Subgroups — C₄×#19151354 #36459592

Order 22784.

954 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083558	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667971	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151354	C₂	1	4	yes	maximal
4	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	2	yes	—
6	C₂×#15667971	C₂²	1	2	yes	—
7	C₄×#13083558	—	2	2	no	—
8	C₄×#13083558	—	2	2	no	—
9	C₂×#15667923	C₄	1	2	yes	complemented
10	C₂×C₄×#10623494	C₄	1	2	yes	complemented
11	C₄×C₄⋊Dic₈₉	—	2	2	no	—
12	C₄×C₂²⋊Dic₈₉	—	2	2	no	—
13	#19151444	C₄	1	2	yes	complemented
14	C₄²×Dic₈₉	C₄	1	2	yes	complemented
15	#19151354	C₄	1	16	yes	direct product factor
16	C₂×#117×C₈₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
19	C₄×C₂²⋊C₄×C₈₉	—	2	2	no	—
20	#376×C₈₉	C₄	1	2	yes	complemented
21	C₄³×C₈₉	C₄	1	2	yes	complemented
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	4	no	—
25	C₂×#13083558	C₂×C₄	1	8	yes	—
26	C₂²×#10623494	C₂×C₄	1	4	yes	—
27	#15667971	—	2	4	no	—
28	#15667971	C₂×C₄	1	8	yes	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
31	C₂×C₄×Dic₈₉	—	2	4	no	—
32	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
33	#15667923	—	2	4	no	—
34	C₄×#10623494	—	2	4	no	—
35	#15667919	—	2	4	no	—
36	C₂³×C₄×C₈₉	D₄	1	2	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	#117×C₈₉	—	2	4	no	—
39	#117×C₈₉	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
42	C₂×C₄²×C₈₉	—	2	4	no	—
43	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	D₄	1	2	yes	—
51	C₂×C₄²×C₈₉	D₄	1	2	yes	—
52	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
53	#95×C₈₉	—	2	4	no	—
54	C₂²×Dic₈₉	—	2	4	no	—
55	#13083558	—	2	8	no	—
56	#13083558	—	2	8	no	—
57	C₂×#10623494	—	2	4	no	—
58	C₂×#10623494	—	2	4	no	—
59	C₂×#10623494	C₄²	1	8	yes	complemented
60	C₄⋊Dic₈₉	—	2	8	no	—
61	C₂²⋊Dic₈₉	—	2	8	no	—
62	C₄×Dic₈₉	—	2	4	no	—
63	C₄×Dic₈₉	—	4	8	no	—
64	C₄×Dic₈₉	C₄²	1	8	yes	complemented
65	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	—	2	4	no	—
67	C₂²×C₄×C₈₉	—	4	4	no	—
68	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	#41	1	4	yes	—
82	C₄⋊C₄×C₈₉	—	2	8	no	—
83	C₂²⋊C₄×C₈₉	—	2	8	no	—
84	C₄²×C₈₉	—	2	4	no	—
85	C₄²×C₈₉	—	4	8	no	—
86	C₄²×C₈₉	C₄²	1	1	yes	characteristic
87	C₄²×C₈₉	C₄²	1	1	yes	characteristic
88	C₄²×C₈₉	C₄²	1	1	yes	characteristic
89	C₄²×C₈₉	C₄²	1	1	yes	characteristic
90	C₄²×C₈₉	C₄²	1	8	yes	complemented
91	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
94	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
95	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
96	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
97	C₂³×C₈₉	—	4	2	no	—
98	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₈₉	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₈₉	—	2	4	no	—
103	C₂×C₄×C₈₉	—	2	4	no	—
104	C₂×C₄×C₈₉	—	2	4	no	—
105	C₂×C₄×C₈₉	—	2	8	no	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	—	4	2	no	—
108	C₂×C₄×C₈₉	—	4	2	no	—
109	C₂×C₄×C₈₉	—	4	8	no	—
110	C₂×C₄×C₈₉	#95	1	2	yes	—
111	C₂×C₄×C₈₉	#95	1	2	yes	—
112	C₂×C₄×C₈₉	#95	1	2	yes	—
113	C₂×C₄×C₈₉	#95	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#117	1	2	yes	—
122	C₂×C₄×C₈₉	#117	1	2	yes	—
123	C₂×C₄×C₈₉	#117	1	2	yes	—
124	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
125	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
127	C₂×Dic₈₉	—	2	8	no	—
128	C₂×Dic₈₉	—	4	8	no	—
129	#10623494	—	2	8	no	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	—	2	2	no	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	4	4	no	—
134	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
135	C₂²×C₈₉	C₄×C₂²⋊C₄	1	2	yes	—
136	C₂²×C₈₉	C₄×C₂²⋊C₄	1	2	yes	—
137	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	—	2	2	no	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	4	8	no	—
142	C₄×C₈₉	H(3,ℤ/4)	1	4	yes	complemented
143	C₄×C₈₉	C₄³	1	1	yes	characteristic
144	C₄×C₈₉	#376	1	1	yes	characteristic
145	C₄×C₈₉	C₄×C₂²⋊C₄	1	2	yes	—
146	C₄×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
147	C₄×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
148	Dic₈₉	—	4	8	no	—
149	C₄×H(3,ℤ/4)	—	89	1	no	maximal, Sylow
150	C₂×C₈₉	—	2	2	no	—
151	C₂×C₈₉	—	2	2	no	—
152	C₂×C₈₉	—	2	2	no	—
153	C₂×C₈₉	C₄×#95	1	1	yes	characteristic
154	C₂×C₈₉	C₂×H(3,ℤ/4)	1	2	yes	—
155	C₂×C₄×M₁₆	—	89	1	no	—
156	C₄×#117	—	89	1	no	—
157	C₄×#117	D₈₉	1	1	yes	characteristic
158	C₂×H(3,ℤ/4)	—	89	4	no	—
159	C₈₉	C₄×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	89	2	no	—
161	C₂×#117	—	89	2	no	—
162	C₂×#117	C₂×D₈₉	1	2	yes	—
163	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
164	C₄×M₁₆	—	178	2	no	—
165	C₄×M₁₆	—	178	2	no	—
166	C₂×#97	—	89	2	no	—
167	C₂×C₄×C₈	—	89	2	no	—
168	C₄×C₄⋊C₄	—	2	2	no	—
169	C₄×C₄⋊C₄	—	178	2	no	—
170	C₄×C₂²⋊C₄	—	2	2	no	—
171	C₄×C₂²⋊C₄	—	178	2	no	—
172	#376	—	89	2	no	—
173	#376	Dic₈₉	1	2	yes	complemented
174	C₄³	—	89	2	no	—
175	C₄³	Dic₈₉	1	2	yes	complemented
176	H(3,ℤ/4)	—	89	16	no	—
177	C₂³×C₄	C₂²⋊D₈₉	1	2	yes	—
178	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
179	C₂×M₁₆	—	89	8	no	—
180	C₂×M₁₆	—	178	2	no	—
181	C₂×M₁₆	—	178	2	no	—
182	C₂×M₁₆	—	178	4	no	—
183	C₂²×C₈	—	89	4	no	—
184	#117	—	2	4	no	—
185	#117	—	89	8	no	—
186	#117	—	178	4	no	—
187	#117	C₄×D₈₉	1	8	yes	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	178	2	no	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	178	2	no	—
192	C₂×C₄²	—	2	4	no	—
193	C₂×C₄²	—	89	4	no	—
194	C₂×C₄²	—	178	4	no	—
195	C₂×C₄²	Dic₁₇₈	1	2	yes	—
196	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
200	C₂×C₄²	D₃₅₆	1	2	yes	—
201	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
202	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
205	#97	—	178	4	no	—
206	C₄×C₈	—	178	4	no	—
207	#95	—	2	4	no	—
208	#95	—	178	4	no	—
209	C₂⁴	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	178	4	no	—
213	C₂²×C₄	#13083551	1	2	yes	—
214	C₂²×C₄	#13083551	1	2	yes	—
215	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
216	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
217	C₂²×C₄	#13083553	1	2	yes	—
218	C₂²×C₄	#13083553	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₈₉	1	4	yes	—
222	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
225	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
226	C₂²×C₄	#13083574	1	4	yes	—
227	M₁₆	—	178	8	no	—
228	M₁₆	—	178	8	no	—
229	C₂×C₈	—	89	8	no	—
230	C₂×C₈	—	178	4	no	—
231	C₂×C₈	—	178	4	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	178	8	no	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	178	8	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	4	8	no	—
238	C₄²	—	89	8	no	—
239	C₄²	—	356	8	no	—
240	C₄²	—	178	4	no	—
241	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
242	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
243	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
244	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
245	C₄²	C₄×Dic₈₉	1	8	yes	complemented
246	C₄²	#13083551	1	2	yes	—
247	C₄²	#13083551	1	2	yes	—
248	C₄²	#13083553	1	2	yes	—
249	C₄²	#13083553	1	2	yes	—
250	C₄²	C₄⋊Dic₈₉	1	2	yes	—
251	C₄²	C₄⋊Dic₈₉	1	2	yes	—
252	C₂³	—	4	2	no	—
253	C₂³	C₂×#13083551	1	1	yes	characteristic
254	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
255	C₂³	C₂×#13083553	1	1	yes	characteristic
256	C₂³	C₄×C₂²⋊D₈₉	1	4	yes	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	356	8	no	—
266	C₂×C₄	—	178	8	no	—
267	C₂×C₄	#15667919	1	2	yes	—
268	C₂×C₄	#15667919	1	2	yes	—
269	C₂×C₄	#15667919	1	2	yes	—
270	C₂×C₄	#15667919	1	2	yes	—
271	C₂×C₄	C₄²×D₈₉	1	2	yes	—
272	C₂×C₄	C₄²×D₈₉	1	2	yes	—
273	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
275	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
276	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
278	C₂×C₄	#15667969	1	2	yes	—
279	C₂×C₄	#15667969	1	2	yes	—
280	C₂×C₄	#15667971	1	2	yes	—
281	C₂×C₄	C₄×C₂²⋊D₈₉	1	4	yes	—
282	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
283	C₂×C₄	C₄×Dic₁₇₈	1	4	yes	—
284	C₈	—	178	8	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂×#15667919	1	1	yes	characteristic
290	C₂²	C₄×C₂²⋊Dic₈₉	1	2	yes	—
291	C₂²	C₄×#13083551	1	2	yes	—
292	C₂²	C₄×#13083553	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	8	no	—
297	C₄	—	356	8	no	—
298	C₄	#19151354	1	4	yes	direct product factor
299	C₄	C₄²×Dic₈₉	1	1	yes	characteristic
300	C₄	#19151444	1	1	yes	characteristic
301	C₄	C₄×#13083551	1	2	yes	—
302	C₄	C₄×C₄⋊Dic₈₉	1	2	yes	—
303	C₄	C₄×#13083553	1	2	yes	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	C₄×#15667919	1	1	yes	characteristic
308	C₂	C₂×#19151354	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₈₉
commutator	C₄×C₈₉	C₄³
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₄×#117×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#95
prime radical	C₂⁴×C₈₉	C₂×D₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₄×#117	D₈₉
hypocenter	C₈₉	C₄×H(3,ℤ/4)
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
9. C₂×#15667923	C₄	semidirect
10. C₂×C₄×#10623494	C₄	semidirect
13. #19151444	C₄	semidirect
14. C₄²×Dic₈₉	C₄	semidirect
15. #19151354	C₄	direct (N × H)
20. #376×C₈₉	C₄	semidirect
21. C₄³×C₈₉	C₄	semidirect
59. C₂×#10623494	C₄²	semidirect
64. C₄×Dic₈₉	C₄²	semidirect
90. C₄²×C₈₉	C₄²	semidirect
142. C₄×C₈₉	H(3,ℤ/4)	semidirect
159. C₈₉	C₄×H(3,ℤ/4)	semidirect
173. #376	Dic₈₉	semidirect
175. C₄³	Dic₈₉	semidirect
245. C₄²	C₄×Dic₈₉	semidirect
298. C₄	#19151354	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×#13083558	2
C₄×#15667971	2
C₂×#19151354	2
C₄×#117×C₈₉	2
C₄×H(3,ℤ/4)	89

Subgroups — C4×#19151354 #36459592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151354 #36459592