Subgroups — C₂×#10588553 #11859924

Order 1152.

732 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#63449	C₂	1	2	yes	maximal
3	C₂×C₉×#107	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#63368	C₂	1	2	yes	maximal
6	#10588553	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4873	—	3	1	no	maximal
8	C₂²×Dic₁₈	C₂²	1	2	yes	—
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×Dic₉⋊C₄	—	2	2	no	—
12	C₂×C₄×Dic₉	—	2	2	no	—
13	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₆	C₄	1	8	yes	complemented
16	C₂×Dic₃₆	C₂²	1	4	yes	—
17	#63449	—	2	4	no	—
18	#63449	C₂²	1	8	yes	—
19	C₉×#107	C₂²	1	4	yes	—
20	C₈×Dic₉	C₂²	1	4	yes	—
21	#63368	C₂²	1	8	yes	—
22	C₂²×Dic₁₂	—	3	1	no	—
23	C₂×#887	—	3	2	no	—
24	C₂×C₃×#107	D₃	1	1	yes	characteristic
25	C₂×C₈×Dic₃	—	3	1	no	—
26	C₂×#810	—	3	2	no	—
27	#4873	—	3	8	no	—
28	C₂²×C₄×C₉	—	2	2	no	—
29	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₉	—	2	2	no	—
31	C₂²×Dic₉	D₄	1	1	yes	characteristic
32	C₂²×Dic₉	D₄	1	1	yes	characteristic
33	C₂×Dic₁₈	—	2	2	no	—
34	C₂×Dic₁₈	—	2	4	no	—
35	C₂×Dic₁₈	C₂×C₄	1	8	yes	—
36	C₂×Dic₁₈	C₂³	1	4	yes	—
37	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
38	C₂×C₈×C₉	C₂³	1	2	yes	—
39	C₉×C₄⋊C₄	—	2	2	no	—
40	C₉×C₄⋊C₄	C₂³	1	4	yes	—
41	Dic₉⋊C₄	—	2	8	no	—
42	C₄×Dic₉	—	2	8	no	—
43	C₄×Dic₉	D₄	1	2	yes	—
44	C₄×Dic₉	C₂³	1	2	yes	—
45	C₂×C₉⋊C₈	—	2	2	no	—
46	C₂×C₉⋊C₈	C₂³	1	2	yes	—
47	Dic₃₆	C₂×C₄	1	16	yes	complemented
48	C₂×C₄×Q₁₆	—	9	1	no	Sylow
49	C₂²×Dic₆	—	3	2	no	—
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
52	C₂×Dic₃⋊C₄	—	6	2	no	—
53	C₂×C₄×Dic₃	—	3	1	no	—
54	C₂×C₄×Dic₃	—	6	2	no	—
55	C₂²×C₃⋊C₈	—	3	1	no	—
56	C₂×Dic₁₂	—	3	8	no	—
57	C₂×Dic₁₂	—	3	4	no	—
58	#887	—	3	8	no	—
59	#887	—	6	4	no	—
60	C₃×#107	C₂×D₃	1	4	yes	—
61	C₈×Dic₃	—	3	4	no	—
62	#810	—	3	8	no	—
63	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉	—	2	4	no	—
65	C₂×C₄×C₉	—	4	4	no	—
66	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₉	#41	1	2	yes	—
69	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
70	C₂×Dic₉	—	2	8	no	—
71	C₂×Dic₉	—	2	2	no	—
72	C₂×Dic₉	—	2	4	no	—
73	C₂×Dic₉	Q₁₆	1	4	yes	complemented
74	C₂×Dic₉	C₂×D₄	1	2	yes	—
75	C₂×Dic₉	C₂×D₄	1	2	yes	—
76	Dic₁₈	—	2	8	no	—
77	Dic₁₈	—	2	4	no	—
78	Dic₁₈	C₂²×C₄	1	8	yes	—
79	C₈×C₉	C₂²×C₄	1	4	yes	—
80	C₉⋊C₈	—	2	2	no	—
81	C₂²×Q₁₆	—	9	1	no	—
82	C₂×C₄×Q₈	—	9	2	no	—
83	C₄×Q₁₆	—	9	8	no	—
84	C₂×#107	D₉	1	1	yes	characteristic
85	C₂×#103	—	9	2	no	—
86	C₂×C₄×C₈	—	9	1	no	—
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂²×Dic₃	—	3	1	no	—
90	C₂²×Dic₃	—	3	1	no	—
91	C₂²×Dic₃	—	6	2	no	—
92	C₂×Dic₆	—	3	8	no	—
93	C₂×Dic₆	—	3	4	no	—
94	C₂×Dic₆	—	6	2	no	—
95	C₂×Dic₆	—	6	4	no	—
96	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
97	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
98	C₃×C₄⋊C₄	—	2	2	no	—
99	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
100	Dic₃⋊C₄	—	6	8	no	—
101	C₄×Dic₃	—	3	2	no	—
102	C₄×Dic₃	—	3	2	no	—
103	C₄×Dic₃	—	6	8	no	—
104	C₂×C₃⋊C₈	—	3	2	no	—
105	C₂×C₃⋊C₈	—	6	2	no	—
106	Dic₁₂	—	3	16	no	—
107	C₂²×C₉	C₄×D₄	1	2	yes	—
108	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
109	C₂²×C₉	#135	1	2	yes	—
110	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₉	—	4	4	no	—
112	C₄×C₉	C₄×D₄	1	2	yes	—
113	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
114	C₄×C₉	C₂×#41	1	1	yes	characteristic
115	Dic₉	—	2	8	no	—
116	Dic₉	—	2	2	no	—
117	Dic₉	C₂×Q₁₆	1	4	yes	complemented
118	C₂²×Q₈	—	9	2	no	—
119	C₂×Q₁₆	—	9	8	no	—
120	C₂×Q₁₆	—	9	4	no	—
121	C₂²×C₈	—	9	1	no	—
122	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
123	C₄×Q₈	—	9	8	no	—
124	C₄×Q₈	—	18	4	no	—
125	C₂×C₄⋊C₄	—	18	2	no	—
126	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
127	C₂×C₄²	—	9	1	no	—
128	C₂×C₄²	—	18	2	no	—
129	#107	C₂×D₉	1	4	yes	—
130	#103	—	9	8	no	—
131	C₄×C₈	—	9	4	no	—
132	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	4	4	no	—
135	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
136	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	#239	1	2	yes	—
138	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
139	C₂×Dic₃	—	3	4	no	—
140	C₂×Dic₃	—	3	2	no	—
141	C₂×Dic₃	—	3	2	no	—
142	C₂×Dic₃	—	6	8	no	—
143	C₂×Dic₃	—	6	2	no	—
144	C₂×Dic₃	—	6	4	no	—
145	Dic₆	—	3	8	no	—
146	Dic₆	—	6	8	no	—
147	Dic₆	—	6	4	no	—
148	C₃×C₈	C₂×C₄×D₃	1	4	yes	—
149	C₃⋊C₈	—	6	2	no	—
150	C₂×C₉	C₄×Q₁₆	1	4	yes	complemented
151	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
153	C₂×C₉	C₂×#135	1	1	yes	characteristic
154	C₂×Q₈	—	9	8	no	—
155	C₂×Q₈	—	9	4	no	—
156	C₂×Q₈	—	18	2	no	—
157	C₂×Q₈	—	18	4	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	9	1	no	—
160	C₂²×C₄	—	9	1	no	—
161	C₂²×C₄	—	18	2	no	—
162	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
163	Q₁₆	—	9	16	no	—
164	C₂×C₈	—	9	2	no	—
165	C₂×C₈	—	18	2	no	—
166	C₂×C₈	C₄×D₉	1	4	yes	—
167	C₂×C₈	C₂²×D₉	1	2	yes	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	18	8	no	—
170	C₄⋊C₄	C₂²×D₉	1	4	yes	—
171	C₄²	—	9	2	no	—
172	C₄²	—	9	2	no	—
173	C₄²	—	18	8	no	—
174	C₂²×C₃	#893	1	2	yes	—
175	C₂²×C₃	#912	1	2	yes	—
176	C₂²×C₃	D₃×Q₁₆	1	2	yes	—
177	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
178	C₃×C₄	—	4	4	no	—
179	C₃×C₄	#893	1	2	yes	—
180	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
181	C₃×C₄	C₂×#239	1	1	yes	characteristic
182	Dic₃	—	3	4	no	—
183	Dic₃	—	6	2	no	—
184	Dic₃	—	6	8	no	—
185	C₉	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
186	C₂³	D₄×D₉	1	1	yes	characteristic
187	Q₈	—	9	8	no	—
188	Q₈	—	18	8	no	—
189	Q₈	—	18	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	9	2	no	—
193	C₂×C₄	—	9	4	no	—
194	C₂×C₄	—	9	2	no	—
195	C₂×C₄	—	18	4	no	—
196	C₂×C₄	—	18	8	no	—
197	C₂×C₄	—	18	2	no	—
198	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
199	C₂×C₄	D₄×D₉	1	1	yes	characteristic
200	C₂×C₄	#3701	1	2	yes	—
201	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
202	C₈	—	18	2	no	—
203	C₈	C₂×C₄×D₉	1	4	yes	—
204	C₂×C₃	#4873	1	4	yes	—
205	C₂×C₃	C₂×#893	1	1	yes	characteristic
206	C₂×C₃	C₂×#912	1	1	yes	characteristic
207	C₂×C₃	C₂×D₃×Q₁₆	1	1	yes	characteristic
208	C₂²	#63455	1	2	yes	—
209	C₂²	#63474	1	2	yes	—
210	C₂²	Q₁₆×D₉	1	2	yes	—
211	C₂²	C₂×D₄×D₉	1	1	yes	characteristic
212	C₄	—	4	4	no	—
213	C₄	—	9	4	no	—
214	C₄	—	18	2	no	—
215	C₄	—	18	8	no	—
216	C₄	#63455	1	2	yes	—
217	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
218	C₄	C₂×#3701	1	1	yes	characteristic
219	C₃	C₂×#4873	1	1	yes	characteristic
220	C₂	#10588553	1	4	yes	direct product factor
221	C₂	C₂×#63455	1	1	yes	characteristic
222	C₂	C₂×#63474	1	1	yes	characteristic
223	C₂	C₂×Q₁₆×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#107	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×#107	D₉
hypocenter	C₉	C₂×C₄×Q₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
6. #10588553	C₂	direct (N × H)
15. C₂×Dic₃₆	C₄	semidirect
47. Dic₃₆	C₂×C₄	semidirect
73. C₂×Dic₉	Q₁₆	semidirect
117. Dic₉	C₂×Q₁₆	semidirect
150. C₂×C₉	C₄×Q₁₆	semidirect
185. C₉	C₂×C₄×Q₁₆	semidirect
220. C₂	#10588553	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×Dic₃₆	2
C₂×#63449	2
C₂×C₉×#107	2
C₂×C₈×Dic₉	2
C₂×#63368	2
#10588553	2
C₂×#4873	3

Subgroups — C2×#10588553 #11859924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588553 #11859924