Subgroups — C₂×#13116610 #16015932

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 304 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	#13116610	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718086	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	—	2	1	no	—
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88180	—	2	1	no	—
9	C₂×C₁₃×#108	—	2	1	no	—
10	C₂×C₁₃×#108	—	2	2	no	—
11	C₂×C₁₃×#108	C₂²	1	2	yes	—
12	C₂×C₁₃×#108	C₂²	1	4	yes	—
13	#11718767	—	2	2	no	—
14	#11718767	C₂²	1	4	yes	—
15	C₂×#88058	—	2	1	no	—
16	C₂×#88057	C₂²	1	2	yes	—
17	C₂×#88052	C₄	1	2	yes	complemented
18	C₂×#88052	C₄	1	4	yes	complemented
19	C₂×#88052	C₂²	1	2	yes	—
20	C₂×#88052	C₂²	1	4	yes	—
21	C₂²×#6090	C₄	1	2	yes	complemented
22	#11718086	C₂²	1	16	yes	complemented
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	—	4	1	no	—
27	C₂×C₁₃×M₁₆	—	4	2	no	—
28	C₂²×C₈×C₁₃	—	4	1	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	2	yes	—
34	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
36	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
40	#88180	—	2	4	no	—
41	#88180	—	4	2	no	—
42	C₂×C₄⋊Dic₁₃	—	2	2	no	—
43	C₂×C₄×Dic₁₃	—	4	1	no	—
44	C₁₃×#108	—	2	2	no	—
45	C₁₃×#108	—	2	4	no	—
46	C₁₃×#108	C₂³	1	4	yes	—
47	#88058	—	2	4	no	—
48	#88057	—	2	2	no	—
49	#88057	C₂³	1	4	yes	—
50	#88052	C₂×C₄	1	4	yes	—
51	#88052	C₂×C₄	1	8	yes	complemented
52	#88052	C₂³	1	4	yes	—
53	C₂×#6090	C₂×C₄	1	4	yes	—
54	C₂×#6090	C₂×C₄	1	8	yes	complemented
55	C₂×#2155	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂²×Dic₁₃	—	4	2	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₁₃×M₁₆	—	4	4	no	—
71	C₂×C₈×C₁₃	—	4	4	no	—
72	C₂×C₈×C₁₃	—	4	2	no	—
73	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×Q₈	1	2	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	C₂²×C₄	1	8	yes	—
96	C₂²×M₃₂	—	13	1	no	—
97	C₂²×#108	D₁₃	1	1	yes	characteristic
98	C₂×#427	—	13	1	no	—
99	#2155	—	13	8	no	—
100	C₂×#365	—	13	4	no	—
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	#106	1	1	yes	characteristic
103	C₂³×C₁₃	#106	1	1	yes	characteristic
104	C₂³×C₁₃	#106	1	2	yes	—
105	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
107	C₂×C₄×C₁₃	#106	1	2	yes	—
108	C₂×C₄×C₁₃	#106	1	2	yes	—
109	C₂×C₄×C₁₃	#106	1	2	yes	—
110	C₂×C₄×C₁₃	#106	1	2	yes	—
111	C₂×C₄×C₁₃	#106	1	2	yes	—
112	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	8	4	no	—
127	C₈×C₁₃	—	4	4	no	—
128	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂²×M₁₆	—	2	1	no	—
135	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
136	C₂×#117	—	26	1	no	—
137	C₂×M₃₂	—	13	2	no	—
138	C₂×M₃₂	—	13	4	no	—
139	C₂×M₃₂	—	13	2	no	—
140	C₂×M₃₂	—	13	4	no	—
141	C₂²×C₁₆	—	13	2	no	—
142	C₂×#108	—	2	1	no	—
143	C₂×#108	—	2	2	no	—
144	C₂×#108	C₂×D₁₃	1	2	yes	—
145	C₂×#108	C₂×D₁₃	1	4	yes	—
146	#427	—	13	4	no	—
147	#427	—	26	2	no	—
148	C₂×#107	—	26	1	no	—
149	C₂×#106	—	13	2	no	—
150	#365	—	13	16	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	C₂×#106	1	2	yes	—
157	C₂²×C₁₃	C₂×#106	1	2	yes	—
158	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
160	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
161	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
162	C₄×C₁₃	C₂×#106	1	2	yes	—
163	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
164	C₄×C₁₃	C₂×#106	1	2	yes	—
165	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
167	Dic₁₃	—	8	4	no	—
168	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	4	1	no	—
172	C₂×M₁₆	—	4	2	no	—
173	C₂²×C₈	—	4	1	no	—
174	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
175	C₂²×C₈	Dic₂₆	1	2	yes	—
176	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₁₃	1	2	yes	—
180	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
184	#117	—	26	4	no	—
185	#117	—	52	2	no	—
186	C₂×C₄⋊C₄	—	26	2	no	—
187	C₂×C₂²⋊C₄	—	52	1	no	—
188	C₂×C₄²	—	52	1	no	—
189	M₃₂	—	13	4	no	—
190	M₃₂	—	13	8	no	—
191	M₃₂	—	13	4	no	—
192	C₂×C₁₆	—	13	4	no	—
193	C₂×C₁₆	—	13	8	no	—
194	#108	—	2	2	no	—
195	#108	—	2	4	no	—
196	#108	C₂²×D₁₃	1	4	yes	—
197	#107	—	26	4	no	—
198	#106	—	13	4	no	—
199	#106	—	26	2	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	#2155	1	2	yes	complemented
205	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
206	C₂⁴	#6106	1	1	yes	characteristic
207	C₂²×C₄	—	52	2	no	—
208	C₂²×C₄	#6101	1	1	yes	characteristic
209	C₂²×C₄	#6101	1	2	yes	—
210	C₂²×C₄	#6101	1	1	yes	characteristic
211	C₂²×C₄	#6105	1	1	yes	characteristic
212	C₂²×C₄	#6105	1	2	yes	—
213	C₂²×C₄	#6105	1	1	yes	characteristic
214	C₂²×C₄	#6106	1	2	yes	—
215	C₂²×C₄	#6106	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	#6101	1	1	yes	characteristic
224	C₂×C₈	#6101	1	2	yes	—
225	C₂×C₈	#6101	1	2	yes	—
226	C₂×C₈	#6101	1	1	yes	characteristic
227	C₂×C₈	#6101	1	2	yes	—
228	C₂×C₈	#6101	1	2	yes	—
229	C₂×C₈	#6101	1	2	yes	—
230	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
231	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
232	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
233	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
240	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
241	C₄⋊C₄	—	26	4	no	—
242	C₄⋊C₄	—	52	2	no	—
243	C₂²⋊C₄	—	52	4	no	—
244	C₄²	—	52	4	no	—
245	C₁₆	—	13	8	no	—
246	C₁₃	C₂×#2155	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	4	2	no	—
248	C₂³	#88048	1	1	yes	characteristic
249	C₂³	#88048	1	1	yes	characteristic
250	C₂³	#88048	1	2	yes	—
251	C₂³	C₂×#6106	1	1	yes	characteristic
252	C₂³	C₂×#6106	1	2	yes	—
253	C₂×C₄	—	52	4	no	—
254	C₂×C₄	—	104	4	no	—
255	C₂×C₄	#88048	1	2	yes	—
256	C₂×C₄	#88048	1	2	yes	—
257	C₂×C₄	#88048	1	1	yes	characteristic
258	C₂×C₄	#88048	1	2	yes	—
259	C₂×C₄	#88048	1	2	yes	—
260	C₂×C₄	#88048	1	2	yes	—
261	C₂×C₄	#88048	1	1	yes	characteristic
262	C₂×C₄	C₂×#6101	1	1	yes	characteristic
263	C₂×C₄	C₂×#6101	1	2	yes	—
264	C₂×C₄	C₂×#6101	1	1	yes	characteristic
265	C₂×C₄	C₂×#6101	1	2	yes	—
266	C₂×C₄	C₂×#6105	1	1	yes	characteristic
267	C₂×C₄	C₂×#6105	1	2	yes	—
268	C₂×C₄	C₂×#6105	1	1	yes	characteristic
269	C₂×C₄	C₂×#6105	1	2	yes	—
270	C₂×C₄	C₂×#6106	1	2	yes	—
271	C₂×C₄	C₂×#6106	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₈	—	4	4	no	—
274	C₈	C₂×#6101	1	1	yes	characteristic
275	C₈	C₂×#6101	1	2	yes	—
276	C₈	C₂×#6101	1	1	yes	characteristic
277	C₈	C₂×#6101	1	2	yes	—
278	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
279	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₂×#88048	1	2	yes	—
286	C₂²	C₂×#88048	1	1	yes	characteristic
287	C₂²	C₂×#88048	1	2	yes	—
288	C₂²	C₂×#88048	1	1	yes	characteristic
289	C₂²	C₂²×#6106	1	1	yes	characteristic
290	C₄	—	104	4	no	—
291	C₄	C₂×#88048	1	1	yes	characteristic
292	C₄	C₂×#88048	1	2	yes	—
293	C₄	C₂×#88048	1	1	yes	characteristic
294	C₄	C₂×#88048	1	2	yes	—
295	C₄	C₂²×#6101	1	1	yes	characteristic
296	C₄	C₂²×#6105	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#13116610	1	2	yes	direct product factor
302	C₂	C₂²×#88048	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88048
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂⁴×C₁₃	SD₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	C₂×#2155
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #13116610	C₂	direct (N × H)
4. C₂×#11718086	C₂	semidirect
17. C₂×#88052	C₄	semidirect
18. C₂×#88052	C₄	semidirect
21. C₂²×#6090	C₄	semidirect
22. #11718086	C₂²	semidirect
51. #88052	C₂×C₄	semidirect
54. C₂×#6090	C₂×C₄	semidirect
204. C₂×C₁₃	#2155	semidirect
246. C₁₃	C₂×#2155	semidirect
301. C₂	#13116610	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
#13116610	2
C₂×#11718086	2
C₂²×C₁₃×#108	2
C₂×#2155	13

Subgroups — C2×#13116610 #16015932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116610 #16015932