Subgroups — C₂×#11719438 #13128217

Order 1664.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88201	C₂	1	2	yes	maximal, complemented
3	C₂×#88200	C₂	1	2	yes	maximal, complemented
4	#11719438	C₂	1	16	yes	maximal, direct product factor
5	C₂×#88152	C₂	1	2	yes	maximal, complemented
6	C₂×#88151	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88149	C₂	1	1	yes	maximal, characteristic
8	C₂×#88148	C₂	1	2	yes	maximal, complemented
9	C₂×#88147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88141	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#88129	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88126	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₂²×D₅₂	C₂²	1	1	yes	characteristic
15	C₂²×D₅₂	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
18	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
24	#88201	C₂²	1	16	yes	complemented
25	#88200	C₂²	1	16	yes	complemented
26	C₂×#6103	C₂²	1	1	yes	characteristic
27	C₂×#6103	C₂²	1	1	yes	characteristic
28	C₂×#6103	C₂²	1	2	yes	—
29	C₂×#6103	C₂²	1	2	yes	—
30	C₂×#6103	C₂²	1	2	yes	—
31	C₂×#6103	C₂²	1	2	yes	—
32	C₂×#6103	C₂²	1	2	yes	—
33	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
34	C₂×#6101	C₂²	1	1	yes	characteristic
35	C₂×#6101	C₂²	1	1	yes	characteristic
36	C₂×#6101	C₂²	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
39	#88152	C₂²	1	16	yes	complemented
40	#88151	C₂²	1	8	yes	complemented
41	#88149	C₂²	1	8	yes	—
42	#88148	C₂²	1	16	yes	complemented
43	#88147	C₂²	1	8	yes	complemented
44	#88141	C₂²	1	8	yes	complemented
45	#88129	C₂²	1	8	yes	complemented
46	#88126	C₂²	1	8	yes	complemented
47	C₂³×D₁₃	C₂³	1	1	yes	characteristic
48	C₂³×D₁₃	C₂³	1	2	yes	—
49	C₂×Q₈×C₁₃	—	4	4	no	—
50	C₂×Q₈×C₁₃	C₂³	1	8	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂³	1	2	yes	—
59	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
60	C₂×D₅₂	—	2	4	no	—
61	C₂×D₅₂	—	4	4	no	—
62	C₂×D₅₂	C₂³	1	4	yes	—
63	C₂×D₅₂	C₂³	1	8	yes	—
64	C₂×C₄×D₁₃	—	2	4	no	—
65	C₂×C₄×D₁₃	—	4	4	no	—
66	C₂×C₄×D₁₃	C₂³	1	4	yes	—
67	C₂×C₄×D₁₃	C₂³	1	8	yes	—
68	C₂×Dic₂₆	—	2	4	no	—
69	C₂×Dic₂₆	C₂³	1	4	yes	—
70	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
71	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
72	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
73	C₄²×C₁₃	C₂³	1	4	yes	—
74	#6103	C₂³	1	4	yes	—
75	#6103	C₂³	1	4	yes	—
76	#6103	C₂³	1	8	yes	—
77	#6103	C₂³	1	8	yes	—
78	#6103	C₂³	1	8	yes	—
79	#6103	C₂³	1	8	yes	—
80	#6103	C₂³	1	8	yes	—
81	C₄⋊Dic₁₃	C₂³	1	8	yes	—
82	#6101	C₂³	1	4	yes	—
83	#6101	C₂³	1	4	yes	—
84	#6101	C₂³	1	8	yes	—
85	C₄×Dic₁₃	C₂³	1	4	yes	—
86	C₄×Dic₁₃	C₂³	1	8	yes	—
87	C₂×#542	—	13	1	no	maximal, Sylow
88	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂²×D₁₃	—	2	2	no	—
90	C₂²×D₁₃	—	2	4	no	—
91	C₂²×D₁₃	—	2	4	no	—
92	C₂²×D₁₃	—	2	2	no	—
93	C₂²×D₁₃	—	2	2	no	—
94	C₂²×D₁₃	—	2	4	no	—
95	C₂²×D₁₃	C₂⁴	1	2	yes	—
96	C₂²×D₁₃	C₂⁴	1	4	yes	—
97	Q₈×C₁₃	—	4	8	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	#41	1	4	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
105	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
106	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
107	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
108	C₂×Dic₁₃	—	2	4	no	—
109	C₂×Dic₁₃	—	2	2	no	—
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	—	2	2	no	—
112	C₂×Dic₁₃	C₂⁴	1	2	yes	—
113	C₂×Dic₁₃	C₂⁴	1	2	yes	—
114	C₂×Dic₁₃	C₂⁴	1	4	yes	—
115	C₂×Dic₁₃	C₂⁴	1	2	yes	—
116	D₅₂	—	2	8	no	—
117	D₅₂	—	4	8	no	—
118	C₄×D₁₃	—	2	8	no	—
119	C₄×D₁₃	—	4	8	no	—
120	Dic₂₆	—	2	8	no	—
121	#542	—	13	16	no	—
122	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
123	C₂×#126	—	13	2	no	—
124	C₂×#124	—	13	2	no	—
125	C₂×#124	—	13	1	no	—
126	C₂×#123	—	13	2	no	—
127	C₂×C₂²⋊Q₈	—	13	2	no	—
128	C₂×C₂²⋊Q₈	—	13	1	no	—
129	C₂×#121	—	13	1	no	—
130	C₂×C₄×Q₈	—	13	1	no	—
131	C₂×C₄×D₄	—	13	1	no	—
132	C₂×#117	—	13	1	no	—
133	C₂²×C₁₃	C₂×#41	1	2	yes	—
134	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
135	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
136	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
137	C₂×D₁₃	—	2	2	no	—
138	C₂×D₁₃	—	2	4	no	—
139	C₂×D₁₃	—	2	4	no	—
140	C₂×D₁₃	—	2	2	no	—
141	C₂×D₁₃	—	2	2	no	—
142	C₂×D₁₃	—	2	4	no	—
143	C₂×D₁₃	—	4	4	no	—
144	C₂×D₁₃	—	4	8	no	—
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	C₂×#41	1	4	yes	—
150	Dic₁₃	—	2	4	no	—
151	Dic₁₃	—	2	2	no	—
152	Dic₁₃	—	2	2	no	—
153	Dic₁₃	—	2	2	no	—
154	C₂²×Q₈	—	13	1	no	—
155	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
156	C₂²×D₄	—	13	1	no	—
157	C₂²×D₄	—	13	2	no	—
158	C₂³×C₄	—	13	1	no	—
159	C₂³×C₄	—	13	2	no	—
160	C₄⋊Q₈	C₂×D₁₃	1	8	yes	complemented
161	#126	—	13	16	no	—
162	#124	—	13	16	no	—
163	#124	—	13	8	no	—
164	#123	—	13	16	no	—
165	C₂²⋊Q₈	—	13	16	no	—
166	C₂²⋊Q₈	—	13	8	no	—
167	#121	—	13	8	no	—
168	C₄×Q₈	—	13	8	no	—
169	C₄×D₄	—	13	8	no	—
170	#117	—	13	8	no	—
171	C₂×C₄⋊C₄	—	13	1	no	—
172	C₂×C₄⋊C₄	—	13	1	no	—
173	C₂×C₄⋊C₄	—	13	2	no	—
174	C₂×C₄⋊C₄	—	13	2	no	—
175	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
178	C₂×C₂²⋊C₄	—	13	1	no	—
179	C₂×C₂²⋊C₄	—	13	1	no	—
180	C₂×C₂²⋊C₄	—	13	2	no	—
181	C₂×C₂²⋊C₄	—	13	2	no	—
182	C₂×C₂²⋊C₄	—	13	2	no	—
183	C₂×C₂²⋊C₄	—	13	2	no	—
184	C₂×C₂²⋊C₄	—	13	2	no	—
185	C₂×C₄²	—	13	1	no	—
186	C₂×C₄²	—	13	2	no	—
187	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
188	C₂×C₁₃	#542	1	4	yes	complemented
189	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
190	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
192	D₁₃	—	4	2	no	—
193	D₁₃	—	4	4	no	—
194	C₂⁴	—	13	2	no	—
195	C₂⁴	—	13	1	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	—	13	4	no	—
198	C₂×Q₈	—	26	4	no	—
199	C₂×Q₈	C₂²×D₁₃	1	8	yes	—
200	C₂×D₄	—	13	4	no	—
201	C₂×D₄	—	13	8	no	—
202	C₂×D₄	—	26	4	no	—
203	C₂×D₄	—	52	4	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	13	2	no	—
206	C₂²×C₄	—	13	1	no	—
207	C₂²×C₄	—	13	1	no	—
208	C₂²×C₄	—	13	4	no	—
209	C₂²×C₄	—	13	8	no	—
210	C₂²×C₄	—	26	4	no	—
211	C₂²×C₄	—	52	4	no	—
212	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
213	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
216	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
217	C₄⋊C₄	—	13	4	no	—
218	C₄⋊C₄	—	13	4	no	—
219	C₄⋊C₄	—	13	8	no	—
220	C₄⋊C₄	—	13	8	no	—
221	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
222	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
223	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
224	C₂²⋊C₄	—	13	4	no	—
225	C₂²⋊C₄	—	13	4	no	—
226	C₂²⋊C₄	—	13	8	no	—
227	C₂²⋊C₄	—	13	8	no	—
228	C₂²⋊C₄	—	13	8	no	—
229	C₂²⋊C₄	—	13	8	no	—
230	C₂²⋊C₄	—	13	8	no	—
231	C₄²	—	13	4	no	—
232	C₄²	—	13	8	no	—
233	C₄²	C₂²×D₁₃	1	4	yes	—
234	C₁₃	C₂×#542	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	13	4	no	—
236	C₂³	—	13	2	no	—
237	C₂³	—	26	4	no	—
238	C₂³	—	26	2	no	—
239	C₂³	—	26	2	no	—
240	C₂³	—	26	4	no	—
241	C₂³	—	26	4	no	—
242	C₂³	—	26	2	no	—
243	C₂³	C₂³×D₁₃	1	1	yes	characteristic
244	Q₈	—	4	8	no	—
245	Q₈	—	26	8	no	—
246	D₄	—	26	8	no	—
247	D₄	—	52	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	13	2	no	—
253	C₂×C₄	—	13	4	no	—
254	C₂×C₄	—	13	2	no	—
255	C₂×C₄	—	13	2	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	4	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	26	8	no	—
261	C₂×C₄	—	52	8	no	—
262	C₂×C₄	#6131	1	4	yes	—
263	C₂×C₄	C₂³×D₁₃	1	4	yes	—
264	C₂×C₄	C₂³×D₁₃	1	2	yes	—
265	C₂×C₄	C₂³×D₁₃	1	2	yes	—
266	C₂×C₄	C₂³×D₁₃	1	4	yes	—
267	C₂×C₄	C₂³×D₁₃	1	2	yes	—
268	C₂²	—	26	4	no	—
269	C₂²	—	26	2	no	—
270	C₂²	—	26	2	no	—
271	C₂²	—	26	4	no	—
272	C₂²	—	26	4	no	—
273	C₂²	—	26	2	no	—
274	C₂²	—	52	8	no	—
275	C₂²	—	52	4	no	—
276	C₂²	#88251	1	2	yes	—
277	C₂²	C₂×#6131	1	2	yes	—
278	C₂²	#88254	1	2	yes	—
279	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	26	2	no	—
285	C₄	—	26	2	no	—
286	C₄	—	26	4	no	—
287	C₄	—	26	2	no	—
288	C₄	C₂×#6131	1	4	yes	—
289	C₂	—	52	4	no	—
290	C₂	—	52	2	no	—
291	C₂	#11719438	1	4	yes	direct product factor
292	C₂	C₂×#88251	1	1	yes	characteristic
293	C₂	C₂²×#6131	1	1	yes	characteristic
294	C₂	C₂×#88254	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂×#88149	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	C₂×#542
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×C₄⋊Q₈	C₂	semidirect
2. C₂×#88201	C₂	semidirect
3. C₂×#88200	C₂	semidirect
4. #11719438	C₂	direct (N × H)
5. C₂×#88152	C₂	semidirect
6. C₂×#88151	C₂	semidirect
8. C₂×#88148	C₂	semidirect
9. C₂×#88147	C₂	semidirect
10. C₂×#88141	C₂	semidirect
11. C₂×#88129	C₂	semidirect
12. C₂×#88126	C₂	semidirect
19. C₁₃×C₄⋊Q₈	C₂²	semidirect
24. #88201	C₂²	semidirect
25. #88200	C₂²	semidirect
39. #88152	C₂²	semidirect
40. #88151	C₂²	semidirect
42. #88148	C₂²	semidirect
43. #88147	C₂²	semidirect
44. #88141	C₂²	semidirect
45. #88129	C₂²	semidirect
46. #88126	C₂²	semidirect
122. C₂×C₄⋊Q₈	D₁₃	semidirect
160. C₄⋊Q₈	C₂×D₁₃	semidirect
188. C₂×C₁₃	#542	semidirect
234. C₁₃	C₂×#542	semidirect
291. C₂	#11719438	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×C₁₃×C₄⋊Q₈	2
C₂×#88201	2
C₂×#88200	2
#11719438	2
C₂×#88152	2
C₂×#88151	2
C₂×#88149	2
C₂×#88148	2
C₂×#88147	2
C₂×#88141	2
C₂×#88129	2
C₂×#88126	2
C₂×#542	13

Subgroups — C2×#11719438 #13128217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719438 #13128217