Subgroups — M₁₆×Dic₁₇₈ #26029205

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#19151728	C₂	1	2	yes	maximal, complemented
3	#19151726	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₇₈	C₂	1	2	yes	maximal, complemented
5	#19151622	C₂	1	1	yes	maximal, characteristic
6	#19151621	C₂	1	2	yes	maximal
7	M₁₆×Dic₈₉	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₈	C₄	1	1	yes	characteristic
10	C₂×#13083558	C₂²	1	2	yes	—
11	C₄×Dic₁₇₈	C₄	1	2	yes	—
12	C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
13	C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
14	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₇₈	C₂²	1	2	yes	—
16	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
18	C₂×#13083553	C₄	1	2	yes	—
19	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
20	C₄⋊#10623494	C₂²	1	2	yes	—
21	C₄⋊#10623494	C₂²	1	2	yes	—
22	#15667941	C₂²	1	4	yes	—
23	#15667941	C₂²	1	4	yes	—
24	#15667922	C₂²	1	4	yes	—
25	C₈×Dic₈₉	C₂²	1	4	yes	—
26	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
29	#97×C₈₉	C₂²	1	2	yes	—
30	C₄×C₈×C₈₉	C₂²	1	2	yes	—
31	C₂×Dic₁₇₈	—	2	4	no	—
32	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
36	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
37	#13083558	—	2	4	no	—
38	C₂×#10623494	C₂³	1	4	yes	—
39	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
40	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
42	#13083553	C₂×C₄	1	2	yes	—
43	#13083553	C₂×C₄	1	2	yes	—
44	#13083553	C₂×C₄	1	2	yes	—
45	#13083553	C₂×C₄	1	2	yes	—
46	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
47	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
48	C₄×Dic₈₉	C₂³	1	2	yes	—
49	C₄×Dic₈₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
53	M₁₆×C₈₉	—	2	2	no	—
54	M₁₆×C₈₉	Q₈	1	4	yes	complemented
55	C₂×C₈×C₈₉	C₂³	1	2	yes	—
56	C₂×C₈×C₈₉	C₂³	1	2	yes	—
57	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	4	1	no	—
64	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₈₉	#41	1	2	yes	—
71	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₈₉	—	2	4	no	—
73	C₂×Dic₈₉	—	4	2	no	—
74	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
77	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
78	Dic₁₇₈	—	2	2	no	—
79	Dic₁₇₈	—	2	4	no	—
80	Dic₁₇₈	M₁₆	1	4	yes	direct product factor
81	C₈×C₈₉	—	2	2	no	—
82	C₈×C₈₉	C₂×Q₈	1	4	yes	complemented
83	#10623494	—	2	4	no	—
84	C₂²×C₈₉	—	2	1	no	—
85	C₂²×C₈₉	—	2	1	no	—
86	C₂²×C₈₉	C₄×Q₈	1	2	yes	—
87	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₈₉	—	2	1	no	—
89	C₄×C₈₉	—	2	1	no	—
90	C₄×C₈₉	—	2	1	no	—
91	C₄×C₈₉	C₄×Q₈	1	2	yes	—
92	C₄×C₈₉	C₂×M₁₆	1	2	yes	—
93	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
94	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
95	Dic₈₉	—	2	2	no	—
96	Dic₈₉	—	2	2	no	—
97	Dic₈₉	—	2	2	no	—
98	Dic₈₉	C₂×M₁₆	1	4	yes	—
99	C₂×C₈₉	—	2	2	no	—
100	C₂×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₈₉	#568	1	1	yes	characteristic
103	Q₈×M₁₆	—	89	1	no	maximal, Sylow
104	C₈₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	89	1	no	—
106	#446	—	89	4	no	—
107	#446	—	89	2	no	—
108	C₈×Q₈	—	89	2	no	—
109	#423	—	89	2	no	—
110	#423	—	89	1	no	—
111	C₄×M₁₆	—	89	2	no	—
112	C₄×M₁₆	D₈₉	1	1	yes	characteristic
113	C₂²×Q₈	—	89	1	no	—
114	C₂×M₁₆	—	89	2	no	—
115	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
116	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
117	C₄×Q₈	—	89	1	no	—
118	C₄×Q₈	—	89	2	no	—
119	C₄×Q₈	—	89	1	no	—
120	C₄×Q₈	—	89	1	no	—
121	C₄×Q₈	—	89	2	no	—
122	C₄×Q₈	—	89	1	no	—
123	C₂×C₄⋊C₄	—	89	2	no	—
124	C₂×C₄⋊C₄	—	89	1	no	—
125	C₂×C₄²	—	89	2	no	—
126	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
127	C₄⋊C₈	—	89	4	no	—
128	C₄⋊C₈	—	89	4	no	—
129	C₄⋊C₈	—	89	2	no	—
130	C₄⋊C₈	—	89	2	no	—
131	#97	—	89	4	no	—
132	#97	C₂×D₈₉	1	2	yes	—
133	C₄×C₈	—	89	4	no	—
134	C₄×C₈	C₂×D₈₉	1	2	yes	—
135	C₂×Q₈	—	89	1	no	—
136	C₂×Q₈	—	89	2	no	—
137	C₂×Q₈	—	89	1	no	—
138	C₂×Q₈	—	178	4	no	—
139	C₂²×C₄	—	89	2	no	—
140	C₂²×C₄	—	89	2	no	—
141	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	178	4	no	—
146	M₁₆	Dic₁₇₈	1	4	yes	direct product factor
147	C₂×C₈	—	89	4	no	—
148	C₂×C₈	C₂²×D₈₉	1	2	yes	—
149	C₂×C₈	C₂²×D₈₉	1	2	yes	—
150	C₄⋊C₄	—	89	2	no	—
151	C₄⋊C₄	—	89	2	no	—
152	C₄⋊C₄	—	89	2	no	—
153	C₄⋊C₄	—	89	2	no	—
154	C₄⋊C₄	—	89	2	no	—
155	C₄⋊C₄	—	89	1	no	—
156	C₄⋊C₄	—	89	1	no	—
157	C₄²	—	89	2	no	—
158	C₄²	—	89	2	no	—
159	C₄²	—	89	2	no	—
160	C₄²	—	89	2	no	—
161	C₄²	C₄×D₈₉	1	1	yes	characteristic
162	C₄²	C₄×D₈₉	1	1	yes	characteristic
163	C₄²	C₂²×D₈₉	1	1	yes	characteristic
164	C₄²	C₂²×D₈₉	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
166	Q₈	—	89	4	no	—
167	Q₈	—	178	4	no	—
168	Q₈	—	178	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	89	2	no	—
172	C₂×C₄	—	89	2	no	—
173	C₂×C₄	—	89	2	no	—
174	C₂×C₄	—	89	2	no	—
175	C₂×C₄	—	356	2	no	—
176	C₂×C₄	—	178	4	no	—
177	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
182	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
183	C₂×C₄	#13083574	1	2	yes	—
184	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	178	4	no	—
187	C₈	C₂×Dic₁₇₈	1	4	yes	complemented
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	C₄×Dic₁₇₈	1	2	yes	—
191	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	89	4	no	—
196	C₄	—	178	2	no	—
197	C₄	—	178	2	no	—
198	C₄	—	178	2	no	—
199	C₄	C₄×Dic₁₇₈	1	2	yes	—
200	C₄	M₁₆×D₈₉	1	2	yes	—
201	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
202	C₄	C₂×#13083574	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₈₉	1	1	yes	characteristic
206	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	C₂³×C₈₉	C₂²×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₄×M₁₆	D₈₉
hypocenter	C₈₉	Q₈×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
2. #19151728	C₂	semidirect
3. #19151726	C₂	semidirect
4. C₈×Dic₁₇₈	C₂	semidirect
54. M₁₆×C₈₉	Q₈	semidirect
80. Dic₁₇₈	M₁₆	direct (N × H)
82. C₈×C₈₉	C₂×Q₈	semidirect
104. C₈₉	Q₈×M₁₆	semidirect
146. M₁₆	Dic₁₇₈	direct (N × H)
187. C₈	C₂×Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₇₈	2
#19151728	2
#19151726	2
C₈×Dic₁₇₈	2
#19151622	2
#19151621	2
M₁₆×Dic₈₉	2
C₄×M₁₆×C₈₉	2
Q₈×M₁₆	89

Subgroups — M16×Dic178 #26029205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₁₇₈ #26029205