Subgroups — #475×D₁₀₉ #28372793

Order 13952.

416 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20295312	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295311	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295310	C₂	1	2	yes	maximal, complemented
6	#20295309	C₂	1	2	yes	maximal, complemented
7	#20295308	C₂	1	1	yes	maximal, characteristic, complemented
8	#106×D₁₀₉	C₂	1	2	yes	maximal
9	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
10	#103×D₁₀₉	C₂	1	2	yes	maximal
11	#475×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₉	—	2	1	no	—
13	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
15	#16812699	C₂²	1	1	yes	characteristic
16	#16812698	—	2	1	no	—
17	#16812698	C₂²	1	2	yes	—
18	#16812697	—	2	1	no	—
19	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₁₈	—	2	1	no	—
21	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
22	#16812664	—	2	1	no	—
23	#16812664	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×D₁₀₉	—	2	1	no	—
25	C₄⋊C₄×D₁₀₉	—	2	1	no	—
26	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
28	C₄²×D₁₀₉	—	2	1	no	—
29	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
30	#16812631	C₂²	1	2	yes	—
31	#16812630	C₂²	1	2	yes	—
32	#16812630	C₂²	1	2	yes	—
33	C₄⋊#11724784	C₂²	1	1	yes	characteristic
34	#16812628	C₂²	1	2	yes	—
35	#16812624	C₂²	1	2	yes	—
36	#16812623	C₂²	1	2	yes	—
37	#16812622	C₂²	1	2	yes	—
38	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	#106×C₁₀₉	C₂²	1	2	yes	—
41	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
42	#103×C₁₀₉	C₂²	1	2	yes	—
43	C₂×Dic₂₁₈	—	2	1	no	—
44	C₂×Dic₂₁₈	—	2	1	no	—
45	C₂×Dic₂₁₈	—	2	2	no	—
46	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
47	Q₈×D₁₀₉	—	2	2	no	—
48	Q₈×D₁₀₉	—	4	4	no	—
49	Q₈×D₁₀₉	Q₈	1	4	yes	complemented
50	C₂×C₄×D₁₀₉	—	2	1	no	—
51	C₂×C₄×D₁₀₉	—	2	1	no	—
52	C₂×C₄×D₁₀₉	—	2	2	no	—
53	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
55	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
56	C₂×#11724784	C₂³	1	2	yes	—
57	C₈×D₁₀₉	—	2	4	no	—
58	C₄⋊Dic₁₀₉	—	2	1	no	—
59	C₄⋊Dic₁₀₉	—	2	1	no	—
60	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
61	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
62	#13182008	—	2	1	no	—
63	#13182008	—	2	1	no	—
64	#13182008	—	2	2	no	—
65	#13182008	—	2	1	no	—
66	#13182008	—	2	2	no	—
67	C₄×Dic₁₀₉	—	2	1	no	—
68	C₄×Dic₁₀₉	—	2	1	no	—
69	C₄×Dic₁₀₉	—	2	1	no	—
70	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
71	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
72	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₂×Q₈×C₁₀₉	—	2	1	no	—
74	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
76	C₄⋊C₄×C₁₀₉	—	2	1	no	—
77	C₄⋊C₄×C₁₀₉	—	2	1	no	—
78	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
79	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₄²×C₁₀₉	—	2	1	no	—
81	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₁₀₉	—	2	1	no	—
84	Q₈×C₁₀₉	—	4	2	no	—
85	Q₈×C₁₀₉	C₂×Q₈	1	2	yes	complemented
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
92	C₂×Dic₁₀₉	—	2	1	no	—
93	C₂×Dic₁₀₉	—	2	1	no	—
94	C₂×Dic₁₀₉	—	2	2	no	—
95	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
98	C₄×D₁₀₉	—	2	4	no	—
99	C₄×D₁₀₉	—	2	2	no	—
100	C₄×D₁₀₉	—	4	4	no	—
101	C₄×D₁₀₉	—	4	2	no	—
102	C₄×D₁₀₉	SD₁₆	1	4	yes	—
103	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
104	C₄×D₁₀₉	#41	1	2	yes	—
105	Dic₂₁₈	—	2	1	no	—
106	Dic₂₁₈	—	2	4	no	—
107	Dic₂₁₈	—	4	2	no	—
108	Dic₂₁₈	—	4	4	no	—
109	Dic₂₁₈	C₂×Q₈	1	2	yes	complemented
110	C₈×C₁₀₉	—	2	2	no	—
111	#11724784	—	2	2	no	—
112	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×D₁₀₉	C₂×SD₁₆	1	2	yes	—
115	C₂×D₁₀₉	#137	1	2	yes	—
116	C₄×C₁₀₉	—	2	2	no	—
117	C₄×C₁₀₉	—	2	1	no	—
118	C₄×C₁₀₉	—	4	2	no	—
119	C₄×C₁₀₉	—	4	1	no	—
120	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
121	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
122	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
123	Dic₁₀₉	—	2	2	no	—
124	Dic₁₀₉	—	2	1	no	—
125	Dic₁₀₉	—	4	2	no	—
126	Dic₁₀₉	—	4	1	no	—
127	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
128	Dic₁₀₉	C₂×SD₁₆	1	2	yes	—
129	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
131	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
132	D₁₀₉	#475	1	4	yes	direct product factor
133	C₂×#475	—	109	1	no	maximal, Sylow
134	C₁₀₉	C₂×#475	1	1	yes	characteristic, complemented, Sylow
135	C₂×C₄⋊Q₈	—	109	1	no	—
136	C₂×C₄×Q₈	—	109	1	no	—
137	#475	—	109	1	no	—
138	#475	—	109	1	no	—
139	#475	—	109	2	no	—
140	#475	—	109	1	no	—
141	#475	—	109	2	no	—
142	#475	D₁₀₉	1	1	yes	characteristic, direct product factor
143	C₂×#106	—	109	2	no	—
144	C₂×C₄⋊C₈	—	109	1	no	—
145	C₂×#103	—	109	2	no	—
146	C₂²×Q₈	—	109	1	no	—
147	C₂²×Q₈	—	218	1	no	—
148	C₂²×C₈	—	109	2	no	—
149	C₄⋊Q₈	—	109	2	no	—
150	C₄⋊Q₈	—	109	1	no	—
151	C₄⋊Q₈	—	218	1	no	—
152	C₄⋊Q₈	—	218	1	no	—
153	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
154	C₄×Q₈	—	109	1	no	—
155	C₄×Q₈	—	109	1	no	—
156	C₄×Q₈	—	109	1	no	—
157	C₄×Q₈	—	218	1	no	—
158	C₄×Q₈	—	218	1	no	—
159	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
160	C₂×C₄⋊C₄	—	109	1	no	—
161	C₂×C₄⋊C₄	—	109	2	no	—
162	C₂×C₄⋊C₄	—	218	1	no	—
163	C₂×C₄⋊C₄	—	218	1	no	—
164	C₂×C₄²	—	109	1	no	—
165	C₂×C₄²	—	218	1	no	—
166	#106	—	109	2	no	—
167	#106	—	109	2	no	—
168	#106	—	109	2	no	—
169	#106	C₂×D₁₀₉	1	2	yes	—
170	C₄⋊C₈	—	109	2	no	—
171	C₄⋊C₈	—	109	1	no	—
172	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
173	#103	—	109	2	no	—
174	#103	—	109	2	no	—
175	#103	—	109	2	no	—
176	#103	C₂×D₁₀₉	1	2	yes	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	—	109	4	no	—
179	C₂×Q₈	—	109	1	no	—
180	C₂×Q₈	—	436	4	no	—
181	C₂×Q₈	—	218	2	no	—
182	C₂×Q₈	—	218	1	no	—
183	C₂×Q₈	—	218	2	no	—
184	C₂×Q₈	—	218	1	no	—
185	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
186	C₂²×C₄	—	109	1	no	—
187	C₂²×C₄	—	109	1	no	—
188	C₂²×C₄	—	109	1	no	—
189	C₂²×C₄	—	218	1	no	—
190	C₂²×C₄	—	218	2	no	—
191	C₂²×C₄	—	218	1	no	—
192	C₂×C₈	—	109	2	no	—
193	C₂×C₈	—	218	4	no	—
194	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	109	2	no	—
198	C₄⋊C₄	—	109	1	no	—
199	C₄⋊C₄	—	218	1	no	—
200	C₄⋊C₄	—	218	2	no	—
201	C₄⋊C₄	—	218	1	no	—
202	C₄⋊C₄	—	218	2	no	—
203	C₄⋊C₄	—	218	1	no	—
204	C₄⋊C₄	—	218	1	no	—
205	C₄⋊C₄	—	218	1	no	—
206	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
207	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
208	C₄²	—	2	1	no	—
209	C₄²	—	109	1	no	—
210	C₄²	—	109	1	no	—
211	C₄²	—	109	1	no	—
212	C₄²	—	218	1	no	—
213	C₄²	—	218	1	no	—
214	C₄²	—	218	1	no	—
215	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
216	C₂³	—	109	1	no	—
217	Q₈	—	2	1	no	—
218	Q₈	—	4	2	no	—
219	Q₈	—	109	2	no	—
220	Q₈	—	436	4	no	—
221	Q₈	—	436	2	no	—
222	Q₈	—	218	4	no	—
223	Q₈	—	218	1	no	—
224	Q₈	Q₈×D₁₀₉	1	2	yes	complemented
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	109	1	no	—
229	C₂×C₄	—	109	1	no	—
230	C₂×C₄	—	109	4	no	—
231	C₂×C₄	—	109	2	no	—
232	C₂×C₄	—	109	2	no	—
233	C₂×C₄	—	109	1	no	—
234	C₂×C₄	—	436	4	no	—
235	C₂×C₄	—	436	2	no	—
236	C₂×C₄	—	218	2	no	—
237	C₂×C₄	—	218	4	no	—
238	C₂×C₄	—	218	1	no	—
239	C₂×C₄	—	218	2	no	—
240	C₂×C₄	—	218	1	no	—
241	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
242	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	218	2	no	—
246	C₂²	—	109	2	no	—
247	C₂²	—	109	2	no	—
248	C₂²	—	109	2	no	—
249	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	109	2	no	—
255	C₄	—	109	2	no	—
256	C₄	—	436	2	no	—
257	C₄	—	436	1	no	—
258	C₄	—	218	2	no	—
259	C₄	—	218	1	no	—
260	C₄	SD₁₆×D₁₀₉	1	2	yes	—
261	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
262	C₄	#41×D₁₀₉	1	1	yes	characteristic
263	C₂	—	109	4	no	—
264	C₂	C₂²⋊Q₈×D₁₀₉	1	1	yes	characteristic
265	C₂	C₂×SD₁₆×D₁₀₉	1	1	yes	characteristic
266	C₂	#137×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#475×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
hypercenter	#475	D₁₀₉
hypocenter	C₁₀₉	C₂×#475
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
3. #20295312	C₂	semidirect
4. #20295311	C₂	semidirect
5. #20295310	C₂	semidirect
6. #20295309	C₂	semidirect
7. #20295308	C₂	semidirect
11. #475×C₁₀₉	C₂	semidirect
49. Q₈×D₁₀₉	Q₈	semidirect
85. Q₈×C₁₀₉	C₂×Q₈	semidirect
109. Dic₂₁₈	C₂×Q₈	semidirect
132. D₁₀₉	#475	direct (N × H)
134. C₁₀₉	C₂×#475	semidirect
142. #475	D₁₀₉	direct (N × H)
224. Q₈	Q₈×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×D₁₀₉	2
C₄×Q₈×D₁₀₉	2
#20295312	2
#20295311	2
#20295310	2
#20295309	2
#20295308	2
#106×D₁₀₉	2
C₄⋊C₈×D₁₀₉	2
#103×D₁₀₉	2
#475×C₁₀₉	2
C₂×#475	109

Subgroups — #475×D109 #28372793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #475×D₁₀₉ #28372793