Subgroups — C₂×D₄×Dic₁₇₈ #26033726

Order 11392.

1568 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₁₇₈	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₁₇₈	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15668012	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15668011	C₂	1	2	yes	maximal, complemented
5	C₂×#15667991	C₂	1	2	yes	maximal, complemented
6	C₂×#15667989	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×Dic₈₉	C₂	1	2	yes	maximal
9	C₂×C₄×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₇₈	—	2	16	no	—
11	C₂²×Dic₁₇₈	C₂²	1	4	yes	—
12	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
13	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₇₈	C₂²	1	16	yes	complemented
15	C₂³×Dic₈₉	C₂²	1	4	yes	—
16	#15668012	C₂²	1	8	yes	complemented
17	#15668011	C₂²	1	16	yes	complemented
18	#15667991	C₂²	1	16	yes	complemented
19	#15667989	C₂²	1	32	yes	complemented
20	C₄×Dic₁₇₈	C₂²	1	8	yes	complemented
21	D₄×Dic₈₉	C₂²	1	16	yes	—
22	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
25	C₂×#13083553	C₂²	1	4	yes	—
26	C₂×#13083553	C₂²	1	2	yes	—
27	C₂×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
28	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
29	C₂²×D₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₈₉	C₂²	1	2	yes	—
31	C₄×D₄×C₈₉	C₂²	1	8	yes	—
32	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
34	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₇₈	—	2	8	no	—
36	C₂×Dic₁₇₈	—	2	16	no	—
37	C₂×Dic₁₇₈	—	2	64	no	—
38	C₂×Dic₁₇₈	D₄	1	8	yes	direct product factor
39	C₂×Dic₁₇₈	C₂³	1	16	yes	—
40	C₂×Dic₁₇₈	C₂³	1	8	yes	—
41	C₂×Dic₁₇₈	C₂³	1	4	yes	—
42	C₂²×Dic₈₉	—	2	16	no	—
43	C₂²×Dic₈₉	C₂³	1	16	yes	—
44	C₂²×Dic₈₉	C₂³	1	4	yes	—
45	C₂²×Dic₈₉	C₂³	1	2	yes	—
46	C₂²×Dic₈₉	C₂³	1	2	yes	—
47	C₄⋊Dic₈₉	C₂³	1	8	yes	—
48	C₄⋊Dic₈₉	C₂³	1	4	yes	—
49	C₄⋊Dic₈₉	C₂³	1	8	yes	—
50	#13083553	C₂³	1	8	yes	—
51	#13083553	C₂³	1	16	yes	—
52	C₂²⋊Dic₈₉	C₂³	1	16	yes	—
53	C₄×Dic₈₉	C₂³	1	8	yes	—
54	C₂⁴×C₈₉	C₂³	1	2	yes	—
55	C₂×D₄×C₈₉	Q₈	1	8	yes	complemented
56	C₂×D₄×C₈₉	C₂³	1	4	yes	—
57	C₂²×C₄×C₈₉	—	2	8	no	—
58	C₂²×C₄×C₈₉	C₂³	1	8	yes	—
59	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
60	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
64	C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
65	C₄²×C₈₉	C₂³	1	4	yes	—
66	C₂³×C₈₉	—	2	4	no	—
67	C₂³×C₈₉	—	2	4	no	—
68	C₂³×C₈₉	C₂×Q₈	1	8	yes	complemented
69	C₂³×C₈₉	C₂⁴	1	4	yes	—
70	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
71	D₄×C₈₉	C₂×Q₈	1	16	yes	complemented
72	C₂×C₄×C₈₉	—	2	4	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	2	16	no	—
75	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
76	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	complemented
77	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
78	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
79	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
80	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
81	C₂×Dic₈₉	—	2	8	no	—
82	C₂×Dic₈₉	—	2	4	no	—
83	C₂×Dic₈₉	—	2	32	no	—
84	C₂×Dic₈₉	C₂×D₄	1	8	yes	—
85	C₂×Dic₈₉	C₂⁴	1	4	yes	—
86	C₂×Dic₈₉	C₂⁴	1	8	yes	—
87	C₂×Dic₈₉	C₂⁴	1	4	yes	—
88	Dic₁₇₈	—	2	16	no	—
89	Dic₁₇₈	—	2	32	no	—
90	Dic₁₇₈	C₂×D₄	1	16	yes	direct product factor
91	C₂²×C₈₉	—	2	16	no	—
92	C₂²×C₈₉	—	2	4	no	—
93	C₂²×C₈₉	—	2	4	no	—
94	C₂²×C₈₉	C₂²×D₄	1	2	yes	—
95	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
96	C₂²×C₈₉	C₂²×Q₈	1	8	yes	complemented
97	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
98	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
99	C₄×C₈₉	—	2	4	no	—
100	C₄×C₈₉	—	2	2	no	—
101	C₄×C₈₉	C₂²×D₄	1	4	yes	—
102	C₄×C₈₉	C₂²×Q₈	1	4	yes	complemented
103	Dic₈₉	—	2	8	no	—
104	Dic₈₉	—	2	4	no	—
105	Dic₈₉	C₂²×D₄	1	8	yes	—
106	C₂×C₈₉	—	2	8	no	—
107	C₂×C₈₉	D₄×Q₈	1	4	yes	complemented
108	C₂×C₈₉	C₂³×D₄	1	1	yes	characteristic
109	C₂×C₈₉	C₂³×Q₈	1	1	yes	characteristic
110	C₂×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×D₄×Q₈	—	89	1	no	maximal, Sylow
112	C₈₉	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
113	C₂³×Q₈	—	89	2	no	—
114	D₄×Q₈	—	89	16	no	—
115	C₂×C₄⋊Q₈	—	89	2	no	—
116	C₂×C₄⋊Q₈	—	89	1	no	—
117	C₂×C₂²⋊Q₈	—	89	2	no	—
118	C₂×C₂²⋊Q₈	—	89	4	no	—
119	C₂×C₄×Q₈	—	89	1	no	—
120	C₂×C₄×D₄	—	89	2	no	—
121	C₂×C₄×D₄	D₈₉	1	1	yes	characteristic
122	C₂²×Q₈	—	89	1	no	—
123	C₂²×Q₈	—	89	4	no	—
124	C₂²×Q₈	—	89	2	no	—
125	C₂²×Q₈	—	89	16	no	—
126	C₂²×Q₈	—	178	16	no	—
127	C₂²×D₄	C₂×D₈₉	1	1	yes	characteristic
128	C₂³×C₄	—	89	4	no	—
129	C₂³×C₄	C₂×D₈₉	1	2	yes	—
130	C₄⋊Q₈	—	89	16	no	—
131	C₄⋊Q₈	—	89	8	no	—
132	C₂²⋊Q₈	—	89	16	no	—
133	C₂²⋊Q₈	—	89	32	no	—
134	C₄×Q₈	—	89	8	no	—
135	C₄×D₄	—	89	16	no	—
136	C₄×D₄	C₂×D₈₉	1	8	yes	—
137	C₂×C₄⋊C₄	—	89	2	no	—
138	C₂×C₄⋊C₄	—	89	2	no	—
139	C₂×C₄⋊C₄	—	89	1	no	—
140	C₂×C₄⋊C₄	—	89	4	no	—
141	C₂×C₄⋊C₄	—	89	2	no	—
142	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	—	89	4	no	—
144	C₂×C₂²⋊C₄	C₂×D₈₉	1	2	yes	—
145	C₂×C₄²	—	89	2	no	—
146	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
147	C₂⁴	C₂²×D₈₉	1	2	yes	—
148	C₂×Q₈	—	89	8	no	—
149	C₂×Q₈	—	89	4	no	—
150	C₂×Q₈	—	89	16	no	—
151	C₂×Q₈	—	89	8	no	—
152	C₂×Q₈	—	178	16	no	—
153	C₂×Q₈	—	178	64	no	—
154	C₂×Q₈	—	178	8	no	—
155	C₂×D₄	Dic₁₇₈	1	8	yes	direct product factor
156	C₂×D₄	C₂²×D₈₉	1	4	yes	—
157	C₂²×C₄	—	2	8	no	—
158	C₂²×C₄	—	89	2	no	—
159	C₂²×C₄	—	89	16	no	—
160	C₂²×C₄	—	89	2	no	—
161	C₂²×C₄	—	89	4	no	—
162	C₂²×C₄	—	178	16	no	—
163	C₂²×C₄	C₂²×D₈₉	1	8	yes	—
164	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
166	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
168	C₄⋊C₄	—	89	8	no	—
169	C₄⋊C₄	—	89	8	no	—
170	C₄⋊C₄	—	89	4	no	—
171	C₄⋊C₄	—	89	16	no	—
172	C₄⋊C₄	—	89	8	no	—
173	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
174	C₂²⋊C₄	—	89	16	no	—
175	C₂²⋊C₄	C₂²×D₈₉	1	8	yes	—
176	C₄²	—	89	8	no	—
177	C₄²	C₂²×D₈₉	1	4	yes	—
178	C₂³	—	2	4	no	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×Dic₁₇₈	1	8	yes	complemented
181	C₂³	C₂³×D₈₉	1	4	yes	—
182	C₂³	C₂³×D₈₉	1	1	yes	characteristic
183	Q₈	—	89	16	no	—
184	Q₈	—	178	32	no	—
185	Q₈	—	178	16	no	—
186	D₄	C₂×Dic₁₇₈	1	16	yes	direct product factor
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	16	no	—
190	C₂×C₄	—	89	8	no	—
191	C₂×C₄	—	89	4	no	—
192	C₂×C₄	—	89	4	no	—
193	C₂×C₄	—	89	8	no	—
194	C₂×C₄	—	178	32	no	—
195	C₂×C₄	—	178	8	no	—
196	C₂×C₄	—	178	4	no	—
197	C₂×C₄	D₄×D₈₉	1	4	yes	—
198	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	complemented
199	C₂×C₄	C₂³×D₈₉	1	2	yes	—
200	C₂×C₄	C₂³×D₈₉	1	4	yes	—
201	C₂×C₄	C₂³×D₈₉	1	2	yes	—
202	C₂×C₄	C₂³×D₈₉	1	2	yes	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	16	no	—
206	C₂²	C₂×D₄×D₈₉	1	2	yes	—
207	C₂²	C₂²×Dic₁₇₈	1	2	yes	—
208	C₂²	C₂²×Dic₁₇₈	1	8	yes	complemented
209	C₂²	#15668059	1	2	yes	—
210	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	2	no	—
213	C₄	—	89	8	no	—
214	C₄	—	178	8	no	—
215	C₄	—	178	4	no	—
216	C₄	C₂×D₄×D₈₉	1	4	yes	—
217	C₄	C₂²×Dic₁₇₈	1	4	yes	complemented
218	C₂	—	2	8	no	—
219	C₂	D₄×Dic₁₇₈	1	4	yes	direct product factor
220	C₂	C₂²×D₄×D₈₉	1	1	yes	characteristic
221	C₂	C₂³×Dic₁₇₈	1	1	yes	characteristic
222	C₂	C₂×#15668059	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₈₉
commutator, cosocle	C₂²×C₈₉	C₂⁵
Frattini	C₂²	C₂⁴×D₈₉
Fitting	C₂×C₄×D₄×C₈₉	C₂
socle	C₂³×C₈₉	C₂⁴
prime radical	C₂²×D₄×C₈₉	C₂²
hypercenter	C₂×C₄×D₄	D₈₉
hypocenter	C₈₉	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₁₇₈	C₂	semidirect
2. D₄×Dic₁₇₈	C₂	direct (N × H)
3. C₂×#15668012	C₂	semidirect
4. C₂×#15668011	C₂	semidirect
5. C₂×#15667991	C₂	semidirect
6. C₂×#15667989	C₂	semidirect
7. C₂×C₄×Dic₁₇₈	C₂	semidirect
14. C₂²×Dic₁₇₈	C₂²	semidirect
16. #15668012	C₂²	semidirect
17. #15668011	C₂²	semidirect
18. #15667991	C₂²	semidirect
19. #15667989	C₂²	semidirect
20. C₄×Dic₁₇₈	C₂²	semidirect
38. C₂×Dic₁₇₈	D₄	direct (N × H)
55. C₂×D₄×C₈₉	Q₈	semidirect
68. C₂³×C₈₉	C₂×Q₈	semidirect
71. D₄×C₈₉	C₂×Q₈	semidirect
76. C₂×C₄×C₈₉	C₂×Q₈	semidirect
90. Dic₁₇₈	C₂×D₄	direct (N × H)
96. C₂²×C₈₉	C₂²×Q₈	semidirect
102. C₄×C₈₉	C₂²×Q₈	semidirect
107. C₂×C₈₉	D₄×Q₈	semidirect
112. C₈₉	C₂×D₄×Q₈	semidirect
155. C₂×D₄	Dic₁₇₈	direct (N × H)
180. C₂³	C₂×Dic₁₇₈	semidirect
186. D₄	C₂×Dic₁₇₈	direct (N × H)
198. C₂×C₄	C₂×Dic₁₇₈	semidirect
208. C₂²	C₂²×Dic₁₇₈	semidirect
217. C₄	C₂²×Dic₁₇₈	semidirect
219. C₂	D₄×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
C₂³×Dic₁₇₈	2
D₄×Dic₁₇₈	2
C₂×#15668012	2
C₂×#15668011	2
C₂×#15667991	2
C₂×#15667989	2
C₂×C₄×Dic₁₇₈	2
C₂×D₄×Dic₈₉	2
C₂×C₄×D₄×C₈₉	2
C₂×D₄×Q₈	89

Subgroups — C2×D4×Dic178 #26033726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×Dic₁₇₈ #26033726