Subgroups — C₂×#13129880 #16789063

Order 3328.

670 conjugacy classes of subgroups.

The subgroup lattice has 264 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
2	#13129880	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	C₂×#88106	—	2	4	no	—
8	C₂×#88106	—	2	2	no	—
9	C₂×#88106	C₄	1	8	yes	complemented
10	C₂×#88106	C₂²	1	4	yes	—
11	C₂²×#6092	—	2	2	no	—
12	C₂×C₄×#1055	—	2	4	no	—
13	C₂×C₄×#1055	C₄	1	2	yes	—
14	C₂×C₄×#1055	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₃	—	2	1	no	—
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	—	2	2	no	—
18	C₂×C₄²×C₁₃	—	2	1	no	—
19	C₂×C₄²×C₁₃	C₈	1	4	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
21	C₂²×#1055	—	2	2	no	—
22	C₂²×#1055	—	2	2	no	—
23	C₂²×#1055	C₈	1	2	yes	—
24	C₂²×#1055	C₈	1	2	yes	—
25	C₂²×#1055	C₂×C₄	1	2	yes	—
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	4	yes	—
28	C₂²×#1055	D₄	1	1	yes	characteristic
29	C₂²×#1055	D₄	1	1	yes	characteristic
30	C₂²×#1055	D₄	1	4	yes	—
31	C₂²×#1055	Q₈	1	1	yes	characteristic
32	C₂²×#1055	Q₈	1	1	yes	characteristic
33	C₂²×#1055	C₂³	1	1	yes	characteristic
34	#88106	—	2	8	no	—
35	#88106	—	2	4	no	—
36	#88106	C₂×C₄	1	8	yes	complemented
37	C₂×#6092	—	2	8	no	—
38	C₂×#6092	—	2	4	no	—
39	C₄×#1055	—	2	4	no	—
40	C₄×#1055	—	2	4	no	—
41	C₂×#1376	—	13	1	no	maximal, Sylow
42	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	—	2	8	no	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
54	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₃	—	2	4	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	C₂×C₈	1	4	yes	—
59	C₂×#1055	—	2	4	no	—
60	C₂×#1055	—	2	4	no	—
61	C₂×#1055	—	2	2	no	—
62	C₂×#1055	—	2	2	no	—
63	C₂×#1055	C₄²	1	2	yes	—
64	C₂×#1055	C₂²⋊C₄	1	2	yes	—
65	C₂×#1055	C₂²⋊C₄	1	4	yes	—
66	C₂×#1055	C₂²⋊C₄	1	4	yes	—
67	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
68	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
69	C₂×#1055	C₄⋊C₄	1	4	yes	—
70	C₂×#1055	C₄⋊C₄	1	4	yes	—
71	C₂×#1055	C₂×C₈	1	4	yes	—
72	C₂×#1055	C₂²×C₄	1	2	yes	—
73	C₂×#1055	C₂×D₄	1	1	yes	characteristic
74	C₂×#1055	C₂×D₄	1	2	yes	—
75	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
76	#6092	—	2	8	no	—
77	C₂²×M₃₂	—	13	2	no	—
78	C₂²×C₄×C₈	—	13	1	no	—
79	#1376	—	13	4	no	—
80	C₂³×C₁₃	—	2	4	no	—
81	C₂³×C₁₃	#97	1	2	yes	—
82	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
83	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
84	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
85	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	16	no	—
87	C₂×C₄×C₁₃	—	2	8	no	—
88	C₂×C₄×C₁₃	—	2	4	no	—
89	C₂×C₄×C₁₃	#95	1	2	yes	—
90	C₂×C₄×C₁₃	#95	1	2	yes	—
91	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
93	C₂×C₄×C₁₃	#97	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	#1055	—	2	4	no	—
108	#1055	#95	1	2	yes	—
109	#1055	#95	1	2	yes	—
110	#1055	C₂×C₂²⋊C₄	1	2	yes	—
111	#1055	C₂×C₄⋊C₄	1	2	yes	—
112	C₂³×C₈	—	13	1	no	—
113	C₂³×C₈	—	13	1	no	—
114	C₂²×C₄²	#261	1	1	yes	characteristic
115	C₂×M₃₂	—	13	4	no	—
116	C₂×M₃₂	—	13	8	no	—
117	C₂×M₃₂	—	26	2	no	—
118	C₂×M₃₂	—	26	4	no	—
119	C₂²×C₁₆	—	26	2	no	—
120	C₂×C₄×C₈	—	13	2	no	—
121	C₂×C₄×C₈	—	13	2	no	—
122	C₂×C₄×C₈	—	26	4	no	—
123	C₂²×C₁₃	—	2	4	no	—
124	C₂²×C₁₃	—	2	4	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	#336	1	2	yes	—
128	C₂²×C₁₃	#336	1	2	yes	—
129	C₂²×C₁₃	#364	1	4	yes	—
130	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
131	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₄×C₁₃	—	2	8	no	—
134	C₄×C₁₃	#336	1	2	yes	—
135	C₄×C₁₃	#336	1	2	yes	—
136	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
137	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₂³×C₄	—	2	1	no	—
141	C₂³×C₄	C₂×#261	1	1	yes	characteristic
142	C₂²×C₈	—	13	4	no	—
143	C₂²×C₈	—	13	1	no	—
144	C₂²×C₈	—	13	1	no	—
145	C₂²×C₈	—	13	1	no	—
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	4	no	—
148	C₂²×C₈	—	13	1	no	—
149	C₂²×C₈	—	13	1	no	—
150	C₂²×C₈	—	13	2	no	—
151	C₂²×C₈	—	13	2	no	—
152	C₂²×C₈	—	13	2	no	—
153	C₂²×C₈	—	26	2	no	—
154	C₂²×C₈	—	26	2	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	#1057	1	4	yes	—
158	C₂×C₄²	C₂×#261	1	2	yes	—
159	M₃₂	—	13	8	no	—
160	M₃₂	—	26	8	no	—
161	M₃₂	—	26	4	no	—
162	C₂×C₁₆	—	26	8	no	—
163	C₂×C₁₆	—	26	4	no	—
164	C₄×C₈	—	26	4	no	—
165	C₄×C₈	—	26	4	no	—
166	C₂×C₁₃	—	2	1	no	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	2	1	no	—
169	C₂×C₁₃	#1376	1	4	yes	complemented
170	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
171	C₂×C₁₃	C₂×#364	1	2	yes	—
172	C₂⁴	#6122	1	1	yes	characteristic
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
177	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
178	C₂²×C₄	#6118	1	2	yes	—
179	C₂²×C₄	C₄×#261	1	2	yes	—
180	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
182	C₂²×C₄	C₂×#1057	1	2	yes	—
183	C₂²×C₄	#6122	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊#261	1	2	yes	—
185	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
186	C₂×C₈	—	13	4	no	—
187	C₂×C₈	—	13	4	no	—
188	C₂×C₈	—	13	4	no	—
189	C₂×C₈	—	13	4	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	13	2	no	—
192	C₂×C₈	—	13	1	no	—
193	C₂×C₈	—	13	1	no	—
194	C₂×C₈	—	13	1	no	—
195	C₂×C₈	—	13	1	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	—	13	4	no	—
199	C₂×C₈	—	26	4	no	—
200	C₂×C₈	—	26	4	no	—
201	C₂×C₈	—	26	2	no	—
202	C₂×C₈	—	26	2	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	C₂×#1057	1	4	yes	—
206	C₁₆	—	26	8	no	—
207	C₁₃	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	2	4	no	—
209	C₂³	#88110	1	2	yes	—
210	C₂³	#88112	1	1	yes	characteristic
211	C₂³	#88112	1	1	yes	characteristic
212	C₂³	C₂²⋊#1057	1	2	yes	—
213	C₂³	C₂×#6122	1	1	yes	characteristic
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	C₄×#1057	1	2	yes	—
218	C₂×C₄	C₄×#1057	1	2	yes	—
219	C₂×C₄	C₄⋊#1057	1	2	yes	—
220	C₂×C₄	C₄⋊#1057	1	2	yes	—
221	C₂×C₄	#88110	1	2	yes	—
222	C₂×C₄	#88111	1	2	yes	—
223	C₂×C₄	#88111	1	2	yes	—
224	C₂×C₄	#88112	1	1	yes	characteristic
225	C₂×C₄	#88112	1	1	yes	characteristic
226	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
227	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
228	C₂×C₄	C₂²⋊#1057	1	2	yes	—
229	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
230	C₂×C₄	C₂×#6118	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
233	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
235	C₈	—	13	2	no	—
236	C₈	—	13	2	no	—
237	C₈	—	13	2	no	—
238	C₈	—	13	2	no	—
239	C₈	—	26	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	#11718233	1	4	yes	—
245	C₂²	#11718290	1	2	yes	—
246	C₂²	#11718290	1	2	yes	—
247	C₂²	C₂×#88110	1	1	yes	characteristic
248	C₂²	C₂×#88112	1	1	yes	characteristic
249	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
250	C₄	—	2	8	no	—
251	C₄	#11718290	1	2	yes	—
252	C₄	#11718290	1	2	yes	—
253	C₄	C₂×C₄×#1057	1	1	yes	characteristic
254	C₄	C₂×C₄⋊#1057	1	1	yes	characteristic
255	C₄	C₂×#88111	1	1	yes	characteristic
256	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	2	1	no	—
260	C₂	#13129880	1	4	yes	direct product factor
261	C₂	C₂×#11718290	1	1	yes	characteristic
262	C₂	C₂×#11718233	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₄⋊#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	C₄⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂²×#1055	C₂³
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₂×#1376
LCS	G ▷ C₄×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. #13129880	C₂	direct (N × H)
9. C₂×#88106	C₄	semidirect
36. #88106	C₂×C₄	semidirect
169. C₂×C₁₃	#1376	semidirect
207. C₁₃	C₂×#1376	semidirect
260. C₂	#13129880	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
#13129880	2
C₂²×C₄×#1055	2
C₂×#1376	13

Subgroups — C2×#13129880 #16789063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13129880 #16789063