Subgroups — #15761533

Order 3328.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88059	C₂	1	1	yes	maximal, characteristic
2	#13113586	C₂	1	2	yes	maximal
3	#13113119	C₂	1	1	yes	maximal, characteristic
4	#13112686	C₂	1	2	yes	maximal
5	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88059	C₄	1	1	yes	characteristic
9	C₂×#88059	C₄	1	1	yes	characteristic
10	#11718586	C₂²	1	1	yes	characteristic
11	#11718586	C₂²	1	1	yes	characteristic
12	C₄×#6099	C₂²	1	2	yes	—
13	C₁₃×#335	—	2	1	no	—
14	C₁₃×#335	—	2	1	no	—
15	#11718144	C₄	1	2	yes	—
16	#11718072	—	2	2	no	—
17	#11718071	—	2	2	no	—
18	#11718047	—	2	1	no	—
19	C₈×#1055	—	2	1	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×#6099	C₂×C₄	1	2	yes	—
25	C₂×#6099	C₂×C₄	1	2	yes	—
26	C₁₃×C₄⋊C₈	—	2	2	no	—
27	C₁₃×C₄⋊C₈	—	2	2	no	—
28	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
32	C₂²⋊#1055	—	2	1	no	—
33	C₂²⋊#1055	Q₈	1	2	yes	—
34	#88059	—	2	2	no	—
35	#88059	C₈	1	4	yes	complemented
36	C₄⋊#1055	—	2	1	no	—
37	C₄⋊#1055	D₄	1	1	yes	characteristic
38	C₄⋊#1055	D₄	1	1	yes	characteristic
39	#88043	—	2	2	no	—
40	C₄×#1055	—	2	1	no	—
41	C₄×#1055	—	2	1	no	—
42	C₄×#1055	—	2	2	no	—
43	#9608	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	4	2	no	—
50	C₂×C₈×C₁₃	—	8	2	no	—
51	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
57	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₁₃	#41	1	1	yes	characteristic
61	#6099	—	2	2	no	—
62	#6099	—	2	2	no	—
63	#6099	—	2	2	no	—
64	#6099	C₂×C₈	1	4	yes	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	2	2	no	—
67	C₂×#1055	—	2	1	no	—
68	C₂×#1055	—	2	1	no	—
69	C₂×#1055	—	2	2	no	—
70	C₄×#108	—	13	1	no	—
71	#1565	—	13	2	no	—
72	C₂×#335	D₁₃	1	1	yes	characteristic
73	#1450	—	13	1	no	—
74	#1283	—	13	2	no	—
75	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	4	1	no	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
89	C₈×C₁₃	—	2	1	no	—
90	C₈×C₁₃	—	2	1	no	—
91	C₈×C₁₃	—	8	2	no	—
92	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
93	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
94	#1055	—	2	4	no	—
95	#1055	—	4	2	no	—
96	#1055	—	4	2	no	—
97	#1055	—	4	1	no	—
98	#1055	—	4	1	no	—
99	#431	—	13	1	no	—
100	#431	—	13	1	no	—
101	C₂×#108	—	13	1	no	—
102	C₂×#108	—	13	1	no	—
103	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
104	C₄×M₁₆	—	13	2	no	—
105	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
106	#341	—	13	2	no	—
107	#335	—	2	1	no	—
108	#335	—	2	1	no	—
109	#335	—	26	2	no	—
110	#334	—	26	2	no	—
111	#322	—	26	1	no	—
112	C₈²	—	26	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	#389	1	1	yes	characteristic
116	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
117	C₂²×C₁₃	#446	1	1	yes	characteristic
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₄×C₁₃	#424	1	1	yes	characteristic
123	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
124	C₄×C₁₃	#436	1	1	yes	characteristic
125	C₄×C₁₃	#472	1	1	yes	characteristic
126	C₄×C₁₃	#473	1	1	yes	characteristic
127	C₂×M₁₆	—	13	2	no	—
128	C₂×M₁₆	—	13	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
131	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
133	#108	—	13	4	no	—
134	#108	—	26	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	13	1	no	—
138	C₄⋊C₈	—	13	1	no	—
139	C₄⋊C₈	—	26	1	no	—
140	C₂²⋊C₈	—	13	2	no	—
141	C₂²⋊C₈	—	26	1	no	—
142	#97	—	26	2	no	—
143	C₄×C₈	—	26	2	no	—
144	C₄×C₈	—	26	1	no	—
145	C₄×C₈	—	26	1	no	—
146	C₄×C₈	Dic₂₆	1	1	yes	characteristic
147	C₄×C₈	Dic₂₆	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂×C₁₃	—	2	1	no	—
151	C₂×C₁₃	—	2	1	no	—
152	C₂×C₁₃	#1939	1	1	yes	characteristic
153	C₂×C₁₃	#1952	1	1	yes	characteristic
154	C₂×C₁₃	#1953	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
156	C₂²×C₄	#6129	1	1	yes	characteristic
157	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
158	M₁₆	—	13	4	no	—
159	M₁₆	—	26	2	no	—
160	M₁₆	—	26	2	no	—
161	M₁₆	—	26	2	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	8	2	no	—
166	C₂×C₈	—	26	2	no	—
167	C₂×C₈	—	26	2	no	—
168	C₂×C₈	—	26	1	no	—
169	C₂×C₈	—	26	1	no	—
170	C₂×C₈	—	26	2	no	—
171	C₂×C₈	C₈×D₁₃	1	2	yes	—
172	C₂×C₈	#6094	1	2	yes	—
173	C₂×C₈	#6101	1	1	yes	characteristic
174	C₂×C₈	#6101	1	1	yes	characteristic
175	C₂×C₈	#6101	1	1	yes	characteristic
176	C₂×C₈	#6101	1	1	yes	characteristic
177	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
178	C₄²	D₄×D₁₃	1	1	yes	characteristic
179	C₄²	#6131	1	1	yes	characteristic
180	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
181	C₁₃	#9608	1	1	yes	characteristic, complemented, Sylow
182	C₂³	#88141	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	#88142	1	1	yes	characteristic
187	C₂×C₄	#88144	1	1	yes	characteristic
188	C₂×C₄	#88147	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
190	C₂×C₄	C₂×#6094	1	1	yes	characteristic
191	C₂×C₄	#88161	1	1	yes	characteristic
192	C₂×C₄	#88161	1	1	yes	characteristic
193	C₂×C₄	C₂×#6101	1	1	yes	characteristic
194	C₂×C₄	#88192	1	1	yes	characteristic
195	C₂×C₄	#88200	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	8	2	no	—
199	C₈	—	26	4	no	—
200	C₈	—	52	2	no	—
201	C₈	—	52	2	no	—
202	C₈	—	52	1	no	—
203	C₈	—	52	1	no	—
204	C₈	#88054	1	2	yes	—
205	C₈	#88054	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#11718491	1	1	yes	characteristic
209	C₂²	#11718492	1	1	yes	characteristic
210	C₂²	#11718634	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	#11718495	1	1	yes	characteristic
215	C₄	#11718499	1	1	yes	characteristic
216	C₄	C₂×#88054	1	1	yes	characteristic
217	C₄	#11718780	1	1	yes	characteristic
218	C₄	#11718821	1	1	yes	characteristic
219	C₄	#11718853	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#13115552	1	1	yes	characteristic
223	C₂	#13115619	1	1	yes	characteristic
224	C₂	#13115624	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88192
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle	C₂²×C₁₃	#389
prime radical	C₂³×C₁₃	C₄×Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	#9608
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
35. #88059	C₈	semidirect
181. C₁₃	#9608	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88059	2
#13113586	2
#13113119	2
#13112686	2
C₂×C₁₃×#335	2
#9608	13