Subgroups — C₂×#13117019 #16017702

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88058	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117019	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718082	C₂	1	2	yes	maximal, complemented
4	C₂×#11718154	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#121	—	2	1	no	—
10	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
11	C₁₃×#466	—	2	2	no	—
12	C₁₃×#466	C₂²	1	4	yes	—
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#102	—	2	1	no	—
15	C₂×#88058	—	2	1	no	—
16	C₂×#88058	—	2	2	no	—
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×#88058	C₂²	1	4	yes	complemented
19	C₂²×#6090	C₂²	1	2	yes	—
20	#11718154	C₂²	1	8	yes	complemented
21	C₂²⋊#6090	C₂²	1	4	yes	complemented
22	#11718082	C₂²	1	8	yes	complemented
23	C₂²×D₄×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	—	4	1	no	—
25	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
26	C₂³×Dic₁₃	—	4	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	4	no	—
29	C₂×C₁₃×D₈	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	—	2	4	no	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#121	—	2	4	no	—
36	C₁₃×#121	—	4	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	—	2	4	no	—
41	C₂×C₄⋊Dic₁₃	—	4	1	no	—
42	C₂×C₄⋊Dic₁₃	—	4	2	no	—
43	C₁₃×#107	—	2	1	no	—
44	C₁₃×#107	C₂³	1	2	yes	—
45	C₁₃×#102	—	2	4	no	—
46	#88058	—	2	2	no	—
47	#88058	—	2	4	no	—
48	#88058	C₂³	1	4	yes	—
49	C₂×#6090	—	2	4	no	—
50	C₂×#6090	C₂³	1	4	yes	—
51	C₂×#2233	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	—	4	1	no	—
53	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	—	4	4	no	—
56	C₂×D₄×C₁₃	—	4	2	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	8	4	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	D₈	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	—	4	4	no	—
68	C₂²×Dic₁₃	—	4	2	no	—
69	C₂²×Dic₁₃	—	8	4	no	—
70	C₁₃×D₈	—	2	4	no	—
71	C₁₃×D₈	—	4	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	4	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	4	1	no	—
81	C₁₃×C₂²⋊C₄	—	4	4	no	—
82	C₄⋊Dic₁₃	—	2	4	no	—
83	C₄⋊Dic₁₃	—	4	4	no	—
84	C₄⋊Dic₁₃	—	4	2	no	—
85	#6090	—	2	4	no	—
86	C₂×#466	D₁₃	1	1	yes	characteristic
87	C₂²×#107	—	13	1	no	—
88	#2233	—	13	8	no	—
89	C₂×#367	—	13	2	no	—
90	C₂×#357	—	13	2	no	—
91	C₂×C₂²⋊C₁₆	—	13	1	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	—	8	2	no	—
96	C₂³×C₁₃	—	8	2	no	—
97	C₂³×C₁₃	—	8	2	no	—
98	C₂³×C₁₃	SD₃₂	1	4	yes	—
99	C₂³×C₁₃	C₂×D₈	1	2	yes	—
100	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
101	D₄×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	8	8	no	—
104	D₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	8	2	no	—
111	C₂×C₄×C₁₃	#123	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#137	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×Dic₁₃	—	4	4	no	—
117	C₂×Dic₁₃	—	8	8	no	—
118	C₂×Dic₁₃	—	8	4	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	#123	1	2	yes	—
121	C₈×C₁₃	C₂×#41	1	2	yes	—
122	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
123	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
124	C₂×#121	—	2	1	no	—
125	C₂²×C₄⋊C₄	—	26	1	no	—
126	C₂²×C₁₆	—	13	2	no	—
127	#466	—	2	2	no	—
128	#466	C₂×D₁₃	1	4	yes	—
129	C₂×#107	—	13	2	no	—
130	C₂×#107	—	13	4	no	—
131	C₂×#107	—	26	1	no	—
132	C₂×#107	—	26	2	no	—
133	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
134	C₂×#102	—	2	1	no	—
135	#367	—	13	8	no	—
136	#357	—	13	8	no	—
137	C₂²⋊C₁₆	—	13	4	no	—
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	2	8	no	—
141	C₂²×C₁₃	—	8	2	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	16	4	no	—
145	C₂²×C₁₃	#480	1	2	yes	—
146	C₂²×C₁₃	C₂×SD₃₂	1	4	yes	—
147	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
148	C₂²×C₁₃	#509	1	2	yes	—
149	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	8	2	no	—
152	C₄×C₁₃	#480	1	2	yes	—
153	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
154	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
155	Dic₁₃	—	8	4	no	—
156	C₂²×D₄	—	2	1	no	—
157	C₂²×D₄	—	4	1	no	—
158	C₂³×C₄	—	52	1	no	—
159	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₂×D₈	—	2	4	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	C₂²×D₁₃	1	2	yes	—
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
166	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
167	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	#121	—	2	4	no	—
169	#121	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	26	4	no	—
172	C₂×C₄⋊C₄	—	26	2	no	—
173	C₂×C₄⋊C₄	—	52	2	no	—
174	C₂×C₄⋊C₄	—	52	1	no	—
175	C₂×C₂²⋊C₄	—	4	1	no	—
176	C₂×C₁₆	—	13	4	no	—
177	C₂×C₁₆	—	26	4	no	—
178	#107	—	2	1	no	—
179	#107	—	13	4	no	—
180	#107	—	26	4	no	—
181	#107	—	26	2	no	—
182	#107	C₂²×D₁₃	1	2	yes	—
183	#102	—	2	4	no	—
184	C₂×C₁₃	—	2	4	no	—
185	C₂×C₁₃	—	16	2	no	—
186	C₂×C₁₃	#2233	1	4	yes	complemented
187	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
188	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
189	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
190	C₂⁴	—	4	1	no	—
191	C₂⁴	#6104	1	1	yes	characteristic
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	4	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	8	4	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	52	4	no	—
202	C₂²×C₄	—	52	2	no	—
203	C₂²×C₄	—	104	4	no	—
204	C₂²×C₄	#6104	1	2	yes	—
205	C₂²×C₄	#6104	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	D₈	—	2	4	no	—
209	D₈	—	4	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	#6129	1	4	yes	—
214	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
215	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	26	4	no	—
220	C₄⋊C₄	—	52	4	no	—
221	C₄⋊C₄	—	52	2	no	—
222	C₂²⋊C₄	—	4	4	no	—
223	C₁₆	—	26	4	no	—
224	C₁₃	C₂×#2233	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	#88067	1	4	yes	—
232	C₂³	C₂×#6104	1	2	yes	—
233	C₂³	C₂×#6104	1	1	yes	characteristic
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	8	8	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	52	4	no	—
245	C₂×C₄	—	104	8	no	—
246	C₂×C₄	—	104	4	no	—
247	C₂×C₄	C₂×#6104	1	2	yes	—
248	C₂×C₄	C₂×#6104	1	1	yes	characteristic
249	C₂×C₄	#88189	1	2	yes	—
250	C₂×C₄	#88205	1	2	yes	—
251	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	#88189	1	2	yes	—
254	C₈	C₂×#6129	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	16	4	no	—
262	C₂²	#11718694	1	2	yes	—
263	C₂²	C₂×#88067	1	4	yes	—
264	C₂²	C₂×#88067	1	2	yes	—
265	C₂²	#11718854	1	2	yes	—
266	C₂²	C₂²×#6104	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	104	4	no	—
270	C₄	#11718694	1	2	yes	—
271	C₄	C₂×#88189	1	1	yes	characteristic
272	C₄	C₂×#88205	1	1	yes	characteristic
273	C₂	—	2	4	no	—
274	C₂	—	16	2	no	—
275	C₂	#13117019	1	4	yes	direct product factor
276	C₂	C₂×#11718694	1	1	yes	characteristic
277	C₂	C₂²×#88067	1	1	yes	characteristic
278	C₂	C₂×#11718854	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₂×C₁₃×#466	C₂
socle	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×#466	D₁₃
hypocenter	C₁₃	C₂×#2233
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂²×#88058	C₂	semidirect
2. #13117019	C₂	direct (N × H)
3. C₂×#11718082	C₂	semidirect
4. C₂×#11718154	C₂	semidirect
5. C₂×C₂²⋊#6090	C₂	semidirect
18. C₂×#88058	C₂²	semidirect
20. #11718154	C₂²	semidirect
21. C₂²⋊#6090	C₂²	semidirect
22. #11718082	C₂²	semidirect
186. C₂×C₁₃	#2233	semidirect
224. C₁₃	C₂×#2233	semidirect
275. C₂	#13117019	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×#88058	2
#13117019	2
C₂×#11718082	2
C₂×#11718154	2
C₂×C₂²⋊#6090	2
C₂×C₁₃×#466	2
C₂×#2233	13

Subgroups — C2×#13117019 #16017702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117019 #16017702