Subgroups — #13123871

Order 1664.

464 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	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	#11718868	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#1055	C₂	1	2	yes	maximal, complemented
4	#11718673	C₂	1	2	yes	maximal, complemented
5	#11718672	C₂	1	4	yes	maximal, complemented
6	D₄×#1055	C₂	1	2	yes	maximal, complemented
7	#11718587	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₃	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
11	C₄×D₄×C₁₃	C₄	1	2	yes	—
12	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
13	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
14	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#6099	C₂²	1	4	yes	complemented
21	C₂²×#1055	C₂²	1	4	yes	complemented
22	C₂²⋊#1055	C₂²	1	1	yes	characteristic
23	C₂²⋊#1055	C₂²	1	1	yes	characteristic
24	C₂²⋊#1055	C₂²	1	2	yes	—
25	C₂²⋊#1055	C₂²	1	4	yes	complemented
26	C₂²⋊#1055	C₂²	1	4	yes	complemented
27	C₄⋊#1055	C₂²	1	2	yes	complemented
28	C₄⋊#1055	C₂²	1	2	yes	complemented
29	#88043	C₂²	1	2	yes	complemented
30	C₄×#1055	C₂²	1	2	yes	complemented
31	C₂⁴×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×D₄×C₁₃	—	2	4	no	—
33	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	—	4	4	no	—
37	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
49	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
50	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
51	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	#6099	—	4	4	no	—
57	C₂×#1055	—	4	4	no	—
58	C₂×#1055	C₂³	1	2	yes	—
59	C₂×#1055	C₂³	1	4	yes	—
60	C₂×#1055	C₂³	1	2	yes	—
61	#2974	—	13	1	no	maximal, Sylow
62	C₂³×C₁₃	—	2	4	no	—
63	C₂³×C₁₃	—	2	4	no	—
64	C₂³×C₁₃	—	4	2	no	—
65	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	D₄×C₁₃	—	2	4	no	—
69	D₄×C₁₃	—	2	2	no	—
70	D₄×C₁₃	M₁₆	1	4	yes	complemented
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	#1055	—	2	2	no	—
88	#1055	—	2	4	no	—
89	#1055	—	2	2	no	—
90	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic
91	#436	—	13	2	no	—
92	#435	—	13	4	no	—
93	C₈×D₄	—	13	2	no	—
94	#432	—	13	1	no	—
95	#431	—	13	1	no	—
96	#407	—	13	2	no	—
97	C₂×C₂²⋊C₈	—	13	2	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	2	2	no	—
100	C₂²×C₁₃	—	2	1	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	—	2	2	no	—
105	C₂²×C₁₃	—	4	8	no	—
106	C₂²×C₁₃	C₂×M₁₆	1	4	yes	—
107	C₂²×C₁₃	#131	1	2	yes	—
108	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	2	1	no	—
113	C₄×C₁₃	—	2	1	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
116	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
117	C₂²×D₄	Dic₁₃	1	1	yes	characteristic
118	C₂³×C₄	C₂×D₁₃	1	2	yes	—
119	C₂×M₁₆	—	13	4	no	—
120	C₂²×C₈	—	13	4	no	—
121	C₄×D₄	Dic₁₃	1	2	yes	—
122	C₄×D₄	Dic₁₃	1	1	yes	characteristic
123	C₄×D₄	Dic₁₃	1	1	yes	characteristic
124	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
125	C₄×D₄	C₂×D₁₃	1	2	yes	—
126	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
127	C₂×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	Dic₁₃	1	2	yes	—
129	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
130	C₄⋊C₈	—	13	2	no	—
131	C₄⋊C₈	—	13	2	no	—
132	C₂²⋊C₈	—	13	1	no	—
133	C₂²⋊C₈	—	13	2	no	—
134	C₂²⋊C₈	—	13	1	no	—
135	C₂²⋊C₈	—	13	4	no	—
136	C₂²⋊C₈	—	13	4	no	—
137	#97	—	13	2	no	—
138	C₄×C₈	—	13	2	no	—
139	C₂×C₁₃	—	2	2	no	—
140	C₂×C₁₃	—	2	4	no	—
141	C₂×C₁₃	—	4	2	no	—
142	C₂×C₁₃	#518	1	1	yes	characteristic
143	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
144	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
145	C₂⁴	C₂×Dic₁₃	1	2	yes	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
148	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
149	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
154	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
155	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
156	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
158	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
159	M₁₆	—	52	4	no	—
160	C₂×C₈	—	13	2	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	4	no	—
163	C₂×C₈	—	52	4	no	—
164	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
165	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
166	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
167	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
168	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
169	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
170	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
171	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
172	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
173	C₄²	C₂²×D₁₃	1	1	yes	characteristic
174	C₄²	C₂²×D₁₃	1	1	yes	characteristic
175	C₁₃	#2974	1	1	yes	characteristic, complemented, Sylow
176	C₂³	—	2	4	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	4	2	no	—
179	C₂³	C₂²×Dic₁₃	1	2	yes	—
180	C₂³	C₂²×Dic₁₃	1	2	yes	—
181	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
182	D₄	—	2	4	no	—
183	D₄	—	2	2	no	—
184	D₄	#6099	1	4	yes	complemented
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
197	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
198	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
199	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
201	C₈	—	26	2	no	—
202	C₈	—	26	4	no	—
203	C₈	—	26	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	8	no	—
212	C₂²	C₂×#6099	1	4	yes	—
213	C₂²	#88202	1	2	yes	—
214	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	C₂×#6099	1	2	yes	—
222	C₄	#88251	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	4	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×#6099	1	1	yes	characteristic
227	C₂	#11719503	1	1	yes	characteristic
228	C₂	C₂×#88202	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₃	C₂
prime radical	C₂²×D₄×C₁₃	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₃
hypocenter	C₁₃	#2974
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #11718868	C₂	semidirect
3. C₂×C₂²⋊#1055	C₂	semidirect
4. #11718673	C₂	semidirect
5. #11718672	C₂	semidirect
6. D₄×#1055	C₂	semidirect
7. #11718587	C₂	semidirect
8. #11718586	C₂	semidirect
20. C₂×#6099	C₂²	semidirect
21. C₂²×#1055	C₂²	semidirect
25. C₂²⋊#1055	C₂²	semidirect
26. C₂²⋊#1055	C₂²	semidirect
27. C₄⋊#1055	C₂²	semidirect
28. C₄⋊#1055	C₂²	semidirect
29. #88043	C₂²	semidirect
30. C₄×#1055	C₂²	semidirect
70. D₄×C₁₃	M₁₆	semidirect
175. C₁₃	#2974	semidirect
184. D₄	#6099	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₁₃	2
#11718868	2
C₂×C₂²⋊#1055	2
#11718673	2
#11718672	2
D₄×#1055	2
#11718587	2
#11718586	2
#2974	13