Subgroups — C₂×#13131642 #16808455

Order 3328.

2566 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131642	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#1057	C₂	1	2	yes	maximal, complemented
3	C₂×#11719073	C₂	1	6	yes	maximal, complemented
4	C₂×#11719051	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×D₁₃⋊C₈	C₂	1	3	yes	maximal, complemented
6	C₂×C₄×#6131	C₂	1	1	yes	maximal, characteristic
7	C₂²×#6131	C₄	1	1	yes	characteristic
8	C₂²×D₁₃⋊C₈	C₂²	1	6	yes	—
9	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₄×#6131	C₄	1	8	yes	—
12	C₄×#6131	C₂²	1	8	yes	—
13	C₂×#88147	C₂²	1	3	yes	—
14	C₂×#88146	C₄	1	3	yes	—
15	C₂×C₄×D₅₂	C₄	1	3	yes	—
16	C₂×C₄²×D₁₃	C₂²	1	3	yes	—
17	Q₈×#1057	C₂²	1	16	yes	complemented
18	C₂×#88111	C₂²	1	6	yes	—
19	C₂×C₄⋊#1057	C₂²	1	6	yes	—
20	C₂×C₄×#1057	C₂²	1	6	yes	—
21	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
22	C₂×C₄×#1057	C₂²	1	3	yes	—
23	#11719073	C₂²	1	48	yes	complemented
24	#11719051	C₂²	1	24	yes	complemented
25	C₄×D₁₃⋊C₈	C₂²	1	24	yes	complemented
26	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#6131	C₈	1	16	yes	complemented
28	C₂×#6131	C₂×C₄	1	8	yes	—
29	C₂²×D₅₂	C₂×C₄	1	3	yes	—
30	C₂²×C₄×D₁₃	C₂×C₄	1	3	yes	—
31	C₂²×C₄×D₁₃	C₂×C₄	1	3	yes	—
32	C₂²×C₄×D₁₃	C₂³	1	3	yes	—
33	C₂²×#1057	C₂³	1	6	yes	—
34	C₂²×#1057	C₂³	1	2	yes	—
35	C₂×D₁₃⋊C₈	—	2	24	no	—
36	C₂×D₁₃⋊C₈	C₂³	1	24	yes	—
37	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
38	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
40	Q₈×Dic₁₃	C₂×C₄	1	4	yes	—
41	Q₈×Dic₁₃	C₂³	1	4	yes	—
42	C₂×#6103	C₂×C₄	1	6	yes	—
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
44	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
45	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
47	#88147	C₂×C₄	1	12	yes	—
48	#88147	C₂³	1	12	yes	—
49	#88146	C₂×C₄	1	24	yes	—
50	C₄×D₅₂	C₂×C₄	1	24	yes	—
51	C₄²×D₁₃	C₂×C₄	1	12	yes	—
52	C₄²×D₁₃	C₂³	1	12	yes	—
53	#88111	C₂³	1	24	yes	—
54	C₄⋊#1057	C₂³	1	24	yes	—
55	C₄×#1057	C₂³	1	4	yes	—
56	C₄×#1057	C₂³	1	12	yes	—
57	C₄×#1057	C₂³	1	24	yes	—
58	C₂×C₈×#41	—	13	1	no	maximal, Sylow
59	C₂³×D₁₃	C₂²×C₄	1	3	yes	—
60	C₂×Q₈×C₁₃	C₂×C₈	1	8	yes	complemented
61	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
66	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
68	#6131	C₂×C₈	1	64	yes	complemented
69	C₂×D₅₂	C₂×C₈	1	24	yes	complemented
70	C₂×D₅₂	C₂²×C₄	1	12	yes	—
71	C₂×C₄×D₁₃	—	2	12	no	—
72	C₂×C₄×D₁₃	—	2	12	no	—
73	C₂×C₄×D₁₃	C₂×C₈	1	24	yes	complemented
74	C₂×C₄×D₁₃	C₂²×C₄	1	6	yes	—
75	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
76	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
77	C₂×C₄×D₁₃	C₂⁴	1	6	yes	—
78	C₂×#1057	—	2	12	no	—
79	C₂×#1057	#41	1	8	yes	complemented
80	C₂×#1057	C₂⁴	1	12	yes	—
81	C₂×#1057	C₂⁴	1	4	yes	—
82	D₁₃⋊C₈	—	2	48	no	—
83	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
85	#6103	C₂²×C₄	1	24	yes	—
86	C₄⋊Dic₁₃	C₂²×C₄	1	12	yes	—
87	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
88	C₄×Dic₁₃	C₂²×C₄	1	6	yes	—
89	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
90	C₄×Dic₁₃	C₂⁴	1	2	yes	—
91	C₄×Dic₁₃	C₂⁴	1	6	yes	—
92	C₂×C₄×#41	—	13	1	no	—
93	C₈×#41	—	13	16	no	—
94	C₂×C₈×Q₈	—	13	2	no	—
95	C₂×C₈×D₄	—	13	6	no	—
96	C₂×#431	—	13	3	no	—
97	C₂²×C₄×C₈	—	13	3	no	—
98	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²×D₁₃	—	2	6	no	—
100	C₂²×D₁₃	—	2	6	no	—
101	C₂²×D₁₃	C₂²×C₈	1	12	yes	—
102	C₂²×D₁₃	C₂³×C₄	1	6	yes	—
103	Q₈×C₁₃	C₂²×C₈	1	16	yes	complemented
104	C₂×C₄×C₁₃	—	2	6	no	—
105	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
106	C₂×C₄×C₁₃	#131	1	4	yes	—
107	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
110	C₂×Dic₁₃	—	2	6	no	—
111	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
112	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
113	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
115	C₂×Dic₁₃	C₂×#41	1	4	yes	—
116	C₂×Dic₁₃	C₂⁵	1	1	yes	characteristic
117	D₅₂	C₂²×C₈	1	48	yes	complemented
118	C₄×D₁₃	—	2	24	no	—
119	C₄×D₁₃	—	2	12	no	—
120	C₄×D₁₃	—	2	12	no	—
121	C₄×D₁₃	C₂²×C₈	1	48	yes	complemented
122	#1057	—	2	12	no	—
123	#1057	C₂×#41	1	8	yes	complemented
124	C₂²×#41	—	13	1	no	—
125	C₂³×C₈	—	13	6	no	—
126	C₄×#41	—	13	8	no	—
127	C₄×#41	—	13	8	no	—
128	C₂×C₄×Q₈	—	13	1	no	—
129	C₂×C₄×Q₈	#261	1	1	yes	characteristic
130	C₂×C₄×D₄	—	13	3	no	—
131	C₂×C₄×D₄	—	13	3	no	—
132	C₂×#117	—	13	3	no	—
133	C₂²×C₄²	—	13	3	no	—
134	C₈×Q₈	—	13	16	no	—
135	C₈×D₄	—	13	48	no	—
136	#431	—	13	24	no	—
137	C₂×C₄⋊C₈	—	13	6	no	—
138	C₂×C₂²⋊C₈	—	13	6	no	—
139	C₂×C₄×C₈	—	13	1	no	—
140	C₂×C₄×C₈	—	13	3	no	—
141	C₂×C₄×C₈	—	13	6	no	—
142	C₂×C₄×C₈	—	13	24	no	—
143	C₂²×C₁₃	C₄×#41	1	2	yes	—
144	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
145	C₂²×C₁₃	C₂×#131	1	2	yes	—
146	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
147	C₂×D₁₃	—	2	6	no	—
148	C₂×D₁₃	—	2	24	no	—
149	C₂×D₁₃	—	2	6	no	—
150	C₂×D₁₃	C₂³×C₈	1	12	yes	—
151	C₄×C₁₃	—	2	6	no	—
152	C₄×C₁₃	C₂³×C₈	1	12	yes	—
153	C₄×C₁₃	C₂×#131	1	4	yes	—
154	Dic₁₃	—	2	6	no	—
155	Dic₁₃	C₄×#41	1	2	yes	—
156	Dic₁₃	C₂³×C₈	1	4	yes	—
157	Dic₁₃	C₂²×#41	1	2	yes	—
158	C₂×#41	—	13	16	no	—
159	C₂×#41	—	13	8	no	—
160	C₂²×Q₈	C₂×#261	1	1	yes	characteristic
161	C₂²×D₄	—	13	3	no	—
162	C₂³×C₄	—	13	3	no	—
163	C₂³×C₄	—	13	3	no	—
164	C₂³×C₄	—	13	3	no	—
165	C₂²×C₈	—	13	2	no	—
166	C₂²×C₈	—	13	6	no	—
167	C₂²×C₈	—	13	24	no	—
168	C₂²×C₈	—	26	24	no	—
169	C₄×Q₈	—	13	4	no	—
170	C₄×Q₈	—	13	4	no	—
171	C₄×Q₈	C₂×#261	1	8	yes	—
172	C₄×D₄	—	13	12	no	—
173	C₄×D₄	—	13	12	no	—
174	C₄×D₄	—	13	24	no	—
175	#117	—	13	24	no	—
176	C₂×C₄⋊C₄	—	13	3	no	—
177	C₂×C₄⋊C₄	C₂×#261	1	3	yes	—
178	C₂×C₂²⋊C₄	—	13	6	no	—
179	C₂×C₄²	—	13	3	no	—
180	C₂×C₄²	—	13	3	no	—
181	C₂×C₄²	—	13	1	no	—
182	C₂×C₄²	—	13	12	no	—
183	C₂×C₄²	—	13	12	no	—
184	C₂×C₄²	C₂×#261	1	3	yes	—
185	C₄⋊C₈	—	13	24	no	—
186	C₂²⋊C₈	—	13	24	no	—
187	C₄×C₈	—	13	4	no	—
188	C₄×C₈	—	13	12	no	—
189	C₄×C₈	—	13	24	no	—
190	C₂×C₁₃	C₈×#41	1	4	yes	complemented
191	C₂×C₁₃	C₂×C₄×#41	1	1	yes	characteristic
192	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
193	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
194	D₁₃	—	2	12	no	—
195	C₂⁴	—	13	3	no	—
196	#41	—	13	64	no	—
197	C₂×Q₈	D₁₃⋊C₈	1	8	yes	complemented
198	C₂×Q₈	C₂²×#261	1	4	yes	—
199	C₂×D₄	—	13	24	no	—
200	C₂×D₄	—	13	12	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	13	24	no	—
204	C₂²×C₄	—	13	12	no	—
205	C₂²×C₄	—	13	3	no	—
206	C₂²×C₄	—	13	6	no	—
207	C₂²×C₄	—	13	6	no	—
208	C₂²×C₄	—	13	12	no	—
209	C₂²×C₄	—	26	12	no	—
210	C₂²×C₄	—	26	12	no	—
211	C₂²×C₄	C₂²×#261	1	3	yes	—
212	C₂²×C₄	C₂²×#261	1	3	yes	—
213	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
214	C₂×C₈	—	13	4	no	—
215	C₂×C₈	—	13	8	no	—
216	C₂×C₈	—	13	12	no	—
217	C₂×C₈	—	26	48	no	—
218	C₂×C₈	—	26	12	no	—
219	C₄⋊C₄	—	13	12	no	—
220	C₄⋊C₄	C₂²×#261	1	12	yes	—
221	C₂²⋊C₄	—	13	24	no	—
222	C₄²	—	13	2	no	—
223	C₄²	—	13	6	no	—
224	C₄²	—	13	2	no	—
225	C₄²	—	13	12	no	—
226	C₄²	—	13	6	no	—
227	C₄²	C₂²×#261	1	12	yes	—
228	C₁₃	C₂×C₈×#41	1	1	yes	characteristic, complemented, Sylow
229	C₂³	—	13	12	no	—
230	C₂³	—	13	6	no	—
231	C₂³	—	26	6	no	—
232	C₂³	—	26	6	no	—
233	C₂³	C₂³×#261	1	1	yes	characteristic
234	Q₈	C₂×D₁₃⋊C₈	1	16	yes	complemented
235	D₄	—	13	48	no	—
236	C₂×C₄	—	2	6	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	13	1	no	—
240	C₂×C₄	—	13	1	no	—
241	C₂×C₄	—	13	2	no	—
242	C₂×C₄	—	13	48	no	—
243	C₂×C₄	—	13	6	no	—
244	C₂×C₄	—	26	12	no	—
245	C₂×C₄	—	26	12	no	—
246	C₂×C₄	—	26	6	no	—
247	C₂×C₄	—	26	24	no	—
248	C₂×C₄	C₂×D₁₃⋊C₈	1	12	yes	—
249	C₂×C₄	#88240	1	4	yes	—
250	C₂×C₄	C₂³×#261	1	6	yes	—
251	C₂×C₄	C₂³×#261	1	6	yes	—
252	C₂×C₄	C₂³×#261	1	2	yes	—
253	C₈	—	13	8	no	—
254	C₈	—	26	12	no	—
255	C₂²	—	13	12	no	—
256	C₂²	—	26	6	no	—
257	C₂²	—	26	6	no	—
258	C₂²	—	26	24	no	—
259	C₂²	C₂²×D₁₃⋊C₈	1	2	yes	—
260	C₂²	C₂×#88240	1	2	yes	—
261	C₂²	#41×#261	1	2	yes	—
262	C₂²	C₂⁴×#261	1	1	yes	characteristic
263	C₄	—	2	6	no	—
264	C₄	—	13	2	no	—
265	C₄	—	13	2	no	—
266	C₄	—	13	4	no	—
267	C₄	—	26	6	no	—
268	C₄	C₂²×D₁₃⋊C₈	1	12	yes	—
269	C₄	C₂×#88240	1	4	yes	—
270	C₂	—	26	12	no	—
271	C₂	#13131642	1	4	yes	direct product factor
272	C₂	C₂×#41×#261	1	1	yes	characteristic
273	C₂	C₂³×D₁₃⋊C₈	1	1	yes	characteristic
274	C₂	C₂²×#88240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₂²	C₂⁴×#261
Fitting	C₂×C₄×Q₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂²×#6131	C₄
cosocle	C₂×Dic₁₃	C₂⁵
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	C₂×C₈×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #13131642	C₂	direct (N × H)
2. C₂×Q₈×#1057	C₂	semidirect
3. C₂×#11719073	C₂	semidirect
4. C₂×#11719051	C₂	semidirect
5. C₂×C₄×D₁₃⋊C₈	C₂	semidirect
17. Q₈×#1057	C₂²	semidirect
23. #11719073	C₂²	semidirect
24. #11719051	C₂²	semidirect
25. C₄×D₁₃⋊C₈	C₂²	semidirect
27. C₂×#6131	C₈	semidirect
60. C₂×Q₈×C₁₃	C₂×C₈	semidirect
68. #6131	C₂×C₈	semidirect
69. C₂×D₅₂	C₂×C₈	semidirect
73. C₂×C₄×D₁₃	C₂×C₈	semidirect
79. C₂×#1057	#41	semidirect
103. Q₈×C₁₃	C₂²×C₈	semidirect
117. D₅₂	C₂²×C₈	semidirect
121. C₄×D₁₃	C₂²×C₈	semidirect
123. #1057	C₂×#41	semidirect
190. C₂×C₁₃	C₈×#41	semidirect
197. C₂×Q₈	D₁₃⋊C₈	semidirect
228. C₁₃	C₂×C₈×#41	semidirect
234. Q₈	C₂×D₁₃⋊C₈	semidirect
271. C₂	#13131642	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13131642	2
C₂×Q₈×#1057	2
C₂×#11719073	2
C₂×#11719051	2
C₂×C₄×D₁₃⋊C₈	2
C₂×C₄×#6131	2
C₂×C₈×#41	13

Subgroups — C2×#13131642 #16808455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131642 #16808455