Subgroups — #28380012

Order 13952.

1540 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296095	C₂	1	8	yes	maximal, complemented
2	#20296094	C₂	1	2	yes	maximal, complemented
3	#20296093	C₂	1	6	yes	maximal, complemented
4	#41×#89137	C₂	1	8	yes	maximal, complemented
5	C₂×#16812778	C₂	1	6	yes	maximal, complemented
6	C₂×#41×D₁₀₉	C₂	1	1	yes	maximal, characteristic
7	Q₈×#89137	C₂²	1	16	yes	complemented
8	D₄×#89137	C₂²	1	48	yes	complemented
9	#16812778	C₂²	1	4	yes	—
10	#16812778	C₂²	1	12	yes	—
11	#16812778	C₂²	1	12	yes	complemented
12	#16812778	C₂²	1	12	yes	complemented
13	C₂×C₄⋊#89137	C₂²	1	12	yes	complemented
14	C₂×C₂²⋊#89137	C₂²	1	12	yes	—
15	C₂×C₄×#89137	C₂²	1	12	yes	complemented
16	C₂×#13182030	C₄	1	1	yes	characteristic, complemented
17	C₂×#13182029	C₄	1	3	yes	complemented
18	C₂×#13182028	C₄	1	3	yes	complemented
19	#41×D₁₀₉	C₄	1	4	yes	complemented
20	#41×D₁₀₉	C₄	1	4	yes	complemented
21	#41×D₁₀₉	C₂²	1	4	yes	—
22	#41×D₁₀₉	C₂²	1	4	yes	—
23	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×D₄×D₁₀₉	C₂²	1	3	yes	—
25	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
26	C₂×#41×C₁₀₉	C₄	1	1	yes	characteristic, complemented
27	C₂²×#89137	C₂³	1	24	yes	—
28	C₄⋊#89137	C₂³	1	8	yes	—
29	C₄⋊#89137	C₂³	1	24	yes	—
30	C₂²⋊#89137	C₂³	1	8	yes	—
31	C₂²⋊#89137	C₂³	1	24	yes	—
32	C₄×#89137	C₂³	1	8	yes	—
33	C₄×#89137	C₂³	1	24	yes	—
34	C₂³×D₁₀₉	C₂³	1	3	yes	—
35	#13182030	C₂×C₄	1	4	yes	complemented
36	#13182030	C₂×C₄	1	4	yes	complemented
37	#13182029	C₂×C₄	1	12	yes	complemented
38	#13182029	C₂×C₄	1	12	yes	complemented
39	#13182028	C₂×C₄	1	12	yes	complemented
40	#13182028	C₂×C₄	1	12	yes	complemented
41	C₂×Dic₂₁₈	C₂×C₄	1	3	yes	complemented
42	Q₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
43	Q₈×D₁₀₉	C₂³	1	4	yes	—
44	C₂×D₄₃₆	C₂×C₄	1	3	yes	complemented
45	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	6	yes	complemented
46	D₄×D₁₀₉	C₂×C₄	1	12	yes	complemented
47	D₄×D₁₀₉	C₂³	1	12	yes	—
48	C₂²×Dic₁₀₉	C₂×C₄	1	3	yes	complemented
49	C₂×C₄×D₁₀₉	C₂×C₄	1	6	yes	complemented
50	C₂×C₄×D₁₀₉	C₂×C₄	1	6	yes	complemented
51	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	complemented
52	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	complemented
53	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
55	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
56	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
57	#41×C₁₀₉	C₂×C₄	1	4	yes	complemented
58	#41×C₁₀₉	C₂×C₄	1	4	yes	complemented
59	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
60	C₂×D₄×C₁₀₉	C₂×C₄	1	3	yes	complemented
61	C₂²×C₄×C₁₀₉	C₂×C₄	1	3	yes	complemented
62	C₂³×C₁₀₉	C₂²×C₄	1	3	yes	—
63	C₂²×D₁₀₉	—	4	6	no	—
64	C₂²×D₁₀₉	C₂²×C₄	1	6	yes	—
65	C₂²×D₁₀₉	C₂²×C₄	1	3	yes	—
66	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
67	C₂²×D₁₀₉	C₂⁴	1	6	yes	—
68	C₂×#89137	—	4	24	no	—
69	C₂×#89137	C₂⁴	1	16	yes	—
70	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	—
71	D₄×C₁₀₉	C₂²×C₄	1	12	yes	—
72	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
74	C₂×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
75	C₂×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
76	C₂²⋊D₁₀₉	C₂²×C₄	1	24	yes	—
77	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
79	C₂×Dic₁₀₉	C₂²×C₄	1	6	yes	—
80	C₂×Dic₁₀₉	C₂²×C₄	1	6	yes	—
81	D₄₃₆	C₂²×C₄	1	12	yes	—
82	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
83	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
85	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
86	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
87	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
88	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
89	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
90	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
91	C₄×D₁₀₉	C₂⁴	1	6	yes	—
92	Dic₂₁₈	C₂²×C₄	1	12	yes	—
93	C₂²×C₁₀₉	—	4	3	no	—
94	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
95	C₂²×C₁₀₉	C₂³×C₄	1	6	yes	—
96	C₂×D₁₀₉	—	2	12	no	—
97	C₂×D₁₀₉	—	2	2	no	—
98	C₂×D₁₀₉	—	4	6	no	—
99	C₂×D₁₀₉	—	4	3	no	—
100	C₂×D₁₀₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₀₉	C₂³×C₄	1	6	yes	—
102	C₂×D₁₀₉	C₂⁵	1	1	yes	characteristic
103	#89137	—	2	16	no	—
104	C₄×C₁₀₉	C₂³×C₄	1	6	yes	—
105	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
107	Dic₁₀₉	C₂³×C₄	1	6	yes	—
108	Dic₁₀₉	C₂³×C₄	1	1	yes	characteristic
109	Dic₁₀₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₁₀₉	—	2	1	no	—
111	C₂×C₁₀₉	—	2	6	no	—
112	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
113	D₁₀₉	—	2	1	no	—
114	D₁₀₉	—	2	6	no	—
115	D₁₀₉	#585	1	2	yes	—
116	#3428	—	109	1	no	maximal, Sylow
117	C₁₀₉	#3428	1	1	yes	characteristic, complemented, Sylow
118	C₂²×#41	—	109	1	no	—
119	#520	—	109	8	no	—
120	#519	—	109	2	no	—
121	#518	—	109	6	no	—
122	C₄×#41	—	109	8	no	—
123	C₂×#117	—	109	6	no	—
124	C₂×#41	—	109	4	no	—
125	C₂×#41	—	109	4	no	—
126	C₂×#41	—	109	4	no	—
127	C₂×#41	—	109	4	no	—
128	C₂×#41	—	109	1	no	—
129	C₂×#41	—	109	3	no	—
130	C₂×#41	—	109	3	no	—
131	C₂×#41	#89137	1	1	yes	characteristic, complemented
132	C₂²×Q₈	—	109	1	no	—
133	C₂²×D₄	—	109	3	no	—
134	C₂³×C₄	—	109	3	no	—
135	C₄×Q₈	—	109	16	no	—
136	C₄×D₄	—	109	48	no	—
137	#117	—	109	4	no	—
138	#117	—	109	12	no	—
139	#117	—	109	12	no	—
140	#117	—	109	12	no	—
141	C₂×C₄⋊C₄	—	109	12	no	—
142	C₂×C₂²⋊C₄	—	109	12	no	—
143	C₂×C₄²	—	109	12	no	—
144	C₂⁴	—	109	3	no	—
145	#41	—	109	4	no	—
146	#41	—	109	4	no	—
147	#41	—	109	12	no	—
148	#41	—	109	12	no	—
149	#41	—	109	12	no	—
150	#41	—	109	12	no	—
151	#41	C₂×#89137	1	4	yes	complemented
152	#41	C₂×#89137	1	4	yes	complemented
153	C₂×Q₈	—	109	4	no	—
154	C₂×Q₈	—	109	4	no	—
155	C₂×Q₈	—	109	3	no	—
156	C₂×Q₈	C₂×#89137	1	1	yes	characteristic, complemented
157	C₂×D₄	—	109	6	no	—
158	C₂×D₄	—	109	12	no	—
159	C₂×D₄	—	109	12	no	—
160	C₂×D₄	—	109	3	no	—
161	C₂×D₄	C₂×#89137	1	3	yes	complemented
162	C₂²×C₄	—	109	3	no	—
163	C₂²×C₄	—	109	3	no	—
164	C₂²×C₄	—	109	3	no	—
165	C₂²×C₄	—	109	6	no	—
166	C₂²×C₄	—	109	6	no	—
167	C₂²×C₄	—	109	6	no	—
168	C₂²×C₄	—	109	6	no	—
169	C₂²×C₄	—	109	1	no	—
170	C₂²×C₄	—	109	3	no	—
171	C₂²×C₄	—	109	24	no	—
172	C₂²×C₄	C₂×#89137	1	3	yes	complemented
173	C₄⋊C₄	—	109	8	no	—
174	C₄⋊C₄	—	109	24	no	—
175	C₂²⋊C₄	—	109	8	no	—
176	C₂²⋊C₄	—	109	24	no	—
177	C₄²	—	109	8	no	—
178	C₄²	—	109	24	no	—
179	C₂³	—	109	6	no	—
180	C₂³	—	109	3	no	—
181	C₂³	—	109	1	no	—
182	C₂³	—	109	6	no	—
183	C₂³	—	436	6	no	—
184	C₂³	C₂²×#89137	1	3	yes	—
185	Q₈	—	109	12	no	—
186	Q₈	C₂²×#89137	1	4	yes	—
187	D₄	—	109	24	no	—
188	D₄	—	109	12	no	—
189	D₄	C₂²×#89137	1	12	yes	—
190	C₂×C₄	—	109	1	no	—
191	C₂×C₄	—	109	3	no	—
192	C₂×C₄	—	109	6	no	—
193	C₂×C₄	—	109	6	no	—
194	C₂×C₄	—	109	6	no	—
195	C₂×C₄	—	109	1	no	—
196	C₂×C₄	—	109	1	no	—
197	C₂×C₄	—	109	6	no	—
198	C₂×C₄	—	109	6	no	—
199	C₂×C₄	—	109	6	no	—
200	C₂×C₄	—	109	6	no	—
201	C₂×C₄	—	109	1	no	—
202	C₂×C₄	—	109	1	no	—
203	C₂×C₄	—	109	6	no	—
204	C₂×C₄	—	109	16	no	—
205	C₂×C₄	—	436	24	no	—
206	C₂×C₄	C₂²×#89137	1	3	yes	—
207	C₂×C₄	C₂²×#89137	1	6	yes	—
208	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
209	C₂×C₄	C₂²×#89137	1	6	yes	—
210	C₂²	—	4	3	no	—
211	C₂²	—	109	1	no	—
212	C₂²	—	109	1	no	—
213	C₂²	—	109	6	no	—
214	C₂²	—	436	3	no	—
215	C₂²	—	436	6	no	—
216	C₂²	—	218	12	no	—
217	C₂²	—	218	2	no	—
218	C₂²	C₂³×#89137	1	1	yes	characteristic
219	C₂²	C₂³×#89137	1	6	yes	—
220	C₄	—	109	1	no	—
221	C₄	—	109	1	no	—
222	C₄	—	109	6	no	—
223	C₄	—	218	16	no	—
224	C₄	C₂³×#89137	1	1	yes	characteristic
225	C₄	C₂³×#89137	1	6	yes	—
226	C₄	C₂³×#89137	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	6	no	—
229	C₂	—	109	2	no	—
230	C₂	—	218	1	no	—
231	C₂	—	218	6	no	—
232	C₂	C₂⁴×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×#89137
commutator, socle	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂	C₂⁴×#89137
Fitting	C₂×#41×C₁₀₉	C₄
prime radical	C₂×#41×D₁₀₉	C₂
cosocle	C₂×D₁₀₉	C₂⁵
hypercenter	C₂×#41	#89137
hypocenter	C₁₀₉	#3428
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20296095	C₂	semidirect
2. #20296094	C₂	semidirect
3. #20296093	C₂	semidirect
4. #41×#89137	C₂	semidirect
5. C₂×#16812778	C₂	semidirect
7. Q₈×#89137	C₂²	semidirect
8. D₄×#89137	C₂²	semidirect
11. #16812778	C₂²	semidirect
12. #16812778	C₂²	semidirect
13. C₂×C₄⋊#89137	C₂²	semidirect
15. C₂×C₄×#89137	C₂²	semidirect
16. C₂×#13182030	C₄	semidirect
17. C₂×#13182029	C₄	semidirect
18. C₂×#13182028	C₄	semidirect
19. #41×D₁₀₉	C₄	semidirect
20. #41×D₁₀₉	C₄	semidirect
26. C₂×#41×C₁₀₉	C₄	semidirect
35. #13182030	C₂×C₄	semidirect
36. #13182030	C₂×C₄	semidirect
37. #13182029	C₂×C₄	semidirect
38. #13182029	C₂×C₄	semidirect
39. #13182028	C₂×C₄	semidirect
40. #13182028	C₂×C₄	semidirect
41. C₂×Dic₂₁₈	C₂×C₄	semidirect
42. Q₈×D₁₀₉	C₂×C₄	semidirect
44. C₂×D₄₃₆	C₂×C₄	semidirect
45. C₂×C₂²⋊D₁₀₉	C₂×C₄	semidirect
46. D₄×D₁₀₉	C₂×C₄	semidirect
48. C₂²×Dic₁₀₉	C₂×C₄	semidirect
49. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
50. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
51. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
52. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
57. #41×C₁₀₉	C₂×C₄	semidirect
58. #41×C₁₀₉	C₂×C₄	semidirect
59. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
60. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
61. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
117. C₁₀₉	#3428	semidirect
131. C₂×#41	#89137	semidirect
151. #41	C₂×#89137	semidirect
152. #41	C₂×#89137	semidirect
156. C₂×Q₈	C₂×#89137	semidirect
161. C₂×D₄	C₂×#89137	semidirect
172. C₂²×C₄	C₂×#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296095	2
#20296094	2
#20296093	2
#41×#89137	2
C₂×#16812778	2
C₂×#41×D₁₀₉	2
#3428	109