Subgroups — #13128639

Order 1664.

1100 conjugacy classes of subgroups.

The subgroup lattice has 228 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	#11719503	C₂	1	2	yes	maximal
3	C₂×#88178	C₂	1	4	yes	maximal, complemented
4	#11719248	C₂	1	4	yes	maximal, complemented
5	#11719244	C₂	1	8	yes	maximal, complemented
6	D₄×Dic₂₆	C₂	1	4	yes	maximal, complemented
7	#11719205	C₂	1	2	yes	maximal, complemented
8	#11719188	C₂	1	4	yes	maximal, complemented
9	#11719175	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
12	C₂²×Dic₂₆	C₂²	1	4	yes	complemented
13	C₄×D₄×C₁₃	C₂²	1	4	yes	—
14	C₄×D₄×C₁₃	C₂²	1	4	yes	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₁₃	C₂²	1	8	yes	—
19	D₄×Dic₁₃	C₂²	1	8	yes	—
20	D₄×Dic₁₃	C₂²	1	8	yes	—
21	#88180	C₂²	1	4	yes	—
22	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	complemented
23	#88178	C₂²	1	4	yes	—
24	#88178	C₂²	1	4	yes	—
25	#88178	C₂²	1	8	yes	—
26	#88178	C₂²	1	8	yes	complemented
27	#88178	C₂²	1	8	yes	complemented
28	C₂×#6101	C₂²	1	8	yes	complemented
29	#88144	C₂²	1	8	yes	complemented
30	#88142	C₂²	1	8	yes	complemented
31	#88132	C₂²	1	16	yes	complemented
32	#88132	C₂²	1	16	yes	complemented
33	#88124	C₂²	1	4	yes	complemented
34	#88123	C₂²	1	2	yes	complemented
35	#88123	C₂²	1	2	yes	complemented
36	C₄×Dic₂₆	C₂²	1	4	yes	complemented
37	C₄×Dic₂₆	C₂²	1	4	yes	complemented
38	C₂⁴×C₁₃	C₂³	1	2	yes	—
39	C₂×D₄×C₁₃	Q₈	1	8	yes	complemented
40	C₂×D₄×C₁₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₃	—	4	4	no	—
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂³	1	8	yes	—
49	C₂²×Dic₁₃	C₂³	1	8	yes	—
50	C₂×Dic₂₆	—	4	8	no	—
51	C₂×Dic₂₆	C₂³	1	4	yes	—
52	C₂×Dic₂₆	C₂³	1	4	yes	—
53	C₂×Dic₂₆	C₂³	1	8	yes	—
54	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
55	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
56	C₄²×C₁₃	C₂³	1	2	yes	—
57	C₄²×C₁₃	C₂³	1	2	yes	—
58	C₂²⋊Dic₁₃	C₂³	1	16	yes	—
59	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
60	C₄⋊Dic₁₃	C₂³	1	4	yes	—
61	C₄⋊Dic₁₃	C₂³	1	4	yes	—
62	C₄⋊Dic₁₃	C₂³	1	4	yes	—
63	C₄⋊Dic₁₃	C₂³	1	4	yes	—
64	C₄⋊Dic₁₃	C₂³	1	8	yes	—
65	#6101	C₂³	1	8	yes	—
66	#6101	C₂³	1	8	yes	—
67	#6101	C₂³	1	16	yes	—
68	C₄×Dic₁₃	C₂³	1	8	yes	—
69	#3501	—	13	1	no	maximal, Sylow
70	C₂³×C₁₃	—	4	2	no	—
71	C₂³×C₁₃	—	4	2	no	—
72	C₂³×C₁₃	C₂×Q₈	1	8	yes	complemented
73	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂³×C₁₃	C₂⁴	1	4	yes	—
75	D₄×C₁₃	C₂×Q₈	1	16	yes	complemented
76	C₂×C₄×C₁₃	—	2	8	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	—	4	4	no	—
80	C₂×C₄×C₁₃	—	4	1	no	—
81	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	complemented
82	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
84	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
85	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
86	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂×Dic₁₃	—	4	8	no	—
88	C₂×Dic₁₃	—	4	8	no	—
89	C₂×Dic₁₃	C₂⁴	1	8	yes	—
90	C₂×Dic₁₃	C₂⁴	1	8	yes	—
91	Dic₂₆	—	2	8	no	—
92	Dic₂₆	—	4	8	no	—
93	Dic₂₆	—	4	4	no	—
94	#554	—	13	8	no	—
95	#554	—	13	4	no	—
96	D₄×Q₈	—	13	4	no	—
97	#544	—	13	2	no	—
98	#543	—	13	4	no	—
99	#533	—	13	2	no	—
100	C₂×C₂²⋊Q₈	—	13	4	no	—
101	#518	—	13	2	no	—
102	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic
103	C₂²×C₁₃	—	2	4	no	—
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	—	2	1	no	—
107	C₂²×C₁₃	—	4	8	no	—
108	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
109	C₂²×C₁₃	C₂²×Q₈	1	8	yes	—
110	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	—	2	1	no	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	C₂²×Q₈	1	4	yes	—
115	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
116	Dic₁₃	—	2	8	no	—
117	Dic₁₃	—	2	8	no	—
118	C₂²×Q₈	—	13	4	no	—
119	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
120	C₂³×C₄	C₂×D₁₃	1	2	yes	—
121	C₄⋊Q₈	—	13	8	no	—
122	C₄⋊Q₈	—	13	2	no	—
123	C₄⋊Q₈	—	13	2	no	—
124	#125	—	13	8	no	—
125	#125	—	13	4	no	—
126	C₂²⋊Q₈	—	13	4	no	—
127	C₂²⋊Q₈	—	13	8	no	—
128	C₂²⋊Q₈	—	13	8	no	—
129	C₂²⋊Q₈	—	13	8	no	—
130	C₂²⋊Q₈	—	13	4	no	—
131	C₂²⋊Q₈	—	13	16	no	—
132	C₂²⋊Q₈	—	13	16	no	—
133	C₄×Q₈	—	13	4	no	—
134	C₄×Q₈	—	13	4	no	—
135	C₄×D₄	—	13	8	no	—
136	C₄×D₄	—	13	8	no	—
137	C₄×D₄	C₂×D₁₃	1	4	yes	—
138	C₄×D₄	C₂×D₁₃	1	4	yes	—
139	#117	—	13	4	no	—
140	C₂×C₄⋊C₄	—	13	8	no	—
141	C₂×C₄⋊C₄	—	13	4	no	—
142	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	—	13	8	no	—
144	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
145	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
146	C₂×C₁₃	—	2	2	no	—
147	C₂×C₁₃	—	2	8	no	—
148	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
149	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₁₃	#585	1	1	yes	characteristic
151	C₂⁴	C₂²×D₁₃	1	2	yes	—
152	C₂×Q₈	—	13	4	no	—
153	C₂×Q₈	—	13	4	no	—
154	C₂×Q₈	—	13	8	no	—
155	C₂×Q₈	—	52	8	no	—
156	C₂×D₄	Dic₂₆	1	8	yes	complemented
157	C₂×D₄	C₂²×D₁₃	1	4	yes	—
158	C₂²×C₄	—	4	4	no	—
159	C₂²×C₄	—	13	8	no	—
160	C₂²×C₄	—	13	8	no	—
161	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
163	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
166	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
167	C₄⋊C₄	—	13	8	no	—
168	C₄⋊C₄	—	13	4	no	—
169	C₄⋊C₄	—	13	8	no	—
170	C₄⋊C₄	—	13	16	no	—
171	C₄⋊C₄	—	13	4	no	—
172	C₄⋊C₄	—	13	4	no	—
173	C₄⋊C₄	—	13	8	no	—
174	C₄⋊C₄	—	13	4	no	—
175	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
176	C₂²⋊C₄	—	13	8	no	—
177	C₂²⋊C₄	—	13	16	no	—
178	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
179	C₄²	—	13	8	no	—
180	C₄²	C₂²×D₁₃	1	2	yes	—
181	C₄²	C₂²×D₁₃	1	2	yes	—
182	C₁₃	#3501	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	4	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂×Dic₂₆	1	8	yes	complemented
186	C₂³	C₂³×D₁₃	1	1	yes	characteristic
187	C₂³	C₂³×D₁₃	1	4	yes	—
188	Q₈	—	26	8	no	—
189	Q₈	—	52	4	no	—
190	Q₈	—	52	8	no	—
191	D₄	C₂×Dic₂₆	1	16	yes	complemented
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	13	8	no	—
198	C₂×C₄	—	13	8	no	—
199	C₂×C₄	—	52	8	no	—
200	C₂×C₄	—	52	8	no	—
201	C₂×C₄	C₂×Dic₂₆	1	4	yes	complemented
202	C₂×C₄	C₂³×D₁₃	1	2	yes	—
203	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
204	C₂×C₄	C₂³×D₁₃	1	4	yes	—
205	C₂×C₄	C₂³×D₁₃	1	2	yes	—
206	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
207	C₂²	—	2	1	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	8	no	—
212	C₂²	C₂²×Dic₂₆	1	2	yes	—
213	C₂²	C₂²×Dic₂₆	1	8	yes	—
214	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	26	8	no	—
219	C₄	—	26	8	no	—
220	C₄	C₂²×Dic₂₆	1	4	yes	—
221	C₄	#88251	1	2	yes	—
222	C₂	—	2	2	no	—
223	C₂	—	2	8	no	—
224	C₂	C₂³×Dic₂₆	1	1	yes	characteristic
225	C₂	C₂×#88251	1	1	yes	characteristic
226	C₂	#11719648	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂×C₄×D₄×C₁₃	C₂
prime radical	C₂²×D₄×C₁₃	C₂²
hypercenter	C₂×C₄×D₄	D₁₃
hypocenter	C₁₃	#3501
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
3. C₂×#88178	C₂	semidirect
4. #11719248	C₂	semidirect
5. #11719244	C₂	semidirect
6. D₄×Dic₂₆	C₂	semidirect
7. #11719205	C₂	semidirect
8. #11719188	C₂	semidirect
9. #11719175	C₂	semidirect
12. C₂²×Dic₂₆	C₂²	semidirect
22. C₂×C₄⋊Dic₁₃	C₂²	semidirect
26. #88178	C₂²	semidirect
27. #88178	C₂²	semidirect
28. C₂×#6101	C₂²	semidirect
29. #88144	C₂²	semidirect
30. #88142	C₂²	semidirect
31. #88132	C₂²	semidirect
32. #88132	C₂²	semidirect
33. #88124	C₂²	semidirect
34. #88123	C₂²	semidirect
35. #88123	C₂²	semidirect
36. C₄×Dic₂₆	C₂²	semidirect
37. C₄×Dic₂₆	C₂²	semidirect
39. C₂×D₄×C₁₃	Q₈	semidirect
72. C₂³×C₁₃	C₂×Q₈	semidirect
75. D₄×C₁₃	C₂×Q₈	semidirect
81. C₂×C₄×C₁₃	C₂×Q₈	semidirect
156. C₂×D₄	Dic₂₆	semidirect
182. C₁₃	#3501	semidirect
185. C₂³	C₂×Dic₂₆	semidirect
191. D₄	C₂×Dic₂₆	semidirect
201. C₂×C₄	C₂×Dic₂₆	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
#11719503	2
C₂×#88178	2
#11719248	2
#11719244	2
D₄×Dic₂₆	2
#11719205	2
#11719188	2
#11719175	2
#3501	13