Subgroups — C₄×#19151405 #36469652

Order 22784.

450 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151405	C₂	1	2	yes	maximal
2	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#15667948	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018347	C₂	1	2	yes	maximal, complemented
5	C₄×#135×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	2	yes	—
7	C₂×#15667948	C₂²	1	2	yes	—
8	C₄×#13083558	—	2	1	no	—
9	#19151405	C₄	1	8	yes	direct product factor
10	C₄×#13083549	—	2	2	no	—
11	#19151363	—	2	1	no	—
12	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
13	C₄×#41×C₈₉	—	2	1	no	—
14	C₄×Q₁₆×C₈₉	C₄	1	2	yes	—
15	C₄×SD₁₆×C₈₉	—	2	2	no	—
16	C₄×D₈×C₈₉	C₄	1	2	yes	—
17	#427×C₈₉	C₂²	1	1	yes	characteristic
18	#416×C₈₉	—	2	1	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083549	—	2	2	no	—
22	C₂×#13083549	—	2	2	no	—
23	C₂×#13083549	C₂×C₄	1	4	yes	complemented
24	#15667948	—	2	2	no	—
25	#15667948	C₂×C₄	1	4	yes	complemented
26	#15667923	—	4	1	no	—
27	C₄×#10623494	—	4	1	no	—
28	C₂×#41×C₈₉	—	2	1	no	—
29	#135×C₈₉	—	2	2	no	—
30	#135×C₈₉	C₂×C₄	1	4	yes	—
31	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×SD₁₆×C₈₉	—	2	1	no	—
33	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₈₉	—	2	2	no	—
37	C₄×D₄×C₈₉	—	2	2	no	—
38	C₄×D₄×C₈₉	—	4	2	no	—
39	#117×C₈₉	—	2	1	no	—
40	#117×C₈₉	—	4	1	no	—
41	C₂×C₄²×C₈₉	—	4	1	no	—
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	#107×C₈₉	C₂×C₄	1	2	yes	—
44	#106×C₈₉	—	2	1	no	—
45	#103×C₈₉	—	2	2	no	—
46	#102×C₈₉	—	2	2	no	—
47	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₈₉	D₄	1	2	yes	—
49	#13083558	—	2	4	no	—
50	#13083558	—	4	2	no	—
51	C₂×#10623494	—	4	2	no	—
52	#13083549	—	2	4	no	—
53	#41×C₈₉	—	2	4	no	—
54	#41×C₈₉	—	4	2	no	—
55	C₂×Q₈×C₈₉	—	2	1	no	—
56	C₂×D₄×C₈₉	—	2	1	no	—
57	C₂×D₄×C₈₉	—	4	1	no	—
58	C₂²×C₄×C₈₉	—	4	1	no	—
59	C₂²×C₄×C₈₉	—	4	2	no	—
60	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₈₉	—	2	1	no	—
63	Q₁₆×C₈₉	C₄²	1	2	yes	—
64	SD₁₆×C₈₉	—	2	4	no	—
65	D₈×C₈₉	—	2	1	no	—
66	D₈×C₈₉	C₄²	1	2	yes	—
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	#41	1	2	yes	—
73	C₄⋊C₄×C₈₉	—	2	2	no	—
74	C₄⋊C₄×C₈₉	—	4	2	no	—
75	C₂²⋊C₄×C₈₉	—	4	1	no	—
76	C₂²⋊C₄×C₈₉	—	4	2	no	—
77	C₄²×C₈₉	—	4	2	no	—
78	C₄²×C₈₉	—	4	1	no	—
79	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₈₉	D₈	1	1	yes	characteristic
81	C₄²×C₈₉	D₈	1	1	yes	characteristic
82	C₂³×C₈₉	—	4	1	no	—
83	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
84	Q₈×C₈₉	—	2	2	no	—
85	Q₈×C₈₉	—	4	1	no	—
86	D₄×C₈₉	—	2	2	no	—
87	D₄×C₈₉	—	4	4	no	—
88	D₄×C₈₉	—	4	1	no	—
89	C₂×C₄×C₈₉	—	2	4	no	—
90	C₂×C₄×C₈₉	—	4	1	no	—
91	C₂×C₄×C₈₉	—	4	4	no	—
92	C₂×C₄×C₈₉	—	4	2	no	—
93	C₂×C₄×C₈₉	—	8	4	no	—
94	C₂×C₄×C₈₉	#102	1	2	yes	—
95	C₂×C₄×C₈₉	#102	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#135	1	2	yes	—
100	C₈×C₈₉	—	2	2	no	—
101	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
102	C₈×C₈₉	#117	1	1	yes	characteristic
103	C₈×C₈₉	C₄×D₄	1	2	yes	—
104	#10623494	—	4	4	no	—
105	C₂²×C₈₉	—	2	2	no	—
106	C₂²×C₈₉	—	4	2	no	—
107	C₂²×C₈₉	—	8	2	no	—
108	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
109	C₂²×C₈₉	C₄×SD₁₆	1	2	yes	—
110	C₄×C₈₉	—	2	2	no	—
111	C₄×C₈₉	—	4	2	no	—
112	C₄×C₈₉	—	8	2	no	—
113	C₄×C₈₉	#359	1	4	yes	complemented
114	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
115	C₄×C₈₉	#416	1	1	yes	characteristic
116	C₄×C₈₉	C₄×D₈	1	2	yes	—
117	C₄×#359	—	89	1	no	maximal, Sylow
118	C₂×C₈₉	—	2	2	no	—
119	C₂×C₈₉	—	8	2	no	—
120	C₂×C₈₉	C₄×#102	1	1	yes	characteristic
121	C₂×C₈₉	C₂×#359	1	2	yes	—
122	C₄×#135	D₈₉	1	1	yes	characteristic
123	C₂×#359	—	89	2	no	—
124	C₂×C₄×C₁₆	—	89	1	no	—
125	C₄×#108	—	89	1	no	—
126	#1382	—	89	2	no	—
127	C₈₉	C₄×#359	1	1	yes	characteristic, complemented, Sylow
128	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
129	C₄×#41	—	2	1	no	—
130	C₂²×C₁₆	—	89	2	no	—
131	C₄×Q₁₆	Dic₈₉	1	2	yes	—
132	C₄×SD₁₆	—	2	2	no	—
133	C₄×D₈	Dic₈₉	1	2	yes	—
134	C₂×#108	—	89	2	no	—
135	#427	C₂×D₈₉	1	1	yes	characteristic
136	#416	—	2	1	no	—
137	C₄×M₁₆	—	178	1	no	—
138	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
139	#359	—	89	8	no	—
140	C₄×C₁₆	—	178	2	no	—
141	#341	—	178	1	no	—
142	C₂×#41	—	2	1	no	—
143	#135	—	2	2	no	—
144	#135	C₄×D₈₉	1	4	yes	—
145	C₂×Q₁₆	C₂×Dic₈₉	1	1	yes	characteristic
146	C₂×SD₁₆	—	2	1	no	—
147	C₂×D₈	C₂×Dic₈₉	1	1	yes	characteristic
148	C₂×M₁₆	—	178	2	no	—
149	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
151	C₄×Q₈	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	4	2	no	—
154	#117	—	2	1	no	—
155	#117	—	4	1	no	—
156	C₂×C₄²	—	4	1	no	—
157	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
158	C₂×C₁₆	—	89	4	no	—
159	C₂×C₁₆	—	178	2	no	—
160	C₂×C₁₆	—	178	2	no	—
161	#108	—	89	4	no	—
162	#108	—	178	2	no	—
163	#107	C₂×Dic₈₉	1	2	yes	—
164	#106	—	2	1	no	—
165	#103	—	2	2	no	—
166	#102	—	2	2	no	—
167	#97	—	356	1	no	—
168	C₄×C₈	—	356	1	no	—
169	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
170	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
171	#41	—	2	4	no	—
172	#41	—	4	2	no	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	#13083562	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
180	Q₁₆	—	2	1	no	—
181	Q₁₆	C₄×Dic₈₉	1	2	yes	—
182	SD₁₆	—	2	4	no	—
183	D₈	—	2	1	no	—
184	D₈	C₄×Dic₈₉	1	2	yes	—
185	M₁₆	—	356	2	no	—
186	M₁₆	—	178	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	356	2	no	—
189	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
190	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
191	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
193	C₂×C₈	#13083574	1	2	yes	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	2	no	—
200	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
201	C₄²	#13083559	1	1	yes	characteristic
202	C₄²	#13083559	1	1	yes	characteristic
203	C₁₆	—	178	4	no	—
204	C₂³	—	4	1	no	—
205	C₂³	C₂×#13083562	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	4	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	1	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	#15667933	1	2	yes	—
217	C₂×C₄	#15667933	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
219	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
220	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
221	C₂×C₄	#15668032	1	2	yes	—
222	C₈	—	2	2	no	—
223	C₈	—	356	4	no	—
224	C₈	C₂×C₄×Dic₈₉	1	1	yes	characteristic
225	C₈	#15667971	1	1	yes	characteristic
226	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	8	2	no	—
230	C₂²	C₂×#15667933	1	1	yes	characteristic
231	C₂²	C₄×#13083562	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	2	no	—
235	C₄	#19151405	1	4	yes	direct product factor
236	C₄	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
237	C₄	#19151594	1	1	yes	characteristic
238	C₄	C₄×#13083559	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	—	8	2	no	—
241	C₂	C₄×#15667933	1	1	yes	characteristic
242	C₂	C₂×#19151405	1	2	yes	—

	subgroup	G/H
center	C₄²	#13083559
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×#135×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂×#135×C₈₉	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×#135	D₈₉
hypocenter	C₈₉	C₄×#359
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#13083549	C₂	semidirect
3. C₄×#15667948	C₂	semidirect
4. #26018347	C₂	semidirect
9. #19151405	C₄	direct (N × H)
23. C₂×#13083549	C₂×C₄	semidirect
25. #15667948	C₂×C₄	semidirect
113. C₄×C₈₉	#359	semidirect
127. C₈₉	C₄×#359	semidirect
235. C₄	#19151405	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×#19151405	2
C₂×C₄×#13083549	2
C₄×#15667948	2
#26018347	2
C₄×#135×C₈₉	2
C₄×#359	89

Subgroups — C4×#19151405 #36469652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151405 #36469652