Subgroups — C₂×#13117004 #16017646

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718361	C₂	1	1	yes	maximal, characteristic
2	#13117004	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718079	C₂	1	2	yes	maximal, complemented
4	C₂×#11718055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718080	C₂	1	2	yes	maximal
6	C₂×C₁₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₂×#88158	C₂²	1	1	yes	characteristic
9	C₂×#88155	—	2	1	no	—
10	C₁₃×#498	—	2	2	no	—
11	C₁₃×#498	C₂²	1	4	yes	complemented
12	C₂×C₁₃×#106	—	2	1	no	—
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88059	C₂²	1	1	yes	characteristic
16	C₂×#88052	C₂²	1	2	yes	—
17	C₂×#88045	—	2	1	no	—
18	#11718361	—	2	2	no	—
19	#11718361	C₂²	1	4	yes	—
20	#11718080	C₂²	1	8	yes	—
21	#11718079	C₂²	1	8	yes	complemented
22	#11718055	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₁₃	—	4	1	no	—
24	C₂×#6127	—	2	1	no	—
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×C₄⋊Q₈	—	2	4	no	—
28	C₁₃×C₄⋊Q₈	—	4	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×#6099	—	2	1	no	—
33	C₂×Dic₅₂	Q₈	1	1	yes	characteristic, complemented
34	#88158	—	2	1	no	—
35	#88158	—	2	1	no	—
36	#88158	Q₈	1	2	yes	complemented
37	#88158	C₂³	1	2	yes	—
38	C₂×D₁₀₄	Q₈	1	1	yes	characteristic, complemented
39	C₂×#6095	—	2	1	no	—
40	#88155	—	2	4	no	—
41	#88155	—	4	2	no	—
42	C₂×#6094	—	4	1	no	—
43	C₂×C₈×D₁₃	—	4	1	no	—
44	C₁₃×#106	—	2	4	no	—
45	C₁₃×#106	—	2	2	no	—
46	C₁₃×#106	C₂³	1	4	yes	—
47	C₄×C₈×C₁₃	D₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	#88059	—	2	1	no	—
50	#88059	C₂³	1	2	yes	—
51	#88052	—	2	2	no	—
52	#88052	C₂³	1	4	yes	—
53	C₂×#6090	—	2	2	no	—
54	#88045	—	2	4	no	—
55	C₂×#2230	—	13	1	no	maximal, Sylow
56	C₂×Q₈×C₁₃	—	4	4	no	—
57	C₂×Q₈×C₁₃	—	8	4	no	—
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	—	4	1	no	—
62	#6127	—	2	2	no	—
63	#6127	—	2	2	no	—
64	#6127	—	4	1	no	—
65	#6127	—	4	1	no	—
66	C₂×D₅₂	—	2	1	no	—
67	C₂×C₄×D₁₃	—	4	1	no	—
68	C₂×Dic₂₆	—	2	1	no	—
69	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
70	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
71	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	4	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₄²×C₁₃	—	2	1	no	—
83	C₄²×C₁₃	C₂×D₄	1	2	yes	—
84	#6099	—	2	2	no	—
85	#6099	—	4	1	no	—
86	C₂×#1055	—	4	1	no	—
87	Dic₅₂	—	2	1	no	—
88	Dic₅₂	C₂×Q₈	1	2	yes	complemented
89	D₁₀₄	—	2	1	no	—
90	D₁₀₄	C₂×Q₈	1	2	yes	complemented
91	#6095	—	2	2	no	—
92	#6095	—	2	2	no	—
93	#6094	—	4	4	no	—
94	C₈×D₁₃	—	4	4	no	—
95	#6090	—	2	4	no	—
96	C₂×#498	D₁₃	1	1	yes	characteristic, complemented
97	C₂×#443	—	13	1	no	—
98	#2230	—	13	8	no	—
99	C₂×#365	—	13	2	no	—
100	C₂×#364	—	13	1	no	—
101	C₂×#360	—	13	2	no	—
102	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂²×D₁₃	—	4	1	no	—
104	Q₈×C₁₃	—	8	8	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	8	4	no	—
109	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²⋊D₁₃	—	4	2	no	—
116	C₂²⋊D₁₃	—	4	2	no	—
117	C₂×Dic₁₃	—	4	1	no	—
118	D₅₂	—	2	2	no	—
119	D₅₂	—	4	1	no	—
120	C₄×D₁₃	—	4	2	no	—
121	C₄×D₁₃	—	4	2	no	—
122	Dic₂₆	—	2	2	no	—
123	Dic₂₆	—	4	1	no	—
124	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
127	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
128	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
129	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
130	#1055	—	4	2	no	—
131	C₂×#135	—	13	1	no	—
132	C₂×#131	—	26	1	no	—
133	C₂×C₄⋊Q₈	—	2	1	no	—
134	C₂×M₃₂	—	13	2	no	—
135	#498	—	2	2	no	—
136	#498	C₂×D₁₃	1	4	yes	complemented
137	#443	—	13	4	no	—
138	#443	—	26	2	no	—
139	C₂×#108	—	13	1	no	—
140	C₂×#106	—	2	1	no	—
141	C₂×#106	C₂×D₁₃	1	2	yes	—
142	C₂×#104	—	26	1	no	—
143	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
144	#365	—	13	8	no	—
145	#364	—	13	4	no	—
146	#360	—	13	8	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	#476	1	1	yes	characteristic
149	C₂²×C₁₃	#476	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×D₁₃	—	4	2	no	—
152	C₂×D₁₃	—	8	2	no	—
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	—	8	4	no	—
155	C₄×C₁₃	#476	1	1	yes	characteristic
156	C₄×C₁₃	#476	1	1	yes	characteristic
157	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₁₃	—	4	2	no	—
160	C₂×#41	—	26	1	no	—
161	C₂²×Q₈	—	4	1	no	—
162	#135	—	13	2	no	—
163	#135	—	13	2	no	—
164	#135	—	26	1	no	—
165	#135	—	26	1	no	—
166	C₂×Q₁₆	—	13	1	no	—
167	C₂×SD₁₆	—	26	1	no	—
168	C₂×D₈	—	13	1	no	—
169	#131	—	26	4	no	—
170	#131	—	52	2	no	—
171	C₂×M₁₆	—	26	1	no	—
172	C₂×M₁₆	—	52	1	no	—
173	C₂²×C₈	—	52	1	no	—
174	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
176	C₄⋊Q₈	—	2	4	no	—
177	C₄⋊Q₈	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	4	1	no	—
180	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
181	M₃₂	—	13	4	no	—
182	M₃₂	—	26	2	no	—
183	C₂×C₁₆	—	26	2	no	—
184	#108	—	13	2	no	—
185	#108	—	26	1	no	—
186	#106	—	2	4	no	—
187	#106	—	2	2	no	—
188	#106	C₂²×D₁₃	1	4	yes	—
189	#104	—	26	4	no	—
190	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
191	C₄×C₈	C₂²×D₁₃	1	2	yes	—
192	C₂×C₁₃	—	2	1	no	—
193	C₂×C₁₃	—	2	1	no	—
194	C₂×C₁₃	#2230	1	2	yes	complemented
195	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
196	D₁₃	—	8	2	no	—
197	#41	—	26	2	no	—
198	#41	—	26	2	no	—
199	#41	—	52	1	no	—
200	#41	—	52	1	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	8	4	no	—
203	C₂×Q₈	—	26	1	no	—
204	C₂×D₄	—	26	1	no	—
205	C₂×D₄	—	52	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	52	1	no	—
209	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
210	Q₁₆	—	13	2	no	—
211	Q₁₆	—	26	1	no	—
212	SD₁₆	—	26	2	no	—
213	SD₁₆	—	26	2	no	—
214	D₈	—	13	2	no	—
215	D₈	—	26	1	no	—
216	M₁₆	—	26	2	no	—
217	M₁₆	—	52	1	no	—
218	M₁₆	—	52	4	no	—
219	C₂×C₈	—	52	1	no	—
220	C₂×C₈	—	52	4	no	—
221	C₂×C₈	#6106	1	1	yes	characteristic
222	C₂×C₈	#6106	1	1	yes	characteristic
223	C₂×C₈	#6106	1	2	yes	—
224	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
225	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
226	C₂×C₈	#6131	1	1	yes	characteristic
227	C₂×C₈	#6131	1	1	yes	characteristic
228	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
229	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
236	C₁₆	—	26	4	no	—
237	C₁₃	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	52	1	no	—
239	C₂³	#88186	1	1	yes	characteristic
240	Q₈	—	8	8	no	—
241	Q₈	—	26	2	no	—
242	Q₈	—	52	1	no	—
243	D₄	—	26	2	no	—
244	D₄	—	52	2	no	—
245	D₄	—	52	2	no	—
246	D₄	—	52	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	52	1	no	—
252	C₂×C₄	—	52	2	no	—
253	C₂×C₄	—	52	2	no	—
254	C₂×C₄	#88186	1	1	yes	characteristic
255	C₂×C₄	C₂×#6106	1	1	yes	characteristic
256	C₂×C₄	C₂×#6106	1	1	yes	characteristic
257	C₂×C₄	#88200	1	1	yes	characteristic
258	C₂×C₄	#88200	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₈	—	52	2	no	—
261	C₈	C₂×#6106	1	2	yes	—
262	C₈	C₂×#6106	1	2	yes	—
263	C₈	#88200	1	1	yes	characteristic
264	C₈	#88200	1	1	yes	characteristic
265	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
266	C₈	C₂×#6131	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	52	2	no	—
269	C₂²	—	104	2	no	—
270	C₂²	#11718745	1	1	yes	characteristic
271	C₂²	#11718745	1	1	yes	characteristic
272	C₂²	C₂×#88186	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	8	4	no	—
275	C₄	—	52	2	no	—
276	C₄	#11718745	1	1	yes	characteristic
277	C₄	#11718745	1	1	yes	characteristic
278	C₄	C₂²×#6106	1	1	yes	characteristic
279	C₄	C₂×#88200	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	104	2	no	—
283	C₂	#13117004	1	2	yes	direct product factor
284	C₂	C₂×#11718745	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718745
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#498	C₂
socle	C₂²×C₁₃	#476
prime radical	C₂×#88158	C₂²
hypercenter	C₂×#498	D₁₃
hypocenter	C₁₃	C₂×#2230
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
2. #13117004	C₂	direct (N × H)
3. C₂×#11718079	C₂	semidirect
4. C₂×#11718055	C₂	semidirect
6. C₂×C₁₃×#498	C₂	semidirect
11. C₁₃×#498	C₂²	semidirect
21. #11718079	C₂²	semidirect
22. #11718055	C₂²	semidirect
33. C₂×Dic₅₂	Q₈	semidirect
36. #88158	Q₈	semidirect
38. C₂×D₁₀₄	Q₈	semidirect
88. Dic₅₂	C₂×Q₈	semidirect
90. D₁₀₄	C₂×Q₈	semidirect
96. C₂×#498	D₁₃	semidirect
136. #498	C₂×D₁₃	semidirect
194. C₂×C₁₃	#2230	semidirect
237. C₁₃	C₂×#2230	semidirect
283. C₂	#13117004	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×#11718361	2
#13117004	2
C₂×#11718079	2
C₂×#11718055	2
C₂×#11718080	2
C₂×C₁₃×#498	2
C₂×#2230	13

Subgroups — C2×#13117004 #16017646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117004 #16017646