Subgroups — C₂×#13131197 #16807045

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131197	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#6092	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#6092	C₂	1	2	yes	maximal
4	C₂×#11718231	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₂²	1	1	yes	characteristic
9	C₂²×#6092	C₂²	1	4	yes	—
10	C₂×C₄×#1055	C₄	1	4	yes	—
11	C₂×C₄×#1055	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₄	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	4	yes	—
16	C₂²⋊#6092	C₂²	1	8	yes	—
17	#11718231	C₂²	1	8	yes	complemented
18	C₄⋊#6092	C₂²	1	8	yes	complemented
19	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₈	1	4	yes	—
24	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	—	2	4	no	—
29	C₂²×#1055	—	2	4	no	—
30	C₂²×#1055	C₂×C₄	1	4	yes	—
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	2	yes	—
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂³	1	2	yes	—
36	C₂²×#1055	C₂³	1	1	yes	characteristic
37	C₂×#6092	—	2	8	no	—
38	C₂×#6092	C₂³	1	8	yes	—
39	C₄×#1055	C₂×C₄	1	4	yes	—
40	C₄×#1055	C₂×C₄	1	2	yes	—
41	C₄×#1055	C₂×C₄	1	4	yes	—
42	C₄×#1055	C₂×C₄	1	2	yes	—
43	C₄×#1055	C₂³	1	2	yes	—
44	C₄×#1055	C₂³	1	2	yes	—
45	C₂×#2163	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	4	no	—
48	C₂²×C₄×C₁₃	—	2	4	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂×C₈	1	4	yes	—
62	C₄²×C₁₃	C₂×C₈	1	4	yes	—
63	C₄²×C₁₃	C₂×C₈	1	2	yes	—
64	C₄²×C₁₃	C₂×C₈	1	2	yes	—
65	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
66	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂×#1055	—	2	2	no	—
68	C₂×#1055	—	2	2	no	—
69	C₂×#1055	—	2	4	no	—
70	C₂×#1055	—	2	8	no	—
71	C₂×#1055	—	2	4	no	—
72	C₂×#1055	M₁₆	1	4	yes	—
73	C₂×#1055	C₂²×C₄	1	2	yes	—
74	C₂×#1055	C₂²×C₄	1	2	yes	—
75	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
76	C₂×#1055	C₂²×C₄	1	2	yes	—
77	C₂×#1055	#41	1	4	yes	—
78	C₂×#1055	C₂⁴	1	1	yes	characteristic
79	#6092	—	2	8	no	—
80	C₂²×C₄×C₈	—	13	1	no	—
81	#2163	—	13	8	no	—
82	C₂×C₄⋊C₁₆	—	13	2	no	—
83	C₂×C₂²⋊C₁₆	—	13	2	no	—
84	C₂×#346	—	13	2	no	—
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	M₃₂	1	4	yes	—
88	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
89	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	8	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	2	8	no	—
99	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
100	C₂×C₄×C₁₃	#117	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	#1055	—	2	2	no	—
110	#1055	—	2	2	no	—
111	#1055	#117	1	2	yes	—
112	#1055	C₂×M₁₆	1	4	yes	—
113	#1055	C₂×#41	1	2	yes	—
114	C₂³×C₈	—	13	1	no	—
115	C₂³×C₈	—	13	1	no	—
116	C₂²×C₄²	#261	1	1	yes	characteristic
117	C₂²×C₁₆	—	13	4	no	—
118	C₂×C₄×C₈	—	13	1	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₂×C₄×C₈	—	13	4	no	—
121	C₂×C₄×C₈	—	13	4	no	—
122	C₂×C₄×C₈	—	13	1	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₄⋊C₁₆	—	13	8	no	—
125	C₂²⋊C₁₆	—	13	8	no	—
126	#346	—	13	8	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	8	no	—
130	C₂²×C₁₃	#431	1	2	yes	—
131	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
132	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
133	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
134	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	#431	1	2	yes	—
140	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
141	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
142	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
143	C₂³×C₄	#1057	1	1	yes	characteristic
144	C₂³×C₄	#1057	1	1	yes	characteristic
145	C₂³×C₄	C₂×#261	1	1	yes	characteristic
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	1	no	—
148	C₂²×C₈	—	13	2	no	—
149	C₂²×C₈	—	13	4	no	—
150	C₂²×C₈	—	13	2	no	—
151	C₂²×C₈	—	13	1	no	—
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	—	26	4	no	—
154	C₂²×C₈	—	26	4	no	—
155	C₂×C₄²	#1057	1	1	yes	characteristic
156	C₂×C₄²	#1057	1	4	yes	—
157	C₂×C₄²	#1057	1	1	yes	characteristic
158	C₂×C₄²	C₂×#261	1	4	yes	—
159	C₂×C₄²	C₂×#261	1	1	yes	characteristic
160	C₂×C₄²	C₂×#261	1	1	yes	characteristic
161	C₂×C₁₆	—	13	8	no	—
162	C₂×C₁₆	—	26	8	no	—
163	C₄×C₈	—	13	2	no	—
164	C₄×C₈	—	13	2	no	—
165	C₄×C₈	—	13	2	no	—
166	C₄×C₈	—	13	2	no	—
167	C₄×C₈	—	13	4	no	—
168	C₄×C₈	—	13	4	no	—
169	C₂×C₁₃	—	2	4	no	—
170	C₂×C₁₃	#2163	1	4	yes	complemented
171	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
172	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
173	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
174	C₂⁴	C₂×#1057	1	1	yes	characteristic
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
180	C₂²×C₄	C₂×#1057	1	2	yes	—
181	C₂²×C₄	C₂×#1057	1	4	yes	—
182	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
183	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
184	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
185	C₂²×C₄	C₂×#1057	1	4	yes	—
186	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
187	C₂²×C₄	C₂²×#261	1	2	yes	—
188	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
189	C₂×C₈	—	13	4	no	—
190	C₂×C₈	—	13	4	no	—
191	C₂×C₈	—	13	1	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	13	1	no	—
194	C₂×C₈	—	13	2	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	26	8	no	—
197	C₂×C₈	—	26	4	no	—
198	C₂×C₈	—	26	4	no	—
199	C₂×C₈	—	26	2	no	—
200	C₂×C₈	—	26	2	no	—
201	C₄²	C₂×#1057	1	4	yes	—
202	C₄²	C₂×#1057	1	4	yes	—
203	C₄²	C₂×#1057	1	2	yes	—
204	C₄²	C₂×#1057	1	2	yes	—
205	C₄²	C₂²×#261	1	2	yes	—
206	C₄²	C₂²×#261	1	2	yes	—
207	C₁₆	—	26	8	no	—
208	C₁₃	C₂×#2163	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	#88106	1	4	yes	—
212	C₂³	C₂²×#1057	1	2	yes	—
213	C₂³	C₂²×#1057	1	1	yes	characteristic
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	#88106	1	4	yes	—
224	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
225	C₂×C₄	#88237	1	2	yes	—
226	C₂×C₄	C₂²×#1057	1	2	yes	—
227	C₂×C₄	C₂²×#1057	1	2	yes	—
228	C₂×C₄	C₂²×#1057	1	2	yes	—
229	C₂×C₄	C₂²×#1057	1	2	yes	—
230	C₂×C₄	C₂²×#1057	1	2	yes	—
231	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
232	C₂×C₄	C₂³×#261	1	1	yes	characteristic
233	C₈	—	13	4	no	—
234	C₈	—	13	2	no	—
235	C₈	—	13	2	no	—
236	C₈	—	26	2	no	—
237	C₈	—	26	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₂×#88106	1	4	yes	—
242	C₂²	C₂×#88106	1	2	yes	—
243	C₂²	C₂×#88106	1	2	yes	—
244	C₂²	#11719119	1	2	yes	—
245	C₂²	C₂³×#1057	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	C₂×#88106	1	4	yes	—
251	C₄	#11719119	1	2	yes	—
252	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
253	C₄	C₂×#88237	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#13131197	1	4	yes	direct product factor
256	C₂	C₂×#11719119	1	1	yes	characteristic
257	C₂	C₂²×#88106	1	1	yes	characteristic
258	C₂	C₂²×#88106	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₂×#2163
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #13131197	C₂	direct (N × H)
2. C₂×C₄⋊#6092	C₂	semidirect
4. C₂×#11718231	C₂	semidirect
17. #11718231	C₂²	semidirect
18. C₄⋊#6092	C₂²	semidirect
170. C₂×C₁₃	#2163	semidirect
208. C₁₃	C₂×#2163	semidirect
255. C₂	#13131197	direct (N × H)

Maximal subgroups

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

subgroup	index
#13131197	2
C₂×C₄⋊#6092	2
C₂×C₂²⋊#6092	2
C₂×#11718231	2
C₂²×C₄×#1055	2
C₂×#2163	13

Subgroups — C2×#13131197 #16807045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131197 #16807045