Subgroups — C₂×C₄²×Dic₁₇₈ #36712624

Order 22784.

3966 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	C₂²×C₄×Dic₁₇₈	C₂	1	3	yes	maximal
2	C₄²×Dic₁₇₈	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×#13083553	C₂	1	6	yes	maximal
4	C₂×C₄×C₄⋊Dic₈₉	C₂	1	3	yes	maximal
5	C₂×C₄²×Dic₈₉	C₂	1	2	yes	maximal
6	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₇₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₈	C₄	1	48	yes	direct product factor
9	C₂×C₄×Dic₁₇₈	C₂²	1	24	yes	—
10	C₂²×C₄⋊Dic₈₉	C₂²	1	3	yes	—
11	C₂²×#13083553	C₂²	1	6	yes	—
12	C₂²×C₄×Dic₈₉	C₂²	1	6	yes	—
13	C₂²×C₄×Dic₈₉	C₂²	1	6	yes	—
14	C₄×#13083553	C₂²	1	48	yes	—
15	C₄×C₄⋊Dic₈₉	C₂²	1	24	yes	—
16	C₂×#15667919	C₂²	1	6	yes	—
17	C₄²×Dic₈₉	C₂²	1	16	yes	—
18	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
19	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
20	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₄³×C₈₉	C₂²	1	8	yes	—
22	C₂²×Dic₁₇₈	C₂×C₄	1	24	yes	—
23	C₂²×Dic₁₇₈	C₂³	1	4	yes	—
24	C₂³×Dic₈₉	C₂³	1	6	yes	—
25	C₂³×Dic₈₉	C₂³	1	2	yes	—
26	C₄×Dic₁₇₈	C₂×C₄	1	192	yes	direct product factor
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	24	yes	—
28	C₂×C₄⋊Dic₈₉	C₂³	1	12	yes	—
29	C₂×#13083553	C₂×C₄	1	48	yes	—
30	C₂×#13083553	C₂³	1	24	yes	—
31	C₂×C₄×Dic₈₉	—	2	24	no	—
32	C₂×C₄×Dic₈₉	C₂×C₄	1	48	yes	—
33	C₂×C₄×Dic₈₉	C₂³	1	24	yes	—
34	C₂×C₄×Dic₈₉	C₂³	1	24	yes	—
35	#15667919	C₂³	1	24	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
39	C₂×C₄²×C₈₉	—	2	12	no	—
40	C₂×C₄²×C₈₉	C₂×C₄	1	24	yes	—
41	C₂×C₄²×C₈₉	Q₈	1	8	yes	complemented
42	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
43	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
44	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
45	C₂×Dic₁₇₈	C₄²	1	64	yes	direct product factor
46	C₂×Dic₁₇₈	C₂²×C₄	1	48	yes	—
47	C₂²×Dic₈₉	—	2	24	no	—
48	C₂²×Dic₈₉	C₂²×C₄	1	24	yes	—
49	C₂²×Dic₈₉	C₂²×C₄	1	24	yes	—
50	C₂²×Dic₈₉	C₂⁴	1	12	yes	—
51	C₂²×Dic₈₉	C₂⁴	1	4	yes	—
52	C₄⋊Dic₈₉	C₂²×C₄	1	48	yes	—
53	#13083553	C₂²×C₄	1	96	yes	—
54	C₄×Dic₈₉	—	2	48	no	—
55	C₄×Dic₈₉	C₂²×C₄	1	96	yes	—
56	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	—	2	12	no	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
60	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₈₉	C₂×Q₈	1	12	yes	—
62	C₂²×C₄×C₈₉	#41	1	12	yes	—
63	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
64	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
66	C₄²×C₈₉	—	2	24	no	—
67	C₄²×C₈₉	C₂²×C₄	1	48	yes	—
68	C₄²×C₈₉	C₂×Q₈	1	16	yes	complemented
69	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
70	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
71	C₂³×C₈₉	C₂×#41	1	6	yes	—
72	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
73	C₂×C₄×C₈₉	—	2	24	no	—
74	C₂×C₄×C₈₉	—	2	12	no	—
75	C₂×C₄×C₈₉	C₂×C₄²	1	16	yes	—
76	C₂×C₄×C₈₉	C₄×Q₈	1	48	yes	complemented
77	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
78	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
79	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
80	C₂×C₄×C₈₉	C₂²×Q₈	1	12	yes	—
81	C₂×C₄×C₈₉	C₂×#41	1	12	yes	—
82	C₂×Dic₈₉	—	2	48	no	—
83	C₂×Dic₈₉	—	2	24	no	—
84	C₂×Dic₈₉	C₂×C₄²	1	32	yes	—
85	C₂×Dic₈₉	C₂³×C₄	1	24	yes	—
86	C₂×Dic₈₉	C₂³×C₄	1	24	yes	—
87	Dic₁₇₈	C₂×C₄²	1	64	yes	direct product factor
88	C₂²×C₈₉	C₂²×C₄²	1	4	yes	—
89	C₂²×C₈₉	C₂×C₄×Q₈	1	12	yes	—
90	C₂²×C₈₉	C₄×#41	1	12	yes	—
91	C₂²×C₈₉	C₂⁴×C₄	1	3	yes	—
92	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
93	C₂²×C₈₉	C₂²×#41	1	3	yes	—
94	C₄×C₈₉	—	2	12	no	—
95	C₄×C₈₉	C₂²×C₄²	1	8	yes	—
96	C₄×C₈₉	C₂×C₄×Q₈	1	24	yes	complemented
97	Dic₈₉	—	2	24	no	—
98	Dic₈₉	C₂²×C₄²	1	16	yes	—
99	C₂×C₄²×Q₈	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	C₄²×Q₈	1	8	yes	complemented
101	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
102	C₂×C₈₉	C₂²×C₄×Q₈	1	3	yes	—
103	C₂×C₈₉	C₂×C₄×#41	1	3	yes	—
104	C₂²×C₄×Q₈	—	89	3	no	—
105	C₄²×Q₈	—	89	16	no	—
106	C₂×C₄×C₄⋊C₄	—	89	3	no	—
107	C₂×C₄×C₄⋊C₄	—	89	6	no	—
108	C₂×C₄³	—	89	2	no	—
109	C₂×C₄³	D₈₉	1	1	yes	characteristic
110	C₈₉	C₂×C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
111	C₂³×Q₈	—	89	1	no	—
112	C₂×C₄×Q₈	—	89	48	no	—
113	C₂×C₄×Q₈	—	89	24	no	—
114	C₂²×C₄⋊C₄	—	89	6	no	—
115	C₂²×C₄⋊C₄	—	89	3	no	—
116	C₂²×C₄²	—	89	6	no	—
117	C₂²×C₄²	—	89	6	no	—
118	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
119	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
120	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
121	C₄×C₄⋊C₄	—	89	24	no	—
122	C₄×C₄⋊C₄	—	89	48	no	—
123	C₂×#95	—	89	6	no	—
124	C₄³	—	89	16	no	—
125	C₄³	C₂×D₈₉	1	8	yes	—
126	C₂²×Q₈	—	89	24	no	—
127	C₂²×Q₈	—	89	4	no	—
128	C₂³×C₄	—	89	2	no	—
129	C₂³×C₄	—	89	6	no	—
130	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
131	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
132	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
133	C₄×Q₈	—	89	192	no	—
134	C₂×C₄⋊C₄	—	89	48	no	—
135	C₂×C₄⋊C₄	—	89	24	no	—
136	C₂×C₄⋊C₄	—	89	24	no	—
137	C₂×C₄⋊C₄	—	89	12	no	—
138	C₂×C₄²	—	2	12	no	—
139	C₂×C₄²	—	89	48	no	—
140	C₂×C₄²	—	89	24	no	—
141	C₂×C₄²	—	89	24	no	—
142	C₂×C₄²	—	178	24	no	—
143	C₂×C₄²	Dic₁₇₈	1	8	yes	direct product factor
144	C₂×C₄²	C₄×D₈₉	1	24	yes	—
145	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
146	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
147	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
148	#95	—	89	24	no	—
149	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
150	C₂×Q₈	—	89	48	no	—
151	C₂×Q₈	—	89	64	no	—
152	C₂²×C₄	—	2	12	no	—
153	C₂²×C₄	—	89	24	no	—
154	C₂²×C₄	—	89	4	no	—
155	C₂²×C₄	—	89	24	no	—
156	C₂²×C₄	—	89	12	no	—
157	C₂²×C₄	—	178	24	no	—
158	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
159	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
160	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
161	C₂²×C₄	C₂×Dic₁₇₈	1	12	yes	—
162	C₂²×C₄	#13083574	1	12	yes	—
163	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
164	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
165	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
166	C₄⋊C₄	—	89	96	no	—
167	C₄⋊C₄	—	89	48	no	—
168	C₄²	—	2	24	no	—
169	C₄²	—	89	96	no	—
170	C₄²	—	178	48	no	—
171	C₄²	C₂×C₄×D₈₉	1	48	yes	—
172	C₄²	C₂×Dic₁₇₈	1	16	yes	direct product factor
173	C₂³	C₂²×C₄×D₈₉	1	6	yes	—
174	C₂³	C₂²×Dic₁₇₈	1	2	yes	—
175	C₂³	C₂×#13083574	1	6	yes	—
176	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
177	Q₈	—	89	64	no	—
178	C₂×C₄	—	2	12	no	—
179	C₂×C₄	—	2	24	no	—
180	C₂×C₄	—	89	24	no	—
181	C₂×C₄	—	89	32	no	—
182	C₂×C₄	—	89	24	no	—
183	C₂×C₄	—	178	24	no	—
184	C₂×C₄	—	178	48	no	—
185	C₂×C₄	C₄²×D₈₉	1	16	yes	—
186	C₂×C₄	C₄×Dic₁₇₈	1	48	yes	direct product factor
187	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
188	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
189	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
190	C₂×C₄	C₂²×Dic₁₇₈	1	12	yes	—
191	C₂×C₄	C₂×#13083574	1	12	yes	—
192	C₂²	C₂×C₄²×D₈₉	1	4	yes	—
193	C₂²	C₂×C₄×Dic₁₇₈	1	12	yes	—
194	C₂²	C₄×#13083574	1	12	yes	—
195	C₂²	C₂³×C₄×D₈₉	1	3	yes	—
196	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
197	C₂²	C₂²×#13083574	1	3	yes	—
198	C₄	—	2	12	no	—
199	C₄	—	89	16	no	—
200	C₄	—	178	24	no	—
201	C₄	C₂×C₄²×D₈₉	1	8	yes	—
202	C₄	C₂×C₄×Dic₁₇₈	1	24	yes	direct product factor
203	C₂	C₄²×Dic₁₇₈	1	8	yes	direct product factor
204	C₂	C₂²×C₄²×D₈₉	1	1	yes	characteristic
205	C₂	C₂²×C₄×Dic₁₇₈	1	3	yes	—
206	C₂	C₂×C₄×#13083574	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄³×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂⁴
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	C₂×C₄²×Q₈
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. C₄²×Dic₁₇₈	C₂	direct (N × H)
8. C₂×C₄×Dic₁₇₈	C₄	direct (N × H)
26. C₄×Dic₁₇₈	C₂×C₄	direct (N × H)
41. C₂×C₄²×C₈₉	Q₈	semidirect
45. C₂×Dic₁₇₈	C₄²	direct (N × H)
68. C₄²×C₈₉	C₂×Q₈	semidirect
76. C₂×C₄×C₈₉	C₄×Q₈	semidirect
87. Dic₁₇₈	C₂×C₄²	direct (N × H)
96. C₄×C₈₉	C₂×C₄×Q₈	semidirect
100. C₂×C₈₉	C₄²×Q₈	semidirect
110. C₈₉	C₂×C₄²×Q₈	semidirect
143. C₂×C₄²	Dic₁₇₈	direct (N × H)
172. C₄²	C₂×Dic₁₇₈	direct (N × H)
186. C₂×C₄	C₄×Dic₁₇₈	direct (N × H)
202. C₄	C₂×C₄×Dic₁₇₈	direct (N × H)
203. C₂	C₄²×Dic₁₇₈	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×C₄×Dic₁₇₈	2
C₄²×Dic₁₇₈	2
C₂×C₄×#13083553	2
C₂×C₄×C₄⋊Dic₈₉	2
C₂×C₄²×Dic₈₉	2
C₂×C₄³×C₈₉	2
C₂×C₄²×Q₈	89

Subgroups — C2×C42×Dic178 #36712624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×Dic₁₇₈ #36712624