Subgroups — C₄×#88129 #13117380

Order 1664.

652 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88129	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718601	C₂	1	4	yes	maximal, complemented
6	#11718600	C₂	1	1	yes	maximal, characteristic
7	C₄×#6103	C₂	1	4	yes	maximal, complemented
8	#11718597	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718590	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₅₂	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
12	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#6103	C₂²	1	2	yes	—
19	C₂×#6103	C₂²	1	4	yes	—
20	C₂×#6103	C₂²	1	4	yes	—
21	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×#6101	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
25	#88129	C₄	1	8	yes	direct product factor
26	C₄×D₅₂	—	2	4	no	—
27	C₄×Dic₂₆	—	2	4	no	—
28	#88071	C₂²	1	4	yes	—
29	#88071	C₂²	1	4	yes	—
30	C₂³×D₁₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₃	C₂³	1	2	yes	—
37	C₂²×Dic₁₃	C₂³	1	2	yes	—
38	C₂²×Dic₁₃	C₂³	1	2	yes	—
39	C₂×D₅₂	—	2	2	no	—
40	C₂×D₅₂	—	2	2	no	—
41	C₂×D₅₂	C₂×C₄	1	4	yes	—
42	C₂×C₄×D₁₃	—	2	4	no	—
43	C₂×C₄×D₁₃	—	2	8	no	—
44	C₂×Dic₂₆	—	2	2	no	—
45	C₂×Dic₂₆	—	2	2	no	—
46	C₂×Dic₂₆	C₂×C₄	1	4	yes	complemented
47	C₄²×C₁₃	—	2	4	no	—
48	C₄²×C₁₃	—	2	2	no	—
49	C₄²×C₁₃	—	2	4	no	—
50	C₄²×C₁₃	C₂×C₄	1	4	yes	complemented
51	C₄²×C₁₃	D₄	1	4	yes	—
52	#6103	—	2	4	no	—
53	#6103	—	2	8	no	—
54	#6103	C₂×C₄	1	16	yes	complemented
55	C₄⋊Dic₁₃	—	2	2	no	—
56	C₄⋊Dic₁₃	—	2	2	no	—
57	#6101	—	2	4	no	—
58	C₄×Dic₁₃	—	2	4	no	—
59	C₄×#124	—	13	1	no	maximal, Sylow
60	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂²×D₁₃	—	2	2	no	—
62	C₂²×D₁₃	—	2	4	no	—
63	C₂²×D₁₃	—	2	4	no	—
64	C₂²×D₁₃	—	2	2	no	—
65	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₃	#41	1	4	yes	—
77	C₂×C₄×C₁₃	#41	1	2	yes	—
78	C₂×C₄×C₁₃	#41	1	4	yes	—
79	C₂×Dic₁₃	—	2	2	no	—
80	C₂×Dic₁₃	—	2	2	no	—
81	C₂×Dic₁₃	—	2	2	no	—
82	C₂×Dic₁₃	—	2	2	no	—
83	C₂×Dic₁₃	—	2	4	no	—
84	C₂×Dic₁₃	—	2	4	no	—
85	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
86	D₅₂	—	2	8	no	—
87	C₄×D₁₃	—	4	8	no	—
88	Dic₂₆	—	2	8	no	—
89	C₂×#124	—	13	1	no	—
90	C₂×C₄×Q₈	—	13	1	no	—
91	C₂×C₄×D₄	—	13	1	no	—
92	#391	—	13	1	no	—
93	#390	—	13	1	no	—
94	#388	—	13	4	no	—
95	#382	—	13	1	no	—
96	C₄×C₂²⋊C₄	—	13	4	no	—
97	C₄³	D₁₃	1	1	yes	characteristic, complemented
98	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×#41	1	2	yes	—
102	C₂²×C₁₃	C₂×#41	1	2	yes	—
103	C₂×D₁₃	—	2	4	no	—
104	C₂×D₁₃	—	2	8	no	—
105	C₂×D₁₃	—	4	2	no	—
106	C₂×D₁₃	—	4	4	no	—
107	C₂×D₁₃	—	4	2	no	—
108	C₄×C₁₃	—	2	4	no	—
109	C₄×C₁₃	—	2	4	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	C₄×D₄	1	4	yes	—
112	C₄×C₁₃	#124	1	4	yes	complemented
113	Dic₁₃	—	2	4	no	—
114	Dic₁₃	—	4	2	no	—
115	Dic₁₃	—	4	2	no	—
116	C₂²×Q₈	—	13	1	no	—
117	C₂²×D₄	—	13	1	no	—
118	C₂³×C₄	—	13	2	no	—
119	#124	—	13	8	no	—
120	C₄×Q₈	—	26	4	no	—
121	C₄×D₄	—	26	4	no	—
122	C₂×C₄⋊C₄	—	13	1	no	—
123	C₂×C₄⋊C₄	—	13	2	no	—
124	C₂×C₄⋊C₄	—	13	1	no	—
125	C₂×C₂²⋊C₄	—	13	4	no	—
126	C₂×C₂²⋊C₄	—	13	2	no	—
127	C₂×C₂²⋊C₄	—	13	4	no	—
128	C₂×C₄²	—	13	2	no	—
129	C₂×C₄²	C₂×D₁₃	1	2	yes	—
130	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₄²	C₂×D₁₃	1	2	yes	—
132	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
133	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
134	#95	—	13	4	no	—
135	#95	—	13	4	no	—
136	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₁₃	C₄×#41	1	2	yes	—
138	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
139	C₂×C₁₃	#529	1	2	yes	—
140	C₂×C₁₃	#532	1	1	yes	characteristic
141	D₁₃	—	4	4	no	—
142	C₂⁴	—	13	2	no	—
143	C₂×Q₈	—	13	4	no	—
144	C₂×Q₈	—	26	2	no	—
145	C₂×Q₈	—	26	2	no	—
146	C₂×D₄	—	13	4	no	—
147	C₂×D₄	—	26	2	no	—
148	C₂×D₄	—	26	2	no	—
149	C₂²×C₄	—	13	2	no	—
150	C₂²×C₄	—	13	2	no	—
151	C₂²×C₄	—	13	2	no	—
152	C₂²×C₄	—	26	4	no	—
153	C₂²×C₄	—	26	8	no	—
154	C₂²×C₄	C₂²×D₁₃	1	2	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	—	26	2	no	—
160	C₄⋊C₄	—	26	4	no	—
161	C₄⋊C₄	—	26	2	no	—
162	C₂²⋊C₄	—	13	16	no	—
163	C₂²⋊C₄	—	26	4	no	—
164	C₂²⋊C₄	—	26	8	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	—	26	4	no	—
169	C₄²	C₄×D₁₃	1	4	yes	complemented
170	C₄²	D₅₂	1	4	yes	—
171	C₁₃	C₄×#124	1	1	yes	characteristic, complemented, Sylow
172	C₂³	—	13	4	no	—
173	C₂³	—	26	2	no	—
174	C₂³	—	26	4	no	—
175	C₂³	—	26	4	no	—
176	C₂³	—	26	2	no	—
177	C₂³	C₂³×D₁₃	1	1	yes	characteristic
178	Q₈	—	26	8	no	—
179	D₄	—	26	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	13	4	no	—
186	C₂×C₄	—	26	2	no	—
187	C₂×C₄	—	26	2	no	—
188	C₂×C₄	—	26	2	no	—
189	C₂×C₄	—	26	2	no	—
190	C₂×C₄	—	26	4	no	—
191	C₂×C₄	—	26	4	no	—
192	C₂×C₄	—	52	8	no	—
193	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
194	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
195	C₂×C₄	C₂×D₅₂	1	2	yes	—
196	C₂×C₄	C₂×D₅₂	1	2	yes	—
197	C₂×C₄	C₂×D₅₂	1	2	yes	—
198	C₂×C₄	#6127	1	4	yes	—
199	C₂×C₄	#6127	1	2	yes	—
200	C₂×C₄	#6127	1	4	yes	—
201	C₂²	—	26	4	no	—
202	C₂²	—	26	8	no	—
203	C₂²	—	52	2	no	—
204	C₂²	—	52	2	no	—
205	C₂²	—	52	4	no	—
206	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
207	C₂²	C₂²×D₅₂	1	1	yes	characteristic
208	C₂²	C₂×#6127	1	1	yes	characteristic
209	C₂²	C₂×#6127	1	2	yes	—
210	C₂²	C₂×#6127	1	2	yes	—
211	C₄	—	2	4	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	4	no	—
214	C₄	—	26	4	no	—
215	C₄	—	52	2	no	—
216	C₄	—	52	2	no	—
217	C₄	C₄×D₅₂	1	4	yes	—
218	C₄	#88129	1	4	yes	direct product factor
219	C₂	—	52	4	no	—
220	C₂	C₂×C₄×D₅₂	1	1	yes	characteristic
221	C₂	C₄×#6127	1	2	yes	—
222	C₂	C₂×#88129	1	1	yes	characteristic
223	C₂	#11719182	1	1	yes	characteristic
224	C₂	#11719184	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Frattini	C₂³	C₂³×D₁₃
Fitting	C₄³×C₁₃	C₂
socle, cosocle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₄³	D₁₃
hypocenter	C₁₃	C₄×#124
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₂₆	C₂	semidirect
4. C₄³×C₁₃	C₂	semidirect
5. #11718601	C₂	semidirect
7. C₄×#6103	C₂	semidirect
8. #11718597	C₂	semidirect
9. #11718590	C₂	semidirect
25. #88129	C₄	direct (N × H)
46. C₂×Dic₂₆	C₂×C₄	semidirect
50. C₄²×C₁₃	C₂×C₄	semidirect
54. #6103	C₂×C₄	semidirect
97. C₄³	D₁₃	semidirect
112. C₄×C₁₃	#124	semidirect
169. C₄²	C₄×D₁₃	semidirect
171. C₁₃	C₄×#124	semidirect
218. C₄	#88129	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×#88129	2
C₂×C₄×D₅₂	2
C₂×C₄×Dic₂₆	2
C₄³×C₁₃	2
#11718601	2
#11718600	2
C₄×#6103	2
#11718597	2
#11718590	2
C₄×#124	13

Subgroups — C4×#88129 #13117380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88129 #13117380