—
182	C₄²	C₂×D₅₂	1	2	yes	—
183	C₄²	#6127	1	2	yes	—
184	C₁₃	#9637	1	1	yes	characteristic, complemented, Sylow
185	C₂³	C₂×#6103	1	1	yes	characteristic
186	Q₈	—	26	8	no	—
187	Q₈	—	52	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	26	4	no	—
192	C₂×C₄	—	52	2	no	—
193	C₂×C₄	—	52	2	no	—
194	C₂×C₄	—	52	4	no	—
195	C₂×C₄	—	52	2	no	—
196	C₂×C₄	#88045	1	4	yes	—
197	C₂×C₄	#88056	1	4	yes	—
198	C₂×C₄	#88056	1	4	yes	—
199	C₂×C₄	#88060	1	4	yes	—
200	C₂×C₄	C₄×D₅₂	1	2	yes	—
201	C₂×C₄	#88128	1	1	yes	characteristic
202	C₂×C₄	#88129	1	1	yes	characteristic
203	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
204	C₂×C₄	C₂×#6094	1	1	yes	characteristic
205	C₂×C₄	#88155	1	2	yes	—
206	C₂×C₄	C₂×#6095	1	2	yes	—
207	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
208	C₂×C₄	#88184	1	2	yes	—
209	C₈	—	2	8	no	—
210	C₈	—	104	2	no	—
211	C₂²	C₂×#88045	1	2	yes	—
212	C₂²	#11718600	1	1	yes	characteristic
213	C₂²	C₂×#88056	1	2	yes	—
214	C₂²	C₂×#88060	1	1	yes	characteristic
215	C₂²	#11718771	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	52	4	no	—
219	C₄	—	104	2	no	—
220	C₄	#11718048	1	8	yes	complemented
221	C₄	#11718342	1	2	yes	—
222	C₄	C₈×D₅₂	1	2	yes	—
223	C₄	#11718349	1	2	yes	—
224	C₄	#11718354	1	1	yes	characteristic
225	C₄	#11718360	1	1	yes	characteristic
226	C₂	C₂×#11718048	1	2	yes	—
227	C₂	#13113292	1	2	yes	—
228	C₂	#13115630	1	1	yes	characteristic
229	C₂	#13115638	1	1	yes	characteristic
230	C₂	#13115664	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	#9637
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
31. #88123	C₈	semidirect
119. C₄×C₁₃	Q₈⋊C₈	semidirect
184. C₁₃	#9637	semidirect
220. C₄	#11718048	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88123	2
#13114988	2
C₂×#11718048	2
C₄²×C₈×C₁₃	2
#9637	13