Subgroups — #28371839

Order 13952.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812727	C₂	1	1	yes	maximal, characteristic
2	#20295494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812661	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295137	C₂	1	2	yes	maximal, complemented
5	#20295125	C₂	1	2	yes	maximal, complemented
6	#20295113	C₂	1	2	yes	maximal, complemented
7	#20295108	C₂	1	2	yes	maximal, complemented
8	#20295030	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295025	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₀₉	—	2	1	no	—
15	#16812727	C₄	1	8	yes	complemented
16	C₂×#13182018	C₂²	1	2	yes	—
17	C₂×D₈₇₂	C₂²	1	2	yes	—
18	C₂×#13182013	C₂²	1	1	yes	characteristic
19	#16812664	C₂²	1	1	yes	characteristic, complemented
20	#16812664	C₂²	1	1	yes	characteristic, complemented
21	#16812661	—	2	1	no	—
22	#16812661	—	2	1	no	—
23	#16812661	C₂²	1	2	yes	—
24	#16812661	C₂²	1	2	yes	—
25	#16812661	C₂²	1	1	yes	characteristic, complemented
26	#16812661	C₂²	1	1	yes	characteristic, complemented
27	#16812659	—	2	1	no	—
28	#16812659	—	2	1	no	—
29	#16812656	—	2	1	no	—
30	C₂×#13182006	—	2	1	no	—
31	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	C₄²×D₁₀₉	—	2	1	no	—
33	#16812622	C₂²	1	2	yes	complemented
34	#16812619	C₂²	1	2	yes	—
35	#16812619	C₂²	1	2	yes	—
36	#16812619	C₂²	1	2	yes	complemented
37	#16812609	C₂²	1	2	yes	complemented
38	C₈×Dic₁₀₉	C₂²	1	2	yes	complemented
39	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
40	#117×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
42	#107×C₁₀₉	C₂²	1	2	yes	complemented
43	#106×C₁₀₉	C₂²	1	2	yes	complemented
44	C₂³×D₁₀₉	—	2	1	no	—
45	#13182029	—	2	2	no	—
46	#13182029	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
48	C₂×D₄₃₆	—	2	1	no	—
49	C₂×D₄₃₆	—	2	2	no	—
50	C₂×D₄₃₆	C₂×C₄	1	4	yes	complemented
51	C₂×D₄₃₆	C₂³	1	2	yes	—
52	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
54	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
55	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
56	C₂×C₄×D₁₀₉	—	2	1	no	—
57	C₂×C₄×D₁₀₉	—	2	1	no	—
58	C₂×C₄×D₁₀₉	—	2	1	no	—
59	C₂×C₄×D₁₀₉	—	2	2	no	—
60	C₂×C₄×D₁₀₉	—	2	2	no	—
61	C₂×C₄×D₁₀₉	—	4	4	no	—
62	#13182018	C₂×C₄	1	8	yes	complemented
63	D₈₇₂	C₂×C₄	1	8	yes	complemented
64	#13182013	—	2	2	no	—
65	C₂×#11724784	C₂³	1	2	yes	—
66	#13182008	—	2	1	no	—
67	#13182008	—	2	1	no	—
68	#13182006	—	2	2	no	—
69	#13182006	—	2	2	no	—
70	#13182006	—	2	1	no	—
71	#13182006	—	2	1	no	—
72	C₄×Dic₁₀₉	—	2	1	no	—
73	C₄×Dic₁₀₉	—	2	1	no	—
74	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
75	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
76	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	—	2	1	no	—
79	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
81	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
82	C₄⋊C₄×C₁₀₉	—	2	1	no	—
83	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
85	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
87	C₄²×C₁₀₉	—	2	1	no	—
88	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×D₁₀₉	—	2	4	no	—
90	C₂²×D₁₀₉	—	2	1	no	—
91	C₂²×D₁₀₉	—	2	2	no	—
92	C₂²×D₁₀₉	—	4	2	no	—
93	C₂²×D₁₀₉	—	4	2	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	4	2	no	—
98	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	#41	1	2	yes	—
101	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₁₀₉	—	2	4	no	—
103	C₂×Dic₁₀₉	—	2	1	no	—
104	C₂×Dic₁₀₉	—	2	2	no	—
105	C₂×Dic₁₀₉	—	4	1	no	—
106	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
108	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
110	D₄₃₆	—	2	4	no	—
111	D₄₃₆	—	2	2	no	—
112	D₄₃₆	—	2	1	no	—
113	D₄₃₆	C₂²×C₄	1	4	yes	—
114	D₄₃₆	C₂²×C₄	1	2	yes	complemented
115	C₄×D₁₀₉	—	2	4	no	—
116	C₄×D₁₀₉	—	4	2	no	—
117	C₄×D₁₀₉	—	4	4	no	—
118	C₄×D₁₀₉	—	4	2	no	—
119	C₄×D₁₀₉	—	4	4	no	—
120	Dic₂₁₈	—	2	1	no	—
121	Dic₂₁₈	C₂²×C₄	1	2	yes	complemented
122	C₈×C₁₀₉	C₂²×C₄	1	4	yes	complemented
123	#11724784	—	2	2	no	—
124	C₂²×C₁₀₉	—	2	1	no	—
125	C₂²×C₁₀₉	—	2	1	no	—
126	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
127	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₁₀₉	—	2	2	no	—
129	C₂×D₁₀₉	—	2	4	no	—
130	C₂×D₁₀₉	—	4	8	no	—
131	C₂×D₁₀₉	—	4	2	no	—
132	C₂×D₁₀₉	—	4	1	no	—
133	C₂×D₁₀₉	—	4	1	no	—
134	C₂×D₁₀₉	—	4	2	no	—
135	C₄×C₁₀₉	—	4	2	no	—
136	C₄×C₁₀₉	—	4	1	no	—
137	C₄×C₁₀₉	—	4	1	no	—
138	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
139	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
141	Dic₁₀₉	—	2	2	no	—
142	Dic₁₀₉	—	2	1	no	—
143	Dic₁₀₉	—	2	1	no	—
144	Dic₁₀₉	—	2	1	no	—
145	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	complemented
146	C₂×C₁₀₉	—	2	2	no	—
147	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₁₀₉	#575	1	1	yes	characteristic
150	D₁₀₉	—	4	2	no	—
151	D₁₀₉	—	4	4	no	—
152	C₄×AGL(1,ℤ/8)	—	109	1	no	maximal, Sylow
153	C₁₀₉	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
154	C₂×AGL(1,ℤ/8)	—	109	1	no	—
155	C₄×#41	—	109	1	no	—
156	C₂×C₄×D₄	—	109	1	no	—
157	#442	—	109	2	no	—
158	#440	—	109	2	no	—
159	C₄×SD₁₆	—	109	2	no	—
160	C₄×D₈	—	109	2	no	—
161	#428	D₁₀₉	1	1	yes	characteristic, complemented
162	#418	—	109	1	no	—
163	#417	—	109	1	no	—
164	C₄×M₁₆	—	109	1	no	—
165	C₂×#41	—	109	1	no	—
166	C₂²×D₄	—	109	1	no	—
167	C₂³×C₄	—	218	1	no	—
168	AGL(1,ℤ/8)	—	109	8	no	—
169	C₂×SD₁₆	—	109	2	no	—
170	C₂×D₈	—	109	2	no	—
171	C₂×M₁₆	—	109	1	no	—
172	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
173	C₄×Q₈	—	109	1	no	—
174	C₄×Q₈	—	109	1	no	—
175	C₄×D₄	—	109	2	no	—
176	C₄×D₄	—	109	2	no	—
177	C₄×D₄	—	109	1	no	—
178	C₄×D₄	—	109	1	no	—
179	C₄×D₄	—	218	1	no	—
180	C₄×D₄	—	218	1	no	—
181	C₄×D₄	—	218	1	no	—
182	C₄×D₄	—	218	1	no	—
183	#117	—	218	1	no	—
184	#117	C₂×D₁₀₉	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	218	1	no	—
187	C₂×C₄²	—	109	1	no	—
188	C₂×C₄²	—	218	1	no	—
189	#107	C₂×D₁₀₉	1	2	yes	complemented
190	#106	C₂×D₁₀₉	1	2	yes	complemented
191	#103	—	109	2	no	—
192	#102	—	109	2	no	—
193	#102	—	109	2	no	—
194	#102	—	109	2	no	—
195	#97	—	109	2	no	—
196	C₄×C₈	—	109	2	no	—
197	C₂⁴	—	218	1	no	—
198	#41	—	109	4	no	—
199	#41	—	218	2	no	—
200	C₂×Q₈	—	109	1	no	—
201	C₂×D₄	—	109	4	no	—
202	C₂×D₄	—	109	1	no	—
203	C₂×D₄	—	109	2	no	—
204	C₂×D₄	—	218	1	no	—
205	C₂×D₄	—	218	1	no	—
206	C₂×D₄	—	218	2	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	109	1	no	—
209	C₂²×C₄	—	109	1	no	—
210	C₂²×C₄	—	436	4	no	—
211	C₂²×C₄	—	218	1	no	—
212	C₂²×C₄	—	218	1	no	—
213	C₂²×C₄	—	218	2	no	—
214	C₂²×C₄	—	218	2	no	—
215	C₂²×C₄	—	218	1	no	—
216	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
217	SD₁₆	—	109	8	no	—
218	D₈	—	109	8	no	—
219	M₁₆	—	218	2	no	—
220	M₁₆	C₄×D₁₀₉	1	4	yes	complemented
221	C₂×C₈	—	109	2	no	—
222	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	218	1	no	—
225	C₄⋊C₄	—	218	1	no	—
226	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
228	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	218	2	no	—
231	C₂²⋊C₄	—	218	2	no	—
232	C₂²⋊C₄	—	218	1	no	—
233	C₂²⋊C₄	—	218	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	109	1	no	—
236	C₄²	—	109	1	no	—
237	C₄²	—	109	1	no	—
238	C₄²	—	109	1	no	—
239	C₄²	—	218	1	no	—
240	C₄²	—	218	1	no	—
241	C₂³	—	436	2	no	—
242	C₂³	—	436	2	no	—
243	C₂³	—	218	4	no	—
244	C₂³	—	218	1	no	—
245	C₂³	—	218	2	no	—
246	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
247	Q₈	—	109	2	no	—
248	Q₈	—	218	1	no	—
249	D₄	—	109	2	no	—
250	D₄	—	109	4	no	—
251	D₄	—	218	4	no	—
252	D₄	—	218	4	no	—
253	D₄	—	218	2	no	—
254	D₄	—	218	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	109	1	no	—
260	C₂×C₄	—	109	1	no	—
261	C₂×C₄	—	109	1	no	—
262	C₂×C₄	—	109	1	no	—
263	C₂×C₄	—	436	4	no	—
264	C₂×C₄	—	436	2	no	—
265	C₂×C₄	—	436	4	no	—
266	C₂×C₄	—	436	2	no	—
267	C₂×C₄	—	436	1	no	—
268	C₂×C₄	—	218	2	no	—
269	C₂×C₄	—	218	1	no	—
270	C₂×C₄	—	218	4	no	—
271	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
272	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
273	C₂×C₄	#13182030	1	2	yes	—
274	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
275	C₈	—	218	2	no	—
276	C₈	C₂×C₄×D₁₀₉	1	4	yes	complemented
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	436	8	no	—
280	C₂²	—	436	2	no	—
281	C₂²	—	436	1	no	—
282	C₂²	—	436	1	no	—
283	C₂²	—	436	2	no	—
284	C₂²	—	218	2	no	—
285	C₂²	—	218	4	no	—
286	C₂²	#16812661	1	2	yes	—
287	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
288	C₄	—	4	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	109	2	no	—
292	C₄	—	218	2	no	—
293	C₄	—	218	1	no	—
294	C₄	—	218	1	no	—
295	C₄	—	218	1	no	—
296	C₄	#16812661	1	2	yes	—
297	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
298	C₄	C₂×#13182030	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	436	4	no	—
301	C₂	—	436	2	no	—
302	C₂	C₂×#16812661	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
304	C₂	#20295825	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×#16812727	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#428	D₁₀₉
hypocenter	C₁₀₉	C₄×AGL(1,ℤ/8)
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
2. #20295494	C₂	semidirect
3. C₂×#16812661	C₂	semidirect
4. #20295137	C₂	semidirect
5. #20295125	C₂	semidirect
6. #20295113	C₂	semidirect
7. #20295108	C₂	semidirect
8. #20295030	C₂	semidirect
9. #20295025	C₂	semidirect
10. M₁₆×Dic₁₀₉	C₂	semidirect
11. #428×C₁₀₉	C₂	semidirect
15. #16812727	C₄	semidirect
19. #16812664	C₂²	semidirect
20. #16812664	C₂²	semidirect
25. #16812661	C₂²	semidirect
26. #16812661	C₂²	semidirect
33. #16812622	C₂²	semidirect
36. #16812619	C₂²	semidirect
37. #16812609	C₂²	semidirect
38. C₈×Dic₁₀₉	C₂²	semidirect
42. #107×C₁₀₉	C₂²	semidirect
43. #106×C₁₀₉	C₂²	semidirect
46. #13182029	C₂×C₄	semidirect
50. C₂×D₄₃₆	C₂×C₄	semidirect
62. #13182018	C₂×C₄	semidirect
63. D₈₇₂	C₂×C₄	semidirect
80. M₁₆×C₁₀₉	C₂×C₄	semidirect
114. D₄₃₆	C₂²×C₄	semidirect
121. Dic₂₁₈	C₂²×C₄	semidirect
122. C₈×C₁₀₉	C₂²×C₄	semidirect
145. Dic₁₀₉	AGL(1,ℤ/8)	semidirect
153. C₁₀₉	C₄×AGL(1,ℤ/8)	semidirect
161. #428	D₁₀₉	semidirect
189. #107	C₂×D₁₀₉	semidirect
190. #106	C₂×D₁₀₉	semidirect
220. M₁₆	C₄×D₁₀₉	semidirect
276. C₈	C₂×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×#16812727	2
#20295494	2
C₂×#16812661	2
#20295137	2
#20295125	2
#20295113	2
#20295108	2
#20295030	2
#20295025	2
M₁₆×Dic₁₀₉	2
#428×C₁₀₉	2
C₄×AGL(1,ℤ/8)	109