Subgroups — C₂×#13127515 #16742524

Order 3328.

1830 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719275	C₂	1	1	yes	maximal, characteristic
3	#13127515	C₂	1	16	yes	maximal, direct product factor
4	C₂×#11718342	C₂	1	3	yes	maximal
5	C₂×#11718511	C₂	1	3	yes	maximal
6	C₂×#11718510	C₂	1	3	yes	maximal
7	C₂×C₄×Dic₅₂	C₂	1	3	yes	maximal
8	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₃	—	2	1	no	—
11	C₂×Q₈×Dic₁₃	—	2	1	no	—
12	C₂²×Dic₅₂	C₂²	1	3	yes	—
13	#11719275	—	2	4	no	—
14	#11719275	C₂²	1	8	yes	—
15	Q₈×Dic₂₆	—	2	4	no	—
16	Q₈×Dic₂₆	C₂²	1	8	yes	—
17	C₂×#88144	—	2	3	no	—
18	C₂×#88142	—	2	3	no	—
19	C₂×#88123	C₂²	1	3	yes	—
20	C₂×#88123	C₂²	1	3	yes	—
21	C₂×C₄×Dic₂₆	C₂²	1	3	yes	—
22	C₂×C₄×Dic₂₆	C₂²	1	3	yes	—
23	C₈×Q₈×C₁₃	C₂²	1	8	yes	—
24	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
26	C₂×#88058	C₂²	1	3	yes	—
27	C₂×#88056	C₂²	1	6	yes	—
28	C₂×#88056	C₂²	1	1	yes	characteristic
29	C₂×#88056	C₂²	1	3	yes	—
30	#11718511	C₂²	1	24	yes	—
31	#11718510	C₂²	1	24	yes	—
32	C₄×Dic₅₂	C₂²	1	24	yes	—
33	#11718342	C₂²	1	24	yes	—
34	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×Dic₂₆	—	2	3	no	—
36	C₂²×Dic₂₆	C₂³	1	3	yes	—
37	C₂²×Dic₂₆	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₁₃	D₄	1	4	yes	—
41	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
42	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
44	Q₈×Dic₁₃	—	2	8	no	—
45	Q₈×Dic₁₃	—	2	8	no	—
46	C₂×C₄⋊Dic₁₃	—	2	3	no	—
47	C₂×C₄⋊Dic₁₃	—	2	3	no	—
48	C₂×C₄⋊Dic₁₃	—	2	3	no	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊Dic₁₃	C₂³	1	3	yes	—
51	C₂×#6101	—	2	3	no	—
52	C₂×#6101	—	2	3	no	—
53	C₂×C₄×Dic₁₃	—	2	3	no	—
54	C₂×C₄×Dic₁₃	—	2	3	no	—
55	C₂×Dic₅₂	—	2	12	no	—
56	C₂×Dic₅₂	C₂³	1	12	yes	—
57	#88144	—	2	24	no	—
58	#88142	—	2	24	no	—
59	#88123	—	2	6	no	—
60	#88123	—	2	6	no	—
61	#88123	C₂³	1	12	yes	—
62	#88123	C₂³	1	12	yes	—
63	C₄×Dic₂₆	—	2	6	no	—
64	C₄×Dic₂₆	—	2	6	no	—
65	C₄×Dic₂₆	C₂³	1	12	yes	—
66	C₄×Dic₂₆	C₂³	1	12	yes	—
67	C₁₃×C₄⋊C₈	C₂³	1	12	yes	—
68	C₄×C₈×C₁₃	C₂³	1	12	yes	—
69	#88058	C₂³	1	12	yes	—
70	#88056	C₂³	1	24	yes	—
71	#88056	C₂³	1	12	yes	—
72	#88056	C₂³	1	4	yes	—
73	C₂×#3363	—	13	1	no	maximal, Sylow
74	C₂×Q₈×C₁₃	Q₁₆	1	8	yes	complemented
75	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	2	1	no	—
80	C₂²×Dic₁₃	—	2	1	no	—
81	C₂²×Dic₁₃	—	2	3	no	—
82	C₂²×Dic₁₃	—	2	3	no	—
83	C₂×Dic₂₆	—	2	1	no	—
84	C₂×Dic₂₆	—	2	2	no	—
85	C₂×Dic₂₆	—	2	6	no	—
86	C₂×Dic₂₆	—	2	6	no	—
87	C₂×Dic₂₆	—	2	3	no	—
88	C₂×Dic₂₆	—	2	12	no	—
89	C₂×Dic₂₆	—	8	6	no	—
90	C₂×Dic₂₆	#41	1	4	yes	—
91	C₂×Dic₂₆	C₂⁴	1	6	yes	—
92	C₂×Dic₂₆	C₂⁴	1	2	yes	—
93	C₂×C₈×C₁₃	—	2	6	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
96	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
97	C₁₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
98	C₄²×C₁₃	C₂×D₄	1	6	yes	—
99	C₄²×C₁₃	C₂⁴	1	6	yes	—
100	C₄⋊Dic₁₃	—	2	1	no	—
101	C₄⋊Dic₁₃	—	2	3	no	—
102	C₄⋊Dic₁₃	—	2	12	no	—
103	C₄⋊Dic₁₃	—	2	12	no	—
104	C₄⋊Dic₁₃	—	2	12	no	—
105	C₄⋊Dic₁₃	C₂⁴	1	2	yes	—
106	C₄⋊Dic₁₃	C₂⁴	1	6	yes	—
107	#6101	—	2	12	no	—
108	#6101	—	2	12	no	—
109	C₄×Dic₁₃	—	2	12	no	—
110	C₄×Dic₁₃	—	2	12	no	—
111	Dic₅₂	—	2	24	no	—
112	C₂×Q₈²	—	13	1	no	—
113	C₂×#557	—	13	1	no	—
114	#3363	—	13	16	no	—
115	C₂×#487	—	13	3	no	—
116	C₂×#477	—	13	3	no	—
117	C₂×#462	—	13	3	no	—
118	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
119	C₂×C₄×Q₁₆	—	13	3	no	—
120	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₁₃	C₂×Q₁₆	1	16	yes	complemented
122	C₂×C₄×C₁₃	—	2	6	no	—
123	C₂×C₄×C₁₃	C₂×Q₁₆	1	12	yes	—
124	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
129	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
130	C₂×Dic₁₃	—	2	4	no	—
131	C₂×Dic₁₃	—	2	2	no	—
132	C₂×Dic₁₃	—	2	2	no	—
133	C₂×Dic₁₃	—	2	6	no	—
134	C₂×Dic₁₃	—	2	6	no	—
135	C₂×Dic₁₃	—	4	6	no	—
136	C₂×Dic₁₃	—	4	6	no	—
137	C₂×Dic₁₃	—	4	2	no	—
138	Dic₂₆	—	2	4	no	—
139	Dic₂₆	—	2	2	no	—
140	Dic₂₆	—	2	6	no	—
141	Dic₂₆	—	2	6	no	—
142	Dic₂₆	—	4	6	no	—
143	Dic₂₆	—	8	12	no	—
144	Dic₂₆	C₂×#41	1	4	yes	—
145	C₈×C₁₃	—	2	6	no	—
146	C₈×C₁₃	—	2	2	no	—
147	C₂²×Q₁₆	—	13	3	no	—
148	Q₈²	—	13	8	no	—
149	Q₈²	—	26	4	no	—
150	#557	—	13	8	no	—
151	#557	—	26	4	no	—
152	C₂×C₄⋊Q₈	—	13	3	no	—
153	C₂×C₄⋊Q₈	—	13	3	no	—
154	C₂×C₄⋊Q₈	—	26	3	no	—
155	C₂×#125	—	26	3	no	—
156	C₂×C₄×Q₈	—	13	3	no	—
157	C₂×C₄×Q₈	—	13	3	no	—
158	C₂×C₄×Q₈	—	26	1	no	—
159	C₂×C₄×Q₈	—	26	1	no	—
160	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
161	#487	—	13	24	no	—
162	#477	—	13	24	no	—
163	#462	—	13	24	no	—
164	C₈×Q₈	C₂×D₁₃	1	8	yes	—
165	C₄×Q₁₆	—	13	24	no	—
166	C₂×#107	—	13	3	no	—
167	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
168	C₂×#103	—	13	6	no	—
169	C₂×#103	—	13	3	no	—
170	C₂×#103	—	13	1	no	—
171	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
172	C₂²×C₁₃	#548	1	2	yes	—
173	C₂²×C₁₃	C₂²×Q₁₆	1	2	yes	—
174	C₂²×C₁₃	#577	1	2	yes	—
175	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
176	C₄×C₁₃	—	2	6	no	—
177	C₄×C₁₃	#548	1	2	yes	—
178	C₄×C₁₃	C₂²×Q₁₆	1	12	yes	—
179	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
180	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
181	Dic₁₃	—	2	4	no	—
182	Dic₁₃	—	4	6	no	—
183	Dic₁₃	—	4	6	no	—
184	Dic₁₃	—	4	2	no	—
185	C₂²×Q₈	—	13	1	no	—
186	C₂²×Q₈	—	13	3	no	—
187	C₂²×Q₈	—	26	3	no	—
188	C₂²×Q₈	D₅₂	1	1	yes	characteristic
189	C₂×Q₁₆	—	13	12	no	—
190	C₂×Q₁₆	—	26	12	no	—
191	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
192	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	13	12	no	—
194	C₄⋊Q₈	—	13	12	no	—
195	C₄⋊Q₈	—	26	24	no	—
196	C₄⋊Q₈	—	26	6	no	—
197	C₄⋊Q₈	—	26	6	no	—
198	#125	—	26	24	no	—
199	C₄×Q₈	—	13	12	no	—
200	C₄×Q₈	—	13	12	no	—
201	C₄×Q₈	—	26	8	no	—
202	C₄×Q₈	—	26	8	no	—
203	C₄×Q₈	—	26	6	no	—
204	C₄×Q₈	—	26	6	no	—
205	C₄×Q₈	D₅₂	1	4	yes	—
206	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
207	C₂×C₄⋊C₄	—	13	1	no	—
208	C₂×C₄⋊C₄	—	13	3	no	—
209	C₂×C₄⋊C₄	—	26	3	no	—
210	C₂×C₄⋊C₄	—	26	3	no	—
211	C₂×C₄⋊C₄	—	26	3	no	—
212	C₂×C₄⋊C₄	—	26	3	no	—
213	C₂×C₄⋊C₄	—	26	3	no	—
214	C₂×C₄⋊C₄	D₅₂	1	3	yes	—
215	C₂×C₄²	—	26	3	no	—
216	C₂×C₄²	—	26	3	no	—
217	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
218	#107	—	13	12	no	—
219	C₄⋊C₈	C₂²×D₁₃	1	12	yes	—
220	#103	—	13	4	no	—
221	#103	—	13	12	no	—
222	#103	—	13	24	no	—
223	C₄×C₈	C₂²×D₁₃	1	12	yes	—
224	C₂×C₁₃	#3363	1	4	yes	complemented
225	C₂×C₁₃	C₂×#548	1	1	yes	characteristic
226	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
227	C₂×C₁₃	C₂×#577	1	1	yes	characteristic
228	C₂×Q₈	—	13	4	no	—
229	C₂×Q₈	—	13	2	no	—
230	C₂×Q₈	—	13	6	no	—
231	C₂×Q₈	—	26	1	no	—
232	C₂×Q₈	—	26	3	no	—
233	C₂×Q₈	—	26	12	no	—
234	C₂×Q₈	—	26	2	no	—
235	C₂×Q₈	—	26	6	no	—
236	C₂×Q₈	—	26	6	no	—
237	C₂×Q₈	—	104	6	no	—
238	C₂×Q₈	Dic₅₂	1	8	yes	complemented
239	C₂×Q₈	C₂×D₅₂	1	4	yes	—
240	C₂²×C₄	—	26	1	no	—
241	C₂²×C₄	—	26	1	no	—
242	C₂²×C₄	—	26	3	no	—
243	C₂²×C₄	—	26	3	no	—
244	C₂²×C₄	C₂×D₅₂	1	3	yes	—
245	C₂²×C₄	C₂×D₅₂	1	3	yes	—
246	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
247	Q₁₆	—	26	24	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	6	no	—
250	C₂×C₈	C₂³×D₁₃	1	2	yes	—
251	C₂×C₈	C₂³×D₁₃	1	6	yes	—
252	C₄⋊C₄	—	13	6	no	—
253	C₄⋊C₄	—	13	2	no	—
254	C₄⋊C₄	—	26	1	no	—
255	C₄⋊C₄	—	26	3	no	—
256	C₄⋊C₄	—	26	12	no	—
257	C₄⋊C₄	—	26	12	no	—
258	C₄⋊C₄	—	26	12	no	—
259	C₄⋊C₄	—	26	12	no	—
260	C₄⋊C₄	—	26	12	no	—
261	C₄⋊C₄	C₂×D₅₂	1	12	yes	—
262	C₄²	—	26	12	no	—
263	C₄²	—	26	12	no	—
264	C₄²	C₂×D₅₂	1	6	yes	—
265	C₄²	C₂³×D₁₃	1	6	yes	—
266	C₁₃	C₂×#3363	1	1	yes	characteristic, complemented, Sylow
267	C₂³	C₂²×D₅₂	1	1	yes	characteristic
268	Q₈	—	13	4	no	—
269	Q₈	—	26	4	no	—
270	Q₈	—	26	2	no	—
271	Q₈	—	26	6	no	—
272	Q₈	—	26	6	no	—
273	Q₈	—	52	6	no	—
274	Q₈	—	104	12	no	—
275	Q₈	C₂×Dic₅₂	1	16	yes	complemented
276	C₂×C₄	—	2	6	no	—
277	C₂×C₄	—	26	4	no	—
278	C₂×C₄	—	26	2	no	—
279	C₂×C₄	—	26	2	no	—
280	C₂×C₄	—	26	6	no	—
281	C₂×C₄	—	26	6	no	—
282	C₂×C₄	—	52	2	no	—
283	C₂×C₄	—	52	6	no	—
284	C₂×C₄	—	52	6	no	—
285	C₂×C₄	C₂×Dic₅₂	1	12	yes	—
286	C₂×C₄	C₂²×D₅₂	1	6	yes	—
287	C₂×C₄	C₂²×D₅₂	1	6	yes	—
288	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
289	C₂×C₄	#88254	1	2	yes	—
290	C₂×C₄	#41×D₁₃	1	2	yes	—
291	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	2	6	no	—
294	C₂²	#11719281	1	2	yes	—
295	C₂²	C₂²×Dic₅₂	1	2	yes	—
296	C₂²	#11719456	1	2	yes	—
297	C₂²	C₂³×D₅₂	1	1	yes	characteristic
298	C₄	—	2	6	no	—
299	C₄	—	26	4	no	—
300	C₄	—	52	6	no	—
301	C₄	—	52	2	no	—
302	C₄	—	52	6	no	—
303	C₄	#11719281	1	2	yes	—
304	C₄	C₂²×Dic₅₂	1	12	yes	—
305	C₄	C₂×#88254	1	1	yes	characteristic
306	C₄	C₂×#41×D₁₃	1	1	yes	characteristic
307	C₂	#13127515	1	4	yes	direct product factor
308	C₂	C₂×#11719281	1	1	yes	characteristic
309	C₂	C₂³×Dic₅₂	1	1	yes	characteristic
310	C₂	C₂×#11719456	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	C₂×#3363
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #13127515	C₂	direct (N × H)
74. C₂×Q₈×C₁₃	Q₁₆	semidirect
121. Q₈×C₁₃	C₂×Q₁₆	semidirect
224. C₂×C₁₃	#3363	semidirect
238. C₂×Q₈	Dic₅₂	semidirect
266. C₁₃	C₂×#3363	semidirect
275. Q₈	C₂×Dic₅₂	semidirect
307. C₂	#13127515	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×Q₈×Dic₂₆	2
C₂×#11719275	2
#13127515	2
C₂×#11718342	2
C₂×#11718511	2
C₂×#11718510	2
C₂×C₄×Dic₅₂	2
C₂×C₈×Q₈×C₁₃	2
C₂×#3363	13

Subgroups — C2×#13127515 #16742524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13127515 #16742524