Subgroups — C₂×#26018222 #36464189

Order 22784.

398 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×#19151374	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151349	C₂	1	1	yes	maximal, characteristic
3	#26018222	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	—	2	1	no	—
6	C₂×C₄⋊#10623494	—	2	1	no	—
7	#19151374	—	2	1	no	—
8	#19151374	C₂²	1	2	yes	—
9	#19151349	—	2	1	no	—
10	#19151349	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₈₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
13	C₄×Q₁₆×C₈₉	—	2	1	no	—
14	C₄×Q₁₆×C₈₉	—	2	1	no	—
15	C₄×Q₁₆×C₈₉	C₄	1	2	yes	—
16	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂×#107×C₈₉	C₄	1	1	yes	characteristic
18	C₂×#103×C₈₉	—	2	1	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×#10623494	—	4	1	no	—
21	C₂×#13083549	—	2	4	no	—
22	C₂×#13083549	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	4	1	no	—
25	C₂²×Q₈×C₈₉	—	2	1	no	—
26	C₂×Q₁₆×C₈₉	—	2	1	no	—
27	C₂×Q₁₆×C₈₉	—	2	2	no	—
28	C₂×Q₁₆×C₈₉	C₈	1	4	yes	complemented
29	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₈₉	—	2	2	no	—
33	C₄×Q₈×C₈₉	—	2	2	no	—
34	C₄×Q₈×C₈₉	—	4	1	no	—
35	C₄×Q₈×C₈₉	—	4	1	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
38	C₂×C₄²×C₈₉	—	4	1	no	—
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	#107×C₈₉	—	2	1	no	—
41	#107×C₈₉	C₂×C₄	1	2	yes	—
42	#103×C₈₉	—	2	2	no	—
43	#103×C₈₉	—	2	2	no	—
44	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×#10623494	—	4	2	no	—
49	C₂×#10623494	—	8	2	no	—
50	#13083549	—	2	4	no	—
51	C₂×Q₈×C₈₉	—	2	4	no	—
52	C₂×Q₈×C₈₉	—	2	2	no	—
53	C₂×Q₈×C₈₉	—	4	1	no	—
54	C₂×Q₈×C₈₉	—	4	2	no	—
55	C₂²×C₄×C₈₉	—	4	1	no	—
56	C₂²×C₄×C₈₉	—	4	1	no	—
57	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
60	Q₁₆×C₈₉	—	2	4	no	—
61	Q₁₆×C₈₉	—	2	2	no	—
62	Q₁₆×C₈₉	C₂×C₈	1	4	yes	complemented
63	C₂×C₈×C₈₉	—	2	2	no	—
64	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
68	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₈₉	—	2	2	no	—
71	C₄⋊C₄×C₈₉	—	4	1	no	—
72	C₄⋊C₄×C₈₉	—	4	2	no	—
73	C₄⋊C₄×C₈₉	—	4	2	no	—
74	C₄²×C₈₉	—	4	2	no	—
75	C₄²×C₈₉	—	4	2	no	—
76	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₈₉	D₈	1	1	yes	characteristic
78	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
79	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂³×C₈₉	#102	1	1	yes	characteristic
81	Q₈×C₈₉	—	2	4	no	—
82	Q₈×C₈₉	—	4	4	no	—
83	Q₈×C₈₉	—	4	2	no	—
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	4	4	no	—
87	C₂×C₄×C₈₉	—	4	2	no	—
88	C₂×C₄×C₈₉	—	4	2	no	—
89	C₂×C₄×C₈₉	—	8	2	no	—
90	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
93	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
94	C₂×C₄×C₈₉	#104	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₈₉	—	2	1	no	—
99	C₈×C₈₉	—	2	1	no	—
100	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
101	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
102	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
103	#10623494	—	8	2	no	—
104	C₂²×C₈₉	D₄⋊C₈	1	2	yes	—
105	C₂²×C₈₉	#360	1	2	yes	—
106	C₂²×C₈₉	#362	1	2	yes	—
107	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
108	C₄×C₈₉	—	2	1	no	—
109	C₄×C₈₉	—	2	1	no	—
110	C₄×C₈₉	—	2	1	no	—
111	C₄×C₈₉	—	2	1	no	—
112	C₄×C₈₉	—	4	4	no	—
113	C₄×C₈₉	—	8	2	no	—
114	C₄×C₈₉	D₄⋊C₈	1	2	yes	—
115	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
117	C₂×#1334	—	89	1	no	maximal, Sylow
118	C₂×C₈₉	#1334	1	4	yes	complemented
119	C₂×C₈₉	C₂×D₄⋊C₈	1	1	yes	characteristic
120	C₂×C₈₉	C₂×#360	1	1	yes	characteristic
121	C₂×C₈₉	C₂×#362	1	1	yes	characteristic
122	C₂×C₄×Q₁₆	D₈₉	1	1	yes	characteristic
123	C₂×#346	—	89	1	no	—
124	C₂×#334	—	89	1	no	—
125	#1334	—	89	4	no	—
126	C₈₉	C₂×#1334	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₁₆	Dic₈₉	1	1	yes	characteristic
128	C₂×C₄×Q₈	—	2	1	no	—
129	C₂²×C₁₆	—	178	1	no	—
130	C₄×Q₁₆	—	2	1	no	—
131	C₄×Q₁₆	—	2	1	no	—
132	C₄×Q₁₆	Dic₈₉	1	2	yes	—
133	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
134	C₂×#107	Dic₈₉	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	178	1	no	—
136	C₂×#103	—	2	1	no	—
137	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
138	#346	—	89	2	no	—
139	#346	—	178	1	no	—
140	#334	—	89	2	no	—
141	#334	—	178	1	no	—
142	C₂²×Q₈	—	2	1	no	—
143	C₂×Q₁₆	—	2	1	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	#10623494	1	4	yes	complemented
146	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
147	C₂²×C₈	—	356	1	no	—
148	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
150	C₄×Q₈	—	2	2	no	—
151	C₄×Q₈	—	2	2	no	—
152	C₄×Q₈	—	4	1	no	—
153	C₄×Q₈	—	4	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	4	1	no	—
156	C₂×C₄²	—	4	1	no	—
157	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
158	C₂×C₁₆	—	178	4	no	—
159	C₂×C₁₆	—	178	2	no	—
160	#107	—	2	1	no	—
161	#107	C₂×Dic₈₉	1	2	yes	—
162	C₄⋊C₈	—	356	1	no	—
163	C₄⋊C₈	—	178	2	no	—
164	#103	—	2	2	no	—
165	#103	—	2	2	no	—
166	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
169	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂²×C₄	—	4	1	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
177	C₂²×C₄	#13083559	1	1	yes	characteristic
178	C₂²×C₄	#13083562	1	1	yes	characteristic
179	Q₁₆	—	2	2	no	—
180	Q₁₆	—	2	4	no	—
181	Q₁₆	C₂×#10623494	1	4	yes	complemented
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	356	2	no	—
184	C₂×C₈	—	712	2	no	—
185	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
187	C₂×C₈	C₂×#10623494	1	2	yes	—
188	C₂×C₈	#13083558	1	2	yes	—
189	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
190	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	—	4	2	no	—
197	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
198	C₄²	#13083559	1	1	yes	characteristic
199	C₄²	#13083562	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
201	C₁₆	—	178	4	no	—
202	C₂³	#15667933	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	4	no	—
205	Q₈	—	4	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
213	C₂×C₄	#15667933	1	1	yes	characteristic
214	C₂×C₄	#15667933	1	1	yes	characteristic
215	C₂×C₄	#15667933	1	1	yes	characteristic
216	C₂×C₄	#15667939	1	2	yes	—
217	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
218	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
219	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	712	2	no	—
223	C₈	C₂²⋊#10623494	1	2	yes	—
224	C₈	C₂²×#10623494	1	1	yes	characteristic
225	C₈	C₂×#13083558	1	1	yes	characteristic
226	C₂²	D₄⋊#10623494	1	2	yes	—
227	C₂²	#19151408	1	2	yes	—
228	C₂²	#19151414	1	2	yes	—
229	C₂²	C₂×#15667933	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	4	no	—
235	C₄	—	8	2	no	—
236	C₄	D₄⋊#10623494	1	2	yes	—
237	C₄	C₂×C₂²⋊#10623494	1	1	yes	characteristic
238	C₄	C₂×#15667939	1	1	yes	characteristic
239	C₂	#26018222	1	4	yes	direct product factor
240	C₂	C₂×D₄⋊#10623494	1	1	yes	characteristic
241	C₂	C₂×#19151408	1	1	yes	characteristic
242	C₂	C₂×#19151414	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667933
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×C₄×Q₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#102
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×Q₁₆	D₈₉
hypocenter	C₈₉	C₂×#1334
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #26018222	C₂	direct (N × H)
28. C₂×Q₁₆×C₈₉	C₈	semidirect
62. Q₁₆×C₈₉	C₂×C₈	semidirect
118. C₂×C₈₉	#1334	semidirect
126. C₈₉	C₂×#1334	semidirect
145. C₂×Q₁₆	#10623494	semidirect
181. Q₁₆	C₂×#10623494	semidirect
239. C₂	#26018222	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×#19151374	2
C₂×#19151349	2
#26018222	2
C₂×C₄×Q₁₆×C₈₉	2
C₂×#1334	89

Subgroups — C2×#26018222 #36464189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018222 #36464189