Subgroups — C₂²×#11718640 #16029109

Order 3328.

6046 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	C₂⁴×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×#6101	C₂	1	2	yes	maximal
3	C₂×#11718640	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#88071	C₂	1	2	yes	maximal
5	C₂²×#88071	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
11	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
12	C₂³×Dic₂₆	—	2	8	no	—
13	C₂³×Dic₂₆	—	2	16	no	—
14	C₂³×Dic₂₆	C₂²	1	12	yes	—
15	C₂²×C₁₃×C₄⋊C₄	—	2	8	no	—
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	12	yes	—
17	C₂²×#6101	—	2	16	no	—
18	C₂²×#6101	C₂²	1	24	yes	—
19	C₂×#88071	C₂²	1	24	yes	—
20	C₂×#88071	C₂²	1	12	yes	—
21	#11718640	C₂²	1	64	yes	direct product factor
22	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	—	2	4	no	—
24	C₂³×C₄×C₁₃	—	2	8	no	—
25	C₂³×C₄×C₁₃	—	2	4	no	—
26	C₂³×C₄×C₁₃	—	2	8	no	—
27	C₂³×C₄×C₁₃	—	2	4	no	—
28	C₂³×C₄×C₁₃	D₄	1	8	yes	—
29	C₂³×C₄×C₁₃	Q₈	1	8	yes	complemented
30	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
33	C₂³×Dic₁₃	—	2	8	no	—
34	C₂³×Dic₁₃	—	2	8	no	—
35	C₂³×Dic₁₃	—	2	8	no	—
36	C₂³×Dic₁₃	—	2	8	no	—
37	C₂³×Dic₁₃	D₄	1	16	yes	—
38	C₂³×Dic₁₃	Q₈	1	16	yes	complemented
39	C₂³×Dic₁₃	C₂³	1	12	yes	—
40	C₂³×Dic₁₃	C₂³	1	12	yes	—
41	C₂²×Dic₂₆	—	2	128	no	—
42	C₂²×Dic₂₆	—	2	48	no	—
43	C₂²×Dic₂₆	—	2	96	no	—
44	C₂²×Dic₂₆	C₂³	1	16	yes	—
45	C₂×C₁₃×C₄⋊C₄	—	2	48	no	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
47	C₂×#6101	—	2	96	no	—
48	C₂×#6101	C₂³	1	32	yes	—
49	#88071	C₂³	1	32	yes	—
50	#88071	C₂³	1	16	yes	—
51	C₂²×#395	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
53	C₂⁴×C₁₃	C₂×D₄	1	4	yes	—
54	C₂⁴×C₁₃	C₂×Q₈	1	4	yes	—
55	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
56	C₂⁴×C₁₃	#41	1	4	yes	—
57	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
58	C₂²×C₄×C₁₃	—	2	32	no	—
59	C₂²×C₄×C₁₃	—	2	32	no	—
60	C₂²×C₄×C₁₃	—	2	12	no	—
61	C₂²×C₄×C₁₃	—	2	24	no	—
62	C₂²×C₄×C₁₃	—	2	12	no	—
63	C₂²×C₄×C₁₃	—	2	24	no	—
64	C₂²×C₄×C₁₃	—	2	12	no	—
65	C₂²×C₄×C₁₃	—	4	16	no	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
67	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	complemented
68	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
71	C₂²×Dic₁₃	—	2	64	no	—
72	C₂²×Dic₁₃	—	2	64	no	—
73	C₂²×Dic₁₃	—	2	24	no	—
74	C₂²×Dic₁₃	—	2	24	no	—
75	C₂²×Dic₁₃	—	2	24	no	—
76	C₂²×Dic₁₃	—	2	24	no	—
77	C₂²×Dic₁₃	C₂×D₄	1	48	yes	—
78	C₂²×Dic₁₃	C₂×Q₈	1	48	yes	complemented
79	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
80	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
81	C₂×Dic₂₆	—	2	384	no	—
82	C₂×Dic₂₆	—	2	32	no	—
83	C₂×Dic₂₆	—	2	64	no	—
84	C₁₃×C₄⋊C₄	—	2	32	no	—
85	#6101	—	2	64	no	—
86	C₂⁴×Q₈	—	13	1	no	—
87	C₂³×C₄⋊C₄	—	13	2	no	—
88	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic
89	C₂×#395	—	13	24	no	—
90	C₂²×#95	—	13	2	no	—
91	C₂²×#95	—	13	1	no	—
92	C₂³×C₁₃	#120	1	16	yes	—
93	C₂³×C₁₃	C₂²⋊Q₈	1	32	yes	—
94	C₂³×C₁₃	C₂²⋊Q₈	1	16	yes	—
95	C₂³×C₁₃	C₄⋊Q₈	1	32	yes	—
96	C₂³×C₁₃	C₄⋊Q₈	1	16	yes	—
97	C₂³×C₁₃	C₂²×D₄	1	12	yes	—
98	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
99	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
100	C₂³×C₁₃	C₂²×Q₈	1	12	yes	—
101	C₂³×C₁₃	C₂×#41	1	6	yes	—
102	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
103	C₂×C₄×C₁₃	—	2	48	no	—
104	C₂×C₄×C₁₃	—	2	48	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	8	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	8	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	4	24	no	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
112	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	complemented
113	C₂×Dic₁₃	—	2	96	no	—
114	C₂×Dic₁₃	—	2	96	no	—
115	C₂×Dic₁₃	—	2	8	no	—
116	C₂×Dic₁₃	—	2	8	no	—
117	C₂×Dic₁₃	—	2	8	no	—
118	C₂×Dic₁₃	—	2	8	no	—
119	C₂×Dic₁₃	C₂²×D₄	1	16	yes	—
120	C₂×Dic₁₃	C₂²×Q₈	1	16	yes	complemented
121	Dic₂₆	—	2	128	no	—
122	C₂³×Q₈	—	13	12	no	—
123	C₂³×Q₈	—	26	16	no	—
124	C₂³×Q₈	—	26	8	no	—
125	C₂⁴×C₄	—	13	2	no	—
126	C₂⁴×C₄	—	13	2	no	—
127	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
128	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
129	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
130	C₂²×C₄⋊C₄	—	2	8	no	—
131	C₂²×C₄⋊C₄	—	13	24	no	—
132	C₂²×C₄⋊C₄	—	26	16	no	—
133	C₂²×C₄⋊C₄	C₂×D₁₃	1	12	yes	—
134	#395	—	13	64	no	—
135	C₂×#95	—	13	24	no	—
136	C₂×#95	—	13	12	no	—
137	C₂²×C₁₃	#395	1	64	yes	complemented
138	C₂²×C₁₃	C₂×#120	1	12	yes	—
139	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	24	yes	—
140	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	12	yes	—
141	C₂²×C₁₃	C₂×C₄⋊Q₈	1	24	yes	—
142	C₂²×C₁₃	C₂×C₄⋊Q₈	1	12	yes	—
143	C₂²×C₁₃	C₂³×D₄	1	2	yes	—
144	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
145	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
146	C₂²×C₁₃	C₂³×Q₈	1	2	yes	—
147	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
148	C₄×C₁₃	—	2	8	no	—
149	C₄×C₁₃	—	2	8	no	—
150	C₄×C₁₃	—	4	4	no	—
151	Dic₁₃	—	2	16	no	—
152	Dic₁₃	—	2	16	no	—
153	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
154	C₂²×Q₈	—	13	16	no	—
155	C₂²×Q₈	—	26	96	no	—
156	C₂²×Q₈	—	26	128	no	—
157	C₂²×Q₈	—	26	48	no	—
158	C₂³×C₄	—	2	4	no	—
159	C₂³×C₄	—	2	8	no	—
160	C₂³×C₄	—	2	4	no	—
161	C₂³×C₄	—	2	8	no	—
162	C₂³×C₄	—	2	4	no	—
163	C₂³×C₄	—	13	16	no	—
164	C₂³×C₄	—	13	16	no	—
165	C₂³×C₄	—	13	12	no	—
166	C₂³×C₄	—	13	12	no	—
167	C₂³×C₄	—	26	8	no	—
168	C₂³×C₄	—	26	8	no	—
169	C₂³×C₄	—	26	8	no	—
170	C₂³×C₄	—	26	8	no	—
171	C₂³×C₄	Dic₂₆	1	8	yes	complemented
172	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
173	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
174	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
175	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
176	C₂×C₄⋊C₄	—	2	48	no	—
177	C₂×C₄⋊C₄	—	13	32	no	—
178	C₂×C₄⋊C₄	—	26	96	no	—
179	C₂×C₄⋊C₄	C₂²×D₁₃	1	16	yes	—
180	#95	—	13	16	no	—
181	#95	—	13	32	no	—
182	C₂×C₁₃	C₂×#395	1	24	yes	complemented
183	C₂×C₁₃	C₂²×#120	1	1	yes	characteristic
184	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	2	yes	—
185	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
186	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₁₃	C₂²×C₄⋊Q₈	1	2	yes	—
188	C₂⁴	C₂×Dic₂₆	1	4	yes	—
189	C₂⁴	#6127	1	4	yes	—
190	C₂⁴	D₄×D₁₃	1	8	yes	—
191	C₂⁴	Q₈×D₁₃	1	8	yes	—
192	C₂⁴	C₂×C₂²⋊D₁₃	1	4	yes	—
193	C₂⁴	C₂³×D₁₃	1	3	yes	—
194	C₂×Q₈	—	26	64	no	—
195	C₂×Q₈	—	26	384	no	—
196	C₂×Q₈	—	26	32	no	—
197	C₂²×C₄	—	2	32	no	—
198	C₂²×C₄	—	2	32	no	—
199	C₂²×C₄	—	2	12	no	—
200	C₂²×C₄	—	2	24	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	—	2	24	no	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	4	16	no	—
205	C₂²×C₄	—	13	48	no	—
206	C₂²×C₄	—	13	48	no	—
207	C₂²×C₄	—	13	8	no	—
208	C₂²×C₄	—	13	8	no	—
209	C₂²×C₄	—	26	64	no	—
210	C₂²×C₄	—	26	64	no	—
211	C₂²×C₄	—	26	24	no	—
212	C₂²×C₄	—	26	24	no	—
213	C₂²×C₄	—	26	24	no	—
214	C₂²×C₄	—	26	24	no	—
215	C₂²×C₄	C₂×Dic₂₆	1	24	yes	complemented
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	24	yes	—
217	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
218	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
219	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
220	C₄⋊C₄	—	2	32	no	—
221	C₄⋊C₄	—	26	64	no	—
222	C₁₃	C₂²×#395	1	1	yes	characteristic, complemented, Sylow
223	C₂³	#88142	1	32	yes	—
224	C₂³	#88150	1	32	yes	—
225	C₂³	#88178	1	16	yes	—
226	C₂³	#88191	1	16	yes	—
227	C₂³	#88198	1	16	yes	—
228	C₂³	C₂²×Dic₂₆	1	6	yes	—
229	C₂³	C₂×#6127	1	6	yes	—
230	C₂³	C₂×D₄×D₁₃	1	12	yes	—
231	C₂³	C₂×Q₈×D₁₃	1	12	yes	—
232	C₂³	C₂²×C₂²⋊D₁₃	1	6	yes	—
233	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
234	Q₈	—	26	128	no	—
235	C₂×C₄	—	2	48	no	—
236	C₂×C₄	—	2	48	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	24	no	—
243	C₂×C₄	—	13	16	no	—
244	C₂×C₄	—	13	16	no	—
245	C₂×C₄	—	26	96	no	—
246	C₂×C₄	—	26	96	no	—
247	C₂×C₄	—	26	8	no	—
248	C₂×C₄	—	26	8	no	—
249	C₂×C₄	—	26	8	no	—
250	C₂×C₄	—	26	8	no	—
251	C₂×C₄	C₂²×Dic₂₆	1	8	yes	complemented
252	C₂×C₄	C₂²×C₂²⋊D₁₃	1	8	yes	—
253	C₂²	#11718640	1	64	yes	direct product factor
254	C₂²	C₂×#88142	1	24	yes	—
255	C₂²	C₂×#88150	1	24	yes	—
256	C₂²	C₂×#88178	1	12	yes	—
257	C₂²	C₂×#88191	1	12	yes	—
258	C₂²	C₂×#88198	1	12	yes	—
259	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
260	C₂²	C₂²×#6127	1	1	yes	characteristic
261	C₂²	C₂²×D₄×D₁₃	1	2	yes	—
262	C₂²	C₂²×Q₈×D₁₃	1	2	yes	—
263	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
264	C₄	—	2	8	no	—
265	C₄	—	2	8	no	—
266	C₄	—	4	4	no	—
267	C₄	—	26	16	no	—
268	C₄	—	26	16	no	—
269	C₂	C₂×#11718640	1	24	yes	direct product factor
270	C₂	C₂²×#88191	1	1	yes	characteristic
271	C₂	C₂²×#88150	1	2	yes	—
272	C₂	C₂²×#88178	1	1	yes	characteristic
273	C₂	C₂²×#88198	1	1	yes	characteristic
274	C₂	C₂²×#88142	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator, cosocle	C₂³×C₁₃	C₂⁵
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂⁵×C₁₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂²×#395
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#11718640	C₂	direct (N × H)
21. #11718640	C₂²	direct (N × H)
29. C₂³×C₄×C₁₃	Q₈	semidirect
38. C₂³×Dic₁₃	Q₈	semidirect
67. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
78. C₂²×Dic₁₃	C₂×Q₈	semidirect
112. C₂×C₄×C₁₃	C₂²×Q₈	semidirect
120. C₂×Dic₁₃	C₂²×Q₈	semidirect
137. C₂²×C₁₃	#395	semidirect
171. C₂³×C₄	Dic₂₆	semidirect
182. C₂×C₁₃	C₂×#395	semidirect
215. C₂²×C₄	C₂×Dic₂₆	semidirect
222. C₁₃	C₂²×#395	semidirect
251. C₂×C₄	C₂²×Dic₂₆	semidirect
253. C₂²	#11718640	direct (N × H)
269. C₂	C₂×#11718640	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₂₆	2
C₂³×#6101	2
C₂×#11718640	2
C₂²×#88071	2
C₂³×C₁₃×C₄⋊C₄	2
C₂²×#395	13

Subgroups — C22×#11718640 #16029109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718640 #16029109