Subgroups — C₂×#13114973 #15863144

Order 3328.

870 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
3	#13114973	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718150	C₂	1	4	yes	maximal
5	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6099	C₂²	1	2	yes	—
9	C₂³×#1055	C₂²	1	2	yes	—
10	C₁₃×#423	—	2	2	no	—
11	C₁₃×#423	C₂²	1	4	yes	—
12	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
13	#11718585	—	2	2	no	—
14	#11718585	C₂²	1	4	yes	—
15	C₂×C₄⋊#1055	—	2	2	no	—
16	C₂×C₄×#1055	C₄	1	4	yes	—
17	C₂×C₄×#1055	C₄	1	2	yes	—
18	C₂×C₄×#1055	C₄	1	2	yes	—
19	C₂×C₄×#1055	C₂²	1	4	yes	—
20	#11718150	C₂²	1	16	yes	—
21	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	8	no	—
25	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
34	C₂×#6099	—	2	2	no	—
35	C₂×#6099	—	2	8	no	—
36	C₂×#6099	C₂³	1	4	yes	—
37	C₂²×#1055	—	2	4	no	—
38	C₂²×#1055	—	2	4	no	—
39	C₂²×#1055	—	2	2	no	—
40	C₂²×#1055	C₂×C₄	1	4	yes	—
41	C₂²×#1055	C₂×C₄	1	4	yes	—
42	C₂²×#1055	C₂³	1	4	yes	—
43	C₁₃×C₄⋊C₈	—	2	8	no	—
44	C₄⋊#1055	—	2	8	no	—
45	C₄×#1055	C₂×C₄	1	8	yes	—
46	C₄×#1055	C₂×C₄	1	8	yes	—
47	C₂×#1837	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	8	no	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	4	yes	—
60	C₂²×C₄×C₁₃	#41	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₁₃×M₁₆	—	2	8	no	—
63	C₁₃×M₁₆	—	4	4	no	—
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	4	4	no	—
66	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
68	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₁₃	C₂×D₄	1	4	yes	—
72	#6099	—	2	8	no	—
73	#6099	—	4	4	no	—
74	C₂×#1055	—	2	8	no	—
75	C₂×#1055	—	2	8	no	—
76	C₂×#1055	—	2	4	no	—
77	C₂×#1055	—	2	4	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	—	4	4	no	—
80	C₂×#1055	C₂²×C₄	1	8	yes	—
81	C₂×#423	—	13	1	no	—
82	C₂×#423	D₁₃	1	1	yes	characteristic
83	C₂²×C₄×C₈	—	13	1	no	—
84	#1837	—	13	8	no	—
85	C₂×#341	—	13	4	no	—
86	C₂³×C₁₃	—	2	4	no	—
87	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂³×C₁₃	#125	1	4	yes	—
90	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	2	8	no	—
93	C₂×C₄×C₁₃	—	2	8	no	—
94	C₂×C₄×C₁₃	#108	1	8	yes	—
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₃	#117	1	4	yes	—
100	C₂×C₄×C₁₃	#117	1	4	yes	—
101	C₂×C₄×C₁₃	#124	1	2	yes	—
102	C₂×C₄×C₁₃	#124	1	2	yes	—
103	C₂×C₄×C₁₃	#124	1	2	yes	—
104	C₂×C₄×C₁₃	#124	1	2	yes	—
105	C₂×C₄×C₁₃	#125	1	4	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
110	C₈×C₁₃	—	4	4	no	—
111	#1055	—	2	8	no	—
112	#1055	—	4	4	no	—
113	C₂²×M₁₆	—	13	2	no	—
114	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
115	C₂³×C₈	—	13	2	no	—
116	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
117	#423	—	2	2	no	—
118	#423	—	13	4	no	—
119	#423	—	26	2	no	—
120	#423	C₂×D₁₃	1	4	yes	—
121	C₂×C₄⋊C₈	—	2	2	no	—
122	C₂×C₄⋊C₈	—	26	2	no	—
123	C₂×C₄×C₈	—	13	2	no	—
124	C₂×C₄×C₈	—	13	4	no	—
125	C₂×C₄×C₈	—	13	2	no	—
126	C₂×C₄×C₈	—	13	4	no	—
127	#341	—	13	16	no	—
128	C₂²×C₁₃	—	2	8	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	#382	1	2	yes	—
132	C₂²×C₁₃	#382	1	2	yes	—
133	C₂²×C₁₃	C₂×#108	1	4	yes	—
134	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#125	1	2	yes	—
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	#382	1	2	yes	—
138	C₄×C₁₃	#382	1	2	yes	—
139	C₄×C₁₃	C₂×#108	1	8	yes	—
140	C₄×C₁₃	C₂×#117	1	2	yes	—
141	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
142	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
143	C₂³×C₄	Dic₂₆	1	2	yes	—
144	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	8	no	—
147	C₂×M₁₆	—	13	4	no	—
148	C₂×M₁₆	—	26	2	no	—
149	C₂×M₁₆	—	26	8	no	—
150	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	13	4	no	—
153	C₂²×C₈	—	13	4	no	—
154	C₂²×C₈	—	13	4	no	—
155	C₂²×C₈	—	26	4	no	—
156	C₂²×C₈	—	26	4	no	—
157	C₂²×C₈	—	26	2	no	—
158	C₂×C₄²	Dic₂₆	1	2	yes	—
159	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
160	C₂×C₄²	C₄×D₁₃	1	2	yes	—
161	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
163	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
164	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
165	C₄⋊C₈	—	2	8	no	—
166	C₄⋊C₈	—	26	8	no	—
167	C₄×C₈	—	13	8	no	—
168	C₄×C₈	—	13	8	no	—
169	C₂×C₁₃	—	2	2	no	—
170	C₂×C₁₃	—	2	2	no	—
171	C₂×C₁₃	#1837	1	4	yes	complemented
172	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
173	C₂×C₁₃	C₂²×#108	1	2	yes	—
174	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	#6101	1	2	yes	—
177	C₂²×C₄	#6101	1	4	yes	—
178	C₂²×C₄	#6101	1	2	yes	—
179	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
180	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
183	C₂²×C₄	#6129	1	4	yes	—
184	C₂²×C₄	#6131	1	4	yes	—
185	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
188	M₁₆	—	2	8	no	—
189	M₁₆	—	4	4	no	—
190	M₁₆	—	26	8	no	—
191	M₁₆	—	52	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	—	13	8	no	—
195	C₂×C₈	—	26	8	no	—
196	C₂×C₈	—	26	8	no	—
197	C₂×C₈	—	26	4	no	—
198	C₂×C₈	—	26	4	no	—
199	C₂×C₈	—	26	4	no	—
200	C₂×C₈	—	52	4	no	—
201	C₄²	#6101	1	2	yes	—
202	C₄²	#6101	1	4	yes	—
203	C₄²	#6101	1	2	yes	—
204	C₄²	C₂×C₄×D₁₃	1	2	yes	—
205	C₄²	C₂×C₄×D₁₃	1	2	yes	—
206	C₄²	C₂×C₂²⋊D₁₃	1	4	yes	—
207	C₁₃	C₂×#1837	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	2	4	no	—
209	C₂³	#88144	1	4	yes	—
210	C₂³	C₂×#6101	1	1	yes	characteristic
211	C₂³	C₂×#6101	1	1	yes	characteristic
212	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	#88062	1	8	yes	—
217	C₂×C₄	#88144	1	4	yes	—
218	C₂×C₄	#88146	1	4	yes	—
219	C₂×C₄	#88146	1	4	yes	—
220	C₂×C₄	C₂×#6101	1	4	yes	—
221	C₂×C₄	C₂×#6101	1	4	yes	—
222	C₂×C₄	C₂×#6101	1	1	yes	characteristic
223	C₂×C₄	C₂×#6101	1	1	yes	characteristic
224	C₂×C₄	#88190	1	2	yes	—
225	C₂×C₄	#88190	1	2	yes	—
226	C₂×C₄	#88201	1	2	yes	—
227	C₂×C₄	#88201	1	2	yes	—
228	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
229	C₂×C₄	C₂×#6129	1	2	yes	—
230	C₂×C₄	C₂×#6131	1	2	yes	—
231	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
232	C₈	—	4	4	no	—
233	C₈	—	26	8	no	—
234	C₈	—	52	4	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	#11718638	1	2	yes	—
239	C₂²	#11718638	1	2	yes	—
240	C₂²	C₂×#88062	1	4	yes	—
241	C₂²	C₂×#88144	1	2	yes	—
242	C₂²	C₂²×#6101	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	#11718638	1	2	yes	—
245	C₄	#11718638	1	2	yes	—
246	C₄	C₂×#88062	1	8	yes	—
247	C₄	C₂×#88146	1	2	yes	—
248	C₄	C₂×#88190	1	1	yes	characteristic
249	C₄	C₂×#88201	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	2	2	no	—
252	C₂	#13114973	1	4	yes	direct product factor
253	C₂	C₂×#11718638	1	1	yes	characteristic
254	C₂	C₂²×#88062	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#423	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#423	D₁₃
hypocenter	C₁₃	C₂×#1837
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #13114973	C₂	direct (N × H)
171. C₂×C₁₃	#1837	semidirect
207. C₁₃	C₂×#1837	semidirect
252. C₂	#13114973	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×#11718585	2
C₂²×C₄×#1055	2
#13114973	2
C₂×#11718150	2
C₂×C₁₃×#423	2
C₂×#1837	13

Subgroups — C2×#13114973 #15863144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114973 #15863144