Subgroups — C₂×#13116556 #16015811

Order 3328.

1150 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
3	#13116556	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718156	C₂	1	4	yes	maximal, complemented
5	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₁₃	—	2	1	no	—
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	—	2	2	no	—
9	C₂²×C₁₃×Q₁₆	—	2	4	no	—
10	C₂²×C₁₃×Q₁₆	C₄	1	8	yes	complemented
11	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
12	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
13	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88180	—	2	1	no	—
15	#11718767	—	2	2	no	—
16	#11718767	C₂²	1	4	yes	—
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×#88057	—	2	1	no	—
19	C₂×#88052	—	2	4	no	—
20	C₂×#88052	C₂²	1	4	yes	—
21	C₂²×#6090	C₂²	1	2	yes	—
22	#11718156	C₂²	1	16	yes	complemented
23	C₂²×Q₈×C₁₃	—	2	8	no	—
24	C₂²×Q₈×C₁₃	—	2	2	no	—
25	C₂²×Q₈×C₁₃	—	2	4	no	—
26	C₂²×Q₈×C₁₃	—	4	1	no	—
27	C₂²×Q₈×C₁₃	—	4	2	no	—
28	C₂²×Q₈×C₁₃	—	4	4	no	—
29	C₂³×C₄×C₁₃	—	4	1	no	—
30	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₁₃×Q₁₆	—	2	16	no	—
32	C₂×C₁₃×Q₁₆	—	2	8	no	—
33	C₂×C₁₃×Q₁₆	—	2	4	no	—
34	C₂×C₁₃×Q₁₆	—	2	2	no	—
35	C₂×C₁₃×Q₁₆	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	—	2	8	no	—
37	C₂×C₁₃×Q₁₆	C₂×C₄	1	8	yes	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	16	yes	complemented
39	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
40	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	D₄	1	2	yes	—
43	C₂²×C₈×C₁₃	D₄	1	4	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
46	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
47	#88180	—	2	4	no	—
48	#88180	—	4	2	no	—
49	C₂×C₄⋊Dic₁₃	—	2	2	no	—
50	C₂×C₄×Dic₁₃	—	4	1	no	—
51	#88058	—	2	2	no	—
52	#88058	C₂³	1	4	yes	—
53	#88057	—	2	4	no	—
54	#88052	—	2	8	no	—
55	C₂×#6090	—	2	4	no	—
56	C₂×#6090	C₂³	1	4	yes	—
57	C₂×#2141	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	8	no	—
60	C₂×Q₈×C₁₃	—	2	16	no	—
61	C₂×Q₈×C₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₃	—	4	16	no	—
63	C₂×Q₈×C₁₃	—	4	8	no	—
64	C₂×Q₈×C₁₃	—	4	4	no	—
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂×Q₈×C₁₃	—	4	8	no	—
67	C₂×Q₈×C₁₃	—	4	4	no	—
68	C₂²×C₄×C₁₃	—	4	8	no	—
69	C₂²×C₄×C₁₃	—	4	4	no	—
70	C₂²×C₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	D₈	1	2	yes	—
74	C₂²×C₄×C₁₃	SD₁₆	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂²×Dic₁₃	—	4	2	no	—
78	C₁₃×Q₁₆	—	2	8	no	—
79	C₁₃×Q₁₆	—	2	16	no	—
80	C₁₃×Q₁₆	—	2	4	no	—
81	C₁₃×Q₁₆	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
83	C₂×C₈×C₁₃	C₂²⋊C₄	1	8	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂²⋊Dic₁₃	—	4	4	no	—
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	—	4	2	no	—
94	C₄×Dic₁₃	—	4	4	no	—
95	#6090	—	2	4	no	—
96	C₂³×Q₁₆	D₁₃	1	1	yes	characteristic
97	C₂²×M₃₂	—	13	1	no	—
98	C₂×#427	—	13	1	no	—
99	#2141	—	13	8	no	—
100	C₂×#358	—	13	4	no	—
101	C₂³×C₁₃	—	2	4	no	—
102	C₂³×C₁₃	#102	1	4	yes	—
103	C₂³×C₁₃	C₂×D₈	1	2	yes	—
104	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₁₃	—	2	8	no	—
106	Q₈×C₁₃	—	4	4	no	—
107	Q₈×C₁₃	—	4	8	no	—
108	Q₈×C₁₃	—	4	16	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	—	4	16	no	—
111	C₂×C₄×C₁₃	—	4	8	no	—
112	C₂×C₄×C₁₃	#102	1	8	yes	—
113	C₂×C₄×C₁₃	#102	1	4	yes	—
114	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
119	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃	—	4	4	no	—
122	C₂×Dic₁₃	—	8	4	no	—
123	C₈×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
124	C₈×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
125	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂³×Q₈	—	2	1	no	—
128	C₂²×Q₁₆	—	2	1	no	—
129	C₂²×Q₁₆	—	2	2	no	—
130	C₂²×Q₁₆	—	2	4	no	—
131	C₂²×Q₁₆	Dic₁₃	1	8	yes	complemented
132	C₂²×Q₁₆	C₂×D₁₃	1	2	yes	—
133	C₂²×Q₁₆	C₂×D₁₃	1	4	yes	—
134	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
135	C₂×#117	—	26	1	no	—
136	C₂×M₃₂	—	13	4	no	—
137	C₂×M₃₂	—	26	4	no	—
138	C₂²×C₁₆	—	13	2	no	—
139	#427	—	13	4	no	—
140	#427	—	26	2	no	—
141	C₂×#107	—	13	2	no	—
142	C₂×#106	—	26	1	no	—
143	#358	—	13	16	no	—
144	C₂²×C₁₃	—	2	4	no	—
145	C₂²×C₁₃	—	2	8	no	—
146	C₂²×C₁₃	C₂×#102	1	4	yes	—
147	C₂²×C₁₃	C₂×#102	1	2	yes	—
148	C₂²×C₁₃	#510	1	4	yes	—
149	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₁₃	—	4	8	no	—
151	C₄×C₁₃	C₂×#102	1	4	yes	—
152	C₄×C₁₃	C₂×#102	1	2	yes	—
153	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
155	Dic₁₃	—	8	4	no	—
156	C₂²×Q₈	—	2	8	no	—
157	C₂²×Q₈	—	2	4	no	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×Q₈	—	4	2	no	—
160	C₂²×Q₈	—	4	1	no	—
161	C₂²×Q₈	—	4	4	no	—
162	C₂³×C₄	—	4	1	no	—
163	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
164	C₂×Q₁₆	—	2	16	no	—
165	C₂×Q₁₆	—	2	8	no	—
166	C₂×Q₁₆	—	2	4	no	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	—	2	4	no	—
169	C₂×Q₁₆	—	2	8	no	—
170	C₂×Q₁₆	C₂×Dic₁₃	1	8	yes	—
171	C₂×Q₁₆	C₂×Dic₁₃	1	16	yes	complemented
172	C₂×Q₁₆	C₂²×D₁₃	1	4	yes	—
173	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
174	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
176	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
179	#117	—	26	4	no	—
180	#117	—	52	2	no	—
181	C₂×C₄⋊C₄	—	26	2	no	—
182	C₂×C₂²⋊C₄	—	52	1	no	—
183	C₂×C₄²	—	52	1	no	—
184	M₃₂	—	26	8	no	—
185	C₂×C₁₆	—	13	4	no	—
186	C₂×C₁₆	—	26	4	no	—
187	#107	—	13	4	no	—
188	#107	—	26	2	no	—
189	#106	—	26	4	no	—
190	C₂×C₁₃	—	2	4	no	—
191	C₂×C₁₃	#2141	1	4	yes	complemented
192	C₂×C₁₃	C₂²×#102	1	1	yes	characteristic
193	C₂×C₁₃	C₂×#510	1	2	yes	—
194	C₂⁴	#6104	1	1	yes	characteristic
195	C₂×Q₈	—	2	8	no	—
196	C₂×Q₈	—	2	16	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	4	16	no	—
199	C₂×Q₈	—	4	8	no	—
200	C₂×Q₈	—	4	4	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	4	8	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂²×C₄	—	4	8	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	52	2	no	—
208	C₂²×C₄	#6104	1	2	yes	—
209	C₂²×C₄	#6104	1	1	yes	characteristic
210	C₂²×C₄	#6105	1	4	yes	—
211	C₂²×C₄	C₂²⋊Dic₁₃	1	4	yes	—
212	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	Q₁₆	—	2	8	no	—
215	Q₁₆	—	2	16	no	—
216	Q₁₆	—	2	4	no	—
217	Q₁₆	C₂²×Dic₁₃	1	8	yes	—
218	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
219	C₂×C₈	C₂²⋊Dic₁₃	1	8	yes	—
220	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
221	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
222	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
223	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
226	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
227	C₄⋊C₄	—	26	4	no	—
228	C₄⋊C₄	—	52	2	no	—
229	C₂²⋊C₄	—	52	4	no	—
230	C₄²	—	52	4	no	—
231	C₁₆	—	26	4	no	—
232	C₁₃	C₂×#2141	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	2	4	no	—
234	C₂³	#88072	1	4	yes	—
235	C₂³	C₂×#6104	1	2	yes	—
236	C₂³	C₂×#6104	1	1	yes	characteristic
237	Q₈	—	2	8	no	—
238	Q₈	—	4	4	no	—
239	Q₈	—	4	8	no	—
240	Q₈	—	4	16	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	16	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	52	4	no	—
245	C₂×C₄	—	104	4	no	—
246	C₂×C₄	#88072	1	8	yes	—
247	C₂×C₄	#88072	1	4	yes	—
248	C₂×C₄	C₂×#6104	1	2	yes	—
249	C₂×C₄	C₂×#6104	1	1	yes	characteristic
250	C₂×C₄	C₂×#6105	1	4	yes	—
251	C₂×C₄	C₂×#6105	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	2	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
255	C₈	C₂×C₂²⋊Dic₁₃	1	2	yes	—
256	C₈	C₂×C₂²⋊Dic₁₃	1	4	yes	—
257	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
258	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂²	—	2	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	#11718840	1	4	yes	—
262	C₂²	C₂×#88072	1	4	yes	—
263	C₂²	C₂×#88072	1	2	yes	—
264	C₂²	C₂²×#6104	1	1	yes	characteristic
265	C₄	—	4	8	no	—
266	C₄	—	104	4	no	—
267	C₄	C₂×#88072	1	4	yes	—
268	C₄	C₂×#88072	1	2	yes	—
269	C₄	C₂²×#6105	1	1	yes	characteristic
270	C₄	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#13116556	1	4	yes	direct product factor
273	C₂	C₂²×#88072	1	1	yes	characteristic
274	C₂	C₂×#11718840	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂³×C₁₃×Q₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂³×Q₁₆	D₁₃
hypocenter	C₁₃	C₂×#2141
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #13116556	C₂	direct (N × H)
4. C₂×#11718156	C₂	semidirect
10. C₂²×C₁₃×Q₁₆	C₄	semidirect
22. #11718156	C₂²	semidirect
38. C₂×C₁₃×Q₁₆	C₂×C₄	semidirect
131. C₂²×Q₁₆	Dic₁₃	semidirect
171. C₂×Q₁₆	C₂×Dic₁₃	semidirect
191. C₂×C₁₃	#2141	semidirect
232. C₁₃	C₂×#2141	semidirect
272. C₂	#13116556	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×#88052	2
C₂×#11718767	2
#13116556	2
C₂×#11718156	2
C₂³×C₁₃×Q₁₆	2
C₂×#2141	13

Subgroups — C2×#13116556 #16015811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116556 #16015811