Subgroups — #15761531

Order 3328.

306 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113588	C₂	1	2	yes	maximal
2	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
3	#13113119	C₂	1	1	yes	maximal, characteristic
4	#13112687	C₂	1	2	yes	maximal
5	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
8	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	#11718586	C₂²	1	1	yes	characteristic
11	#11718586	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₁₃×#341	C₄	1	2	yes	—
14	#11718050	—	2	2	no	—
15	#11718049	—	2	2	no	—
16	#11718049	—	2	1	no	—
17	#11718049	—	2	1	no	—
18	#11718047	—	2	1	no	—
19	C₈×#1055	—	2	1	no	—
20	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×#1055	—	2	2	no	—
26	C₁₃×#108	—	2	2	no	—
27	C₁₃×#108	C₈	1	4	yes	complemented
28	C₁₃×#97	—	2	2	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
34	C₂²⋊#1055	—	2	1	no	—
35	C₂²⋊#1055	Q₈	1	2	yes	—
36	C₄⋊#1055	—	2	1	no	—
37	C₄⋊#1055	—	2	2	no	—
38	C₄⋊#1055	—	2	2	no	—
39	C₄⋊#1055	D₄	1	2	yes	—
40	C₄×#1055	—	2	1	no	—
41	C₄×#1055	—	2	1	no	—
42	#9607	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
46	C₁₃×M₁₆	—	2	2	no	—
47	C₁₃×M₁₆	—	2	2	no	—
48	C₁₃×M₁₆	—	2	2	no	—
49	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
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	2	no	—
54	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
60	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₁₃	#41	1	1	yes	characteristic
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	—	8	2	no	—
68	C₄×#108	D₁₃	1	1	yes	characteristic
69	#1565	—	13	2	no	—
70	C₂×#334	—	13	1	no	—
71	#1450	—	13	1	no	—
72	#1283	—	13	2	no	—
73	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	4	1	no	—
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
86	C₈×C₁₃	—	2	1	no	—
87	C₈×C₁₃	—	2	1	no	—
88	C₈×C₁₃	—	2	4	no	—
89	C₈×C₁₃	—	4	2	no	—
90	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
91	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
92	#1055	—	4	2	no	—
93	#1055	—	4	2	no	—
94	#1055	—	8	2	no	—
95	#431	—	13	1	no	—
96	#431	—	13	1	no	—
97	C₂×#108	Dic₁₃	1	1	yes	characteristic
98	C₂×#108	Dic₁₃	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	13	2	no	—
100	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
102	#341	Dic₁₃	1	2	yes	—
103	#335	—	26	2	no	—
104	#334	—	26	2	no	—
105	#334	—	26	1	no	—
106	#334	—	26	1	no	—
107	#322	—	26	1	no	—
108	C₈²	—	26	1	no	—
109	C₂²×C₁₃	—	2	1	no	—
110	C₂²×C₁₃	—	2	1	no	—
111	C₂²×C₁₃	#389	1	1	yes	characteristic
112	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
113	C₂²×C₁₃	#446	1	1	yes	characteristic
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₄×C₁₃	#424	1	1	yes	characteristic
119	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
120	C₄×C₁₃	#436	1	1	yes	characteristic
121	C₄×C₁₃	#471	1	2	yes	—
122	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
123	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
124	C₂²×C₈	—	26	2	no	—
125	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
126	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
128	#108	—	2	2	no	—
129	#108	#1055	1	4	yes	complemented
130	C₄⋊C₈	—	13	2	no	—
131	C₄⋊C₈	—	26	2	no	—
132	C₄⋊C₈	—	26	2	no	—
133	C₄⋊C₈	—	26	1	no	—
134	C₂²⋊C₈	—	13	2	no	—
135	C₂²⋊C₈	—	26	1	no	—
136	#97	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	26	1	no	—
139	C₄×C₈	—	26	1	no	—
140	C₄×C₈	Dic₂₆	1	1	yes	characteristic
141	C₄×C₈	Dic₂₆	1	1	yes	characteristic
142	C₄×C₈	D₅₂	1	1	yes	characteristic
143	C₄×C₈	D₅₂	1	1	yes	characteristic
144	C₂×C₁₃	—	2	1	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	#1939	1	1	yes	characteristic
147	C₂×C₁₃	#1951	1	1	yes	characteristic
148	C₂×C₁₃	#1953	1	1	yes	characteristic
149	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
150	C₂²×C₄	#6131	1	1	yes	characteristic
151	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	2	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	C₂×#1055	1	4	yes	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	26	2	no	—
161	C₂×C₈	—	26	2	no	—
162	C₂×C₈	—	52	2	no	—
163	C₂×C₈	—	104	2	no	—
164	C₂×C₈	C₂×#1055	1	2	yes	—
165	C₂×C₈	#6099	1	2	yes	—
166	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
168	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
169	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
170	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
171	C₄²	C₂×D₅₂	1	1	yes	characteristic
172	C₄²	D₄×D₁₃	1	1	yes	characteristic
173	C₄²	#6129	1	1	yes	characteristic
174	C₁₃	#9607	1	1	yes	characteristic, complemented, Sylow
175	C₂³	Q₈×Dic₁₃	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	#88142	1	1	yes	characteristic
179	C₂×C₄	#88144	1	1	yes	characteristic
180	C₂×C₄	#88149	1	1	yes	characteristic
181	C₂×C₄	#88151	1	1	yes	characteristic
182	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
183	C₂×C₄	C₂×#6099	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
185	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
186	C₂×C₄	#88202	1	1	yes	characteristic
187	C₂×C₄	#88202	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	4	no	—
191	C₈	—	4	2	no	—
192	C₈	—	52	2	no	—
193	C₈	—	52	2	no	—
194	C₈	—	104	2	no	—
195	C₈	C₄⋊#1055	1	2	yes	—
196	C₈	C₄⋊#1055	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#11718646	1	1	yes	characteristic
200	C₂²	Q₈×#1055	1	1	yes	characteristic
201	C₂²	#11718685	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	#11718559	1	2	yes	—
206	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
207	C₄	#11718660	1	1	yes	characteristic
208	C₄	D₄×#1055	1	1	yes	characteristic
209	C₄	#11718673	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#13115555	1	1	yes	characteristic
213	C₂	#13115617	1	1	yes	characteristic
214	C₂	#13115627	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88149
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	#389
prime radical	C₂³×C₁₃	C₄×Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	#9607
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
27. C₁₃×#108	C₈	semidirect
129. #108	#1055	semidirect
174. C₁₃	#9607	semidirect

Maximal subgroups

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

subgroup	index
#13113588	2
C₂×#11718049	2
#13113119	2
#13112687	2
C₄×C₁₃×#108	2
#9607	13