Subgroups — C₄×#19151415 #36469687

Order 22784.

434 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151415	C₂	1	2	yes	maximal
3	C₄×#15667948	C₂	1	1	yes	maximal, characteristic
4	#26018361	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₈	—	2	1	no	—
8	C₄×Dic₃₅₆	—	2	1	no	—
9	C₄×Dic₃₅₆	—	2	1	no	—
10	C₄×Dic₃₅₆	C₄	1	2	yes	—
11	C₄×Dic₃₅₆	C₄	1	2	yes	—
12	C₂×#15667948	C₂²	1	2	yes	—
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667937	—	2	1	no	—
15	C₄×#13083558	—	2	1	no	—
16	#19151415	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×Dic₁₇₈	—	2	1	no	—
23	C₂×Dic₃₅₆	—	2	1	no	—
24	C₂×Dic₃₅₆	—	2	2	no	—
25	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
26	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
27	C₂×#13083558	—	2	2	no	—
28	C₄×Dic₁₇₈	—	2	2	no	—
29	C₄×Dic₁₇₈	—	2	2	no	—
30	C₄×Dic₁₇₈	—	4	1	no	—
31	C₄×Dic₁₇₈	—	4	1	no	—
32	C₂×C₄⋊Dic₈₉	—	2	1	no	—
33	C₂×#13083553	—	4	1	no	—
34	C₂×C₄×Dic₈₉	—	4	1	no	—
35	#15667948	—	2	2	no	—
36	#15667948	C₂×C₄	1	4	yes	—
37	#15667946	—	2	1	no	—
38	#15667946	C₂×C₄	1	2	yes	—
39	#15667937	—	2	2	no	—
40	#15667937	—	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	—
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	C₂×Dic₁₇₈	—	2	2	no	—
54	C₂×Dic₁₇₈	—	2	4	no	—
55	C₂×Dic₁₇₈	—	4	1	no	—
56	C₂×Dic₁₇₈	—	4	2	no	—
57	C₂²×Dic₈₉	—	4	1	no	—
58	C₂²×Dic₈₉	—	4	1	no	—
59	Dic₃₅₆	—	2	2	no	—
60	Dic₃₅₆	—	2	4	no	—
61	Dic₃₅₆	C₄²	1	4	yes	—
62	#13083558	—	2	4	no	—
63	#13083558	—	4	2	no	—
64	C₂×#10623494	—	4	2	no	—
65	C₄⋊Dic₈₉	—	2	2	no	—
66	C₄⋊Dic₈₉	—	4	1	no	—
67	#13083553	—	4	2	no	—
68	#13083553	—	4	2	no	—
69	C₄×Dic₈₉	—	4	2	no	—
70	C₄×Dic₈₉	—	4	2	no	—
71	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	—	2	2	no	—
75	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
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	2	yes	—
79	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	#41	1	2	yes	—
81	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₈₉	D₈	1	1	yes	characteristic
84	C₄²×C₈₉	D₈	1	1	yes	characteristic
85	C₁₆×C₈₉	—	2	4	no	—
86	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	4	1	no	—
89	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#135	1	2	yes	—
97	C₂×Dic₈₉	—	4	2	no	—
98	C₂×Dic₈₉	—	4	4	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	8	2	no	—
101	Dic₁₇₈	—	2	4	no	—
102	Dic₁₇₈	—	4	2	no	—
103	Dic₁₇₈	—	4	4	no	—
104	C₈×C₈₉	—	2	1	no	—
105	C₈×C₈₉	—	2	1	no	—
106	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
107	C₈×C₈₉	#117	1	1	yes	characteristic
108	C₈×C₈₉	C₄×D₄	1	2	yes	—
109	#10623494	—	4	4	no	—
110	C₂²×C₈₉	—	2	1	no	—
111	C₂²×C₈₉	—	2	1	no	—
112	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
113	C₂²×C₈₉	C₄×SD₁₆	1	2	yes	—
114	C₄×C₈₉	—	2	1	no	—
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	—	2	1	no	—
117	C₄×C₈₉	#362	1	2	yes	complemented
118	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₈₉	#416	1	1	yes	characteristic
120	C₄×C₈₉	C₄×D₈	1	2	yes	—
121	Dic₈₉	—	4	4	no	—
122	Dic₈₉	—	8	2	no	—
123	C₄×#362	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	—	2	2	no	—
125	C₂×C₈₉	C₄×#102	1	1	yes	characteristic
126	C₂×C₈₉	C₂×#362	1	1	yes	characteristic
127	C₂×C₈₉	#2148	1	1	yes	characteristic
128	C₂×C₄×Q₁₆	—	89	1	no	—
129	C₂×#362	—	89	2	no	—
130	C₄×M₃₂	D₈₉	1	1	yes	characteristic
131	C₄×#108	—	89	1	no	—
132	#1387	—	89	2	no	—
133	C₈₉	C₄×#362	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₁₆	—	89	1	no	—
135	C₂×C₄×Q₈	—	178	1	no	—
136	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
137	C₄×Q₁₆	—	89	2	no	—
138	C₄×Q₁₆	—	89	2	no	—
139	C₄×Q₁₆	—	178	1	no	—
140	C₄×Q₁₆	—	178	1	no	—
141	C₂×#108	—	89	2	no	—
142	C₂×#107	—	89	1	no	—
143	C₂×#103	—	178	1	no	—
144	C₄×M₁₆	—	178	1	no	—
145	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
146	#362	—	89	8	no	—
147	#346	—	2	1	no	—
148	C₄×C₁₆	—	2	1	no	—
149	#341	—	178	1	no	—
150	C₂²×Q₈	—	178	1	no	—
151	C₂×Q₁₆	—	89	4	no	—
152	C₂×Q₁₆	—	89	2	no	—
153	C₂×Q₁₆	—	178	1	no	—
154	C₂×Q₁₆	—	178	2	no	—
155	C₂×M₁₆	—	178	2	no	—
156	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
158	C₄×Q₈	—	356	1	no	—
159	C₄×Q₈	—	356	1	no	—
160	C₄×Q₈	—	178	2	no	—
161	C₄×Q₈	—	178	2	no	—
162	C₂×C₄⋊C₄	—	356	1	no	—
163	C₂×C₄⋊C₄	—	178	1	no	—
164	C₂×C₄²	—	356	1	no	—
165	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
166	M₃₂	—	2	2	no	—
167	M₃₂	C₄×D₈₉	1	4	yes	—
168	C₂×C₁₆	—	2	2	no	—
169	#108	—	89	4	no	—
170	#108	—	178	2	no	—
171	#107	—	89	2	no	—
172	#107	—	178	1	no	—
173	#103	—	178	2	no	—
174	#103	—	178	2	no	—
175	#97	—	356	1	no	—
176	C₄×C₈	—	356	1	no	—
177	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
178	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂×Q₈	—	356	1	no	—
182	C₂×Q₈	—	356	2	no	—
183	C₂×Q₈	—	178	4	no	—
184	C₂×Q₈	—	178	2	no	—
185	C₂²×C₄	—	356	1	no	—
186	C₂²×C₄	—	356	1	no	—
187	C₂²×C₄	#13083563	1	1	yes	characteristic
188	C₂²×C₄	#13083563	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
190	Q₁₆	—	89	4	no	—
191	Q₁₆	—	178	2	no	—
192	Q₁₆	—	178	4	no	—
193	M₁₆	—	356	2	no	—
194	M₁₆	—	178	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	356	2	no	—
197	C₂×C₈	#13083551	1	1	yes	characteristic
198	C₂×C₈	#13083551	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
202	C₂×C₈	#13083574	1	2	yes	—
203	C₄⋊C₄	—	356	1	no	—
204	C₄⋊C₄	—	356	2	no	—
205	C₄⋊C₄	—	356	2	no	—
206	C₄⋊C₄	—	178	2	no	—
207	C₄²	—	356	2	no	—
208	C₄²	—	356	2	no	—
209	C₄²	#13083551	1	1	yes	characteristic
210	C₄²	#13083551	1	1	yes	characteristic
211	C₄²	D₇₁₂	1	1	yes	characteristic
212	C₄²	D₇₁₂	1	1	yes	characteristic
213	C₁₆	—	2	4	no	—
214	C₂³	C₂×#13083563	1	1	yes	characteristic
215	Q₈	—	356	4	no	—
216	Q₈	—	356	2	no	—
217	Q₈	—	178	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	356	4	no	—
221	C₂×C₄	—	356	2	no	—
222	C₂×C₄	—	356	2	no	—
223	C₂×C₄	—	712	2	no	—
224	C₂×C₄	#15667934	1	1	yes	characteristic
225	C₂×C₄	#15667934	1	1	yes	characteristic
226	C₂×C₄	#15667934	1	1	yes	characteristic
227	C₂×C₄	#15667934	1	1	yes	characteristic
228	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
229	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
230	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
231	C₂×C₄	#15668034	1	2	yes	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	356	4	no	—
235	C₈	C₄²×D₈₉	1	1	yes	characteristic
236	C₈	#15667969	1	1	yes	characteristic
237	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₂×#15667934	1	1	yes	characteristic
241	C₂²	C₄×#13083563	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	356	4	no	—
246	C₄	—	712	2	no	—
247	C₄	#19151415	1	2	yes	direct product factor
248	C₄	C₄×#13083551	1	1	yes	characteristic
249	C₄	#19151595	1	1	yes	characteristic
250	C₄	C₄×D₇₁₂	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	C₄×#15667934	1	1	yes	characteristic
253	C₂	C₂×#19151415	1	1	yes	characteristic
254	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	C₂³×C₈₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	C₄×#362
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
16. #19151415	C₄	direct (N × H)
117. C₄×C₈₉	#362	semidirect
133. C₈₉	C₄×#362	semidirect
247. C₄	#19151415	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×Dic₃₅₆	2
C₂×#19151415	2
C₄×#15667948	2
#26018361	2
C₄×M₃₂×C₈₉	2
C₄×#362	89

Subgroups — C4×#19151415 #36469687

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151415 #36469687