Subgroups — #476×D₁₀₉ #28372779

Order 13952.

480 conjugacy classes of subgroups.

The subgroup lattice has 296 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, complemented
2	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20295317	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295316	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295315	C₂	1	2	yes	maximal, complemented
6	#20295314	C₂	1	2	yes	maximal, complemented
7	#20295313	C₂	1	1	yes	maximal, characteristic, complemented
8	#106×D₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₁₀₉	C₂	1	2	yes	maximal
11	#476×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₉	—	2	1	no	—
13	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₀₉	—	2	1	no	—
15	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
16	#16812699	C₂²	1	1	yes	characteristic, complemented
17	#16812698	—	2	1	no	—
18	#16812698	C₂²	1	2	yes	complemented
19	#16812697	—	2	1	no	—
20	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
21	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
22	#16812661	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
24	#16812659	—	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	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
29	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
30	#16812631	C₂²	1	2	yes	complemented
31	#16812630	C₂²	1	2	yes	complemented
32	#16812630	C₂²	1	2	yes	complemented
33	C₄⋊#11724784	C₂²	1	1	yes	characteristic, complemented
34	#16812628	C₂²	1	2	yes	complemented
35	#16812621	C₂²	1	2	yes	—
36	#16812620	C₂²	1	2	yes	—
37	#16812619	C₂²	1	2	yes	—
38	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
40	#106×C₁₀₉	C₂²	1	2	yes	complemented
41	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
42	#102×C₁₀₉	C₂²	1	2	yes	—
43	C₂³×D₁₀₉	—	2	1	no	—
44	C₂×Dic₂₁₈	—	2	1	no	—
45	C₂×Dic₂₁₈	—	2	2	no	—
46	Q₈×D₁₀₉	—	4	4	no	—
47	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
49	D₄×D₁₀₉	—	2	2	no	—
50	D₄×D₁₀₉	Q₈	1	4	yes	complemented
51	C₂²×Dic₁₀₉	—	2	1	no	—
52	C₂×C₄×D₁₀₉	—	2	1	no	—
53	C₂×C₄×D₁₀₉	—	2	2	no	—
54	C₂×C₄×D₁₀₉	—	2	1	no	—
55	C₂×C₄×D₁₀₉	—	2	1	no	—
56	C₂×C₄×D₁₀₉	—	4	4	no	—
57	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
58	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
59	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
60	C₂×#11724784	C₂³	1	2	yes	—
61	C₈×D₁₀₉	—	2	4	no	—
62	C₄⋊Dic₁₀₉	—	2	1	no	—
63	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
64	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
65	#13182008	—	2	2	no	—
66	#13182008	—	2	1	no	—
67	#13182008	—	2	2	no	—
68	C₂²⋊Dic₁₀₉	—	2	1	no	—
69	#13182006	—	2	1	no	—
70	#13182006	—	2	1	no	—
71	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
72	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
73	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₁₀₉	—	2	1	no	—
75	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₉	—	2	1	no	—
77	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
78	C₄⋊C₄×C₁₀₉	—	2	1	no	—
79	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
80	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
82	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₂³×C₁₀₉	—	2	1	no	—
84	C₂²×D₁₀₉	—	2	1	no	—
85	C₂²×D₁₀₉	—	2	4	no	—
86	C₂²×D₁₀₉	—	4	2	no	—
87	C₂²×D₁₀₉	—	4	2	no	—
88	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₁₀₉	—	4	2	no	—
90	D₄×C₁₀₉	—	2	1	no	—
91	D₄×C₁₀₉	C₂×Q₈	1	2	yes	complemented
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	4	2	no	—
95	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₁₀₉	—	2	4	no	—
99	C₂×Dic₁₀₉	—	2	1	no	—
100	C₂×Dic₁₀₉	—	2	2	no	—
101	C₂×Dic₁₀₉	—	4	2	no	—
102	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
105	D₄₃₆	—	2	1	no	—
106	D₄₃₆	C₂×Q₈	1	2	yes	complemented
107	C₄×D₁₀₉	—	2	2	no	—
108	C₄×D₁₀₉	—	4	2	no	—
109	C₄×D₁₀₉	—	4	4	no	—
110	C₄×D₁₀₉	—	4	2	no	—
111	C₄×D₁₀₉	—	4	2	no	—
112	C₄×D₁₀₉	SD₁₆	1	4	yes	—
113	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
114	C₄×D₁₀₉	#41	1	2	yes	—
115	Dic₂₁₈	—	4	4	no	—
116	Dic₂₁₈	—	4	2	no	—
117	C₈×C₁₀₉	—	2	2	no	—
118	#11724784	—	2	2	no	—
119	C₂²×C₁₀₉	—	2	2	no	—
120	C₂²×C₁₀₉	—	4	1	no	—
121	C₂²×C₁₀₉	—	4	1	no	—
122	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₀₉	—	2	2	no	—
124	C₂×D₁₀₉	—	4	1	no	—
125	C₂×D₁₀₉	—	4	1	no	—
126	C₂×D₁₀₉	—	4	8	no	—
127	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
128	C₂×D₁₀₉	C₂×SD₁₆	1	2	yes	—
129	C₂×D₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
130	C₄×C₁₀₉	—	2	1	no	—
131	C₄×C₁₀₉	—	4	1	no	—
132	C₄×C₁₀₉	—	4	2	no	—
133	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
134	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
135	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
136	Dic₁₀₉	—	2	1	no	—
137	Dic₁₀₉	—	4	1	no	—
138	Dic₁₀₉	—	4	2	no	—
139	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
140	Dic₁₀₉	C₂×SD₁₆	1	2	yes	—
141	C₂×C₁₀₉	—	4	2	no	—
142	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
144	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
145	D₁₀₉	—	4	2	no	—
146	D₁₀₉	#476	1	4	yes	direct product factor
147	C₂×#476	—	109	1	no	maximal, Sylow
148	C₁₀₉	C₂×#476	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊Q₈	—	109	1	no	—
150	C₂×C₄×D₄	—	109	1	no	—
151	#476	—	109	1	no	—
152	#476	—	109	1	no	—
153	#476	—	109	2	no	—
154	#476	—	109	1	no	—
155	#476	—	109	2	no	—
156	#476	D₁₀₉	1	1	yes	characteristic, direct product factor
157	C₂×#106	—	109	2	no	—
158	C₂×C₄⋊C₈	—	109	1	no	—
159	C₂×#102	—	109	2	no	—
160	C₂²×Q₈	—	218	1	no	—
161	C₂²×D₄	—	109	1	no	—
162	C₂³×C₄	—	218	1	no	—
163	C₂²×C₈	—	109	2	no	—
164	C₄⋊Q₈	—	109	2	no	—
165	C₄⋊Q₈	—	109	1	no	—
166	C₄⋊Q₈	—	218	1	no	—
167	C₄⋊Q₈	—	218	1	no	—
168	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
169	C₄×D₄	—	109	1	no	—
170	C₄×D₄	—	109	1	no	—
171	C₄×D₄	—	109	1	no	—
172	C₄×D₄	—	218	1	no	—
173	C₄×D₄	—	218	1	no	—
174	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	109	1	no	—
176	C₂×C₄⋊C₄	—	109	2	no	—
177	C₂×C₄⋊C₄	—	218	1	no	—
178	C₂×C₂²⋊C₄	—	218	1	no	—
179	C₂×C₄²	—	109	1	no	—
180	#106	—	109	2	no	—
181	#106	—	109	2	no	—
182	#106	—	109	2	no	—
183	#106	C₂×D₁₀₉	1	2	yes	complemented
184	C₄⋊C₈	—	109	2	no	—
185	C₄⋊C₈	—	109	1	no	—
186	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
187	#102	—	109	2	no	—
188	#102	—	109	2	no	—
189	#102	—	109	2	no	—
190	#102	C₂×D₁₀₉	1	2	yes	—
191	C₂⁴	—	218	1	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	436	4	no	—
194	C₂×Q₈	—	218	2	no	—
195	C₂×Q₈	—	218	1	no	—
196	C₂×D₄	—	109	4	no	—
197	C₂×D₄	—	109	1	no	—
198	C₂×D₄	—	218	2	no	—
199	C₂×D₄	—	218	1	no	—
200	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	109	1	no	—
203	C₂²×C₄	—	109	1	no	—
204	C₂²×C₄	—	109	1	no	—
205	C₂²×C₄	—	436	4	no	—
206	C₂²×C₄	—	218	1	no	—
207	C₂²×C₄	—	218	1	no	—
208	C₂²×C₄	—	218	2	no	—
209	C₂²×C₄	—	218	1	no	—
210	C₂²×C₄	—	218	1	no	—
211	C₂×C₈	—	109	2	no	—
212	C₂×C₈	—	218	4	no	—
213	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	109	2	no	—
216	C₄⋊C₄	—	109	1	no	—
217	C₄⋊C₄	—	218	1	no	—
218	C₄⋊C₄	—	218	2	no	—
219	C₄⋊C₄	—	218	2	no	—
220	C₄⋊C₄	—	218	1	no	—
221	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
222	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	218	1	no	—
225	C₂²⋊C₄	—	218	1	no	—
226	C₂²⋊C₄	—	218	1	no	—
227	C₄²	—	109	1	no	—
228	C₄²	—	109	1	no	—
229	C₄²	—	109	1	no	—
230	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	—	109	1	no	—
233	C₂³	—	436	2	no	—
234	C₂³	—	436	2	no	—
235	C₂³	—	218	4	no	—
236	C₂³	—	218	1	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	436	2	no	—
239	Q₈	—	436	4	no	—
240	D₄	—	2	1	no	—
241	D₄	—	109	2	no	—
242	D₄	—	218	4	no	—
243	D₄	—	218	1	no	—
244	D₄	Q₈×D₁₀₉	1	2	yes	complemented
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	109	1	no	—
249	C₂×C₄	—	109	1	no	—
250	C₂×C₄	—	109	4	no	—
251	C₂×C₄	—	109	2	no	—
252	C₂×C₄	—	109	2	no	—
253	C₂×C₄	—	109	1	no	—
254	C₂×C₄	—	436	2	no	—
255	C₂×C₄	—	436	2	no	—
256	C₂×C₄	—	436	4	no	—
257	C₂×C₄	—	436	2	no	—
258	C₂×C₄	—	436	2	no	—
259	C₂×C₄	—	218	2	no	—
260	C₂×C₄	—	218	1	no	—
261	C₂×C₄	—	218	2	no	—
262	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
263	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
264	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	218	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	109	2	no	—
271	C₂²	—	109	2	no	—
272	C₂²	—	109	2	no	—
273	C₂²	—	436	8	no	—
274	C₂²	—	436	1	no	—
275	C₂²	—	436	1	no	—
276	C₂²	—	218	2	no	—
277	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	109	2	no	—
282	C₄	—	109	2	no	—
283	C₄	—	436	1	no	—
284	C₄	—	436	2	no	—
285	C₄	—	218	1	no	—
286	C₄	SD₁₆×D₁₀₉	1	2	yes	—
287	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
288	C₄	#41×D₁₀₉	1	1	yes	characteristic
289	C₂	—	4	2	no	—
290	C₂	—	109	4	no	—
291	C₂	—	436	2	no	—
292	C₂	C₂²⋊Q₈×D₁₀₉	1	1	yes	characteristic
293	C₂	C₂×SD₁₆×D₁₀₉	1	1	yes	characteristic
294	C₂	AGL(1,ℤ/8)×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	#476×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×D₁₀₉	C₂²
hypercenter	#476	D₁₀₉
hypocenter	C₁₀₉	C₂×#476
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×D₁₀₉	C₂	semidirect
3. #20295317	C₂	semidirect
4. #20295316	C₂	semidirect
5. #20295315	C₂	semidirect
6. #20295314	C₂	semidirect
7. #20295313	C₂	semidirect
8. #106×D₁₀₉	C₂	semidirect
9. C₄⋊C₈×D₁₀₉	C₂	semidirect
11. #476×C₁₀₉	C₂	semidirect
16. #16812699	C₂²	semidirect
18. #16812698	C₂²	semidirect
30. #16812631	C₂²	semidirect
31. #16812630	C₂²	semidirect
32. #16812630	C₂²	semidirect
33. C₄⋊#11724784	C₂²	semidirect
34. #16812628	C₂²	semidirect
38. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
40. #106×C₁₀₉	C₂²	semidirect
41. C₄⋊C₈×C₁₀₉	C₂²	semidirect
50. D₄×D₁₀₉	Q₈	semidirect
91. D₄×C₁₀₉	C₂×Q₈	semidirect
106. D₄₃₆	C₂×Q₈	semidirect
146. D₁₀₉	#476	direct (N × H)
148. C₁₀₉	C₂×#476	semidirect
156. #476	D₁₀₉	direct (N × H)
168. C₄⋊Q₈	C₂×D₁₀₉	semidirect
183. #106	C₂×D₁₀₉	semidirect
186. C₄⋊C₈	C₂×D₁₀₉	semidirect
244. D₄	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×D₄×D₁₀₉	2
#20295317	2
#20295316	2
#20295315	2
#20295314	2
#20295313	2
#106×D₁₀₉	2
C₄⋊C₈×D₁₀₉	2
#102×D₁₀₉	2
#476×C₁₀₉	2
C₂×#476	109

Subgroups — #476×D109 #28372779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #476×D₁₀₉ #28372779