Subgroups — C₂×#13131063 #16806765

Order 3328.

1406 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal, complemented
2	#13131063	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₂²⋊#6092	C₂	1	4	yes	maximal, complemented
4	C₂⁴×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×#1055	—	2	8	no	—
7	C₂³×#1055	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₄	1	4	yes	—
9	C₂³×#1055	C₄	1	2	yes	—
10	C₂³×#1055	C₂²	1	2	yes	—
11	C₂³×#1055	C₂²	1	4	yes	—
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₂×#88106	—	2	8	no	—
14	C₂×#88106	C₂²	1	8	yes	complemented
15	C₂²×#6092	C₂²	1	4	yes	complemented
16	C₂²⋊#6092	C₂²	1	16	yes	complemented
17	C₂⁵×C₁₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	—	2	4	no	—
19	C₂³×C₄×C₁₃	—	2	4	no	—
20	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₈	1	4	yes	—
22	C₂³×C₄×C₁₃	C₈	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	—	2	4	no	—
27	C₂²×#1055	—	2	16	no	—
28	C₂²×#1055	—	2	8	no	—
29	C₂²×#1055	—	2	8	no	—
30	C₂²×#1055	—	2	16	no	—
31	C₂²×#1055	C₂×C₄	1	4	yes	—
32	C₂²×#1055	C₂×C₄	1	4	yes	—
33	C₂²×#1055	C₂×C₄	1	8	yes	—
34	C₂²×#1055	C₂×C₄	1	2	yes	—
35	C₂²×#1055	C₂×C₄	1	2	yes	—
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂²×#1055	D₄	1	8	yes	—
38	C₂²×#1055	C₂³	1	4	yes	—
39	C₂²×#1055	C₂³	1	1	yes	characteristic
40	C₂²×#1055	C₂³	1	2	yes	—
41	#88106	—	2	16	no	—
42	C₂×#6092	—	2	8	no	—
43	C₂×#6092	C₂³	1	8	yes	complemented
44	C₂×#2112	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	—	2	4	no	—
46	C₂⁴×C₁₃	—	2	8	no	—
47	C₂⁴×C₁₃	C₂×C₈	1	4	yes	—
48	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂⁴×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	2	16	no	—
52	C₂²×C₄×C₁₃	—	2	8	no	—
53	C₂²×C₄×C₁₃	—	2	8	no	—
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	2	8	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
61	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×#1055	—	2	8	no	—
69	C₂×#1055	—	2	4	no	—
70	C₂×#1055	—	2	8	no	—
71	C₂×#1055	—	2	16	no	—
72	C₂×#1055	—	2	8	no	—
73	C₂×#1055	C₂²⋊C₄	1	8	yes	—
74	C₂×#1055	C₂²⋊C₄	1	4	yes	—
75	C₂×#1055	C₂²×C₄	1	4	yes	—
76	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
77	C₂×#1055	C₂²×C₄	1	2	yes	—
78	C₂×#1055	C₂×D₄	1	4	yes	—
79	C₂×#1055	C₂×D₄	1	8	yes	—
80	C₂×#1055	C₂⁴	1	1	yes	characteristic
81	#6092	—	2	8	no	—
82	C₂⁴×C₈	—	13	1	no	—
83	C₂²×M₃₂	—	13	2	no	—
84	#2112	—	13	8	no	—
85	C₂×C₂²⋊C₁₆	—	13	4	no	—
86	C₂³×C₁₃	—	2	32	no	—
87	C₂³×C₁₃	—	2	8	no	—
88	C₂³×C₁₃	—	2	4	no	—
89	C₂³×C₁₃	—	2	4	no	—
90	C₂³×C₁₃	—	2	4	no	—
91	C₂³×C₁₃	—	2	16	no	—
92	C₂³×C₁₃	C₂²⋊C₈	1	4	yes	—
93	C₂³×C₁₃	M₃₂	1	8	yes	complemented
94	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
95	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
96	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	—	2	8	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	16	no	—
102	C₂×C₄×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	C₂²⋊C₈	1	8	yes	—
104	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	#1055	—	2	4	no	—
114	#1055	C₂×C₂²⋊C₄	1	2	yes	—
115	#1055	C₂×C₂²⋊C₄	1	4	yes	—
116	#1055	C₂²×D₄	1	2	yes	—
117	C₂⁴×C₄	#261	1	1	yes	characteristic
118	C₂³×C₈	—	13	1	no	—
119	C₂³×C₈	—	13	4	no	—
120	C₂³×C₈	—	13	2	no	—
121	C₂³×C₈	—	13	1	no	—
122	C₂³×C₈	—	13	2	no	—
123	C₂³×C₈	—	13	4	no	—
124	C₂³×C₈	—	26	8	no	—
125	C₂×M₃₂	—	13	8	no	—
126	C₂×M₃₂	—	26	8	no	—
127	C₂²×C₁₆	—	13	4	no	—
128	C₂²⋊C₁₆	—	13	16	no	—
129	C₂²×C₁₃	—	2	32	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	—	2	8	no	—
132	C₂²×C₁₃	—	2	16	no	—
133	C₂²×C₁₃	—	2	4	no	—
134	C₂²×C₁₃	—	2	4	no	—
135	C₂²×C₁₃	C₂×C₂²⋊C₈	1	4	yes	—
136	C₂²×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
137	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
138	C₂²×C₁₃	C₂×M₃₂	1	8	yes	complemented
139	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	C₂×C₂²⋊C₈	1	4	yes	—
142	C₄×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
143	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
145	C₂⁵	#1057	1	1	yes	characteristic
146	C₂³×C₄	—	2	4	no	—
147	C₂³×C₄	—	2	4	no	—
148	C₂³×C₄	#1057	1	1	yes	characteristic
149	C₂³×C₄	#1057	1	4	yes	—
150	C₂³×C₄	#1057	1	2	yes	—
151	C₂³×C₄	C₂×#261	1	4	yes	—
152	C₂³×C₄	C₂×#261	1	2	yes	—
153	C₂³×C₄	C₂×#261	1	1	yes	characteristic
154	C₂²×C₈	—	13	8	no	—
155	C₂²×C₈	—	13	2	no	—
156	C₂²×C₈	—	13	2	no	—
157	C₂²×C₈	—	13	1	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	4	no	—
160	C₂²×C₈	—	13	4	no	—
161	C₂²×C₈	—	13	8	no	—
162	C₂²×C₈	—	13	4	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	26	8	no	—
165	C₂²×C₈	—	26	16	no	—
166	C₂²×C₈	—	26	16	no	—
167	C₂²×C₈	—	26	8	no	—
168	C₂²×C₈	—	26	4	no	—
169	M₃₂	—	26	16	no	—
170	C₂×C₁₆	—	13	8	no	—
171	C₂×C₁₆	—	26	8	no	—
172	C₂×C₁₃	—	2	4	no	—
173	C₂×C₁₃	—	2	8	no	—
174	C₂×C₁₃	#2112	1	4	yes	complemented
175	C₂×C₁₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
176	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
177	C₂⁴	—	2	8	no	—
178	C₂⁴	—	2	4	no	—
179	C₂⁴	C₂×#1057	1	4	yes	—
180	C₂⁴	C₂×#1057	1	2	yes	—
181	C₂⁴	C₂×#1057	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	16	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	C₂×#1057	1	4	yes	—
190	C₂²×C₄	C₂×#1057	1	2	yes	—
191	C₂²×C₄	C₂×#1057	1	8	yes	—
192	C₂²×C₄	C₂×#1057	1	4	yes	—
193	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
194	C₂²×C₄	C₂×#1057	1	2	yes	—
195	C₂²×C₄	#6122	1	4	yes	—
196	C₂²×C₄	C₂²⋊#261	1	4	yes	—
197	C₂²×C₄	C₂²×#261	1	4	yes	—
198	C₂²×C₄	C₂²×#261	1	2	yes	—
199	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
200	C₂×C₈	—	13	4	no	—
201	C₂×C₈	—	13	8	no	—
202	C₂×C₈	—	13	8	no	—
203	C₂×C₈	—	13	4	no	—
204	C₂×C₈	—	13	1	no	—
205	C₂×C₈	—	13	1	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	13	4	no	—
208	C₂×C₈	—	26	8	no	—
209	C₂×C₈	—	26	8	no	—
210	C₂×C₈	—	26	16	no	—
211	C₂×C₈	—	26	8	no	—
212	C₂×C₈	—	26	4	no	—
213	C₁₆	—	26	8	no	—
214	C₁₃	C₂×#2112	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	2	32	no	—
216	C₂³	—	2	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	16	no	—
221	C₂³	#88106	1	8	yes	complemented
222	C₂³	C₂²⋊#1057	1	4	yes	—
223	C₂³	C₂²×#1057	1	2	yes	—
224	C₂³	C₂²×#1057	1	4	yes	—
225	C₂³	C₂²×#1057	1	1	yes	characteristic
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	C₂²⋊#1057	1	8	yes	—
233	C₂×C₄	C₂²⋊#1057	1	4	yes	—
234	C₂×C₄	C₂²×#1057	1	4	yes	—
235	C₂×C₄	C₂²×#1057	1	2	yes	—
236	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
237	C₂×C₄	C₂×#6122	1	4	yes	—
238	C₂×C₄	C₂×#6122	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊#261	1	4	yes	—
240	C₂×C₄	C₂×C₂²⋊#261	1	2	yes	—
241	C₂×C₄	C₂³×#261	1	1	yes	characteristic
242	C₈	—	13	2	no	—
243	C₈	—	13	2	no	—
244	C₈	—	13	4	no	—
245	C₈	—	26	4	no	—
246	C₂²	—	2	32	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	16	no	—
250	C₂²	—	2	8	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₂×#88106	1	4	yes	—
253	C₂²	C₂×#88106	1	8	yes	complemented
254	C₂²	C₂×C₂²⋊#1057	1	4	yes	—
255	C₂²	C₂×C₂²⋊#1057	1	2	yes	—
256	C₂²	C₂³×#1057	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	C₂×C₂²⋊#1057	1	4	yes	—
259	C₄	C₂×C₂²⋊#1057	1	2	yes	—
260	C₄	C₂²×#6122	1	1	yes	characteristic
261	C₄	C₂²×C₂²⋊#261	1	1	yes	characteristic
262	C₂	—	2	4	no	—
263	C₂	—	2	8	no	—
264	C₂	#13131063	1	4	yes	direct product factor
265	C₂	C₂²×C₂²⋊#1057	1	1	yes	characteristic
266	C₂	C₂²×#88106	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂⁴×C₄×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁵×C₁₃	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂⁴×C₄	#261
hypocenter	C₁₃	C₂×#2112
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#88106	C₂	semidirect
2. #13131063	C₂	direct (N × H)
3. C₂×C₂²⋊#6092	C₂	semidirect
14. C₂×#88106	C₂²	semidirect
15. C₂²×#6092	C₂²	semidirect
16. C₂²⋊#6092	C₂²	semidirect
43. C₂×#6092	C₂³	semidirect
93. C₂³×C₁₃	M₃₂	semidirect
138. C₂²×C₁₃	C₂×M₃₂	semidirect
174. C₂×C₁₃	#2112	semidirect
214. C₁₃	C₂×#2112	semidirect
221. C₂³	#88106	semidirect
253. C₂²	C₂×#88106	semidirect
264. C₂	#13131063	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88106	2
#13131063	2
C₂×C₂²⋊#6092	2
C₂⁴×#1055	2
C₂×#2112	13

Subgroups — C2×#13131063 #16806765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131063 #16806765