Subgroups — #13123484

Order 1664.

384 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719227	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718661	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718500	C₂	1	2	yes	maximal, complemented
5	#11718496	C₂	1	2	yes	maximal, complemented
6	#11718388	C₂	1	2	yes	maximal
7	#11718380	C₂	1	2	yes	maximal, complemented
8	#11718370	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718351	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6094	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6127	C₂²	1	1	yes	characteristic
13	C₁₃×#117	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#6094	C₂²	1	2	yes	—
16	C₂×C₈×D₁₃	C₂²	1	2	yes	—
17	#88147	C₄	1	2	yes	—
18	#88141	C₄	1	2	yes	—
19	#88135	C₄	1	2	yes	—
20	#88135	C₄	1	2	yes	—
21	#88126	C₂²	1	1	yes	characteristic
22	#88126	C₂²	1	1	yes	characteristic
23	C₄²×D₁₃	C₂²	1	1	yes	characteristic
24	C₄²×D₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
26	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₁₃×#97	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₂²⋊#1055	C₂²	1	2	yes	—
30	#88060	C₂²	1	2	yes	—
31	#88060	C₂²	1	2	yes	—
32	#88055	C₂²	1	2	yes	—
33	#88055	C₂²	1	2	yes	—
34	#88054	C₂²	1	2	yes	—
35	#88054	C₂²	1	2	yes	—
36	C₄⋊#1055	C₂²	1	2	yes	—
37	#88043	C₂²	1	1	yes	characteristic
38	C₄×#1055	C₂²	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
43	#6127	—	2	4	no	—
44	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂×C₈×C₁₃	C₂³	1	2	yes	—
52	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
54	C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	#6103	C₂×C₄	1	2	yes	—
57	#6103	C₂×C₄	1	2	yes	—
58	#6101	C₂×C₄	1	2	yes	—
59	#6101	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
64	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
65	C₂×#1055	C₂³	1	2	yes	—
66	C₂×#1055	C₂³	1	2	yes	—
67	#6094	—	2	4	no	—
68	C₈×D₁₃	—	2	4	no	—
69	#2923	—	13	1	no	maximal, Sylow
70	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₁₃	—	2	2	no	—
79	C₂²⋊D₁₃	—	2	2	no	—
80	C₂×Dic₁₃	—	4	2	no	—
81	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
85	D₅₂	—	2	2	no	—
86	C₄×D₁₃	—	2	2	no	—
87	C₄×D₁₃	—	4	2	no	—
88	C₄×D₁₃	—	4	2	no	—
89	C₄×D₁₃	#41	1	4	yes	—
90	Dic₂₆	—	2	2	no	—
91	C₈×C₁₃	—	2	2	no	—
92	C₈×C₁₃	—	2	2	no	—
93	#1055	—	2	2	no	—
94	#1055	—	2	2	no	—
95	C₄×#41	—	13	1	no	—
96	#433	—	13	1	no	—
97	#433	—	13	1	no	—
98	#433	—	13	1	no	—
99	#433	D₁₃	1	1	yes	characteristic, complemented
100	#432	—	13	2	no	—
101	#432	—	13	2	no	—
102	#424	—	13	2	no	—
103	#408	—	13	2	no	—
104	C₄×M₁₆	—	13	1	no	—
105	C₂×#97	—	13	1	no	—
106	C₂²×C₁₃	—	2	1	no	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×D₁₃	—	2	2	no	—
110	C₂×D₁₃	—	2	1	no	—
111	C₂×D₁₃	—	2	2	no	—
112	C₂×D₁₃	#117	1	2	yes	—
113	C₄×C₁₃	—	2	1	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	C₂×#41	1	2	yes	—
117	Dic₁₃	—	2	1	no	—
118	Dic₁₃	—	2	2	no	—
119	Dic₁₃	—	2	2	no	—
120	Dic₁₃	#117	1	2	yes	—
121	C₂×#41	—	13	1	no	—
122	C₂×M₁₆	—	13	2	no	—
123	C₂²×C₈	—	13	2	no	—
124	C₄×Q₈	—	13	2	no	—
125	C₄×D₄	—	13	2	no	—
126	C₄×D₄	—	13	2	no	—
127	C₄×D₄	—	13	2	no	—
128	#117	—	13	1	no	—
129	#117	—	13	1	no	—
130	#117	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₄²	—	13	1	no	—
132	C₂×C₄²	—	13	1	no	—
133	C₂×C₄²	—	13	1	no	—
134	C₄⋊C₈	—	13	2	no	—
135	C₄⋊C₈	—	13	2	no	—
136	C₄⋊C₈	—	13	2	no	—
137	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
138	C₂²⋊C₈	—	13	2	no	—
139	C₂²⋊C₈	—	13	2	no	—
140	C₂²⋊C₈	—	13	2	no	—
141	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
142	#97	—	13	2	no	—
143	#97	—	13	2	no	—
144	#97	—	13	1	no	—
145	#97	C₂×D₁₃	1	1	yes	characteristic
146	C₄×C₈	—	13	1	no	—
147	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
148	C₂×C₁₃	—	4	1	no	—
149	C₂×C₁₃	C₂×#117	1	1	yes	characteristic
150	C₂×C₁₃	#568	1	2	yes	—
151	D₁₃	—	2	2	no	—
152	D₁₃	—	4	1	no	—
153	#41	—	26	4	no	—
154	C₂×Q₈	—	13	1	no	—
155	C₂×D₄	—	13	1	no	—
156	C₂×D₄	—	13	1	no	—
157	C₂×D₄	—	13	1	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	—	13	2	no	—
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	2	no	—
162	C₂²×C₄	—	13	2	no	—
163	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
164	M₁₆	—	26	4	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	—	13	2	no	—
167	C₂×C₈	—	26	4	no	—
168	C₂×C₈	C₂²×D₁₃	1	2	yes	—
169	C₂×C₈	C₂²×D₁₃	1	2	yes	—
170	C₄⋊C₄	—	13	2	no	—
171	C₄⋊C₄	—	13	2	no	—
172	C₄⋊C₄	C₄×D₁₃	1	2	yes	—
173	C₂²⋊C₄	—	13	2	no	—
174	C₂²⋊C₄	—	13	2	no	—
175	C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
176	C₄²	—	13	2	no	—
177	C₄²	—	13	2	no	—
178	C₄²	—	13	2	no	—
179	C₄²	—	13	1	no	—
180	C₄²	—	13	1	no	—
181	C₄²	C₂²×D₁₃	1	1	yes	characteristic
182	C₄²	C₂²×D₁₃	1	1	yes	characteristic
183	C₁₃	#2923	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	13	1	no	—
185	C₂³	—	13	1	no	—
186	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
187	Q₈	—	26	2	no	—
188	D₄	—	26	2	no	—
189	D₄	—	26	2	no	—
190	D₄	—	26	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	4	no	—
194	C₂×C₄	—	13	2	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	1	no	—
197	C₂×C₄	—	26	2	no	—
198	C₂×C₄	—	52	2	no	—
199	C₂×C₄	—	52	2	no	—
200	C₂×C₄	—	52	2	no	—
201	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
204	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	26	2	no	—
208	C₈	—	26	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	13	2	no	—
212	C₂²	—	26	2	no	—
213	C₂²	—	26	2	no	—
214	C₂²	—	26	1	no	—
215	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	13	2	no	—
220	C₄	—	26	2	no	—
221	C₄	—	26	2	no	—
222	C₄	—	26	1	no	—
223	C₄	#41×D₁₃	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	26	2	no	—
226	C₂	—	52	1	no	—
227	C₂	D₁₃×#117	1	1	yes	characteristic
228	C₂	#11719455	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#433	C₂
prime radical	C₂×#6127	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#433	D₁₃
hypocenter	C₁₃	#2923
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. C₁₃×#433	C₂	semidirect
3. #11718661	C₂	semidirect
4. #11718500	C₂	semidirect
5. #11718496	C₂	semidirect
7. #11718380	C₂	semidirect
8. #11718370	C₂	semidirect
9. D₁₃×#97	C₂	semidirect
10. #11718351	C₂	semidirect
11. C₄×#6094	C₂	semidirect
99. #433	D₁₃	semidirect
183. C₁₃	#2923	semidirect

Maximal subgroups

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

subgroup	index
#11719227	2
C₁₃×#433	2
#11718661	2
#11718500	2
#11718496	2
#11718388	2
#11718380	2
#11718370	2
D₁₃×#97	2
#11718351	2
C₄×#6094	2
#2923	13