Subgroups — C₂²×#11718808 #16769411

Order 3328.

2334 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88158	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718808	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88067	C₂	1	2	yes	maximal, complemented
4	C₂²×#88066	C₂	1	2	yes	maximal, complemented
5	C₂²×#88052	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₁₃	—	2	1	no	—
8	C₂²×#6127	—	2	1	no	—
9	C₂²×C₁₃×D₈	—	2	1	no	—
10	C₂²×C₁₃×D₈	—	2	6	no	—
11	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
12	C₂²×C₁₃×D₈	C₂²	1	12	yes	complemented
13	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic, complemented
15	C₂×#88158	—	2	6	no	—
16	C₂×#88158	C₂²	1	12	yes	complemented
17	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic, complemented
18	C₂²×#6095	—	2	1	no	—
19	C₂×#88067	C₂²	1	24	yes	complemented
20	#11718808	C₂²	1	64	yes	direct product factor
21	C₂×#88066	C₂²	1	24	yes	complemented
22	C₂×#88052	C₂²	1	12	yes	complemented
23	C₂²×#6090	C₂²	1	2	yes	complemented
24	C₂⁵×C₁₃	—	4	1	no	—
25	C₂²×D₄×C₁₃	—	2	2	no	—
26	C₂²×D₄×C₁₃	—	2	12	no	—
27	C₂²×D₄×C₁₃	—	4	1	no	—
28	C₂²×D₄×C₁₃	—	4	6	no	—
29	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₂²⋊D₁₃	—	4	1	no	—
31	C₂×#6127	—	2	12	no	—
32	C₂×#6127	—	4	6	no	—
33	C₂²×D₅₂	—	2	1	no	—
34	C₂²×C₄×D₁₃	—	4	1	no	—
35	C₂²×Dic₂₆	—	2	1	no	—
36	C₂×C₁₃×D₈	—	2	6	no	—
37	C₂×C₁₃×D₈	—	2	12	no	—
38	C₂×C₁₃×D₈	—	2	24	no	—
39	C₂×C₁₃×D₈	C₂³	1	12	yes	—
40	C₂×C₁₃×D₈	C₂³	1	16	yes	complemented
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	D₄	1	6	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
45	C₂×Dic₅₂	—	2	3	no	—
46	C₂×Dic₅₂	C₂³	1	6	yes	complemented
47	#88158	—	2	24	no	—
48	#88158	C₂³	1	16	yes	complemented
49	C₂×D₁₀₄	—	2	3	no	—
50	C₂×D₁₀₄	C₂³	1	6	yes	complemented
51	C₂×#6095	—	2	12	no	—
52	#88067	C₂³	1	32	yes	complemented
53	#88066	C₂³	1	32	yes	complemented
54	#88052	C₂³	1	16	yes	complemented
55	C₂×#6090	C₂³	1	12	yes	complemented
56	C₂²×#509	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	—	4	12	no	—
58	C₂⁴×C₁₃	—	4	2	no	—
59	C₂⁴×C₁₃	—	8	8	no	—
60	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
61	C₂³×D₁₃	—	4	1	no	—
62	C₂×D₄×C₁₃	—	2	12	no	—
63	C₂×D₄×C₁₃	—	2	16	no	—
64	C₂×D₄×C₁₃	—	4	6	no	—
65	C₂×D₄×C₁₃	—	4	12	no	—
66	C₂×D₄×C₁₃	—	4	24	no	—
67	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	D₈	1	6	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
71	C₂×C₂²⋊D₁₃	—	4	12	no	—
72	C₂²×Dic₁₃	—	4	1	no	—
73	#6127	—	2	16	no	—
74	#6127	—	4	24	no	—
75	C₂×D₅₂	—	2	6	no	—
76	C₂×D₅₂	—	4	3	no	—
77	C₂×C₄×D₁₃	—	4	12	no	—
78	C₂×Dic₂₆	—	2	6	no	—
79	C₂×Dic₂₆	—	4	3	no	—
80	C₁₃×D₈	—	2	12	no	—
81	C₁₃×D₈	—	2	16	no	—
82	C₁₃×D₈	C₂⁴	1	8	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	12	yes	—
86	C₂×C₈×C₁₃	C₂⁴	1	3	yes	—
87	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
88	Dic₅₂	—	2	6	no	—
89	Dic₅₂	C₂⁴	1	4	yes	complemented
90	D₁₀₄	—	2	6	no	—
91	D₁₀₄	C₂⁴	1	4	yes	complemented
92	#6095	—	2	16	no	—
93	#6090	C₂⁴	1	8	yes	complemented
94	C₂²×#135	—	13	1	no	—
95	C₂³×D₈	D₁₃	1	1	yes	characteristic, complemented
96	C₂×#509	—	13	24	no	—
97	C₂²×SD₃₂	—	13	2	no	—
98	C₂²×D₁₆	—	13	2	no	—
99	C₂²×M₃₂	—	13	1	no	—
100	C₂³×C₁₃	—	2	4	no	—
101	C₂³×C₁₃	—	4	16	no	—
102	C₂³×C₁₃	—	4	12	no	—
103	C₂³×C₁₃	—	8	48	no	—
104	C₂³×C₁₃	—	8	8	no	—
105	C₂³×C₁₃	C₂×D₈	1	6	yes	—
106	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₁₃	—	4	6	no	—
108	C₂²×D₁₃	—	8	4	no	—
109	D₄×C₁₃	—	2	8	no	—
110	D₄×C₁₃	—	4	12	no	—
111	D₄×C₁₃	—	4	16	no	—
112	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	12	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
117	C₂²⋊D₁₃	—	4	16	no	—
118	C₂×Dic₁₃	—	4	6	no	—
119	D₅₂	—	2	4	no	—
120	D₅₂	—	4	6	no	—
121	C₄×D₁₃	—	4	16	no	—
122	Dic₂₆	—	2	4	no	—
123	Dic₂₆	—	4	6	no	—
124	C₈×C₁₃	C₂²×D₄	1	4	yes	—
125	C₈×C₁₃	C₂²×D₄	1	3	yes	—
126	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
127	C₂²×#41	—	26	1	no	—
128	C₂³×D₄	—	2	1	no	—
129	C₂×#135	—	13	12	no	—
130	C₂×#135	—	26	6	no	—
131	C₂²×Q₁₆	—	13	1	no	—
132	C₂²×SD₁₆	—	26	1	no	—
133	C₂²×D₈	—	2	1	no	—
134	C₂²×D₈	—	2	6	no	—
135	C₂²×D₈	—	13	1	no	—
136	C₂²×D₈	C₂×D₁₃	1	2	yes	—
137	C₂²×D₈	C₂×D₁₃	1	12	yes	complemented
138	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#509	—	13	64	no	—
140	C₂×SD₃₂	—	13	24	no	—
141	C₂×D₁₆	—	13	24	no	—
142	C₂×M₃₂	—	13	12	no	—
143	C₂²×C₁₆	—	13	2	no	—
144	C₂²×C₁₃	—	2	12	no	—
145	C₂²×C₁₃	—	4	8	no	—
146	C₂²×C₁₃	—	8	32	no	—
147	C₂²×C₁₃	—	8	24	no	—
148	C₂²×C₁₃	#509	1	4	yes	complemented
149	C₂²×C₁₃	C₂²×D₈	1	4	yes	—
150	C₂²×C₁₃	C₂²×D₈	1	3	yes	—
151	C₂×D₁₃	—	4	4	no	—
152	C₂×D₁₃	—	8	12	no	—
153	C₄×C₁₃	C₂²×D₈	1	3	yes	—
154	C₄×C₁₃	C₂²×D₈	1	4	yes	—
155	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
156	Dic₁₃	—	4	4	no	—
157	C₂⁵	—	4	1	no	—
158	C₂×#41	—	26	12	no	—
159	C₂×#41	—	52	6	no	—
160	C₂²×Q₈	—	26	1	no	—
161	C₂²×D₄	—	2	2	no	—
162	C₂²×D₄	—	2	12	no	—
163	C₂²×D₄	—	4	6	no	—
164	C₂²×D₄	—	4	1	no	—
165	C₂²×D₄	—	26	1	no	—
166	C₂²×D₄	—	52	1	no	—
167	C₂³×C₄	—	52	1	no	—
168	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
169	#135	—	13	16	no	—
170	#135	—	26	24	no	—
171	C₂×Q₁₆	—	13	6	no	—
172	C₂×Q₁₆	—	26	3	no	—
173	C₂×SD₁₆	—	26	12	no	—
174	C₂×D₈	—	2	6	no	—
175	C₂×D₈	—	2	12	no	—
176	C₂×D₈	—	2	24	no	—
177	C₂×D₈	—	13	6	no	—
178	C₂×D₈	—	26	3	no	—
179	C₂×D₈	C₂²×D₁₃	1	12	yes	—
180	C₂×D₈	C₂²×D₁₃	1	16	yes	complemented
181	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₁₃	1	6	yes	—
183	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
185	SD₃₂	—	13	32	no	—
186	D₁₆	—	13	32	no	—
187	M₃₂	—	13	16	no	—
188	C₂×C₁₆	—	13	12	no	—
189	C₂×C₁₃	—	2	4	no	—
190	C₂×C₁₃	—	8	8	no	—
191	C₂×C₁₃	C₂×#509	1	6	yes	complemented
192	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
193	D₁₃	—	8	4	no	—
194	C₂⁴	—	4	12	no	—
195	C₂⁴	—	4	2	no	—
196	C₂⁴	—	8	8	no	—
197	C₂⁴	—	52	1	no	—
198	C₂⁴	#6104	1	1	yes	characteristic
199	#41	—	26	16	no	—
200	#41	—	52	24	no	—
201	C₂×Q₈	—	26	6	no	—
202	C₂×Q₈	—	52	3	no	—
203	C₂×D₄	—	2	12	no	—
204	C₂×D₄	—	2	16	no	—
205	C₂×D₄	—	4	12	no	—
206	C₂×D₄	—	4	6	no	—
207	C₂×D₄	—	4	24	no	—
208	C₂×D₄	—	26	6	no	—
209	C₂×D₄	—	52	12	no	—
210	C₂×D₄	—	52	3	no	—
211	C₂²×C₄	—	52	1	no	—
212	C₂²×C₄	—	52	12	no	—
213	C₂²×C₄	#6104	1	6	yes	—
214	C₂²×C₄	#6104	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	Q₁₆	—	13	4	no	—
218	Q₁₆	—	26	6	no	—
219	SD₁₆	—	26	16	no	—
220	D₈	—	2	12	no	—
221	D₈	—	2	16	no	—
222	D₈	—	13	4	no	—
223	D₈	—	26	6	no	—
224	D₈	C₂³×D₁₃	1	8	yes	—
225	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
226	C₂×C₈	C₂×C₂²⋊D₁₃	1	6	yes	—
227	C₂×C₈	C₂×C₂²⋊D₁₃	1	12	yes	—
228	C₂×C₈	C₂³×D₁₃	1	3	yes	—
229	C₂×C₈	C₂³×D₁₃	1	4	yes	—
230	C₁₆	—	13	8	no	—
231	C₁₃	C₂²×#509	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	2	4	no	—
233	C₂³	—	4	16	no	—
234	C₂³	—	4	12	no	—
235	C₂³	—	8	48	no	—
236	C₂³	—	8	8	no	—
237	C₂³	—	52	6	no	—
238	C₂³	—	104	4	no	—
239	C₂³	C₂×#6104	1	6	yes	—
240	C₂³	C₂×#6104	1	1	yes	characteristic
241	Q₈	—	26	4	no	—
242	Q₈	—	52	6	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	16	no	—
245	D₄	—	4	12	no	—
246	D₄	—	26	4	no	—
247	D₄	—	52	16	no	—
248	D₄	—	52	6	no	—
249	C₂×C₄	—	52	6	no	—
250	C₂×C₄	—	52	16	no	—
251	C₂×C₄	C₂×#6104	1	6	yes	—
252	C₂×C₄	C₂×#6104	1	3	yes	—
253	C₂×C₄	C₂×#6104	1	12	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
255	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
256	C₈	C₂²×C₂²⋊D₁₃	1	3	yes	—
257	C₈	C₂²×C₂²⋊D₁₃	1	4	yes	—
258	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
259	C₂²	—	2	12	no	—
260	C₂²	—	4	8	no	—
261	C₂²	—	8	32	no	—
262	C₂²	—	8	24	no	—
263	C₂²	—	52	4	no	—
264	C₂²	—	104	12	no	—
265	C₂²	#11718808	1	4	yes	direct product factor
266	C₂²	C₂²×#6104	1	4	yes	—
267	C₂²	C₂²×#6104	1	3	yes	—
268	C₄	—	52	4	no	—
269	C₄	C₂²×#6104	1	4	yes	—
270	C₄	C₂²×#6104	1	3	yes	—
271	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	—	8	8	no	—
274	C₂	—	104	4	no	—
275	C₂	C₂×#11718808	1	6	yes	direct product factor
276	C₂	C₂³×#6104	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₈×C₁₃	C₂⁵
Frattini	C₈	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×D₈	C₂
socle	C₂³×C₁₃	C₂×D₈
hypercenter	C₂³×D₈	D₁₃
hypocenter	C₁₃	C₂²×#509
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#88158	C₂	semidirect
2. C₂×#11718808	C₂	direct (N × H)
3. C₂²×#88067	C₂	semidirect
4. C₂²×#88066	C₂	semidirect
5. C₂²×#88052	C₂	semidirect
6. C₂³×C₁₃×D₈	C₂	semidirect
12. C₂²×C₁₃×D₈	C₂²	semidirect
14. C₂²×Dic₅₂	C₂²	semidirect
16. C₂×#88158	C₂²	semidirect
17. C₂²×D₁₀₄	C₂²	semidirect
19. C₂×#88067	C₂²	semidirect
20. #11718808	C₂²	direct (N × H)
21. C₂×#88066	C₂²	semidirect
22. C₂×#88052	C₂²	semidirect
23. C₂²×#6090	C₂²	semidirect
40. C₂×C₁₃×D₈	C₂³	semidirect
46. C₂×Dic₅₂	C₂³	semidirect
48. #88158	C₂³	semidirect
50. C₂×D₁₀₄	C₂³	semidirect
52. #88067	C₂³	semidirect
53. #88066	C₂³	semidirect
54. #88052	C₂³	semidirect
55. C₂×#6090	C₂³	semidirect
89. Dic₅₂	C₂⁴	semidirect
91. D₁₀₄	C₂⁴	semidirect
93. #6090	C₂⁴	semidirect
95. C₂³×D₈	D₁₃	semidirect
137. C₂²×D₈	C₂×D₁₃	semidirect
148. C₂²×C₁₃	#509	semidirect
180. C₂×D₈	C₂²×D₁₃	semidirect
191. C₂×C₁₃	C₂×#509	semidirect
231. C₁₃	C₂²×#509	semidirect
265. C₂²	#11718808	direct (N × H)
275. C₂	C₂×#11718808	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×#88158	2
C₂×#11718808	2
C₂²×#88067	2
C₂²×#88066	2
C₂²×#88052	2
C₂³×C₁₃×D₈	2
C₂²×#509	13

Subgroups — C22×#11718808 #16769411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718808 #16769411