Subgroups — C₄²×#13083574 #36712629

Order 22784.

4086 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083574	C₂	1	3	yes	maximal
2	C₄²×Dic₁₇₈	C₂	1	4	yes	maximal, complemented
3	C₄²×D₃₅₆	C₂	1	4	yes	maximal, complemented
4	C₄²×C₂²⋊D₈₉	C₂	1	8	yes	maximal, complemented
5	C₄×#15667969	C₂	1	6	yes	maximal, complemented
6	C₄×#15667971	C₂	1	3	yes	maximal, complemented
7	C₄³×D₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13083574	C₂²	1	1	yes	characteristic
10	C₄×#13083574	C₄	1	48	yes	direct product factor
11	C₂×C₄×Dic₁₇₈	C₂²	1	6	yes	—
12	C₂×C₄×D₃₅₆	C₂²	1	6	yes	—
13	C₂×C₄×C₂²⋊D₈₉	C₂²	1	12	yes	—
14	C₂×#15667971	C₂²	1	3	yes	—
15	C₂×#15667969	C₂²	1	6	yes	—
16	C₂×C₄²×D₈₉	C₂²	1	8	yes	—
17	C₂×C₄²×D₈₉	C₂²	1	6	yes	—
18	C₄×#13083553	C₂²	1	24	yes	—
19	C₄×C₄⋊Dic₈₉	C₂²	1	12	yes	—
20	C₄×#13083551	C₂²	1	24	yes	—
21	C₄×C₂²⋊Dic₈₉	C₂²	1	12	yes	—
22	#19151444	C₂²	1	6	yes	—
23	C₄²×Dic₈₉	C₂²	1	6	yes	—
24	C₄²×Dic₈₉	C₂²	1	8	yes	—
25	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
26	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
27	C₄³×C₈₉	C₂²	1	3	yes	—
28	C₄³×C₈₉	C₂²	1	4	yes	—
29	C₄³×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#13083574	C₂×C₄	1	24	yes	—
31	C₂²×Dic₁₇₈	C₂³	1	1	yes	characteristic
32	C₂²×D₃₅₆	C₂³	1	1	yes	characteristic
33	C₂²×C₂²⋊D₈₉	C₂³	1	2	yes	—
34	C₂²×C₄×D₈₉	C₂³	1	12	yes	—
35	C₂²×C₄×D₈₉	C₂³	1	2	yes	—
36	C₄×Dic₁₇₈	C₂×C₄	1	48	yes	complemented
37	C₄×D₃₅₆	C₂×C₄	1	48	yes	complemented
38	C₄×C₂²⋊D₈₉	C₂×C₄	1	96	yes	complemented
39	#15667971	C₂×C₄	1	24	yes	complemented
40	#15667969	C₂×C₄	1	48	yes	complemented
41	C₂×C₄⋊Dic₈₉	C₂³	1	6	yes	—
42	C₂×#13083553	C₂³	1	12	yes	—
43	C₂×C₂²⋊Dic₈₉	C₂³	1	6	yes	—
44	C₂×#13083551	C₂³	1	12	yes	—
45	C₂×C₄×Dic₈₉	C₂³	1	24	yes	—
46	C₂×C₄×Dic₈₉	C₂³	1	6	yes	—
47	C₂×C₄×Dic₈₉	C₂³	1	12	yes	—
48	C₄²×D₈₉	—	2	32	no	—
49	C₄²×D₈₉	C₂×C₄	1	48	yes	complemented
50	#15667919	C₂³	1	24	yes	—
51	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
52	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	—	2	16	no	—
54	C₂×C₄²×C₈₉	C₂×C₄	1	24	yes	complemented
55	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
56	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
57	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
58	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
59	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
60	C₂³×D₈₉	C₂⁴	1	2	yes	—
61	#13083574	C₄²	1	64	yes	direct product factor
62	C₂×Dic₁₇₈	C₂²×C₄	1	12	yes	—
63	C₂×D₃₅₆	C₂²×C₄	1	12	yes	—
64	C₂×C₂²⋊D₈₉	C₂²×C₄	1	24	yes	—
65	C₂²×Dic₈₉	C₂⁴	1	12	yes	—
66	C₂²×Dic₈₉	C₂⁴	1	2	yes	—
67	C₂×C₄×D₈₉	—	2	48	no	—
68	C₂×C₄×D₈₉	C₂²×C₄	1	48	yes	—
69	C₂×C₄×D₈₉	C₂²×C₄	1	24	yes	—
70	C₄⋊Dic₈₉	C₂²×C₄	1	24	yes	—
71	#13083553	C₂²×C₄	1	48	yes	—
72	C₂²⋊Dic₈₉	C₂²×C₄	1	24	yes	—
73	#13083551	C₂²×C₄	1	48	yes	—
74	C₄×Dic₈₉	—	2	48	no	—
75	C₄×Dic₈₉	C₂²×C₄	1	24	yes	—
76	C₄×Dic₈₉	C₂²×C₄	1	48	yes	—
77	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	—	2	24	no	—
79	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
80	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
81	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
82	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
85	C₄²×C₈₉	—	2	24	no	—
86	C₄²×C₈₉	C₂²×C₄	1	12	yes	—
87	C₄²×C₈₉	C₂²×C₄	1	24	yes	—
88	C₄²×C₈₉	C₂²×C₄	1	12	yes	—
89	C₄²×C₈₉	#41	1	16	yes	complemented
90	C₂³×C₈₉	—	2	4	no	—
91	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
92	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
93	C₂²×D₈₉	—	2	8	no	—
94	C₂²×D₈₉	C₂³×C₄	1	12	yes	—
95	C₂×C₄×C₈₉	—	2	12	no	—
96	C₂×C₄×C₈₉	—	2	48	no	—
97	C₂×C₄×C₈₉	C₂×C₄²	1	16	yes	complemented
98	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
99	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
100	C₂×C₄×C₈₉	C₂³×C₄	1	6	yes	—
101	C₂×C₄×C₈₉	C₂³×C₄	1	6	yes	—
102	C₂×C₄×C₈₉	C₂×#41	1	24	yes	—
103	C₂²⋊D₈₉	C₂×C₄²	1	32	yes	complemented
104	C₂×Dic₈₉	—	2	24	no	—
105	C₂×Dic₈₉	C₂³×C₄	1	24	yes	—
106	C₂×Dic₈₉	C₂³×C₄	1	12	yes	—
107	D₃₅₆	C₂×C₄²	1	16	yes	complemented
108	C₄×D₈₉	—	2	96	no	—
109	C₄×D₈₉	C₂×C₄²	1	32	yes	complemented
110	Dic₁₇₈	C₂×C₄²	1	16	yes	complemented
111	C₂²×C₈₉	—	2	12	no	—
112	C₂²×C₈₉	C₂²×C₄²	1	4	yes	—
113	C₂²×C₈₉	C₂⁴×C₄	1	3	yes	—
114	C₂²×C₈₉	C₂²×#41	1	4	yes	—
115	C₂×D₈₉	—	2	24	no	—
116	C₂×D₈₉	C₂²×C₄²	1	8	yes	—
117	C₄×C₈₉	—	2	12	no	—
118	C₄×C₈₉	C₂²×C₄²	1	4	yes	—
119	C₄×C₈₉	C₂²×C₄²	1	4	yes	—
120	C₄×C₈₉	C₄×#41	1	24	yes	complemented
121	Dic₈₉	—	2	24	no	—
122	Dic₈₉	C₂²×C₄²	1	8	yes	—
123	C₄²×#41	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	—	2	4	no	—
125	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
126	C₂×C₈₉	C₂×C₄×#41	1	6	yes	—
127	D₈₉	—	2	8	no	—
128	C₂×C₄×#41	—	89	3	no	—
129	C₄²×Q₈	—	89	4	no	—
130	C₄²×D₄	—	89	8	no	—
131	C₄²×D₄	—	89	4	no	—
132	C₄×#117	—	89	3	no	—
133	C₄×#117	—	89	6	no	—
134	C₂×C₄³	—	89	2	no	—
135	C₂×C₄³	D₈₉	1	1	yes	characteristic, complemented
136	C₈₉	C₄²×#41	1	1	yes	characteristic, complemented, Sylow
137	C₂²×#41	—	89	1	no	—
138	C₄×#41	—	89	48	no	—
139	C₂×C₄×Q₈	—	89	6	no	—
140	C₂×C₄×D₄	—	89	12	no	—
141	C₂×C₄×D₄	—	89	6	no	—
142	C₂×#117	—	89	3	no	—
143	C₂×#117	—	89	6	no	—
144	C₂²×C₄²	—	89	6	no	—
145	C₂²×C₄²	—	89	8	no	—
146	C₂²×C₄²	C₂×D₈₉	1	4	yes	—
147	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
148	C₄×C₄⋊C₄	—	89	12	no	—
149	C₄×C₄⋊C₄	—	89	24	no	—
150	C₄×C₂²⋊C₄	—	89	12	no	—
151	C₄×C₂²⋊C₄	—	89	24	no	—
152	#376	—	89	6	no	—
153	C₄³	—	89	6	no	—
154	C₄³	—	89	8	no	—
155	C₄³	C₂×D₈₉	1	3	yes	—
156	C₄³	C₂×D₈₉	1	4	yes	—
157	C₄³	C₂×D₈₉	1	1	yes	characteristic
158	C₂×#41	—	89	24	no	—
159	C₂²×Q₈	—	89	1	no	—
160	C₂²×D₄	—	89	2	no	—
161	C₂²×D₄	—	89	1	no	—
162	C₂³×C₄	—	89	2	no	—
163	C₂³×C₄	—	89	12	no	—
164	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
165	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
166	C₄×Q₈	—	89	48	no	—
167	C₄×D₄	—	89	96	no	—
168	C₄×D₄	—	89	48	no	—
169	#117	—	89	24	no	—
170	#117	—	89	48	no	—
171	C₂×C₄⋊C₄	—	89	12	no	—
172	C₂×C₄⋊C₄	—	89	6	no	—
173	C₂×C₂²⋊C₄	—	89	6	no	—
174	C₂×C₂²⋊C₄	—	89	12	no	—
175	C₂×C₄²	—	2	16	no	—
176	C₂×C₄²	—	89	6	no	—
177	C₂×C₄²	—	89	12	no	—
178	C₂×C₄²	—	89	48	no	—
179	C₂×C₄²	—	89	24	no	—
180	C₂×C₄²	—	178	32	no	—
181	C₂×C₄²	C₄×D₈₉	1	24	yes	complemented
182	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
183	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
184	C₂×C₄²	C₂²×D₈₉	1	6	yes	—
185	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
186	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
187	#95	—	89	24	no	—
188	C₂⁴	—	89	2	no	—
189	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
190	#41	—	89	64	no	—
191	C₂×Q₈	—	89	12	no	—
192	C₂×D₄	—	89	24	no	—
193	C₂×D₄	—	89	12	no	—
194	C₂²×C₄	—	2	24	no	—
195	C₂²×C₄	—	89	2	no	—
196	C₂²×C₄	—	89	24	no	—
197	C₂²×C₄	—	89	12	no	—
198	C₂²×C₄	—	89	48	no	—
199	C₂²×C₄	—	178	48	no	—
200	C₂²×C₄	C₂×C₄×D₈₉	1	24	yes	—
201	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
202	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
203	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
206	C₄⋊C₄	—	89	48	no	—
207	C₄⋊C₄	—	89	24	no	—
208	C₂²⋊C₄	—	89	24	no	—
209	C₂²⋊C₄	—	89	48	no	—
210	C₄²	—	2	24	no	—
211	C₄²	—	89	24	no	—
212	C₄²	—	89	48	no	—
213	C₄²	—	178	48	no	—
214	C₄²	C₂×C₄×D₈₉	1	12	yes	—
215	C₄²	C₂×C₄×D₈₉	1	12	yes	—
216	C₄²	C₂×C₄×D₈₉	1	24	yes	—
217	C₄²	#13083574	1	16	yes	direct product factor
218	C₂³	—	2	4	no	—
219	C₂³	—	89	12	no	—
220	C₂³	—	178	8	no	—
221	C₂³	C₂²×C₄×D₈₉	1	6	yes	—
222	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
223	Q₈	—	89	16	no	—
224	D₄	—	89	32	no	—
225	D₄	—	89	16	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	2	48	no	—
228	C₂×C₄	—	89	12	no	—
229	C₂×C₄	—	89	32	no	—
230	C₂×C₄	—	89	24	no	—
231	C₂×C₄	—	178	24	no	—
232	C₂×C₄	—	178	96	no	—
233	C₂×C₄	C₄²×D₈₉	1	16	yes	complemented
234	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
235	C₂×C₄	C₂²×C₄×D₈₉	1	6	yes	—
236	C₂×C₄	C₂²×C₄×D₈₉	1	6	yes	—
237	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
238	C₂×C₄	C₂×#13083574	1	24	yes	—
239	C₂²	—	2	12	no	—
240	C₂²	—	89	8	no	—
241	C₂²	—	178	24	no	—
242	C₂²	C₂×C₄²×D₈₉	1	4	yes	—
243	C₂²	C₂³×C₄×D₈₉	1	3	yes	—
244	C₂²	C₂²×#13083574	1	4	yes	—
245	C₄	—	2	12	no	—
246	C₄	—	89	8	no	—
247	C₄	—	178	24	no	—
248	C₄	C₂×C₄²×D₈₉	1	4	yes	—
249	C₄	C₂×C₄²×D₈₉	1	4	yes	—
250	C₄	C₄×#13083574	1	24	yes	direct product factor
251	C₂	—	2	4	no	—
252	C₂	—	178	8	no	—
253	C₂	C₂²×C₄²×D₈₉	1	1	yes	characteristic
254	C₂	C₂×C₄×#13083574	1	6	yes	—

	subgroup	G/H
center	C₄³	C₂×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄³×C₈₉	C₂
socle, cosocle	C₂³×C₈₉	C₂⁵
prime radical	C₂²×#13083574	C₂²
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	C₄²×#41
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄³ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. C₄²×Dic₁₇₈	C₂	semidirect
3. C₄²×D₃₅₆	C₂	semidirect
4. C₄²×C₂²⋊D₈₉	C₂	semidirect
5. C₄×#15667969	C₂	semidirect
6. C₄×#15667971	C₂	semidirect
7. C₄³×D₈₉	C₂	semidirect
8. C₂×C₄³×C₈₉	C₂	semidirect
10. C₄×#13083574	C₄	direct (N × H)
36. C₄×Dic₁₇₈	C₂×C₄	semidirect
37. C₄×D₃₅₆	C₂×C₄	semidirect
38. C₄×C₂²⋊D₈₉	C₂×C₄	semidirect
39. #15667971	C₂×C₄	semidirect
40. #15667969	C₂×C₄	semidirect
49. C₄²×D₈₉	C₂×C₄	semidirect
54. C₂×C₄²×C₈₉	C₂×C₄	semidirect
61. #13083574	C₄²	direct (N × H)
89. C₄²×C₈₉	#41	semidirect
97. C₂×C₄×C₈₉	C₂×C₄²	semidirect
103. C₂²⋊D₈₉	C₂×C₄²	semidirect
107. D₃₅₆	C₂×C₄²	semidirect
109. C₄×D₈₉	C₂×C₄²	semidirect
110. Dic₁₇₈	C₂×C₄²	semidirect
120. C₄×C₈₉	C₄×#41	semidirect
135. C₂×C₄³	D₈₉	semidirect
136. C₈₉	C₄²×#41	semidirect
181. C₂×C₄²	C₄×D₈₉	semidirect
217. C₄²	#13083574	direct (N × H)
233. C₂×C₄	C₄²×D₈₉	semidirect
250. C₄	C₄×#13083574	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×C₄×#13083574	2
C₄²×Dic₁₇₈	2
C₄²×D₃₅₆	2
C₄²×C₂²⋊D₈₉	2
C₄×#15667969	2
C₄×#15667971	2
C₄³×D₈₉	2
C₂×C₄³×C₈₉	2
C₄²×#41	89

Subgroups — C42×#13083574 #36712629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#13083574 #36712629