Subgroups — C₄×C₈×Dic₁₇₈ #36553034

Order 22784.

1070 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₇₈	C₂	1	2	yes	maximal
2	C₄²×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
3	C₈×#13083553	C₂	1	4	yes	maximal
4	C₈×C₄⋊Dic₈₉	C₂	1	2	yes	maximal
5	C₄×#15667941	C₂	1	2	yes	maximal
6	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
7	C₄×C₈×Dic₈₉	C₂	1	2	yes	maximal
8	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₇₈	C₄	1	16	yes	direct product factor
12	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
13	C₂×#15667941	C₂²	1	4	yes	—
14	C₂×C₄×#10623494	C₂²	1	2	yes	—
15	C₂×C₈×Dic₈₉	C₂²	1	4	yes	—
16	C₂×C₈×Dic₈₉	C₂²	1	4	yes	—
17	C₄×#13083553	C₄	1	4	yes	—
18	C₄×#13083553	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₈₉	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
21	C₄²×Dic₈₉	C₂²	1	2	yes	—
22	#19151353	C₂²	1	2	yes	—
23	#19151352	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₄³×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10623494	C₂³	1	4	yes	—
31	C₄×Dic₁₇₈	C₈	1	16	yes	direct product factor
32	C₄×Dic₁₇₈	C₂×C₄	1	8	yes	—
33	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
35	C₂×#13083553	C₂×C₄	1	2	yes	—
36	C₂×#13083553	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
40	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
41	C₄⋊#10623494	C₂×C₄	1	8	yes	—
42	#15667941	C₂×C₄	1	16	yes	—
43	C₄×#10623494	—	2	4	no	—
44	C₈×Dic₈₉	—	2	8	no	—
45	C₈×Dic₈₉	C₂×C₄	1	16	yes	—
46	#15667919	C₂×C₄	1	4	yes	—
47	#15667919	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
49	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₈₉	—	2	4	no	—
56	C₄×C₈×C₈₉	—	2	2	no	—
57	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₈₉	Q₈	1	4	yes	complemented
59	C₂×Dic₁₇₈	C₄²	1	4	yes	—
60	C₂×Dic₁₇₈	C₂×C₈	1	8	yes	—
61	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
62	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
63	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
64	C₂×#10623494	—	2	4	no	—
65	C₂×#10623494	—	2	4	no	—
66	C₂×#10623494	C₂²×C₄	1	8	yes	—
67	C₄⋊Dic₈₉	C₄²	1	4	yes	—
68	C₄⋊Dic₈₉	C₂×C₈	1	8	yes	—
69	#13083553	C₄²	1	8	yes	—
70	#13083553	C₂×C₈	1	16	yes	—
71	C₄×Dic₈₉	—	2	8	no	—
72	C₄×Dic₈₉	—	2	4	no	—
73	C₄×Dic₈₉	C₂×C₈	1	16	yes	—
74	C₄×Dic₈₉	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₈₉	#41	1	4	yes	—
86	C₄²×C₈₉	—	2	4	no	—
87	C₄²×C₈₉	—	2	2	no	—
88	C₄²×C₈₉	C₂×C₈	1	8	yes	—
89	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
90	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
91	C₄²×C₈₉	#41	1	2	yes	—
92	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
99	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
102	C₂×C₄×C₈₉	#131	1	4	yes	—
103	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
104	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
107	C₂×Dic₈₉	—	2	4	no	—
108	C₂×Dic₈₉	—	2	4	no	—
109	C₂×Dic₈₉	—	2	4	no	—
110	C₂×Dic₈₉	C₂×C₄²	1	4	yes	—
111	C₂×Dic₈₉	C₂×C₄²	1	4	yes	—
112	C₂×Dic₈₉	C₂²×C₈	1	8	yes	—
113	C₂×Dic₈₉	C₂²×C₈	1	8	yes	—
114	Dic₁₇₈	C₄×C₈	1	16	yes	direct product factor
115	C₈×C₈₉	—	2	4	no	—
116	C₈×C₈₉	C₄×Q₈	1	8	yes	complemented
117	#10623494	—	2	8	no	—
118	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
121	C₂²×C₈₉	C₂³×C₈	1	2	yes	—
122	C₂²×C₈₉	C₂×#131	1	2	yes	—
123	C₄×C₈₉	—	2	2	no	—
124	C₄×C₈₉	—	2	4	no	—
125	C₄×C₈₉	C₂×C₄×C₈	1	4	yes	—
126	C₄×C₈₉	C₈×Q₈	1	8	yes	complemented
127	C₄×C₈₉	C₂×C₄×Q₈	1	2	yes	—
128	C₄×C₈₉	C₄×#41	1	2	yes	—
129	Dic₈₉	—	2	4	no	—
130	Dic₈₉	—	2	8	no	—
131	Dic₈₉	C₂×C₄×C₈	1	8	yes	—
132	C₄×C₈×Q₈	—	89	1	no	maximal, Sylow
133	C₂×C₈₉	C₄²×Q₈	1	1	yes	characteristic
134	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
135	C₂×C₈₉	C₄×#131	1	1	yes	characteristic
136	C₂×C₈₉	C₂×C₈×Q₈	1	2	yes	—
137	C₂×C₈₉	C₈×#41	1	2	yes	—
138	C₂×C₈×Q₈	—	89	2	no	—
139	C₄²×Q₈	—	89	1	no	—
140	C₈×C₄⋊C₄	—	89	2	no	—
141	C₈×C₄⋊C₄	—	89	4	no	—
142	C₄×C₄⋊C₈	—	89	2	no	—
143	C₄×C₄⋊C₈	—	89	1	no	—
144	C₄²×C₈	—	89	2	no	—
145	C₄²×C₈	D₈₉	1	1	yes	characteristic
146	C₈₉	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄×Q₈	—	89	2	no	—
148	C₂×C₄×Q₈	—	89	1	no	—
149	C₈×Q₈	—	89	16	no	—
150	C₂×C₄⋊C₈	—	89	4	no	—
151	C₂×C₄⋊C₈	—	89	2	no	—
152	C₂×C₄×C₈	—	89	4	no	—
153	C₂×C₄×C₈	—	89	4	no	—
154	C₂×C₄×C₈	—	89	2	no	—
155	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
157	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
158	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
159	C₄×C₄⋊C₄	—	89	2	no	—
160	C₄×C₄⋊C₄	—	89	4	no	—
161	C₄×C₄⋊C₄	—	89	1	no	—
162	C₄×C₄⋊C₄	—	89	2	no	—
163	C₄³	—	89	2	no	—
164	C₄³	C₂×D₈₉	1	1	yes	characteristic
165	#336	—	89	4	no	—
166	#336	—	89	2	no	—
167	C₂²×Q₈	—	89	1	no	—
168	C₂²×C₈	—	89	4	no	—
169	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
170	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
171	C₄×Q₈	—	89	16	no	—
172	C₄×Q₈	—	89	8	no	—
173	C₂×C₄⋊C₄	—	89	4	no	—
174	C₂×C₄⋊C₄	—	89	2	no	—
175	C₂×C₄⋊C₄	—	89	2	no	—
176	C₂×C₄⋊C₄	—	89	1	no	—
177	C₂×C₄²	—	89	4	no	—
178	C₂×C₄²	—	89	2	no	—
179	C₂×C₄²	—	89	4	no	—
180	C₂×C₄²	—	89	2	no	—
181	C₂×C₄²	C₄×D₈₉	1	2	yes	—
182	C₂×C₄²	C₄×D₈₉	1	2	yes	—
183	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
186	C₄⋊C₈	—	89	16	no	—
187	C₄⋊C₈	—	89	8	no	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	89	16	no	—
191	C₄×C₈	—	178	8	no	—
192	C₄×C₈	—	178	4	no	—
193	C₄×C₈	Dic₁₇₈	1	4	yes	direct product factor
194	C₄×C₈	C₄×D₈₉	1	8	yes	—
195	#95	—	89	4	no	—
196	#95	—	89	2	no	—
197	C₂×Q₈	—	89	8	no	—
198	C₂×Q₈	—	89	4	no	—
199	C₂²×C₄	—	89	2	no	—
200	C₂²×C₄	—	89	2	no	—
201	C₂²×C₄	—	89	4	no	—
202	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
205	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	89	8	no	—
210	C₂×C₈	—	178	4	no	—
211	C₂×C₈	—	178	4	no	—
212	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
213	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
214	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
215	C₂×C₈	#13083574	1	4	yes	—
216	C₄⋊C₄	—	89	16	no	—
217	C₄⋊C₄	—	89	8	no	—
218	C₄⋊C₄	—	89	8	no	—
219	C₄⋊C₄	—	89	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	89	16	no	—
223	C₄²	—	89	8	no	—
224	C₄²	—	178	8	no	—
225	C₄²	—	178	4	no	—
226	C₄²	C₈×D₈₉	1	8	yes	—
227	C₄²	C₂×C₄×D₈₉	1	4	yes	—
228	C₄²	C₂×Dic₁₇₈	1	2	yes	—
229	C₄²	#13083574	1	2	yes	—
230	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
231	Q₈	—	89	16	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	89	4	no	—
236	C₂×C₄	—	89	8	no	—
237	C₂×C₄	—	89	4	no	—
238	C₂×C₄	—	89	8	no	—
239	C₂×C₄	—	178	4	no	—
240	C₂×C₄	—	178	4	no	—
241	C₂×C₄	—	178	4	no	—
242	C₂×C₄	C₄²×D₈₉	1	2	yes	—
243	C₂×C₄	C₄²×D₈₉	1	2	yes	—
244	C₂×C₄	C₄×Dic₁₇₈	1	4	yes	—
245	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
246	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
247	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
248	C₂×C₄	#15668018	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
250	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
251	C₂×C₄	C₂×#13083574	1	2	yes	—
252	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₈	—	178	8	no	—
255	C₈	C₄×Dic₁₇₈	1	8	yes	direct product factor
256	C₂²	C₂×C₄²×D₈₉	1	1	yes	characteristic
257	C₂²	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
258	C₂²	C₄×#13083574	1	1	yes	characteristic
259	C₂²	C₂²×C₈×D₈₉	1	2	yes	—
260	C₂²	C₂×#15668018	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	89	8	no	—
264	C₄	—	178	4	no	—
265	C₄	—	178	8	no	—
266	C₄	C₄×C₈×D₈₉	1	4	yes	—
267	C₄	C₈×Dic₁₇₈	1	8	yes	direct product factor
268	C₄	C₂×C₄×Dic₁₇₈	1	2	yes	—
269	C₄	C₄×#13083574	1	2	yes	—
270	C₂	C₄²×Dic₁₇₈	1	1	yes	characteristic
271	C₂	C₂×C₄×C₈×D₈₉	1	1	yes	characteristic
272	C₂	C₄×#15668018	1	1	yes	characteristic
273	C₂	C₂×C₈×Dic₁₇₈	1	2	yes	—
274	C₂	C₈×#13083574	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	C₄×C₈×Q₈
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₁₇₈	C₄	direct (N × H)
31. C₄×Dic₁₇₈	C₈	direct (N × H)
58. C₄×C₈×C₈₉	Q₈	semidirect
114. Dic₁₇₈	C₄×C₈	direct (N × H)
116. C₈×C₈₉	C₄×Q₈	semidirect
126. C₄×C₈₉	C₈×Q₈	semidirect
146. C₈₉	C₄×C₈×Q₈	semidirect
193. C₄×C₈	Dic₁₇₈	direct (N × H)
255. C₈	C₄×Dic₁₇₈	direct (N × H)
267. C₄	C₈×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
C₂×C₈×Dic₁₇₈	2
C₄²×Dic₁₇₈	2
C₈×#13083553	2
C₈×C₄⋊Dic₈₉	2
C₄×#15667941	2
C₄×C₄⋊#10623494	2
C₄×C₈×Dic₈₉	2
C₄²×C₈×C₈₉	2
C₄×C₈×Q₈	89

Subgroups — C4×C8×Dic178 #36553034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×Dic₁₇₈ #36553034