Subgroups — C₂×#13116341 #15892027

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718656	C₂	1	2	yes	maximal
2	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
3	#13116341	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718146	C₂	1	2	yes	maximal
5	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#424	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₁₃×#424	—	2	2	no	—
13	C₁₃×#424	C₂²	1	4	yes	—
14	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
15	C₂×#88058	—	2	2	no	—
16	C₂×#88058	—	2	2	no	—
17	C₂×#88058	C₂²	1	4	yes	—
18	#11718656	—	2	4	no	—
19	#11718656	C₂²	1	8	yes	—
20	C₂×#88048	—	2	2	no	—
21	C₂×#88047	—	2	2	no	—
22	#11718146	C₂²	1	8	yes	—
23	#11718144	C₂²	1	4	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	1	no	—
27	C₂×C₁₃×M₁₆	—	2	4	no	—
28	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	—	2	4	no	—
30	C₂²×C₈×C₁₃	—	2	1	no	—
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₁₃×#117	—	2	1	no	—
35	C₁₃×#117	—	2	1	no	—
36	C₁₃×#117	D₄	1	2	yes	—
37	C₁₃×#117	C₂³	1	2	yes	—
38	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
41	C₂×C₄²×C₁₃	—	2	1	no	—
42	C₂×C₄⋊Dic₁₃	—	2	2	no	—
43	C₂×C₄⋊Dic₁₃	—	2	2	no	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄⋊Dic₁₃	—	2	2	no	—
46	C₂×C₄⋊Dic₁₃	—	2	2	no	—
47	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	4	no	—
50	C₂×#6099	Q₈	1	8	yes	—
51	C₂×#6099	C₂³	1	4	yes	—
52	C₂²×#1055	—	2	2	no	—
53	C₁₃×C₄⋊C₈	—	2	8	no	—
54	#88058	—	2	8	no	—
55	#88048	—	2	8	no	—
56	#88047	—	2	8	no	—
57	C₂×#2089	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	#41	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	4	no	—
70	C₂²×Dic₁₃	—	4	2	no	—
71	C₂²×Dic₁₃	—	4	2	no	—
72	C₂²×Dic₁₃	—	8	4	no	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	—	4	2	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
81	C₁₃×C₄⋊C₄	—	2	1	no	—
82	C₁₃×C₄⋊C₄	—	2	1	no	—
83	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
84	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
85	C₁₃×C₂²⋊C₄	—	2	2	no	—
86	C₁₃×C₂²⋊C₄	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	—	2	4	no	—
91	C₄⋊Dic₁₃	—	4	2	no	—
92	C₄⋊Dic₁₃	—	4	2	no	—
93	#6099	—	2	8	no	—
94	#6099	—	2	4	no	—
95	#6099	C₂×Q₈	1	8	yes	—
96	C₂×#1055	—	2	8	no	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#428	—	13	2	no	—
99	C₂²×#107	—	13	1	no	—
100	C₂×#424	D₁₃	1	1	yes	characteristic
101	#2089	—	13	8	no	—
102	C₂×#343	—	13	2	no	—
103	C₂×#341	—	13	1	no	—
104	C₂³×C₁₃	—	2	2	no	—
105	C₂³×C₁₃	—	2	2	no	—
106	C₂³×C₁₃	#121	1	1	yes	characteristic
107	C₂³×C₁₃	#121	1	1	yes	characteristic
108	C₂³×C₁₃	#125	1	4	yes	—
109	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
118	C₂×C₄×C₁₃	#123	1	2	yes	—
119	C₂×C₄×C₁₃	#123	1	2	yes	—
120	C₂×C₄×C₁₃	#125	1	4	yes	—
121	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	8	4	no	—
129	C₂×Dic₁₃	—	8	4	no	—
130	C₂×Dic₁₃	—	8	4	no	—
131	C₈×C₁₃	—	2	4	no	—
132	C₈×C₁₃	—	4	2	no	—
133	#1055	—	2	8	no	—
134	C₂²×M₁₆	—	13	2	no	—
135	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
136	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
138	C₂²×C₄⋊C₄	—	13	2	no	—
139	#428	—	13	8	no	—
140	#428	—	26	4	no	—
141	C₂×#107	—	13	4	no	—
142	C₂×#107	—	26	2	no	—
143	C₂×#107	—	26	2	no	—
144	C₂×#107	—	26	2	no	—
145	C₂×#106	—	26	2	no	—
146	#424	—	2	2	no	—
147	#424	C₂×D₁₃	1	4	yes	—
148	C₂×C₄⋊C₈	—	2	2	no	—
149	#343	—	13	8	no	—
150	#341	—	13	4	no	—
151	C₂²×C₁₃	—	2	8	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	2	1	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	#398	1	2	yes	—
157	C₂²×C₁₃	#398	1	2	yes	—
158	C₂²×C₁₃	#472	1	2	yes	—
159	C₂²×C₁₃	#473	1	2	yes	—
160	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#125	1	2	yes	—
162	C₄×C₁₃	—	4	2	no	—
163	C₄×C₁₃	—	4	2	no	—
164	C₄×C₁₃	#398	1	2	yes	—
165	C₄×C₁₃	#398	1	2	yes	—
166	C₄×C₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
167	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
168	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
169	Dic₁₃	—	8	4	no	—
170	C₂³×C₄	—	26	2	no	—
171	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	4	no	—
174	C₂×M₁₆	—	13	8	no	—
175	C₂×M₁₆	—	13	4	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	—	26	4	no	—
178	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	26	2	no	—
182	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
183	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
184	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
185	#117	—	2	1	no	—
186	#117	—	2	1	no	—
187	#117	D₅₂	1	2	yes	—
188	#117	C₂²×D₁₃	1	2	yes	—
189	C₂×C₄⋊C₄	—	13	4	no	—
190	C₂×C₄⋊C₄	—	26	2	no	—
191	C₂×C₄⋊C₄	—	26	2	no	—
192	C₂×C₄⋊C₄	—	26	2	no	—
193	C₂×C₄⋊C₄	—	26	2	no	—
194	C₂×C₄⋊C₄	—	26	2	no	—
195	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
196	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	C₂×C₄²	—	2	1	no	—
199	#107	—	26	8	no	—
200	#107	—	26	8	no	—
201	#106	—	26	8	no	—
202	C₄⋊C₈	—	2	8	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	#2089	1	4	yes	complemented
206	C₂×C₁₃	C₂×#398	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
208	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
209	C₂⁴	#6127	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	26	4	no	—
213	C₂²×C₄	—	52	2	no	—
214	C₂²×C₄	—	52	2	no	—
215	C₂²×C₄	—	104	4	no	—
216	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
218	C₂²×C₄	#6127	1	1	yes	characteristic
219	C₂²×C₄	#6129	1	4	yes	—
220	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
221	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	13	8	no	—
227	M₁₆	—	26	8	no	—
228	M₁₆	—	26	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	26	8	no	—
235	C₂×C₈	—	26	4	no	—
236	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	26	4	no	—
240	C₄⋊C₄	—	26	4	no	—
241	C₄⋊C₄	—	52	2	no	—
242	C₄⋊C₄	—	52	2	no	—
243	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
244	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₁₃	C₂×#2089	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	#88143	1	4	yes	—
253	C₂³	#88184	1	1	yes	characteristic
254	C₂³	#88184	1	1	yes	characteristic
255	C₂³	C₂×#6127	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	52	4	no	—
261	C₂×C₄	—	104	4	no	—
262	C₂×C₄	—	104	4	no	—
263	C₂×C₄	—	104	4	no	—
264	C₂×C₄	#88143	1	4	yes	—
265	C₂×C₄	#88151	1	4	yes	—
266	C₂×C₄	#88151	1	4	yes	—
267	C₂×C₄	#88184	1	1	yes	characteristic
268	C₂×C₄	#88184	1	1	yes	characteristic
269	C₂×C₄	#88189	1	2	yes	—
270	C₂×C₄	#88189	1	2	yes	—
271	C₂×C₄	#88198	1	2	yes	—
272	C₂×C₄	#88198	1	2	yes	—
273	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
274	C₂×C₄	C₂×#6129	1	2	yes	—
275	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
276	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₈	—	4	2	no	—
279	C₈	—	26	8	no	—
280	C₂²	—	2	8	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#11718644	1	2	yes	—
286	C₂²	#11718644	1	2	yes	—
287	C₂²	#11718803	1	2	yes	—
288	C₂²	#11718804	1	2	yes	—
289	C₂²	C₂×#88143	1	2	yes	—
290	C₂²	C₂×#88184	1	1	yes	characteristic
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	104	4	no	—
294	C₄	#11718644	1	2	yes	—
295	C₄	#11718644	1	2	yes	—
296	C₄	C₂×#88151	1	2	yes	—
297	C₄	C₂×#88189	1	1	yes	characteristic
298	C₄	C₂×#88198	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#13116341	1	4	yes	direct product factor
302	C₂	C₂×#11718644	1	1	yes	characteristic
303	C₂	C₂×#11718803	1	1	yes	characteristic
304	C₂	C₂×#11718804	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88184
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#424	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂⁴×C₁₃	#41
hypercenter	C₂×#424	D₁₃
hypocenter	C₁₃	C₂×#2089
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
3. #13116341	C₂	direct (N × H)
205. C₂×C₁₃	#2089	semidirect
249. C₁₃	C₂×#2089	semidirect
301. C₂	#13116341	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×#11718656	2
C₂²×#88058	2
#13116341	2
C₂×#11718146	2
C₂×#11718144	2
C₂×C₁₃×#424	2
C₂×#2089	13

Subgroups — C2×#13116341 #15892027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116341 #15892027