Subgroups — C₂×#11719248 #13128367

Order 1664.

1200 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	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Dic₁₃	C₂	1	2	yes	maximal
3	C₂²×C₄⋊Dic₁₃	C₂	1	2	yes	maximal, complemented
4	C₂×#88178	C₂	1	2	yes	maximal, complemented
5	#11719248	C₂	1	16	yes	maximal, direct product factor
6	C₂×#88144	C₂	1	2	yes	maximal, complemented
7	C₂×#88132	C₂	1	4	yes	maximal, complemented
8	C₂×#88123	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
12	C₂³×Dic₁₃	C₂²	1	4	yes	—
13	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₆	C₂²	1	2	yes	—
15	C₄×D₄×C₁₃	C₂²	1	8	yes	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
20	D₄×Dic₁₃	C₂²	1	16	yes	—
21	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
25	C₂×C₄⋊Dic₁₃	C₂²	1	16	yes	complemented
26	#88178	C₂²	1	16	yes	complemented
27	C₂×#6101	C₂²	1	2	yes	—
28	C₂×#6101	C₂²	1	4	yes	—
29	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
30	#88144	C₂²	1	16	yes	complemented
31	#88132	C₂²	1	32	yes	complemented
32	#88123	C₂²	1	8	yes	complemented
33	C₄×Dic₂₆	C₂²	1	8	yes	complemented
34	C₂⁴×C₁₃	C₂³	1	2	yes	—
35	C₂×D₄×C₁₃	Q₈	1	8	yes	complemented
36	C₂×D₄×C₁₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₁₃	—	2	8	no	—
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	8	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	4	8	no	—
44	C₂²×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×Dic₁₃	C₂³	1	2	yes	—
46	C₂²×Dic₁₃	C₂³	1	4	yes	—
47	C₂²×Dic₁₃	C₂³	1	16	yes	—
48	C₂×Dic₂₆	—	2	4	no	—
49	C₂×Dic₂₆	—	4	4	no	—
50	C₂×Dic₂₆	C₂³	1	4	yes	—
51	C₂×Dic₂₆	C₂³	1	8	yes	—
52	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
53	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
54	C₄²×C₁₃	C₂³	1	4	yes	—
55	C₂²⋊Dic₁₃	C₂³	1	16	yes	—
56	C₄⋊Dic₁₃	C₂³	1	4	yes	—
57	C₄⋊Dic₁₃	C₂³	1	8	yes	—
58	C₄⋊Dic₁₃	C₂³	1	8	yes	—
59	C₄⋊Dic₁₃	C₂³	1	16	yes	—
60	#6101	C₂³	1	8	yes	—
61	#6101	C₂³	1	16	yes	—
62	C₄×Dic₁₃	C₂³	1	8	yes	—
63	C₂×#554	—	13	1	no	maximal, Sylow
64	C₂³×C₁₃	—	2	4	no	—
65	C₂³×C₁₃	—	2	4	no	—
66	C₂³×C₁₃	C₂×Q₈	1	8	yes	—
67	C₂³×C₁₃	C₂⁴	1	4	yes	—
68	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
69	D₄×C₁₃	C₂×Q₈	1	16	yes	complemented
70	C₂×C₄×C₁₃	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	16	no	—
73	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
74	C₂×C₄×C₁₃	#41	1	4	yes	—
75	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
76	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
77	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
78	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
79	C₂×Dic₁₃	—	2	4	no	—
80	C₂×Dic₁₃	—	2	8	no	—
81	C₂×Dic₁₃	—	2	4	no	—
82	C₂×Dic₁₃	—	4	16	no	—
83	C₂×Dic₁₃	C₂⁴	1	4	yes	—
84	C₂×Dic₁₃	C₂⁴	1	4	yes	—
85	C₂×Dic₁₃	C₂⁴	1	8	yes	—
86	Dic₂₆	—	2	8	no	—
87	Dic₂₆	—	4	8	no	—
88	#554	—	13	16	no	—
89	C₂×C₄⋊Q₈	—	13	1	no	—
90	C₂×#125	—	13	2	no	—
91	C₂×C₂²⋊Q₈	—	13	2	no	—
92	C₂×C₂²⋊Q₈	—	13	4	no	—
93	C₂×C₄×Q₈	—	13	1	no	—
94	C₂×C₄×D₄	—	13	2	no	—
95	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic
96	C₂²×C₄⋊C₄	—	13	2	no	—
97	C₂²×C₁₃	—	2	16	no	—
98	C₂²×C₁₃	—	2	4	no	—
99	C₂²×C₁₃	—	2	4	no	—
100	C₂²×C₁₃	C₂²×Q₈	1	8	yes	—
101	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
102	C₂²×C₁₃	C₂×#41	1	2	yes	—
103	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
104	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
105	C₄×C₁₃	—	2	4	no	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	C₂²×Q₈	1	4	yes	—
108	C₄×C₁₃	C₂×#41	1	4	yes	—
109	Dic₁₃	—	2	4	no	—
110	Dic₁₃	—	2	8	no	—
111	Dic₁₃	—	2	4	no	—
112	C₂²×Q₈	—	13	1	no	—
113	C₂²×Q₈	—	13	2	no	—
114	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
115	C₂³×C₄	—	13	4	no	—
116	C₂³×C₄	C₂×D₁₃	1	2	yes	—
117	C₄⋊Q₈	—	13	8	no	—
118	#125	—	13	16	no	—
119	C₂²⋊Q₈	—	13	16	no	—
120	C₂²⋊Q₈	—	13	32	no	—
121	C₄×Q₈	—	13	8	no	—
122	C₄×D₄	—	13	16	no	—
123	C₄×D₄	C₂×D₁₃	1	8	yes	—
124	C₂×C₄⋊C₄	—	13	1	no	—
125	C₂×C₄⋊C₄	—	13	4	no	—
126	C₂×C₄⋊C₄	—	13	2	no	—
127	C₂×C₄⋊C₄	—	13	16	no	—
128	C₂×C₄⋊C₄	—	13	2	no	—
129	C₂×C₄⋊C₄	—	13	2	no	—
130	C₂×C₄⋊C₄	—	13	4	no	—
131	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	—	13	4	no	—
133	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
134	C₂×C₄²	—	13	2	no	—
135	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
136	C₂×C₁₃	—	2	8	no	—
137	C₂×C₁₃	#554	1	4	yes	complemented
138	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
139	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
140	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	C₂⁴	C₂²×D₁₃	1	2	yes	—
142	C₂×Q₈	—	13	4	no	—
143	C₂×Q₈	—	13	8	no	—
144	C₂×Q₈	—	26	4	no	—
145	C₂×Q₈	—	52	4	no	—
146	C₂×D₄	Dic₂₆	1	8	yes	complemented
147	C₂×D₄	C₂²×D₁₃	1	4	yes	—
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	—	13	2	no	—
150	C₂²×C₄	—	13	2	no	—
151	C₂²×C₄	—	13	4	no	—
152	C₂²×C₄	—	13	16	no	—
153	C₂²×C₄	—	52	8	no	—
154	C₂²×C₄	C₂²×D₁₃	1	8	yes	—
155	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
156	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
157	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
159	C₄⋊C₄	—	13	4	no	—
160	C₄⋊C₄	—	13	16	no	—
161	C₄⋊C₄	—	13	8	no	—
162	C₄⋊C₄	—	13	8	no	—
163	C₄⋊C₄	—	13	8	no	—
164	C₄⋊C₄	—	13	16	no	—
165	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
166	C₂²⋊C₄	—	13	16	no	—
167	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
168	C₄²	—	13	8	no	—
169	C₄²	C₂²×D₁₃	1	4	yes	—
170	C₁₃	C₂×#554	1	1	yes	characteristic, complemented, Sylow
171	C₂³	—	2	4	no	—
172	C₂³	—	2	4	no	—
173	C₂³	C₂×Dic₂₆	1	8	yes	—
174	C₂³	C₂³×D₁₃	1	4	yes	—
175	C₂³	C₂³×D₁₃	1	1	yes	characteristic
176	Q₈	—	26	8	no	—
177	Q₈	—	52	8	no	—
178	D₄	C₂×Dic₂₆	1	16	yes	complemented
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	13	4	no	—
183	C₂×C₄	—	13	4	no	—
184	C₂×C₄	—	13	8	no	—
185	C₂×C₄	—	26	4	no	—
186	C₂×C₄	—	26	4	no	—
187	C₂×C₄	—	26	8	no	—
188	C₂×C₄	—	52	16	no	—
189	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
190	C₂×C₄	#6129	1	4	yes	—
191	C₂×C₄	C₂³×D₁₃	1	2	yes	—
192	C₂×C₄	C₂³×D₁₃	1	4	yes	—
193	C₂×C₄	C₂³×D₁₃	1	2	yes	—
194	C₂×C₄	C₂³×D₁₃	1	2	yes	—
195	C₂²	—	2	16	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₂²×Dic₂₆	1	8	yes	—
199	C₂²	C₂²×Dic₂₆	1	2	yes	—
200	C₂²	C₂×#6129	1	2	yes	—
201	C₂²	#88256	1	2	yes	—
202	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	26	4	no	—
206	C₄	—	26	4	no	—
207	C₄	—	26	8	no	—
208	C₄	C₂²×Dic₂₆	1	4	yes	—
209	C₄	C₂×#6129	1	4	yes	—
210	C₂	—	2	8	no	—
211	C₂	#11719248	1	4	yes	direct product factor
212	C₂	C₂³×Dic₂₆	1	1	yes	characteristic
213	C₂	C₂²×#6129	1	1	yes	characteristic
214	C₂	C₂×#88256	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₄×D₄×C₁₃	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×D₄×C₁₃	C₂²
hypercenter	C₂×C₄×D₄	D₁₃
hypocenter	C₁₃	C₂×#554
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊Dic₁₃	C₂	semidirect
4. C₂×#88178	C₂	semidirect
5. #11719248	C₂	direct (N × H)
6. C₂×#88144	C₂	semidirect
7. C₂×#88132	C₂	semidirect
8. C₂×#88123	C₂	semidirect
9. C₂×C₄×Dic₂₆	C₂	semidirect
25. C₂×C₄⋊Dic₁₃	C₂²	semidirect
26. #88178	C₂²	semidirect
30. #88144	C₂²	semidirect
31. #88132	C₂²	semidirect
32. #88123	C₂²	semidirect
33. C₄×Dic₂₆	C₂²	semidirect
35. C₂×D₄×C₁₃	Q₈	semidirect
69. D₄×C₁₃	C₂×Q₈	semidirect
137. C₂×C₁₃	#554	semidirect
146. C₂×D₄	Dic₂₆	semidirect
170. C₁₃	C₂×#554	semidirect
178. D₄	C₂×Dic₂₆	semidirect
211. C₂	#11719248	direct (N × H)

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
C₂×D₄×Dic₁₃	2
C₂²×C₄⋊Dic₁₃	2
C₂×#88178	2
#11719248	2
C₂×#88144	2
C₂×#88132	2
C₂×#88123	2
C₂×C₄×Dic₂₆	2
C₂×#554	13

Subgroups — C2×#11719248 #13128367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719248 #13128367