Subgroups — C₄×#19151412 #36469682

Order 22784.

498 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151412	C₂	1	2	yes	maximal
3	C₄×#15667948	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018361	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₃₅₆	—	2	1	no	—
8	C₄×D₇₁₂	—	2	1	no	—
9	C₄×D₇₁₂	—	2	1	no	—
10	C₄×D₇₁₂	C₄	1	2	yes	—
11	C₄×D₇₁₂	C₄	1	2	yes	—
12	C₂×#15667948	C₂²	1	2	yes	—
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667934	—	2	1	no	—
15	C₄×#13083558	—	2	1	no	—
16	#19151412	C₄	1	8	yes	direct product factor
17	#19151363	—	2	1	no	—
18	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#346×C₈₉	—	2	1	no	—
21	C₄×C₁₆×C₈₉	—	2	1	no	—
22	C₂²×D₃₅₆	—	2	1	no	—
23	C₂²×C₄×D₈₉	—	4	1	no	—
24	C₂×D₇₁₂	—	2	1	no	—
25	C₂×D₇₁₂	—	2	2	no	—
26	C₂×D₇₁₂	C₂×C₄	1	4	yes	—
27	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
28	C₂×#13083558	—	2	2	no	—
29	C₄×D₃₅₆	—	2	2	no	—
30	C₄×D₃₅₆	—	2	2	no	—
31	C₄×D₃₅₆	—	4	1	no	—
32	C₄×D₃₅₆	—	4	1	no	—
33	C₂×C₄⋊Dic₈₉	—	2	1	no	—
34	C₂×#13083551	—	4	1	no	—
35	#15667948	—	2	2	no	—
36	#15667948	C₂×C₄	1	4	yes	complemented
37	#15667946	—	2	1	no	—
38	#15667946	C₂×C₄	1	2	yes	—
39	#15667934	—	2	2	no	—
40	#15667934	—	2	2	no	—
41	#15667923	—	4	1	no	—
42	C₄×#10623494	—	4	1	no	—
43	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
46	M₃₂×C₈₉	—	2	2	no	—
47	M₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
48	C₂×C₁₆×C₈₉	—	2	2	no	—
49	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₂³×D₈₉	—	4	1	no	—
54	C₂×D₃₅₆	—	2	2	no	—
55	C₂×D₃₅₆	—	2	4	no	—
56	C₂×D₃₅₆	—	4	1	no	—
57	C₂×D₃₅₆	—	4	2	no	—
58	C₂²×Dic₈₉	—	4	1	no	—
59	C₂×C₄×D₈₉	—	4	2	no	—
60	C₂×C₄×D₈₉	—	4	2	no	—
61	C₂×C₄×D₈₉	—	8	4	no	—
62	D₇₁₂	—	2	2	no	—
63	D₇₁₂	—	2	4	no	—
64	D₇₁₂	C₄²	1	4	yes	—
65	#13083558	—	2	4	no	—
66	#13083558	—	4	2	no	—
67	C₂×#10623494	—	4	2	no	—
68	C₄⋊Dic₈₉	—	2	2	no	—
69	C₄⋊Dic₈₉	—	4	1	no	—
70	#13083551	—	4	2	no	—
71	#13083551	—	4	2	no	—
72	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	—	2	2	no	—
76	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	#41	1	2	yes	—
82	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₈₉	D₈	1	1	yes	characteristic
85	C₄²×C₈₉	D₈	1	1	yes	characteristic
86	C₁₆×C₈₉	—	2	4	no	—
87	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
88	C₂²×D₈₉	—	4	2	no	—
89	C₂²×D₈₉	—	4	4	no	—
90	C₂²×D₈₉	—	8	2	no	—
91	C₂²×D₈₉	—	8	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	4	1	no	—
94	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#135	1	2	yes	—
102	C₂×Dic₈₉	—	4	2	no	—
103	C₂×Dic₈₉	—	8	2	no	—
104	D₃₅₆	—	2	4	no	—
105	D₃₅₆	—	4	2	no	—
106	D₃₅₆	—	4	4	no	—
107	C₄×D₈₉	—	8	4	no	—
108	C₄×D₈₉	—	8	4	no	—
109	C₈×C₈₉	—	2	1	no	—
110	C₈×C₈₉	—	2	1	no	—
111	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
112	C₈×C₈₉	#117	1	1	yes	characteristic
113	C₈×C₈₉	C₄×D₄	1	2	yes	—
114	#10623494	—	4	4	no	—
115	C₂²×C₈₉	—	2	1	no	—
116	C₂²×C₈₉	—	2	1	no	—
117	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
118	C₂²×C₈₉	C₄×SD₁₆	1	2	yes	—
119	C₂×D₈₉	—	4	4	no	—
120	C₂×D₈₉	—	8	8	no	—
121	C₂×D₈₉	—	8	2	no	—
122	C₂×D₈₉	—	8	2	no	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	—	2	1	no	—
125	C₄×C₈₉	—	2	1	no	—
126	C₄×C₈₉	#361	1	2	yes	complemented
127	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₈₉	#416	1	1	yes	characteristic
129	C₄×C₈₉	C₄×D₈	1	2	yes	—
130	Dic₈₉	—	8	2	no	—
131	C₄×#361	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	—	2	2	no	—
133	C₂×C₈₉	C₄×#102	1	1	yes	characteristic
134	C₂×C₈₉	C₂×#361	1	1	yes	characteristic
135	C₂×C₈₉	#2148	1	1	yes	characteristic
136	D₈₉	—	8	4	no	—
137	C₂×C₄×D₈	—	89	1	no	—
138	C₂×#361	—	89	2	no	—
139	C₄×M₃₂	D₈₉	1	1	yes	characteristic, complemented
140	C₄×#108	—	89	1	no	—
141	#1387	—	89	2	no	—
142	C₈₉	C₄×#361	1	1	yes	characteristic, complemented, Sylow
143	C₂²×D₈	—	89	1	no	—
144	C₂×C₄×D₄	—	178	1	no	—
145	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
146	C₄×D₈	—	89	2	no	—
147	C₄×D₈	—	89	2	no	—
148	C₄×D₈	—	178	1	no	—
149	C₄×D₈	—	178	1	no	—
150	C₂×#108	—	89	2	no	—
151	C₂×#107	—	89	1	no	—
152	C₂×#102	—	178	1	no	—
153	C₄×M₁₆	—	178	1	no	—
154	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
155	#361	—	89	8	no	—
156	#346	—	2	1	no	—
157	C₄×C₁₆	—	2	1	no	—
158	#341	—	178	1	no	—
159	C₂²×D₄	—	178	1	no	—
160	C₂³×C₄	—	356	1	no	—
161	C₂×D₈	—	89	4	no	—
162	C₂×D₈	—	89	2	no	—
163	C₂×D₈	—	178	1	no	—
164	C₂×D₈	—	178	2	no	—
165	C₂×M₁₆	—	178	2	no	—
166	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
168	C₄×D₄	—	356	1	no	—
169	C₄×D₄	—	356	1	no	—
170	C₄×D₄	—	178	2	no	—
171	C₄×D₄	—	178	2	no	—
172	C₂×C₄⋊C₄	—	178	1	no	—
173	C₂×C₂²⋊C₄	—	356	1	no	—
174	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
175	M₃₂	—	2	2	no	—
176	M₃₂	C₄×D₈₉	1	4	yes	complemented
177	C₂×C₁₆	—	2	2	no	—
178	#108	—	89	4	no	—
179	#108	—	178	2	no	—
180	#107	—	89	2	no	—
181	#107	—	178	1	no	—
182	#102	—	178	2	no	—
183	#102	—	178	2	no	—
184	#97	—	356	1	no	—
185	C₄×C₈	—	356	1	no	—
186	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
187	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
190	C₂⁴	—	356	1	no	—
191	C₂×D₄	—	356	1	no	—
192	C₂×D₄	—	356	2	no	—
193	C₂×D₄	—	178	4	no	—
194	C₂×D₄	—	178	2	no	—
195	C₂²×C₄	—	356	1	no	—
196	C₂²×C₄	—	356	2	no	—
197	C₂²×C₄	—	356	2	no	—
198	C₂²×C₄	—	712	4	no	—
199	C₂²×C₄	#13083563	1	1	yes	characteristic
200	C₂²×C₄	#13083563	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
202	D₈	—	89	4	no	—
203	D₈	—	178	2	no	—
204	D₈	—	178	4	no	—
205	M₁₆	—	356	2	no	—
206	M₁₆	—	178	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	356	2	no	—
209	C₂×C₈	#13083551	1	1	yes	characteristic
210	C₂×C₈	#13083551	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
213	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
214	C₂×C₈	#13083574	1	2	yes	—
215	C₄⋊C₄	—	356	1	no	—
216	C₄⋊C₄	—	178	2	no	—
217	C₂²⋊C₄	—	356	2	no	—
218	C₂²⋊C₄	—	356	2	no	—
219	C₄²	#13083551	1	1	yes	characteristic
220	C₄²	#13083551	1	1	yes	characteristic
221	C₄²	D₇₁₂	1	1	yes	characteristic
222	C₄²	D₇₁₂	1	1	yes	characteristic
223	C₁₆	—	2	4	no	—
224	C₂³	—	356	4	no	—
225	C₂³	—	356	2	no	—
226	C₂³	—	712	2	no	—
227	C₂³	—	712	2	no	—
228	C₂³	C₂×#13083563	1	1	yes	characteristic
229	D₄	—	356	4	no	—
230	D₄	—	356	2	no	—
231	D₄	—	178	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	356	2	no	—
235	C₂×C₄	—	712	4	no	—
236	C₂×C₄	—	712	2	no	—
237	C₂×C₄	—	712	4	no	—
238	C₂×C₄	#15667934	1	1	yes	characteristic
239	C₂×C₄	#15667934	1	1	yes	characteristic
240	C₂×C₄	#15667934	1	1	yes	characteristic
241	C₂×C₄	#15667934	1	1	yes	characteristic
242	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
243	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
244	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
245	C₂×C₄	#15668034	1	2	yes	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	356	4	no	—
249	C₈	C₄²×D₈₉	1	1	yes	characteristic
250	C₈	#15667969	1	1	yes	characteristic
251	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	356	4	no	—
255	C₂²	—	712	8	no	—
256	C₂²	—	712	2	no	—
257	C₂²	—	712	2	no	—
258	C₂²	C₂×#15667934	1	1	yes	characteristic
259	C₂²	C₄×#13083563	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	712	2	no	—
264	C₄	#19151412	1	2	yes	direct product factor
265	C₄	C₄×#13083551	1	1	yes	characteristic
266	C₄	#19151595	1	1	yes	characteristic
267	C₄	C₄×D₇₁₂	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	712	4	no	—
270	C₂	C₄×#15667934	1	1	yes	characteristic
271	C₂	C₂×#19151412	1	1	yes	characteristic
272	C₂	#26022002	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667934
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	C₄×#361
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₄×#15667948	C₂	semidirect
4. #26018361	C₂	semidirect
5. C₄×M₃₂×C₈₉	C₂	semidirect
16. #19151412	C₄	direct (N × H)
36. #15667948	C₂×C₄	semidirect
47. M₃₂×C₈₉	C₂×C₄	semidirect
126. C₄×C₈₉	#361	semidirect
139. C₄×M₃₂	D₈₉	semidirect
142. C₈₉	C₄×#361	semidirect
176. M₃₂	C₄×D₈₉	semidirect
264. C₄	#19151412	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×C₄×D₇₁₂	2
C₂×#19151412	2
C₄×#15667948	2
#26018361	2
C₄×M₃₂×C₈₉	2
C₄×#361	89

Subgroups — C4×#19151412 #36469682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151412 #36469682