Subgroups — M₁₆×D₃₅₆ #26028990

Order 11392.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₈₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#19151677	C₂	1	2	yes	maximal, complemented
4	#19151671	C₂	1	4	yes	maximal, complemented
5	C₈×D₃₅₆	C₂	1	2	yes	maximal, complemented
6	#19151622	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151561	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₅₆	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₈₉	C₂²	1	2	yes	—
11	C₂×#13083558	C₂²	1	2	yes	—
12	C₂×#13083557	C₂²	1	4	yes	complemented
13	M₁₆×D₈₉	—	2	8	no	—
14	C₂×C₈×D₈₉	C₂²	1	4	yes	complemented
15	C₄×D₃₅₆	C₄	1	2	yes	—
16	C₄×D₃₅₆	C₄	1	1	yes	characteristic
17	C₄×D₃₅₆	C₄	1	1	yes	characteristic
18	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
19	C₄×D₃₅₆	C₂²	1	2	yes	—
20	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
22	C₂×#13083551	C₄	1	2	yes	—
23	C₄⋊#10623494	C₂²	1	2	yes	complemented
24	C₄⋊#10623494	C₂²	1	2	yes	complemented
25	#15667924	C₂²	1	4	yes	complemented
26	#15667924	C₂²	1	4	yes	complemented
27	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
30	#97×C₈₉	C₂²	1	2	yes	complemented
31	C₄×C₈×C₈₉	C₂²	1	2	yes	complemented
32	C₂³×D₈₉	C₂×C₄	1	2	yes	—
33	C₂×D₃₅₆	—	2	4	no	—
34	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
35	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
36	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₈₉	—	2	4	no	—
39	C₂×C₄×D₈₉	—	2	4	no	—
40	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₈₉	C₂³	1	2	yes	—
43	C₂×C₄×D₈₉	C₂³	1	2	yes	—
44	#13083558	—	2	4	no	—
45	#13083557	—	2	8	no	—
46	C₂×#10623494	C₂³	1	4	yes	—
47	C₈×D₈₉	—	2	8	no	—
48	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
51	#13083551	C₂×C₄	1	2	yes	—
52	#13083551	C₂×C₄	1	2	yes	—
53	#13083551	C₂×C₄	1	2	yes	—
54	#13083551	C₂×C₄	1	2	yes	—
55	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
58	M₁₆×C₈₉	—	2	2	no	—
59	M₁₆×C₈₉	D₄	1	4	yes	complemented
60	C₂×C₈×C₈₉	C₂³	1	2	yes	—
61	C₂×C₈×C₈₉	C₂³	1	2	yes	—
62	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₈₉	—	2	4	no	—
68	C₂²×D₈₉	—	2	4	no	—
69	C₂²×D₈₉	—	4	2	no	—
70	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
71	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₈₉	—	2	2	no	—
73	C₂×C₄×C₈₉	—	4	1	no	—
74	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₈₉	#41	1	2	yes	—
81	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₈₉	—	2	4	no	—
83	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
84	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
85	D₃₅₆	—	2	4	no	—
86	D₃₅₆	—	2	2	no	—
87	D₃₅₆	M₁₆	1	4	yes	direct product factor
88	C₄×D₈₉	—	2	4	no	—
89	C₄×D₈₉	—	2	4	no	—
90	C₄×D₈₉	—	2	2	no	—
91	C₄×D₈₉	—	2	2	no	—
92	C₄×D₈₉	—	2	2	no	—
93	C₄×D₈₉	—	2	2	no	—
94	C₈×C₈₉	—	2	2	no	—
95	C₈×C₈₉	C₂×D₄	1	4	yes	complemented
96	#10623494	—	2	4	no	—
97	C₂²×C₈₉	—	2	1	no	—
98	C₂²×C₈₉	—	2	1	no	—
99	C₂²×C₈₉	C₄×D₄	1	2	yes	—
100	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₈₉	—	2	2	no	—
102	C₂×D₈₉	—	2	2	no	—
103	C₂×D₈₉	—	2	2	no	—
104	C₂×D₈₉	—	2	2	no	—
105	C₂×D₈₉	—	2	2	no	—
106	C₂×D₈₉	—	4	8	no	—
107	C₂×D₈₉	C₂×M₁₆	1	4	yes	complemented
108	C₄×C₈₉	—	2	1	no	—
109	C₄×C₈₉	—	2	1	no	—
110	C₄×C₈₉	—	2	1	no	—
111	C₄×C₈₉	C₄×D₄	1	2	yes	—
112	C₄×C₈₉	C₂×M₁₆	1	2	yes	complemented
113	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
115	Dic₈₉	—	2	2	no	—
116	Dic₈₉	—	2	2	no	—
117	C₂×C₈₉	—	2	2	no	—
118	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
120	C₂×C₈₉	#568	1	1	yes	characteristic
121	D₈₉	—	2	4	no	—
122	D₈₉	—	4	2	no	—
123	D₄×M₁₆	—	89	1	no	maximal, Sylow
124	C₈₉	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
125	C₂²×M₁₆	—	89	2	no	—
126	C₂×C₄×D₄	—	89	1	no	—
127	#436	—	89	2	no	—
128	#435	—	89	4	no	—
129	C₈×D₄	—	89	2	no	—
130	#423	—	89	1	no	—
131	#407	—	89	2	no	—
132	C₄×M₁₆	D₈₉	1	1	yes	characteristic, complemented
133	C₂²×D₄	—	89	1	no	—
134	C₂³×C₄	—	89	2	no	—
135	C₂×M₁₆	—	89	2	no	—
136	C₂×M₁₆	—	89	4	no	—
137	C₂×M₁₆	—	178	8	no	—
138	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
139	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
140	C₂²×C₈	—	89	4	no	—
141	C₄×D₄	—	89	1	no	—
142	C₄×D₄	—	89	2	no	—
143	C₄×D₄	—	89	1	no	—
144	C₄×D₄	—	89	1	no	—
145	C₄×D₄	—	89	2	no	—
146	C₄×D₄	—	89	1	no	—
147	C₂×C₄⋊C₄	—	89	1	no	—
148	C₂×C₂²⋊C₄	—	89	2	no	—
149	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
150	C₄⋊C₈	—	89	2	no	—
151	C₄⋊C₈	—	89	2	no	—
152	C₂²⋊C₈	—	89	4	no	—
153	C₂²⋊C₈	—	89	4	no	—
154	#97	C₂×D₈₉	1	2	yes	complemented
155	C₄×C₈	C₂×D₈₉	1	2	yes	complemented
156	C₂⁴	—	89	2	no	—
157	C₂×D₄	—	89	1	no	—
158	C₂×D₄	—	89	2	no	—
159	C₂×D₄	—	89	1	no	—
160	C₂×D₄	—	178	4	no	—
161	C₂²×C₄	—	89	2	no	—
162	C₂²×C₄	—	89	2	no	—
163	C₂²×C₄	—	89	2	no	—
164	C₂²×C₄	—	89	2	no	—
165	C₂²×C₄	—	89	2	no	—
166	C₂²×C₄	—	178	4	no	—
167	C₂²×C₄	—	178	4	no	—
168	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	178	4	no	—
173	M₁₆	—	178	8	no	—
174	M₁₆	D₃₅₆	1	4	yes	direct product factor
175	C₂×C₈	—	89	4	no	—
176	C₂×C₈	—	178	8	no	—
177	C₂×C₈	C₂²×D₈₉	1	2	yes	—
178	C₂×C₈	C₂²×D₈₉	1	2	yes	—
179	C₄⋊C₄	—	89	2	no	—
180	C₄⋊C₄	—	89	1	no	—
181	C₄⋊C₄	—	89	1	no	—
182	C₂²⋊C₄	—	89	2	no	—
183	C₂²⋊C₄	—	89	2	no	—
184	C₂²⋊C₄	—	89	2	no	—
185	C₂²⋊C₄	—	89	2	no	—
186	C₄²	C₄×D₈₉	1	1	yes	characteristic
187	C₄²	C₄×D₈₉	1	1	yes	characteristic
188	C₄²	C₂²×D₈₉	1	1	yes	characteristic
189	C₄²	C₂²×D₈₉	1	1	yes	characteristic
190	C₂³	—	89	2	no	—
191	C₂³	—	89	2	no	—
192	C₂³	—	356	2	no	—
193	C₂³	—	178	4	no	—
194	C₂³	—	178	4	no	—
195	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
196	D₄	—	89	4	no	—
197	D₄	—	178	4	no	—
198	D₄	—	178	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	89	2	no	—
202	C₂×C₄	—	89	2	no	—
203	C₂×C₄	—	178	4	no	—
204	C₂×C₄	—	178	4	no	—
205	C₂×C₄	—	178	4	no	—
206	C₂×C₄	—	178	2	no	—
207	C₂×C₄	—	178	2	no	—
208	C₂×C₄	—	178	2	no	—
209	C₂×C₄	—	178	2	no	—
210	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
215	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
216	C₂×C₄	#13083574	1	2	yes	—
217	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	178	4	no	—
220	C₈	C₂×D₃₅₆	1	4	yes	complemented
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	89	4	no	—
224	C₂²	—	356	8	no	—
225	C₂²	—	178	2	no	—
226	C₂²	—	178	2	no	—
227	C₂²	—	178	2	no	—
228	C₂²	—	178	2	no	—
229	C₂²	—	178	2	no	—
230	C₂²	C₄×D₃₅₆	1	2	yes	—
231	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	178	2	no	—
236	C₄	—	178	2	no	—
237	C₄	C₄×D₃₅₆	1	2	yes	—
238	C₄	M₁₆×D₈₉	1	2	yes	complemented
239	C₄	C₂²×D₃₅₆	1	1	yes	characteristic
240	C₄	C₂×#13083574	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	356	2	no	—
243	C₂	—	178	4	no	—
244	C₂	C₂×C₄×D₃₅₆	1	1	yes	characteristic
245	C₂	C₂×M₁₆×D₈₉	1	1	yes	characteristic
246	C₂	#19152364	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₄×M₁₆×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₄×M₁₆	D₈₉
hypocenter	C₈₉	D₄×M₁₆
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₈₉	C₂	semidirect
3. #19151677	C₂	semidirect
4. #19151671	C₂	semidirect
5. C₈×D₃₅₆	C₂	semidirect
6. #19151622	C₂	semidirect
7. #19151561	C₂	semidirect
8. C₄×M₁₆×C₈₉	C₂	semidirect
12. C₂×#13083557	C₂²	semidirect
14. C₂×C₈×D₈₉	C₂²	semidirect
23. C₄⋊#10623494	C₂²	semidirect
24. C₄⋊#10623494	C₂²	semidirect
25. #15667924	C₂²	semidirect
26. #15667924	C₂²	semidirect
30. #97×C₈₉	C₂²	semidirect
31. C₄×C₈×C₈₉	C₂²	semidirect
59. M₁₆×C₈₉	D₄	semidirect
87. D₃₅₆	M₁₆	direct (N × H)
95. C₈×C₈₉	C₂×D₄	semidirect
107. C₂×D₈₉	C₂×M₁₆	semidirect
112. C₄×C₈₉	C₂×M₁₆	semidirect
124. C₈₉	D₄×M₁₆	semidirect
132. C₄×M₁₆	D₈₉	semidirect
154. #97	C₂×D₈₉	semidirect
155. C₄×C₈	C₂×D₈₉	semidirect
174. M₁₆	D₃₅₆	direct (N × H)
220. C₈	C₂×D₃₅₆	semidirect
238. C₄	M₁₆×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₈₉	2
C₂×C₄×D₃₅₆	2
#19151677	2
#19151671	2
C₈×D₃₅₆	2
#19151622	2
#19151561	2
C₄×M₁₆×C₈₉	2
D₄×M₁₆	89

Subgroups — M16×D356 #26028990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×D₃₅₆ #26028990