Subgroups — C₂×#13115007 #15863374

Order 3328.

1046 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	C₂²×#88062	C₂	1	4	yes	maximal
2	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
4	#13115007	C₂	1	8	yes	maximal, direct product factor
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	C₂×#88062	—	2	16	no	—
14	C₂×#88062	C₂²	1	16	yes	—
15	#11718585	—	2	2	no	—
16	#11718585	C₂²	1	4	yes	—
17	C₂×C₄⋊#1055	—	2	2	no	—
18	C₂×C₄×#1055	C₄	1	4	yes	—
19	C₂×C₄×#1055	C₄	1	2	yes	—
20	C₂×C₄×#1055	C₄	1	2	yes	—
21	C₂×C₄×#1055	C₂²	1	4	yes	—
22	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	8	no	—
26	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
35	C₂×#6099	—	2	2	no	—
36	C₂×#6099	—	2	8	no	—
37	C₂×#6099	C₂³	1	4	yes	—
38	C₂²×#1055	—	2	2	no	—
39	C₂²×#1055	C₂×C₄	1	4	yes	—
40	C₂²×#1055	C₂×C₄	1	4	yes	—
41	C₂²×#1055	D₄	1	8	yes	—
42	C₂²×#1055	Q₈	1	8	yes	—
43	C₂²×#1055	C₂³	1	4	yes	—
44	C₁₃×C₄⋊C₈	—	2	8	no	—
45	#88062	—	2	32	no	—
46	C₄⋊#1055	—	2	8	no	—
47	C₄×#1055	C₂×C₄	1	8	yes	—
48	C₄×#1055	C₂×C₄	1	8	yes	—
49	C₂×#1848	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	—	2	8	no	—
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₁₃×M₁₆	—	2	8	no	—
65	C₁₃×M₁₆	—	4	4	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	—	4	4	no	—
68	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
70	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₁₃	C₂×D₄	1	4	yes	—
74	#6099	—	2	8	no	—
75	#6099	—	4	4	no	—
76	C₂×#1055	—	2	4	no	—
77	C₂×#1055	—	4	4	no	—
78	C₂×#1055	C₄⋊C₄	1	16	yes	—
79	C₂×#1055	C₄⋊C₄	1	16	yes	—
80	C₂×#1055	C₂²×C₄	1	8	yes	—
81	C₂×#1055	C₂×D₄	1	8	yes	—
82	C₂×#1055	C₂×Q₈	1	8	yes	—
83	C₂²×#108	—	13	4	no	—
84	C₂×#423	—	13	1	no	—
85	C₂×#423	D₁₃	1	1	yes	characteristic
86	C₂²×C₄×C₈	—	13	1	no	—
87	#1848	—	13	8	no	—
88	C₂³×C₁₃	—	2	4	no	—
89	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
92	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	8	no	—
95	C₂×C₄×C₁₃	—	2	8	no	—
96	C₂×C₄×C₁₃	#108	1	8	yes	complemented
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₁₃	#127	1	2	yes	—
104	C₂×C₄×C₁₃	#127	1	2	yes	—
105	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
112	C₈×C₁₃	—	4	4	no	—
113	#1055	—	4	4	no	—
114	#1055	C₂×C₄⋊C₄	1	16	yes	—
115	C₂²×M₁₆	—	13	2	no	—
116	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
117	C₂³×C₈	—	13	2	no	—
118	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
119	C₂×#108	—	13	16	no	—
120	C₂×#108	—	26	16	no	—
121	#423	—	2	2	no	—
122	#423	—	13	4	no	—
123	#423	—	26	2	no	—
124	#423	C₂×D₁₃	1	4	yes	—
125	C₂×C₄⋊C₈	—	2	2	no	—
126	C₂×C₄⋊C₈	—	26	2	no	—
127	C₂×C₄×C₈	—	13	2	no	—
128	C₂×C₄×C₈	—	13	4	no	—
129	C₂×C₄×C₈	—	13	2	no	—
130	C₂×C₄×C₈	—	13	4	no	—
131	C₂²×C₁₃	—	2	8	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	#384	1	2	yes	—
135	C₂²×C₁₃	#384	1	2	yes	—
136	C₂²×C₁₃	C₂×#108	1	4	yes	—
137	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	#384	1	2	yes	—
141	C₄×C₁₃	#384	1	2	yes	—
142	C₄×C₁₃	C₂×#108	1	8	yes	complemented
143	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
144	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
145	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂³×C₄	Dic₂₆	1	2	yes	—
147	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	8	no	—
150	C₂×M₁₆	—	13	4	no	—
151	C₂×M₁₆	—	26	2	no	—
152	C₂×M₁₆	—	26	8	no	—
153	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	13	4	no	—
156	C₂²×C₈	—	13	4	no	—
157	C₂²×C₈	—	13	8	no	—
158	C₂²×C₈	—	13	8	no	—
159	C₂²×C₈	—	13	4	no	—
160	C₂²×C₈	—	26	2	no	—
161	C₂×C₄²	Dic₂₆	1	2	yes	—
162	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₁₃	1	2	yes	—
164	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
166	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
167	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
168	#108	—	26	32	no	—
169	C₄⋊C₈	—	2	8	no	—
170	C₄⋊C₈	—	26	8	no	—
171	C₄×C₈	—	13	8	no	—
172	C₄×C₈	—	13	8	no	—
173	C₂×C₁₃	—	2	2	no	—
174	C₂×C₁₃	—	2	2	no	—
175	C₂×C₁₃	#1848	1	4	yes	complemented
176	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
177	C₂×C₁₃	C₂²×#108	1	2	yes	—
178	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	#6101	1	2	yes	—
181	C₂²×C₄	#6101	1	4	yes	—
182	C₂²×C₄	#6101	1	2	yes	—
183	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
184	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
187	C₂²×C₄	D₄×D₁₃	1	4	yes	—
188	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
189	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
192	M₁₆	—	2	8	no	—
193	M₁₆	—	4	4	no	—
194	M₁₆	—	26	8	no	—
195	M₁₆	—	52	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	—	13	16	no	—
199	C₂×C₈	—	13	16	no	—
200	C₂×C₈	—	13	8	no	—
201	C₂×C₈	—	13	8	no	—
202	C₂×C₈	—	13	8	no	—
203	C₂×C₈	—	26	4	no	—
204	C₂×C₈	—	52	4	no	—
205	C₄²	#6101	1	2	yes	—
206	C₄²	#6101	1	4	yes	—
207	C₄²	#6101	1	2	yes	—
208	C₄²	C₂×C₄×D₁₃	1	2	yes	—
209	C₄²	C₂×C₄×D₁₃	1	2	yes	—
210	C₄²	C₂×C₂²⋊D₁₃	1	4	yes	—
211	C₁₃	C₂×#1848	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	4	no	—
213	C₂³	#88142	1	4	yes	—
214	C₂³	C₂×#6101	1	1	yes	characteristic
215	C₂³	C₂×#6101	1	1	yes	characteristic
216	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	#88062	1	8	yes	complemented
221	C₂×C₄	#88142	1	4	yes	—
222	C₂×C₄	C₄⋊C₄×D₁₃	1	4	yes	—
223	C₂×C₄	C₄⋊C₄×D₁₃	1	4	yes	—
224	C₂×C₄	C₂×#6101	1	4	yes	—
225	C₂×C₄	C₂×#6101	1	4	yes	—
226	C₂×C₄	C₂×#6101	1	1	yes	characteristic
227	C₂×C₄	C₂×#6101	1	1	yes	characteristic
228	C₂×C₄	#88194	1	2	yes	—
229	C₂×C₄	#88194	1	2	yes	—
230	C₂×C₄	#88198	1	2	yes	—
231	C₂×C₄	#88198	1	2	yes	—
232	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
234	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
235	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
236	C₈	—	4	4	no	—
237	C₈	—	13	16	no	—
238	C₈	—	52	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	#11718637	1	2	yes	—
243	C₂²	#11718637	1	2	yes	—
244	C₂²	C₂×#88062	1	4	yes	—
245	C₂²	C₂×#88142	1	2	yes	—
246	C₂²	C₂²×#6101	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	#11718637	1	2	yes	—
249	C₄	#11718637	1	2	yes	—
250	C₄	C₂×#88062	1	8	yes	complemented
251	C₄	C₂×C₄⋊C₄×D₁₃	1	2	yes	—
252	C₄	C₂×#88194	1	1	yes	characteristic
253	C₄	C₂×#88198	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	#13115007	1	4	yes	direct product factor
257	C₂	C₂×#11718637	1	1	yes	characteristic
258	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×#1848
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
4. #13115007	C₂	direct (N × H)
96. C₂×C₄×C₁₃	#108	semidirect
142. C₄×C₁₃	C₂×#108	semidirect
175. C₂×C₁₃	#1848	semidirect
211. C₁₃	C₂×#1848	semidirect
220. C₂×C₄	#88062	semidirect
250. C₄	C₂×#88062	semidirect
256. C₂	#13115007	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×#88062	2
C₂×#11718585	2
C₂²×C₄×#1055	2
#13115007	2
C₂×C₁₃×#423	2
C₂×#1848	13

Subgroups — C2×#13115007 #15863374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13115007 #15863374