Subgroups — C₂²×#11718840 #16769417

Order 3328.

2078 conjugacy classes of subgroups.

The subgroup lattice has 266 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
2	C₂×#11718840	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88069	C₂	1	2	yes	maximal, complemented
4	C₂²×#88068	C₂	1	2	yes	maximal, complemented
5	C₂²×#88052	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₃	—	2	1	no	—
8	C₂²×#6127	—	2	1	no	—
9	C₂²×C₁₃×Q₁₆	—	2	1	no	—
10	C₂²×C₁₃×Q₁₆	—	2	6	no	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
12	C₂²×C₁₃×Q₁₆	C₂²	1	12	yes	complemented
13	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
15	C₂×#88158	—	2	6	no	—
16	C₂×#88158	C₂²	1	12	yes	—
17	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
18	C₂²×#6095	—	2	1	no	—
19	C₂×#88069	C₂²	1	24	yes	complemented
20	#11718840	C₂²	1	64	yes	direct product factor
21	C₂×#88068	C₂²	1	24	yes	complemented
22	C₂×#88052	C₂²	1	12	yes	complemented
23	C₂²×#6090	C₂²	1	2	yes	—
24	C₂²×Q₈×C₁₃	—	2	2	no	—
25	C₂²×Q₈×C₁₃	—	2	12	no	—
26	C₂²×Q₈×C₁₃	—	4	1	no	—
27	C₂²×Q₈×C₁₃	—	4	6	no	—
28	C₂³×C₄×C₁₃	—	4	1	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×Q₁₆	—	2	6	no	—
37	C₂×C₁₃×Q₁₆	—	2	12	no	—
38	C₂×C₁₃×Q₁₆	—	2	24	no	—
39	C₂×C₁₃×Q₁₆	C₂³	1	12	yes	—
40	C₂×C₁₃×Q₁₆	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	—
47	#88158	—	2	24	no	—
48	#88158	C₂³	1	16	yes	—
49	C₂×D₁₀₄	—	2	3	no	—
50	C₂×D₁₀₄	C₂³	1	6	yes	—
51	C₂×#6095	—	2	12	no	—
52	#88069	C₂³	1	32	yes	complemented
53	#88068	C₂³	1	32	yes	complemented
54	#88052	C₂³	1	16	yes	complemented
55	C₂×#6090	C₂³	1	12	yes	—
56	C₂²×#510	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
58	C₂³×D₁₃	—	4	1	no	—
59	C₂×Q₈×C₁₃	—	2	12	no	—
60	C₂×Q₈×C₁₃	—	2	16	no	—
61	C₂×Q₈×C₁₃	—	4	6	no	—
62	C₂×Q₈×C₁₃	—	4	12	no	—
63	C₂×Q₈×C₁₃	—	4	24	no	—
64	C₂²×C₄×C₁₃	—	4	12	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	D₈	1	6	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
70	C₂×C₂²⋊D₁₃	—	4	12	no	—
71	C₂²×Dic₁₃	—	4	1	no	—
72	#6127	—	2	16	no	—
73	#6127	—	4	24	no	—
74	C₂×D₅₂	—	2	6	no	—
75	C₂×D₅₂	—	4	3	no	—
76	C₂×C₄×D₁₃	—	4	12	no	—
77	C₂×Dic₂₆	—	2	6	no	—
78	C₂×Dic₂₆	—	4	3	no	—
79	C₁₃×Q₁₆	—	2	12	no	—
80	C₁₃×Q₁₆	—	2	16	no	—
81	C₁₃×Q₁₆	C₂⁴	1	8	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	12	yes	—
85	C₂×C₈×C₁₃	C₂⁴	1	3	yes	—
86	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
87	Dic₅₂	—	2	6	no	—
88	Dic₅₂	C₂⁴	1	4	yes	—
89	D₁₀₄	—	2	6	no	—
90	D₁₀₄	C₂⁴	1	4	yes	—
91	#6095	—	2	16	no	—
92	#6090	C₂⁴	1	8	yes	—
93	C₂²×#135	—	13	1	no	—
94	C₂³×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
95	C₂×#510	—	13	24	no	—
96	C₂²×Q₃₂	—	13	2	no	—
97	C₂²×SD₃₂	—	13	2	no	—
98	C₂²×M₃₂	—	13	1	no	—
99	C₂³×C₁₃	—	2	4	no	—
100	C₂³×C₁₃	C₂×D₈	1	6	yes	—
101	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₁₃	—	4	6	no	—
103	C₂²×D₁₃	—	8	4	no	—
104	Q₈×C₁₃	—	2	8	no	—
105	Q₈×C₁₃	—	4	12	no	—
106	Q₈×C₁₃	—	4	16	no	—
107	C₂×C₄×C₁₃	—	4	16	no	—
108	C₂×C₄×C₁₃	—	4	12	no	—
109	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
110	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
111	C₂×C₄×C₁₃	C₂×D₈	1	12	yes	—
112	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
113	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
114	C₂²⋊D₁₃	—	4	16	no	—
115	C₂×Dic₁₃	—	4	6	no	—
116	D₅₂	—	2	4	no	—
117	D₅₂	—	4	6	no	—
118	C₄×D₁₃	—	4	16	no	—
119	Dic₂₆	—	2	4	no	—
120	Dic₂₆	—	4	6	no	—
121	C₈×C₁₃	C₂²×D₄	1	4	yes	—
122	C₈×C₁₃	C₂²×D₄	1	3	yes	—
123	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
124	C₂²×#41	—	26	1	no	—
125	C₂³×Q₈	—	2	1	no	—
126	C₂×#135	—	13	12	no	—
127	C₂×#135	—	26	6	no	—
128	C₂²×Q₁₆	—	2	1	no	—
129	C₂²×Q₁₆	—	2	6	no	—
130	C₂²×Q₁₆	—	13	1	no	—
131	C₂²×Q₁₆	C₂×D₁₃	1	2	yes	—
132	C₂²×Q₁₆	C₂×D₁₃	1	12	yes	complemented
133	C₂²×SD₁₆	—	26	1	no	—
134	C₂²×D₈	—	13	1	no	—
135	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
136	#510	—	13	64	no	—
137	C₂×Q₃₂	—	13	24	no	—
138	C₂×SD₃₂	—	13	24	no	—
139	C₂×M₃₂	—	13	12	no	—
140	C₂²×C₁₆	—	13	2	no	—
141	C₂²×C₁₃	—	2	12	no	—
142	C₂²×C₁₃	#510	1	4	yes	complemented
143	C₂²×C₁₃	C₂²×D₈	1	4	yes	—
144	C₂²×C₁₃	C₂²×D₈	1	3	yes	—
145	C₂×D₁₃	—	4	4	no	—
146	C₂×D₁₃	—	8	12	no	—
147	C₄×C₁₃	—	4	8	no	—
148	C₄×C₁₃	C₂²×D₈	1	3	yes	—
149	C₄×C₁₃	C₂²×D₈	1	4	yes	—
150	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
151	Dic₁₃	—	4	4	no	—
152	C₂×#41	—	26	12	no	—
153	C₂×#41	—	52	6	no	—
154	C₂²×Q₈	—	2	2	no	—
155	C₂²×Q₈	—	2	12	no	—
156	C₂²×Q₈	—	4	6	no	—
157	C₂²×Q₈	—	4	1	no	—
158	C₂²×Q₈	—	26	1	no	—
159	C₂²×D₄	—	26	1	no	—
160	C₂²×D₄	—	52	1	no	—
161	C₂³×C₄	—	4	1	no	—
162	C₂³×C₄	—	52	1	no	—
163	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
164	#135	—	13	16	no	—
165	#135	—	26	24	no	—
166	C₂×Q₁₆	—	2	6	no	—
167	C₂×Q₁₆	—	2	12	no	—
168	C₂×Q₁₆	—	2	24	no	—
169	C₂×Q₁₆	—	13	6	no	—
170	C₂×Q₁₆	—	26	3	no	—
171	C₂×Q₁₆	C₂²×D₁₃	1	12	yes	—
172	C₂×Q₁₆	C₂²×D₁₃	1	16	yes	complemented
173	C₂×SD₁₆	—	26	12	no	—
174	C₂×D₈	—	13	6	no	—
175	C₂×D₈	—	26	3	no	—
176	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²⋊D₁₃	1	6	yes	—
178	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
180	Q₃₂	—	13	32	no	—
181	SD₃₂	—	13	32	no	—
182	M₃₂	—	13	16	no	—
183	C₂×C₁₆	—	13	12	no	—
184	C₂×C₁₃	—	2	4	no	—
185	C₂×C₁₃	C₂×#510	1	6	yes	complemented
186	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
187	D₁₃	—	8	4	no	—
188	C₂⁴	—	52	1	no	—
189	C₂⁴	#6104	1	1	yes	characteristic
190	#41	—	26	16	no	—
191	#41	—	52	24	no	—
192	C₂×Q₈	—	2	12	no	—
193	C₂×Q₈	—	2	16	no	—
194	C₂×Q₈	—	4	12	no	—
195	C₂×Q₈	—	4	6	no	—
196	C₂×Q₈	—	4	24	no	—
197	C₂×Q₈	—	26	6	no	—
198	C₂×Q₈	—	52	3	no	—
199	C₂×D₄	—	26	6	no	—
200	C₂×D₄	—	52	12	no	—
201	C₂×D₄	—	52	3	no	—
202	C₂²×C₄	—	4	12	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	52	1	no	—
205	C₂²×C₄	—	52	12	no	—
206	C₂²×C₄	#6104	1	6	yes	—
207	C₂²×C₄	#6104	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
210	Q₁₆	—	2	12	no	—
211	Q₁₆	—	2	16	no	—
212	Q₁₆	—	13	4	no	—
213	Q₁₆	—	26	6	no	—
214	Q₁₆	C₂³×D₁₃	1	8	yes	—
215	SD₁₆	—	26	16	no	—
216	D₈	—	13	4	no	—
217	D₈	—	26	6	no	—
218	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
219	C₂×C₈	C₂×C₂²⋊D₁₃	1	6	yes	—
220	C₂×C₈	C₂×C₂²⋊D₁₃	1	12	yes	—
221	C₂×C₈	C₂³×D₁₃	1	3	yes	—
222	C₂×C₈	C₂³×D₁₃	1	4	yes	—
223	C₁₆	—	13	8	no	—
224	C₁₃	C₂²×#510	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	4	no	—
226	C₂³	—	52	6	no	—
227	C₂³	—	104	4	no	—
228	C₂³	C₂×#6104	1	6	yes	—
229	C₂³	C₂×#6104	1	1	yes	characteristic
230	Q₈	—	2	8	no	—
231	Q₈	—	4	16	no	—
232	Q₈	—	4	12	no	—
233	Q₈	—	26	4	no	—
234	Q₈	—	52	6	no	—
235	D₄	—	26	4	no	—
236	D₄	—	52	16	no	—
237	D₄	—	52	6	no	—
238	C₂×C₄	—	4	16	no	—
239	C₂×C₄	—	4	12	no	—
240	C₂×C₄	—	52	6	no	—
241	C₂×C₄	—	52	16	no	—
242	C₂×C₄	C₂×#6104	1	6	yes	—
243	C₂×C₄	C₂×#6104	1	3	yes	—
244	C₂×C₄	C₂×#6104	1	12	yes	—
245	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
247	C₈	C₂²×C₂²⋊D₁₃	1	3	yes	—
248	C₈	C₂²×C₂²⋊D₁₃	1	4	yes	—
249	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
250	C₂²	—	2	12	no	—
251	C₂²	—	52	4	no	—
252	C₂²	—	104	12	no	—
253	C₂²	#11718840	1	4	yes	direct product factor
254	C₂²	C₂²×#6104	1	4	yes	—
255	C₂²	C₂²×#6104	1	3	yes	—
256	C₄	—	4	8	no	—
257	C₄	—	52	4	no	—
258	C₄	C₂²×#6104	1	4	yes	—
259	C₄	C₂²×#6104	1	3	yes	—
260	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₂	—	2	4	no	—
262	C₂	—	104	4	no	—
263	C₂	C₂×#11718840	1	6	yes	direct product factor
264	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×Q₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×#88158	C₂
hypercenter	C₂³×Q₁₆	D₁₃
hypocenter	C₁₃	C₂²×#510
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#11718840	C₂	direct (N × H)
3. C₂²×#88069	C₂	semidirect
4. C₂²×#88068	C₂	semidirect
5. C₂²×#88052	C₂	semidirect
6. C₂³×C₁₃×Q₁₆	C₂	semidirect
12. C₂²×C₁₃×Q₁₆	C₂²	semidirect
19. C₂×#88069	C₂²	semidirect
20. #11718840	C₂²	direct (N × H)
21. C₂×#88068	C₂²	semidirect
22. C₂×#88052	C₂²	semidirect
40. C₂×C₁₃×Q₁₆	C₂³	semidirect
52. #88069	C₂³	semidirect
53. #88068	C₂³	semidirect
54. #88052	C₂³	semidirect
94. C₂³×Q₁₆	D₁₃	semidirect
132. C₂²×Q₁₆	C₂×D₁₃	semidirect
142. C₂²×C₁₃	#510	semidirect
172. C₂×Q₁₆	C₂²×D₁₃	semidirect
185. C₂×C₁₃	C₂×#510	semidirect
224. C₁₃	C₂²×#510	semidirect
253. C₂²	#11718840	direct (N × H)
263. C₂	C₂×#11718840	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×#11718840	2
C₂²×#88069	2
C₂²×#88068	2
C₂²×#88052	2
C₂³×C₁₃×Q₁₆	2
C₂²×#510	13

Subgroups — C22×#11718840 #16769417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718840 #16769417