Subgroups — C₂×#41×#89137 #28380008

Order 13952.

1728 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#89137	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#89137	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×#89137	C₂	1	6	yes	maximal, complemented
4	C₂×#16812778	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄×#89137	C₂	1	3	yes	maximal, complemented
6	C₂×#41×D₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×#89137	C₂²	1	6	yes	—
8	Q₈×#89137	C₂²	1	16	yes	complemented
9	D₄×#89137	C₂²	1	48	yes	complemented
10	#16812778	C₂²	1	24	yes	complemented
11	C₂×C₄⋊#89137	C₂²	1	6	yes	—
12	C₂×C₂²⋊#89137	C₂²	1	6	yes	—
13	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
14	C₂×C₄×#89137	C₂²	1	3	yes	—
15	C₂×C₄×#89137	C₂²	1	6	yes	—
16	C₂×C₄×#89137	C₂²	1	24	yes	complemented
17	C₂×#13182030	C₄	1	1	yes	characteristic, complemented
18	C₂×#13182029	C₄	1	3	yes	complemented
19	C₂×#13182028	C₄	1	3	yes	complemented
20	#41×D₁₀₉	C₄	1	8	yes	complemented
21	#41×D₁₀₉	C₂²	1	8	yes	—
22	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×D₄×D₁₀₉	C₂²	1	3	yes	—
24	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
25	C₂×#41×C₁₀₉	C₄	1	1	yes	characteristic, complemented
26	C₂²×#89137	—	2	24	no	—
27	C₂²×#89137	C₂³	1	6	yes	—
28	C₂²×#89137	C₂³	1	24	yes	—
29	C₂²×#89137	C₂³	1	2	yes	—
30	C₄⋊#89137	C₂³	1	24	yes	—
31	C₂²⋊#89137	C₂³	1	24	yes	—
32	C₄×#89137	C₂³	1	4	yes	—
33	C₄×#89137	C₂³	1	12	yes	—
34	C₄×#89137	C₂³	1	24	yes	—
35	C₂³×D₁₀₉	C₂³	1	3	yes	—
36	#13182030	C₂×C₄	1	8	yes	complemented
37	#13182029	C₂×C₄	1	24	yes	complemented
38	#13182028	C₂×C₄	1	24	yes	complemented
39	C₂×Dic₂₁₈	C₂×C₄	1	3	yes	complemented
40	Q₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
41	Q₈×D₁₀₉	C₂³	1	4	yes	—
42	C₂×D₄₃₆	C₂×C₄	1	3	yes	complemented
43	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	6	yes	complemented
44	D₄×D₁₀₉	C₂×C₄	1	12	yes	complemented
45	D₄×D₁₀₉	C₂³	1	12	yes	—
46	C₂²×Dic₁₀₉	C₂×C₄	1	3	yes	complemented
47	C₂×C₄×D₁₀₉	C₂×C₄	1	12	yes	complemented
48	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	complemented
49	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	complemented
50	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
52	C₂×C₄×D₁₀₉	C₂³	1	12	yes	—
53	#41×C₁₀₉	C₂×C₄	1	8	yes	complemented
54	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂×D₄×C₁₀₉	C₂×C₄	1	3	yes	complemented
56	C₂²×C₄×C₁₀₉	C₂×C₄	1	3	yes	complemented
57	C₂³×C₁₀₉	C₂²×C₄	1	3	yes	—
58	C₂²×D₁₀₉	—	2	12	no	—
59	C₂²×D₁₀₉	C₂²×C₄	1	6	yes	—
60	C₂²×D₁₀₉	C₂²×C₄	1	3	yes	—
61	C₂²×D₁₀₉	C₂⁴	1	6	yes	—
62	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
63	C₂×#89137	—	2	12	no	—
64	C₂×#89137	—	2	48	no	—
65	C₂×#89137	#41	1	8	yes	direct product factor
66	C₂×#89137	C₂⁴	1	12	yes	—
67	C₂×#89137	C₂⁴	1	4	yes	—
68	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	complemented
69	D₄×C₁₀₉	C₂²×C₄	1	12	yes	complemented
70	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
71	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂²×C₄	1	12	yes	complemented
73	C₂²⋊D₁₀₉	C₂²×C₄	1	24	yes	complemented
74	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
76	C₂×Dic₁₀₉	C₂²×C₄	1	12	yes	complemented
77	D₄₃₆	C₂²×C₄	1	12	yes	complemented
78	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
79	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
80	C₄×D₁₀₉	C₂²×C₄	1	12	yes	complemented
81	C₄×D₁₀₉	C₂²×C₄	1	12	yes	complemented
82	C₄×D₁₀₉	C₂⁴	1	2	yes	—
83	C₄×D₁₀₉	C₂⁴	1	6	yes	—
84	Dic₂₁₈	C₂²×C₄	1	12	yes	complemented
85	C₂²×C₁₀₉	—	2	6	no	—
86	C₂²×C₁₀₉	C₂³×C₄	1	6	yes	—
87	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
88	C₂×D₁₀₉	—	2	6	no	—
89	C₂×D₁₀₉	—	2	12	no	—
90	C₂×D₁₀₉	—	2	12	no	—
91	C₂×D₁₀₉	C₂³×C₄	1	6	yes	—
92	C₂×D₁₀₉	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₁₀₉	C₂×#41	1	4	yes	—
94	C₂×D₁₀₉	C₂⁵	1	1	yes	characteristic
95	#89137	—	2	12	no	—
96	#89137	C₂×#41	1	8	yes	direct product factor
97	C₄×C₁₀₉	C₂³×C₄	1	6	yes	—
98	C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
99	Dic₁₀₉	C₂³×C₄	1	6	yes	—
100	Dic₁₀₉	C₂³×C₄	1	2	yes	—
101	C₂×C₁₀₉	—	2	6	no	—
102	C₂×C₁₀₉	C₄×#41	1	2	yes	complemented
103	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
104	D₁₀₉	—	2	6	no	—
105	D₁₀₉	C₄×#41	1	2	yes	complemented
106	D₁₀₉	C₂²×#41	1	2	yes	—
107	C₂×C₄×#41	—	109	1	no	maximal, Sylow
108	C₁₀₉	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
109	C₂²×#41	—	109	1	no	—
110	C₄×#41	—	109	16	no	—
111	C₂×C₄×Q₈	—	109	2	no	—
112	C₂×C₄×D₄	—	109	6	no	—
113	C₂×#117	—	109	3	no	—
114	C₂²×C₄²	—	109	3	no	—
115	C₂×#41	—	109	8	no	—
116	C₂×#41	—	109	8	no	—
117	C₂×#41	—	109	1	no	—
118	C₂×#41	—	109	3	no	—
119	C₂×#41	—	109	3	no	—
120	C₂×#41	#89137	1	1	yes	characteristic, direct product factor
121	C₂²×Q₈	—	109	1	no	—
122	C₂²×D₄	—	109	3	no	—
123	C₂³×C₄	—	109	3	no	—
124	C₂³×C₄	—	109	6	no	—
125	C₄×Q₈	—	109	16	no	—
126	C₄×D₄	—	109	48	no	—
127	#117	—	109	24	no	—
128	C₂×C₄⋊C₄	—	109	6	no	—
129	C₂×C₂²⋊C₄	—	109	6	no	—
130	C₂×C₄²	—	109	3	no	—
131	C₂×C₄²	—	109	6	no	—
132	C₂×C₄²	—	109	24	no	—
133	C₂×C₄²	—	109	1	no	—
134	C₂⁴	—	109	3	no	—
135	#41	—	109	8	no	—
136	#41	—	109	24	no	—
137	#41	—	109	24	no	—
138	#41	C₂×#89137	1	8	yes	direct product factor
139	C₂×Q₈	—	109	4	no	—
140	C₂×Q₈	—	109	4	no	—
141	C₂×Q₈	—	109	3	no	—
142	C₂×Q₈	C₂×#89137	1	1	yes	characteristic, complemented
143	C₂×D₄	—	109	6	no	—
144	C₂×D₄	—	109	12	no	—
145	C₂×D₄	—	109	12	no	—
146	C₂×D₄	—	109	3	no	—
147	C₂×D₄	C₂×#89137	1	3	yes	complemented
148	C₂²×C₄	—	109	2	no	—
149	C₂²×C₄	—	109	3	no	—
150	C₂²×C₄	—	109	3	no	—
151	C₂²×C₄	—	109	3	no	—
152	C₂²×C₄	—	109	3	no	—
153	C₂²×C₄	—	109	1	no	—
154	C₂²×C₄	—	109	12	no	—
155	C₂²×C₄	—	109	12	no	—
156	C₂²×C₄	—	109	6	no	—
157	C₂²×C₄	—	109	24	no	—
158	C₂²×C₄	—	218	24	no	—
159	C₂²×C₄	C₂×#89137	1	3	yes	complemented
160	C₄⋊C₄	—	109	24	no	—
161	C₂²⋊C₄	—	109	24	no	—
162	C₄²	—	109	4	no	—
163	C₄²	—	109	12	no	—
164	C₄²	—	109	24	no	—
165	C₂³	—	109	1	no	—
166	C₂³	—	109	6	no	—
167	C₂³	—	109	6	no	—
168	C₂³	—	109	3	no	—
169	C₂³	—	218	12	no	—
170	C₂³	C₂²×#89137	1	3	yes	—
171	Q₈	—	109	12	no	—
172	Q₈	C₂²×#89137	1	4	yes	complemented
173	D₄	—	109	24	no	—
174	D₄	—	109	12	no	—
175	D₄	C₂²×#89137	1	12	yes	complemented
176	C₂×C₄	—	109	4	no	—
177	C₂×C₄	—	109	8	no	—
178	C₂×C₄	—	109	1	no	—
179	C₂×C₄	—	109	3	no	—
180	C₂×C₄	—	109	12	no	—
181	C₂×C₄	—	109	2	no	—
182	C₂×C₄	—	109	6	no	—
183	C₂×C₄	—	109	2	no	—
184	C₂×C₄	—	109	6	no	—
185	C₂×C₄	—	109	12	no	—
186	C₂×C₄	—	109	12	no	—
187	C₂×C₄	—	109	12	no	—
188	C₂×C₄	—	218	48	no	—
189	C₂×C₄	—	218	12	no	—
190	C₂×C₄	C₂²×#89137	1	3	yes	—
191	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
192	C₂×C₄	C₂²×#89137	1	12	yes	complemented
193	C₂²	—	2	6	no	—
194	C₂²	—	109	4	no	—
195	C₂²	—	109	1	no	—
196	C₂²	—	109	1	no	—
197	C₂²	—	109	6	no	—
198	C₂²	—	218	12	no	—
199	C₂²	—	218	12	no	—
200	C₂²	—	218	6	no	—
201	C₂²	C₂³×#89137	1	6	yes	—
202	C₂²	C₂³×#89137	1	1	yes	characteristic
203	C₄	—	109	8	no	—
204	C₄	—	109	2	no	—
205	C₄	—	109	6	no	—
206	C₄	—	218	12	no	—
207	C₄	C₂³×#89137	1	2	yes	—
208	C₄	C₂³×#89137	1	6	yes	—
209	C₂	—	2	6	no	—
210	C₂	—	109	2	no	—
211	C₂	—	109	2	no	—
212	C₂	—	218	6	no	—
213	C₂	#41×#89137	1	2	yes	direct product factor
214	C₂	C₂⁴×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#89137
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂	C₂⁴×#89137
Fitting	C₂×#41×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂×#41×D₁₀₉	C₂
cosocle	C₂×D₁₀₉	C₂⁵
hypercenter	C₂×#41	#89137
hypocenter	C₁₀₉	C₂×C₄×#41
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #41×#89137	C₂	direct (N × H)
2. C₂×Q₈×#89137	C₂	semidirect
3. C₂×D₄×#89137	C₂	semidirect
4. C₂×#16812778	C₂	semidirect
5. C₂²×C₄×#89137	C₂	semidirect
8. Q₈×#89137	C₂²	semidirect
9. D₄×#89137	C₂²	semidirect
10. #16812778	C₂²	semidirect
16. C₂×C₄×#89137	C₂²	semidirect
17. C₂×#13182030	C₄	semidirect
18. C₂×#13182029	C₄	semidirect
19. C₂×#13182028	C₄	semidirect
20. #41×D₁₀₉	C₄	semidirect
25. C₂×#41×C₁₀₉	C₄	semidirect
36. #13182030	C₂×C₄	semidirect
37. #13182029	C₂×C₄	semidirect
38. #13182028	C₂×C₄	semidirect
39. C₂×Dic₂₁₈	C₂×C₄	semidirect
40. Q₈×D₁₀₉	C₂×C₄	semidirect
42. C₂×D₄₃₆	C₂×C₄	semidirect
43. C₂×C₂²⋊D₁₀₉	C₂×C₄	semidirect
44. D₄×D₁₀₉	C₂×C₄	semidirect
46. C₂²×Dic₁₀₉	C₂×C₄	semidirect
47. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
48. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
49. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
53. #41×C₁₀₉	C₂×C₄	semidirect
54. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
55. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
56. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
65. C₂×#89137	#41	direct (N × H)
68. Q₈×C₁₀₉	C₂²×C₄	semidirect
69. D₄×C₁₀₉	C₂²×C₄	semidirect
72. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
73. C₂²⋊D₁₀₉	C₂²×C₄	semidirect
76. C₂×Dic₁₀₉	C₂²×C₄	semidirect
77. D₄₃₆	C₂²×C₄	semidirect
80. C₄×D₁₀₉	C₂²×C₄	semidirect
81. C₄×D₁₀₉	C₂²×C₄	semidirect
84. Dic₂₁₈	C₂²×C₄	semidirect
96. #89137	C₂×#41	direct (N × H)
102. C₂×C₁₀₉	C₄×#41	semidirect
105. D₁₀₉	C₄×#41	semidirect
108. C₁₀₉	C₂×C₄×#41	semidirect
120. C₂×#41	#89137	direct (N × H)
138. #41	C₂×#89137	direct (N × H)
142. C₂×Q₈	C₂×#89137	semidirect
147. C₂×D₄	C₂×#89137	semidirect
159. C₂²×C₄	C₂×#89137	semidirect
172. Q₈	C₂²×#89137	semidirect
175. D₄	C₂²×#89137	semidirect
192. C₂×C₄	C₂²×#89137	semidirect
213. C₂	#41×#89137	direct (N × H)

Maximal subgroups

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

subgroup	index
#41×#89137	2
C₂×Q₈×#89137	2
C₂×D₄×#89137	2
C₂×#16812778	2
C₂²×C₄×#89137	2
C₂×#41×D₁₀₉	2
C₂×C₄×#41	109

Subgroups — C2×#41×#89137 #28380008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#41×#89137 #28380008