Subgroups — C₂×#13129772 #16786768

Order 3328.

466 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#6092	C₂	1	2	yes	maximal
2	#13129772	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×#1055	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂²×#6092	—	2	2	no	—
8	C₂×C₄⋊#1055	—	2	2	no	—
9	C₂×C₄⋊#1055	—	2	2	no	—
10	C₂×C₄⋊#1055	C₄	1	2	yes	—
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	C₂²⋊#6092	—	2	2	no	—
13	C₂²⋊#6092	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	—	2	1	no	—
18	C₂×C₄²×C₁₃	—	2	1	no	—
19	C₂×C₄²×C₁₃	—	2	1	no	—
20	C₂×C₄²×C₁₃	—	2	1	no	—
21	C₂×C₄²×C₁₃	C₈	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×#1055	—	2	1	no	—
24	C₂²×#1055	—	2	1	no	—
25	C₂²×#1055	—	4	2	no	—
26	C₂²×#1055	—	4	2	no	—
27	C₂²×#1055	C₂×C₄	1	2	yes	—
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	D₄	1	2	yes	—
30	C₂²×#1055	D₄	1	2	yes	—
31	C₂²×#1055	C₂³	1	1	yes	characteristic
32	C₂×#6092	—	2	4	no	—
33	C₂×#6092	—	4	4	no	—
34	C₄⋊#1055	—	2	4	no	—
35	C₄⋊#1055	—	2	4	no	—
36	C₂×#1322	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	—	2	1	no	—
39	C₂²×C₄×C₁₃	—	2	1	no	—
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	C₁₆	1	4	yes	complemented
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	—	2	2	no	—
56	C₄²×C₁₃	—	2	2	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₂×#1055	—	2	4	no	—
60	C₂×#1055	—	2	2	no	—
61	C₂×#1055	—	4	1	no	—
62	C₂×#1055	—	4	2	no	—
63	C₂×#1055	—	4	1	no	—
64	C₂×#1055	—	4	2	no	—
65	C₂×#1055	—	4	2	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	C₂²⋊C₄	1	2	yes	—
68	C₂×#1055	C₂×D₄	1	2	yes	—
69	#6092	—	4	4	no	—
70	C₂²×C₄⋊C₈	—	13	1	no	—
71	C₂×C₂²⋊C₁₆	—	13	2	no	—
72	#1322	—	13	4	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
76	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
77	C₂³×C₁₃	C₂×C₁₆	1	2	yes	—
78	C₂³×C₁₃	M₃₂	1	2	yes	—
79	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	2	4	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#99	1	2	yes	—
98	C₂×C₄×C₁₃	#101	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
102	#1055	—	2	2	no	—
103	#1055	—	2	2	no	—
104	#1055	—	4	2	no	—
105	C₂³×C₈	—	13	1	no	—
106	C₂³×C₈	—	13	1	no	—
107	C₂²×C₄²	#261	1	1	yes	characteristic
108	C₂²×C₁₆	—	26	2	no	—
109	C₂×C₄⋊C₈	—	13	2	no	—
110	C₂×C₄⋊C₈	—	13	2	no	—
111	C₂×C₄⋊C₈	—	26	2	no	—
112	C₂×C₄⋊C₈	—	26	2	no	—
113	C₂²⋊C₁₆	—	13	4	no	—
114	C₂²⋊C₁₆	—	26	2	no	—
115	C₂²×C₁₃	—	2	4	no	—
116	C₂²×C₁₃	—	2	4	no	—
117	C₂²×C₁₃	—	2	1	no	—
118	C₂²×C₁₃	—	2	1	no	—
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	#324	1	2	yes	—
122	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
123	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
124	C₂²×C₁₃	#349	1	2	yes	—
125	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
128	C₄×C₁₃	—	2	4	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	4	2	no	—
132	C₄×C₁₃	#324	1	2	yes	—
133	C₄×C₁₃	C₂×#99	1	1	yes	characteristic
134	C₄×C₁₃	C₂×#101	1	1	yes	characteristic
135	C₂³×C₄	#1057	1	1	yes	characteristic
136	C₂³×C₄	#1057	1	1	yes	characteristic
137	C₂³×C₄	C₂×#261	1	1	yes	characteristic
138	C₂²×C₈	—	13	2	no	—
139	C₂²×C₈	—	13	2	no	—
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	—	13	1	no	—
142	C₂²×C₈	—	13	2	no	—
143	C₂²×C₈	—	26	1	no	—
144	C₂²×C₈	—	26	1	no	—
145	C₂²×C₈	—	52	2	no	—
146	C₂²×C₈	—	52	2	no	—
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	#1057	1	2	yes	—
152	C₂×C₄²	C₂×#261	1	2	yes	—
153	C₂×C₁₆	—	26	4	no	—
154	C₂×C₁₆	—	52	4	no	—
155	C₄⋊C₈	—	26	4	no	—
156	C₄⋊C₈	—	26	4	no	—
157	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	—	2	1	no	—
159	C₂×C₁₃	—	2	1	no	—
160	C₂×C₁₃	#1322	1	4	yes	complemented
161	C₂×C₁₃	C₂×#324	1	1	yes	characteristic
162	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
163	C₂×C₁₃	C₂×#349	1	1	yes	characteristic
164	C₂⁴	C₂×#1057	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	#6092	1	4	yes	complemented
174	C₂²×C₄	C₂×#1057	1	2	yes	—
175	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
176	C₂²×C₄	C₂×#1057	1	2	yes	—
177	C₂²×C₄	#6122	1	1	yes	characteristic
178	C₂²×C₄	#6122	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
181	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
182	C₂×C₈	—	13	2	no	—
183	C₂×C₈	—	13	2	no	—
184	C₂×C₈	—	26	4	no	—
185	C₂×C₈	—	26	2	no	—
186	C₂×C₈	—	52	1	no	—
187	C₂×C₈	—	52	2	no	—
188	C₂×C₈	—	52	1	no	—
189	C₂×C₈	—	52	2	no	—
190	C₂×C₈	—	52	2	no	—
191	C₂×C₈	—	52	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₁₆	—	52	4	no	—
197	C₁₃	C₂×#1322	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	C₂×#6092	1	2	yes	—
201	C₂³	#88106	1	2	yes	—
202	C₂³	C₂²⋊#1057	1	1	yes	characteristic
203	C₂³	C₂²⋊#1057	1	1	yes	characteristic
204	C₂³	C₂²×#1057	1	1	yes	characteristic
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₂×#6092	1	4	yes	complemented
221	C₂×C₄	#88107	1	2	yes	—
222	C₂×C₄	#88113	1	2	yes	—
223	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
224	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
225	C₂×C₄	C₂×#6122	1	1	yes	characteristic
226	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
227	C₈	—	26	2	no	—
228	C₈	—	26	2	no	—
229	C₈	—	52	2	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	C₂²⋊#6092	1	2	yes	—
237	C₂²	C₂²⋊#6092	1	2	yes	—
238	C₂²	#11718284	1	2	yes	—
239	C₂²	#11718285	1	2	yes	—
240	C₂²	C₂²×#6092	1	1	yes	characteristic
241	C₂²	C₂×#88106	1	1	yes	characteristic
242	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	#11718285	1	2	yes	—
248	C₄	C₂×#88107	1	1	yes	characteristic
249	C₄	C₂×#88113	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	#13129772	1	4	yes	direct product factor
254	C₂	C₂×#11718285	1	1	yes	characteristic
255	C₂	C₂×C₂²⋊#6092	1	1	yes	characteristic
256	C₂	C₂×#11718284	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₂×#1322
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #13129772	C₂	direct (N × H)
46. C₂²×C₄×C₁₃	C₁₆	semidirect
99. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
160. C₂×C₁₃	#1322	semidirect
173. C₂²×C₄	#6092	semidirect
197. C₁₃	C₂×#1322	semidirect
220. C₂×C₄	C₂×#6092	semidirect
253. C₂	#13129772	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×C₂²⋊#6092	2
#13129772	2
C₂²×C₄⋊#1055	2
C₂×#1322	13

Subgroups — C2×#13129772 #16786768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13129772 #16786768