Subgroups — #28371867

Order 13952.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812724	C₂	1	1	yes	maximal, characteristic
2	#20295509	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295500	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295135	C₂	1	2	yes	maximal, complemented
5	#20295129	C₂	1	2	yes	maximal, complemented
6	#20295111	C₂	1	2	yes	maximal, complemented
7	#20295106	C₂	1	2	yes	maximal, complemented
8	#20295069	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812621	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294981	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
13	#16812724	C₄	1	8	yes	complemented
14	C₂×#13182016	C₂²	1	2	yes	—
15	C₂×#13182014	C₂²	1	2	yes	—
16	C₂×#13182013	C₂²	1	1	yes	characteristic
17	Q₈×Dic₁₀₉	C₂²	1	2	yes	complemented
18	C₄×D₄₃₆	—	2	1	no	—
19	C₄×D₄₃₆	—	2	1	no	—
20	D₄×Dic₁₀₉	—	2	2	no	—
21	D₄×Dic₁₀₉	C₂²	1	2	yes	complemented
22	#16812661	C₂²	1	2	yes	—
23	#16812661	C₂²	1	2	yes	—
24	#16812658	—	2	1	no	—
25	#16812658	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
27	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×D₁₀₉	—	2	2	no	—
29	#16812632	C₂²	1	2	yes	complemented
30	#16812630	C₂²	1	2	yes	complemented
31	#16812621	C₂²	1	2	yes	—
32	#16812621	C₂²	1	2	yes	—
33	#16812609	C₂²	1	2	yes	complemented
34	C₈×Dic₁₀₉	C₂²	1	2	yes	complemented
35	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
37	#117×C₁₀₉	C₂²	1	1	yes	characteristic
38	#103×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
39	#103×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
40	#102×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
41	#102×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₁₀₉	—	2	1	no	—
43	C₂×D₄₃₆	—	2	1	no	—
44	C₂×D₄₃₆	—	2	2	no	—
45	C₂×D₄₃₆	C₂×C₄	1	4	yes	complemented
46	C₂×D₄₃₆	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₉	—	2	1	no	—
48	C₂²×Dic₁₀₉	—	2	2	no	—
49	C₂×C₄×D₁₀₉	—	2	2	no	—
50	C₂×C₄×D₁₀₉	—	2	2	no	—
51	#13182016	C₂×C₄	1	8	yes	complemented
52	#13182014	C₂×C₄	1	8	yes	complemented
53	#13182013	C₂×C₄	1	4	yes	complemented
54	C₂×#11724784	C₂³	1	2	yes	—
55	C₄⋊Dic₁₀₉	—	2	1	no	—
56	C₄⋊Dic₁₀₉	—	2	2	no	—
57	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
58	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
59	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
60	C₂²⋊Dic₁₀₉	—	2	2	no	—
61	C₂²⋊Dic₁₀₉	D₄	1	2	yes	—
62	#13182006	—	2	2	no	—
63	#13182006	—	2	2	no	—
64	C₄×Dic₁₀₉	—	2	2	no	—
65	C₄×Dic₁₀₉	C₂³	1	2	yes	—
66	#41×C₁₀₉	—	2	2	no	—
67	#41×C₁₀₉	C₂×C₄	1	4	yes	complemented
68	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₁₀₉	—	2	1	no	—
70	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₉	—	2	1	no	—
72	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₀₉	—	2	2	no	—
74	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
79	C₄²×C₁₀₉	—	2	1	no	—
80	C₂³×C₁₀₉	—	2	1	no	—
81	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₁₀₉	—	2	2	no	—
83	C₂²×D₁₀₉	—	2	4	no	—
84	C₂²×D₁₀₉	—	8	1	no	—
85	C₂²×D₁₀₉	—	8	1	no	—
86	Q₈×C₁₀₉	—	2	1	no	—
87	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
88	D₄×C₁₀₉	—	2	4	no	—
89	D₄×C₁₀₉	—	2	1	no	—
90	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	2	4	no	—
93	C₂×C₄×C₁₀₉	—	2	1	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	#41	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₁₀₉	—	2	2	no	—
100	C₂×Dic₁₀₉	—	8	2	no	—
101	C₂×Dic₁₀₉	—	8	1	no	—
102	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
103	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
104	D₄₃₆	—	2	4	no	—
105	D₄₃₆	—	2	2	no	—
106	D₄₃₆	C₂²×C₄	1	4	yes	—
107	C₄×D₁₀₉	—	8	2	no	—
108	C₄×D₁₀₉	—	8	2	no	—
109	C₈×C₁₀₉	—	2	1	no	—
110	C₈×C₁₀₉	—	2	1	no	—
111	#11724784	C₂²×C₄	1	4	yes	—
112	C₂²×C₁₀₉	—	2	1	no	—
113	C₂²×C₁₀₉	—	2	1	no	—
114	C₂²×C₁₀₉	—	2	2	no	—
115	C₂²×C₁₀₉	—	4	1	no	—
116	C₂²×C₁₀₉	—	4	1	no	—
117	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
118	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₁₀₉	—	2	4	no	—
120	C₂×D₁₀₉	—	4	2	no	—
121	C₂×D₁₀₉	—	4	2	no	—
122	C₂×D₁₀₉	—	8	4	no	—
123	C₄×C₁₀₉	—	2	2	no	—
124	C₄×C₁₀₉	—	4	1	no	—
125	C₄×C₁₀₉	—	4	1	no	—
126	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
127	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
129	Dic₁₀₉	—	2	2	no	—
130	Dic₁₀₉	—	2	2	no	—
131	Dic₁₀₉	—	4	2	no	—
132	C₂×C₁₀₉	—	2	2	no	—
133	C₂×C₁₀₉	—	4	2	no	—
134	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₁₀₉	#576	1	1	yes	characteristic
136	C₂×C₁₀₉	#577	1	1	yes	characteristic
137	D₁₀₉	—	4	4	no	—
138	#2946	—	109	1	no	maximal, Sylow
139	C₁₀₉	#2946	1	1	yes	characteristic, complemented, Sylow
140	C₂×AGL(1,ℤ/8)	—	109	1	no	—
141	#520	—	109	1	no	—
142	#518	—	109	1	no	—
143	#442	—	109	2	no	—
144	#440	—	109	2	no	—
145	C₄×SD₁₆	—	109	2	no	—
146	C₄×D₈	—	109	2	no	—
147	#428	—	109	1	no	—
148	#416	D₁₀₉	1	1	yes	characteristic, complemented
149	C₂×#102	—	109	1	no	—
150	#405	—	109	1	no	—
151	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
152	C₂²×D₄	—	109	1	no	—
153	AGL(1,ℤ/8)	—	109	8	no	—
154	C₂×SD₁₆	—	109	2	no	—
155	C₂×D₈	—	109	2	no	—
156	C₂×M₁₆	—	109	1	no	—
157	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
158	C₄×Q₈	—	109	2	no	—
159	C₄×D₄	—	109	2	no	—
160	C₄×D₄	—	109	2	no	—
161	C₄×D₄	—	109	2	no	—
162	C₄×D₄	—	218	1	no	—
163	C₄×D₄	—	218	1	no	—
164	C₄×D₄	—	218	2	no	—
165	#117	—	109	1	no	—
166	#117	—	218	1	no	—
167	#117	C₂×D₁₀₉	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	109	1	no	—
169	C₂×C₄⋊C₄	—	218	1	no	—
170	C₂×C₂²⋊C₄	—	218	2	no	—
171	#107	—	109	2	no	—
172	#106	—	109	2	no	—
173	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
174	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
175	#102	—	109	2	no	—
176	#102	—	109	2	no	—
177	#102	C₂×D₁₀₉	1	1	yes	characteristic, complemented
178	#102	C₂×D₁₀₉	1	1	yes	characteristic, complemented
179	#97	—	109	2	no	—
180	C₄×C₈	—	109	2	no	—
181	C₂⁴	—	218	1	no	—
182	#41	—	2	2	no	—
183	#41	C₄×D₁₀₉	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	109	4	no	—
187	C₂×D₄	—	109	2	no	—
188	C₂×D₄	—	218	1	no	—
189	C₂×D₄	—	218	2	no	—
190	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	218	2	no	—
193	C₂²×C₄	—	218	2	no	—
194	C₂²×C₄	—	218	2	no	—
195	C₂²×C₄	—	218	1	no	—
196	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
197	SD₁₆	—	109	8	no	—
198	D₈	—	109	8	no	—
199	M₁₆	—	109	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	109	2	no	—
202	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
204	C₄⋊C₄	—	109	1	no	—
205	C₄⋊C₄	—	109	1	no	—
206	C₄⋊C₄	—	109	2	no	—
207	C₄⋊C₄	—	218	2	no	—
208	C₄⋊C₄	—	218	1	no	—
209	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	109	2	no	—
213	C₂²⋊C₄	—	218	2	no	—
214	C₂²⋊C₄	—	218	2	no	—
215	C₂²⋊C₄	—	218	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	109	2	no	—
218	C₄²	—	218	2	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	872	1	no	—
221	C₂³	—	872	1	no	—
222	C₂³	—	218	4	no	—
223	C₂³	—	218	2	no	—
224	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	1	no	—
229	D₄	—	109	4	no	—
230	D₄	—	218	4	no	—
231	D₄	—	218	2	no	—
232	D₄	C₂×C₄×D₁₀₉	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	872	2	no	—
238	C₂×C₄	—	872	2	no	—
239	C₂×C₄	—	872	1	no	—
240	C₂×C₄	—	872	2	no	—
241	C₂×C₄	—	109	2	no	—
242	C₂×C₄	—	109	2	no	—
243	C₂×C₄	—	218	2	no	—
244	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
245	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
246	C₂×C₄	#13182028	1	2	yes	—
247	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	109	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	872	4	no	—
257	C₂²	—	436	2	no	—
258	C₂²	—	436	2	no	—
259	C₂²	—	218	4	no	—
260	C₂²	#16812659	1	2	yes	—
261	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	436	2	no	—
266	C₄	—	218	2	no	—
267	C₄	—	218	2	no	—
268	C₄	#16812659	1	2	yes	—
269	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
270	C₄	C₂×#13182028	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	2	no	—
273	C₂	—	436	4	no	—
274	C₂	C₂×#16812659	1	1	yes	characteristic
275	C₂	#20295828	1	1	yes	characteristic
276	C₂	#20295833	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#416×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812724	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#416	D₁₀₉
hypocenter	C₁₀₉	#2946
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #20295509	C₂	semidirect
3. #20295500	C₂	semidirect
4. #20295135	C₂	semidirect
5. #20295129	C₂	semidirect
6. #20295111	C₂	semidirect
7. #20295106	C₂	semidirect
8. #20295069	C₂	semidirect
9. C₂×#16812621	C₂	semidirect
10. #20294981	C₂	semidirect
11. #416×C₁₀₉	C₂	semidirect
13. #16812724	C₄	semidirect
17. Q₈×Dic₁₀₉	C₂²	semidirect
21. D₄×Dic₁₀₉	C₂²	semidirect
29. #16812632	C₂²	semidirect
30. #16812630	C₂²	semidirect
33. #16812609	C₂²	semidirect
34. C₈×Dic₁₀₉	C₂²	semidirect
38. #103×C₁₀₉	C₂²	semidirect
39. #103×C₁₀₉	C₂²	semidirect
40. #102×C₁₀₉	C₂²	semidirect
41. #102×C₁₀₉	C₂²	semidirect
45. C₂×D₄₃₆	C₂×C₄	semidirect
51. #13182016	C₂×C₄	semidirect
52. #13182014	C₂×C₄	semidirect
53. #13182013	C₂×C₄	semidirect
67. #41×C₁₀₉	C₂×C₄	semidirect
139. C₁₀₉	#2946	semidirect
148. #416	D₁₀₉	semidirect
173. #103	C₂×D₁₀₉	semidirect
174. #103	C₂×D₁₀₉	semidirect
177. #102	C₂×D₁₀₉	semidirect
178. #102	C₂×D₁₀₉	semidirect
183. #41	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×#16812724	2
#20295509	2
#20295500	2
#20295135	2
#20295129	2
#20295111	2
#20295106	2
#20295069	2
C₂×#16812621	2
#20294981	2
#416×C₁₀₉	2
#2946	109