Subgroups — #16016298

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#13123705	C₂	1	2	yes	maximal
3	#13116708	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₁₀₄	C₂	1	4	yes	maximal, complemented
5	#13116620	C₂	1	1	yes	maximal, characteristic
6	#13116557	C₂	1	2	yes	maximal
7	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₂	C₂²	1	2	yes	—
9	#11719221	—	2	2	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#405	C₂²	1	1	yes	characteristic
13	#11718781	—	2	2	no	—
14	#11718767	C₂²	1	2	yes	—
15	C₂×Dic₁₀₄	C₄	1	8	yes	complemented
16	C₂×Dic₁₀₄	C₂²	1	2	yes	—
17	C₂×Dic₁₀₄	C₂²	1	2	yes	—
18	#11718377	—	2	4	no	—
19	C₄×Dic₅₂	C₂²	1	8	yes	—
20	C₁₃×#346	C₂²	1	2	yes	—
21	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
22	#11718097	C₂²	1	2	yes	—
23	#11718096	C₂²	1	2	yes	—
24	#11718095	C₂²	1	8	yes	—
25	C₂²×Dic₂₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×#117	D₄	1	1	yes	characteristic
29	#88180	—	2	2	no	—
30	C₂×Dic₅₂	—	2	2	no	—
31	C₂×Dic₅₂	—	2	4	no	—
32	C₂×Dic₅₂	C₂×C₄	1	8	yes	—
33	C₂×Dic₅₂	C₂³	1	4	yes	—
34	#88141	—	4	4	no	—
35	#88131	—	4	4	no	—
36	C₄×Dic₂₆	—	2	8	no	—
37	C₁₃×M₃₂	—	2	2	no	—
38	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
39	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
42	C₁₃×#97	D₄	1	2	yes	—
43	C₄×C₈×C₁₃	C₂³	1	2	yes	—
44	#88058	C₂³	1	4	yes	—
45	#88057	—	2	2	no	—
46	#88056	—	2	8	no	—
47	Dic₁₀₄	C₂×C₄	1	8	yes	—
48	Dic₁₀₄	C₂×C₄	1	8	yes	—
49	#33458	—	13	1	no	maximal, Sylow
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	—	4	2	no	—
52	C₂×Dic₂₆	—	2	8	no	—
53	C₂×Dic₂₆	—	2	4	no	—
54	C₂×Dic₂₆	—	4	2	no	—
55	C₂×Dic₂₆	—	4	4	no	—
56	C₁₃×M₁₆	—	2	2	no	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	#41	1	2	yes	—
61	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
63	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
64	C₄²×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²⋊Dic₁₃	—	4	2	no	—
66	C₄⋊Dic₁₃	—	2	4	no	—
67	#6101	—	4	8	no	—
68	C₄×Dic₁₃	—	4	8	no	—
69	C₄×Dic₁₃	—	4	2	no	—
70	Dic₅₂	—	2	8	no	—
71	Dic₅₂	—	2	4	no	—
72	Dic₅₂	C₂²×C₄	1	8	yes	—
73	C₁₃×C₁₆	—	2	2	no	—
74	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
75	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
76	C₂²×Q₃₂	—	13	1	no	—
77	#2950	—	13	2	no	—
78	#2176	—	13	4	no	—
79	C₄×Q₃₂	—	13	4	no	—
80	#2158	—	13	1	no	—
81	#2141	—	13	2	no	—
82	#2108	D₁₃	1	1	yes	characteristic
83	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
86	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#135	1	2	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×Dic₁₃	—	4	4	no	—
90	C₂×Dic₁₃	—	4	8	no	—
91	C₂×Dic₁₃	—	4	4	no	—
92	Dic₂₆	—	2	8	no	—
93	Dic₂₆	—	4	4	no	—
94	Dic₂₆	—	4	8	no	—
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	C₄×D₄	1	2	yes	—
97	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
99	C₂²×Q₁₆	—	13	2	no	—
100	#519	—	26	2	no	—
101	C₂×Q₃₂	—	13	8	no	—
102	C₂×Q₃₂	—	13	2	no	—
103	C₂×Q₃₂	—	13	2	no	—
104	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
105	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
106	#441	—	26	4	no	—
107	C₄×Q₁₆	—	13	8	no	—
108	#427	—	13	2	no	—
109	#419	—	26	2	no	—
110	#405	C₂×D₁₃	1	1	yes	characteristic
111	#367	—	13	2	no	—
112	#366	—	13	2	no	—
113	#358	—	13	8	no	—
114	#346	C₂×D₁₃	1	2	yes	—
115	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	C₄×D₈	1	2	yes	—
118	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₁₃	—	2	4	no	—
120	C₄×C₁₃	C₄×D₈	1	2	yes	—
121	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
123	Dic₁₃	—	4	8	no	—
124	Dic₁₃	—	8	4	no	—
125	C₂²×Q₈	—	26	2	no	—
126	C₂×Q₁₆	—	13	8	no	—
127	C₂×Q₁₆	—	13	4	no	—
128	C₂×Q₁₆	—	26	2	no	—
129	C₂×Q₁₆	—	26	4	no	—
130	C₂×M₁₆	D₅₂	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
132	C₄×Q₈	—	26	8	no	—
133	C₄×Q₈	—	52	4	no	—
134	#117	—	26	2	no	—
135	#117	—	52	4	no	—
136	#117	D₅₂	1	1	yes	characteristic
137	Q₃₂	—	13	8	no	—
138	Q₃₂	—	13	8	no	—
139	M₃₂	—	2	2	no	—
140	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
141	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
142	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
143	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
144	#107	—	13	4	no	—
145	#106	—	26	2	no	—
146	#103	—	26	8	no	—
147	#97	D₅₂	1	2	yes	—
148	C₄×C₈	C₂²×D₁₃	1	2	yes	—
149	C₂×C₁₃	—	2	2	no	—
150	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
151	C₂×C₁₃	#3417	1	2	yes	—
152	C₂×Q₈	—	26	8	no	—
153	C₂×Q₈	—	26	4	no	—
154	C₂×Q₈	—	52	2	no	—
155	C₂×Q₈	—	52	4	no	—
156	C₂²×C₄	—	52	2	no	—
157	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
158	Q₁₆	—	13	8	no	—
159	Q₁₆	—	26	8	no	—
160	Q₁₆	—	26	4	no	—
161	M₁₆	—	2	2	no	—
162	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
163	C₂×C₈	C₂×D₅₂	1	2	yes	—
164	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
165	C₂×C₈	#6127	1	2	yes	—
166	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
167	C₄⋊C₄	—	26	4	no	—
168	C₄⋊C₄	—	52	8	no	—
169	C₄⋊C₄	D₁₀₄	1	2	yes	—
170	C₂²⋊C₄	—	52	2	no	—
171	C₂²⋊C₄	D₁₀₄	1	2	yes	—
172	C₄²	—	52	2	no	—
173	C₄²	—	52	8	no	—
174	C₄²	C₂×D₅₂	1	2	yes	—
175	C₁₆	—	2	2	no	—
176	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
177	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
178	C₁₃	#33458	1	1	yes	characteristic, complemented, Sylow
179	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
180	Q₈	—	26	8	no	—
181	Q₈	—	52	4	no	—
182	Q₈	—	52	8	no	—
183	C₂×C₄	—	52	4	no	—
184	C₂×C₄	—	52	8	no	—
185	C₂×C₄	—	52	4	no	—
186	C₂×C₄	C₄×D₅₂	1	2	yes	—
187	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
188	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
189	C₂×C₄	#88158	1	2	yes	—
190	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	C₄×D₅₂	1	2	yes	—
193	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
194	C₈	C₂×#6127	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	C₄×D₁₀₄	1	2	yes	—
197	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	52	8	no	—
200	C₄	—	104	4	no	—
201	C₄	C₄×D₁₀₄	1	2	yes	—
202	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
203	C₄	C₂×#88158	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×C₄×D₁₀₄	1	1	yes	characteristic
206	C₂	#13127989	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2108	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2108	D₁₃
hypocenter	C₁₃	#33458
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #13116708	C₂	semidirect
4. C₄×Dic₁₀₄	C₂	semidirect
15. C₂×Dic₁₀₄	C₄	semidirect
178. C₁₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₀₄	2
#13123705	2
#13116708	2
C₄×Dic₁₀₄	2
#13116620	2
#13116557	2
C₁₃×#2108	2
#33458	13