Subgroups — #28371626

Order 13952.

476 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812707	C₂	1	1	yes	maximal, characteristic
2	#20295507	C₂	1	1	yes	maximal, characteristic
3	#20295506	C₂	1	1	yes	maximal, characteristic
4	#20295068	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295067	C₂	1	2	yes	maximal, complemented
6	#20295065	C₂	1	2	yes	maximal, complemented
7	#20295063	C₂	1	2	yes	maximal, complemented
8	#20295062	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812630	C₂	1	1	yes	maximal, characteristic, complemented
10	#106×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	#428×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	#16812707	C₄	1	8	yes	complemented
14	C₂×#13182012	C₂²	1	2	yes	—
15	C₂²×#11724784	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
17	C₄×Dic₂₁₈	—	2	1	no	—
18	#16812664	—	2	1	no	—
19	C₄×D₄₃₆	—	2	1	no	—
20	#16812661	—	2	1	no	—
21	C₄×C₂²⋊D₁₀₉	—	2	2	no	—
22	#16812659	—	2	2	no	—
23	#16812658	C₂²	1	1	yes	characteristic
24	#16812657	C₂²	1	2	yes	—
25	#16812657	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
28	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
29	#16812633	C₂²	1	2	yes	—
30	#16812632	C₂²	1	1	yes	characteristic
31	#16812632	C₂²	1	1	yes	characteristic
32	#16812631	C₂²	1	2	yes	—
33	#16812630	C₂²	1	1	yes	characteristic
34	#16812630	C₂²	1	2	yes	—
35	#16812630	C₂²	1	2	yes	—
36	#16812630	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
38	#117×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
40	#107×C₁₀₉	C₂²	1	2	yes	—
41	#106×C₁₀₉	C₂²	1	2	yes	—
42	#13182029	C₂×C₄	1	4	yes	—
43	#13182029	Q₈	1	4	yes	—
44	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
45	C₂×D₄₃₆	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₉	D₄	1	2	yes	—
47	C₂²×Dic₁₀₉	—	2	1	no	—
48	C₂×C₄×D₁₀₉	—	2	2	no	—
49	C₂×C₄×D₁₀₉	—	2	2	no	—
50	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
51	#13182012	C₂×C₄	1	8	yes	complemented
52	C₂×#11724784	C₂×C₄	1	4	yes	complemented
53	C₂×#11724784	C₂³	1	1	yes	characteristic
54	C₂×#11724784	C₂³	1	1	yes	characteristic
55	C₈×D₁₀₉	C₂×C₄	1	8	yes	complemented
56	C₄⋊Dic₁₀₉	—	2	1	no	—
57	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
58	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
60	#13182008	—	2	2	no	—
61	#13182008	—	2	2	no	—
62	C₂²⋊Dic₁₀₉	—	2	1	no	—
63	#13182006	—	2	2	no	—
64	#13182006	—	2	2	no	—
65	C₄×Dic₁₀₉	—	2	1	no	—
66	C₄×Dic₁₀₉	—	2	2	no	—
67	C₄×Dic₁₀₉	—	2	2	no	—
68	C₂²×C₄×C₁₀₉	—	2	1	no	—
69	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
70	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
71	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
72	C₄⋊C₄×C₁₀₉	—	2	1	no	—
73	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
75	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
77	C₄²×C₁₀₉	—	2	1	no	—
78	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×D₁₀₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₀₉	—	2	1	no	—
81	C₂×C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	8	1	no	—
84	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
88	C₂²⋊D₁₀₉	C₄⋊C₄	1	8	yes	—
89	C₂×Dic₁₀₉	—	2	1	no	—
90	C₂×Dic₁₀₉	—	2	2	no	—
91	C₂×Dic₁₀₉	—	2	1	no	—
92	C₂×Dic₁₀₉	—	8	1	no	—
93	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
94	D₄₃₆	C₄⋊C₄	1	4	yes	—
95	C₄×D₁₀₉	—	8	2	no	—
96	C₄×D₁₀₉	—	8	2	no	—
97	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
98	C₄×D₁₀₉	C₂×Q₈	1	4	yes	—
99	Dic₂₁₈	C₄⋊C₄	1	4	yes	—
100	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
101	#11724784	C₂²×C₄	1	2	yes	—
102	#11724784	C₂²×C₄	1	2	yes	—
103	C₂²×C₁₀₉	—	2	1	no	—
104	C₂²×C₁₀₉	—	2	1	no	—
105	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₁₀₉	—	2	2	no	—
108	C₂×D₁₀₉	—	2	2	no	—
109	C₂×D₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
110	C₄×C₁₀₉	—	4	1	no	—
111	C₄×C₁₀₉	—	4	2	no	—
112	C₄×C₁₀₉	—	4	1	no	—
113	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
114	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
115	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
116	Dic₁₀₉	—	4	1	no	—
117	Dic₁₀₉	—	4	2	no	—
118	Dic₁₀₉	—	4	1	no	—
119	Dic₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₁₀₉	—	2	2	no	—
121	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₁₀₉	#577	1	1	yes	characteristic
123	C₂×C₁₀₉	#577	1	1	yes	characteristic
124	D₁₀₉	—	2	4	no	—
125	#2912	—	109	1	no	maximal, Sylow
126	C₁₀₉	#2912	1	1	yes	characteristic, complemented, Sylow
127	C₂×#131	—	109	1	no	—
128	#520	—	109	1	no	—
129	#520	—	109	1	no	—
130	#428	—	109	1	no	—
131	#428	—	109	2	no	—
132	#428	—	109	2	no	—
133	#428	D₁₀₉	1	1	yes	characteristic, complemented
134	#427	—	109	1	no	—
135	#427	—	109	2	no	—
136	C₂×#106	—	109	1	no	—
137	C₂×#106	—	109	2	no	—
138	C₂×#41	—	109	1	no	—
139	#131	—	109	8	no	—
140	C₂×M₁₆	—	109	2	no	—
141	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
142	C₂²×C₈	—	109	1	no	—
143	C₂²×C₈	—	109	2	no	—
144	C₄×Q₈	—	218	1	no	—
145	C₄×Q₈	—	218	1	no	—
146	C₄×D₄	—	218	1	no	—
147	C₄×D₄	—	218	2	no	—
148	C₄×D₄	—	218	1	no	—
149	C₄×D₄	—	218	2	no	—
150	#117	—	109	1	no	—
151	#117	—	109	2	no	—
152	#117	—	109	2	no	—
153	#117	C₂×D₁₀₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	109	1	no	—
155	C₂×C₄⋊C₄	—	109	2	no	—
156	C₂×C₄⋊C₄	—	109	2	no	—
157	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
158	#107	—	109	2	no	—
159	#107	—	109	1	no	—
160	#107	—	109	1	no	—
161	#107	C₂×D₁₀₉	1	2	yes	—
162	#106	—	109	2	no	—
163	#106	—	109	2	no	—
164	#106	—	109	1	no	—
165	#106	—	109	2	no	—
166	#106	—	109	1	no	—
167	#106	C₂×D₁₀₉	1	2	yes	—
168	#41	—	109	4	no	—
169	#41	—	109	4	no	—
170	C₂×Q₈	—	109	1	no	—
171	C₂×D₄	—	109	2	no	—
172	C₂×D₄	—	109	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	109	2	no	—
175	C₂²×C₄	—	218	2	no	—
176	C₂²×C₄	—	218	2	no	—
177	C₂²×C₄	—	218	1	no	—
178	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
179	M₁₆	—	109	8	no	—
180	M₁₆	C₄×D₁₀₉	1	4	yes	complemented
181	C₂×C₈	—	109	4	no	—
182	C₂×C₈	—	109	1	no	—
183	C₂×C₈	—	109	1	no	—
184	C₂×C₈	—	109	8	no	—
185	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	109	2	no	—
188	C₄⋊C₄	—	109	1	no	—
189	C₄⋊C₄	—	109	1	no	—
190	C₄⋊C₄	—	218	2	no	—
191	C₄⋊C₄	—	218	2	no	—
192	C₄⋊C₄	—	218	1	no	—
193	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
194	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
195	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	218	2	no	—
198	C₂²⋊C₄	—	218	2	no	—
199	C₂²⋊C₄	—	218	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	218	2	no	—
202	C₄²	—	218	2	no	—
203	C₄²	—	218	1	no	—
204	C₂³	—	109	2	no	—
205	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
206	Q₈	—	109	4	no	—
207	D₄	—	109	8	no	—
208	D₄	—	109	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	872	2	no	—
214	C₂×C₄	—	872	2	no	—
215	C₂×C₄	—	872	1	no	—
216	C₂×C₄	—	109	2	no	—
217	C₂×C₄	—	109	4	no	—
218	C₂×C₄	—	109	4	no	—
219	C₂×C₄	—	218	1	no	—
220	C₂×C₄	—	218	2	no	—
221	C₂×C₄	—	218	1	no	—
222	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
223	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
224	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
225	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
226	C₈	—	109	2	no	—
227	C₈	—	109	2	no	—
228	C₈	C₂×C₄×D₁₀₉	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	109	4	no	—
232	C₂²	—	218	2	no	—
233	C₂²	—	218	2	no	—
234	C₂²	C₄⋊C₄×D₁₀₉	1	2	yes	—
235	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
236	C₄	—	4	1	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	109	4	no	—
240	C₄	—	436	2	no	—
241	C₄	—	436	1	no	—
242	C₄	—	436	1	no	—
243	C₄	C₄⋊C₄×D₁₀₉	1	2	yes	—
244	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
245	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	218	4	no	—
248	C₂	C₂×C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
249	C₂	#20295835	1	1	yes	characteristic
250	C₂	#20295836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#428×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182029	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#428	D₁₀₉
hypocenter	C₁₀₉	#2912
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #20295068	C₂	semidirect
5. #20295067	C₂	semidirect
6. #20295065	C₂	semidirect
7. #20295063	C₂	semidirect
8. #20295062	C₂	semidirect
9. C₂×#16812630	C₂	semidirect
10. #106×D₁₀₉	C₂	semidirect
11. #428×C₁₀₉	C₂	semidirect
13. #16812707	C₄	semidirect
51. #13182012	C₂×C₄	semidirect
52. C₂×#11724784	C₂×C₄	semidirect
55. C₈×D₁₀₉	C₂×C₄	semidirect
70. M₁₆×C₁₀₉	C₂×C₄	semidirect
126. C₁₀₉	#2912	semidirect
133. #428	D₁₀₉	semidirect
180. M₁₆	C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812707	2
#20295507	2
#20295506	2
#20295068	2
#20295067	2
#20295065	2
#20295063	2
#20295062	2
C₂×#16812630	2
#106×D₁₀₉	2
#428×C₁₀₉	2
#2912	109