Subgroups — #28345295

Order 13696.

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×#16812137	C₂	1	1	yes	maximal, characteristic
2	#20293107	C₂	1	1	yes	maximal, characteristic, complemented
3	#20293101	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292740	C₂	1	2	yes	maximal, complemented
5	#20292729	C₂	1	2	yes	maximal, complemented
6	C₄×#13133807	C₂	1	2	yes	maximal, complemented
7	C₄×D₈₅₆	C₂	1	2	yes	maximal, complemented
8	#20292669	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292624	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16812031	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133818	C₂²	1	1	yes	characteristic
13	C₂²×D₄₂₈	C₂²	1	1	yes	characteristic
14	#16812137	C₄	1	8	yes	complemented
15	C₂×#13133807	C₂²	1	2	yes	—
16	C₂×D₈₅₆	C₂²	1	2	yes	—
17	C₄×Dic₂₁₄	C₂²	1	2	yes	complemented
18	C₄×D₄₂₈	C₂²	1	2	yes	—
19	C₄×D₄₂₈	C₂²	1	2	yes	—
20	C₄×D₄₂₈	C₂²	1	2	yes	complemented
21	#16812071	—	2	1	no	—
22	#16812071	—	2	1	no	—
23	(C₂²⋊D₁₀₇)⋊C₄	—	2	2	no	—
24	#16812068	C₂²	1	1	yes	characteristic
25	#16812067	—	2	1	no	—
26	C₂×C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₀₇	—	2	1	no	—
28	C₂²⋊C₄×D₁₀₇	—	2	2	no	—
29	#16812043	C₂²	1	2	yes	complemented
30	#16812041	C₂²	1	2	yes	complemented
31	#16812034	C₂²	1	1	yes	characteristic, complemented
32	#16812034	C₂²	1	1	yes	characteristic, complemented
33	#16812031	C₂²	1	2	yes	—
34	#16812031	C₂²	1	2	yes	—
35	#16812031	C₂²	1	1	yes	characteristic, complemented
36	#16812031	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
39	#117×C₁₀₇	C₂²	1	1	yes	characteristic
40	#97×C₁₀₇	C₂²	1	2	yes	complemented
41	C₄×C₈×C₁₀₇	C₂²	1	2	yes	complemented
42	C₂³×D₁₀₇	—	2	1	no	—
43	#13133818	—	2	2	no	—
44	#13133818	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₂₁₄	C₂³	1	1	yes	characteristic
46	C₂×D₄₂₈	—	2	1	no	—
47	C₂×D₄₂₈	—	2	2	no	—
48	C₂×D₄₂₈	C₂×C₄	1	4	yes	complemented
49	C₂×D₄₂₈	C₂³	1	2	yes	—
50	C₂×D₄₂₈	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₀₇	—	2	1	no	—
52	C₂²×Dic₁₀₇	—	2	1	no	—
53	C₂×C₄×D₁₀₇	—	2	1	no	—
54	C₂×C₄×D₁₀₇	—	2	2	no	—
55	C₂×C₄×D₁₀₇	—	2	2	no	—
56	C₂×C₄×D₁₀₇	—	2	2	no	—
57	#13133807	C₂×C₄	1	8	yes	complemented
58	D₈₅₆	C₂×C₄	1	8	yes	complemented
59	C₄⋊Dic₁₀₇	—	2	1	no	—
60	C₄⋊Dic₁₀₇	C₂³	1	1	yes	characteristic
61	C₄⋊Dic₁₀₇	C₂³	1	1	yes	characteristic
62	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
63	Dic₁₀₇⋊C₄	—	2	2	no	—
64	C₂²⋊Dic₁₀₇	—	2	1	no	—
65	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
66	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
67	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
68	C₄×Dic₁₀₇	—	2	2	no	—
69	C₄×Dic₁₀₇	—	2	1	no	—
70	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
71	M₁₆×C₁₀₇	C₂×C₄	1	4	yes	complemented
72	C₂×C₈×C₁₀₇	—	2	2	no	—
73	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
76	C₄⋊C₄×C₁₀₇	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₇	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₀₇	D₄	1	2	yes	—
79	C₄²×C₁₀₇	C₂³	1	2	yes	—
80	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₁₀₇	—	2	1	no	—
82	C₂²×D₁₀₇	—	2	2	no	—
83	C₂²×D₁₀₇	—	2	4	no	—
84	C₂²×D₁₀₇	—	8	1	no	—
85	C₂²×D₁₀₇	—	8	1	no	—
86	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₇	#41	1	2	yes	—
91	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
92	C₂²⋊D₁₀₇	—	2	4	no	—
93	C₂×Dic₁₀₇	—	2	1	no	—
94	C₂×Dic₁₀₇	—	2	1	no	—
95	C₂×Dic₁₀₇	—	2	1	no	—
96	C₂×Dic₁₀₇	—	2	2	no	—
97	C₂×Dic₁₀₇	—	8	1	no	—
98	D₄₂₈	—	2	4	no	—
99	D₄₂₈	—	2	2	no	—
100	D₄₂₈	—	2	1	no	—
101	D₄₂₈	C₂²×C₄	1	2	yes	—
102	D₄₂₈	C₂²×C₄	1	4	yes	—
103	C₄×D₁₀₇	—	2	4	no	—
104	C₄×D₁₀₇	—	8	2	no	—
105	C₄×D₁₀₇	—	8	2	no	—
106	C₄×D₁₀₇	—	8	2	no	—
107	Dic₂₁₄	—	2	1	no	—
108	Dic₂₁₄	C₂²×C₄	1	2	yes	—
109	C₈×C₁₀₇	—	2	1	no	—
110	C₈×C₁₀₇	—	2	1	no	—
111	C₈×C₁₀₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₁₀₇	—	2	1	no	—
113	C₂²×C₁₀₇	—	2	1	no	—
114	C₂²×C₁₀₇	C₄×D₄	1	2	yes	—
115	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₁₀₇	—	2	2	no	—
117	C₂×D₁₀₇	—	2	4	no	—
118	C₂×D₁₀₇	—	4	1	no	—
119	C₂×D₁₀₇	—	4	2	no	—
120	C₂×D₁₀₇	—	4	2	no	—
121	C₂×D₁₀₇	—	4	1	no	—
122	C₂×D₁₀₇	—	8	4	no	—
123	C₄×C₁₀₇	—	2	2	no	—
124	C₄×C₁₀₇	—	2	2	no	—
125	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
126	C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
127	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
128	Dic₁₀₇	—	2	2	no	—
129	Dic₁₀₇	—	4	1	no	—
130	Dic₁₀₇	—	4	2	no	—
131	Dic₁₀₇	—	4	1	no	—
132	C₂×C₁₀₇	—	2	2	no	—
133	C₂×C₁₀₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₁₀₇	#576	1	1	yes	characteristic
135	C₂×C₁₀₇	#577	1	1	yes	characteristic
136	D₁₀₇	—	4	2	no	—
137	D₁₀₇	—	4	4	no	—
138	#2946	—	107	1	no	maximal, Sylow
139	C₁₀₇	#2946	1	1	yes	characteristic, complemented, Sylow
140	C₂×AGL(1,ℤ/8)	—	107	1	no	—
141	#520	—	107	1	no	—
142	#518	—	107	1	no	—
143	#442	—	107	2	no	—
144	#440	—	107	2	no	—
145	C₄×SD₁₆	—	107	2	no	—
146	C₄×D₈	—	107	2	no	—
147	#428	—	107	1	no	—
148	#416	—	107	1	no	—
149	C₂×#102	—	107	1	no	—
150	#405	D₁₀₇	1	1	yes	characteristic, complemented
151	C₂×#41	—	107	1	no	—
152	C₂²×D₄	—	107	1	no	—
153	AGL(1,ℤ/8)	—	107	8	no	—
154	C₂×SD₁₆	—	107	2	no	—
155	C₂×D₈	—	107	2	no	—
156	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
158	C₄×Q₈	—	107	2	no	—
159	C₄×D₄	—	107	2	no	—
160	C₄×D₄	—	107	2	no	—
161	C₄×D₄	—	107	2	no	—
162	C₄×D₄	—	214	1	no	—
163	C₄×D₄	—	214	1	no	—
164	C₄×D₄	—	214	2	no	—
165	#117	—	107	1	no	—
166	#117	—	214	1	no	—
167	#117	C₂×D₁₀₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	107	1	no	—
169	C₂×C₄⋊C₄	—	214	1	no	—
170	C₂×C₂²⋊C₄	—	214	2	no	—
171	#107	—	107	2	no	—
172	#106	—	107	2	no	—
173	#103	—	107	1	no	—
174	#103	—	107	1	no	—
175	#102	—	107	2	no	—
176	#102	—	107	2	no	—
177	#102	—	107	1	no	—
178	#102	—	107	1	no	—
179	#97	C₂×D₁₀₇	1	2	yes	complemented
180	C₄×C₈	C₂×D₁₀₇	1	2	yes	complemented
181	C₂⁴	—	214	1	no	—
182	#41	—	107	4	no	—
183	#41	—	214	2	no	—
184	C₂×Q₈	—	107	1	no	—
185	C₂×D₄	—	107	4	no	—
186	C₂×D₄	—	107	1	no	—
187	C₂×D₄	—	107	2	no	—
188	C₂×D₄	—	214	1	no	—
189	C₂×D₄	—	214	1	no	—
190	C₂×D₄	—	214	2	no	—
191	C₂²×C₄	—	214	1	no	—
192	C₂²×C₄	—	214	2	no	—
193	C₂²×C₄	—	214	2	no	—
194	C₂²×C₄	—	214	2	no	—
195	C₂²×C₄	—	214	1	no	—
196	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
197	SD₁₆	—	107	8	no	—
198	D₈	—	107	8	no	—
199	M₁₆	C₄×D₁₀₇	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
204	C₄⋊C₄	—	107	2	no	—
205	C₄⋊C₄	—	107	1	no	—
206	C₄⋊C₄	—	107	1	no	—
207	C₄⋊C₄	—	214	1	no	—
208	C₄⋊C₄	—	214	2	no	—
209	C₄⋊C₄	D₄₂₈	1	1	yes	characteristic
210	C₄⋊C₄	D₄₂₈	1	1	yes	characteristic
211	C₂²⋊C₄	—	214	1	no	—
212	C₂²⋊C₄	—	214	2	no	—
213	C₂²⋊C₄	—	214	2	no	—
214	C₂²⋊C₄	—	214	2	no	—
215	C₂²⋊C₄	D₄₂₈	1	2	yes	—
216	C₄²	—	214	1	no	—
217	C₄²	—	214	2	no	—
218	C₄²	C₂²×D₁₀₇	1	2	yes	—
219	C₂³	—	856	1	no	—
220	C₂³	—	856	1	no	—
221	C₂³	—	214	4	no	—
222	C₂³	—	214	1	no	—
223	C₂³	—	214	2	no	—
224	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
225	Q₈	—	107	2	no	—
226	Q₈	—	214	1	no	—
227	D₄	—	107	2	no	—
228	D₄	—	107	4	no	—
229	D₄	—	214	4	no	—
230	D₄	—	214	4	no	—
231	D₄	—	214	2	no	—
232	D₄	—	214	1	no	—
233	C₂×C₄	—	856	2	no	—
234	C₂×C₄	—	856	2	no	—
235	C₂×C₄	—	856	2	no	—
236	C₂×C₄	—	856	1	no	—
237	C₂×C₄	—	214	1	no	—
238	C₂×C₄	—	214	1	no	—
239	C₂×C₄	—	214	2	no	—
240	C₂×C₄	—	214	1	no	—
241	C₂×C₄	—	214	4	no	—
242	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
243	C₂×C₄	C₂×D₄₂₈	1	2	yes	—
244	C₂×C₄	C₂×D₄₂₈	1	2	yes	—
245	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
246	C₂×C₄	#13133818	1	2	yes	—
247	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂×C₄×D₁₀₇	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	856	4	no	—
254	C₂²	—	428	1	no	—
255	C₂²	—	428	2	no	—
256	C₂²	—	428	2	no	—
257	C₂²	—	428	1	no	—
258	C₂²	—	214	2	no	—
259	C₂²	—	214	4	no	—
260	C₂²	C₄×D₄₂₈	1	2	yes	—
261	C₂²	C₂²×D₄₂₈	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	428	1	no	—
265	C₄	—	428	2	no	—
266	C₄	—	428	1	no	—
267	C₄	—	214	2	no	—
268	C₄	C₄×D₄₂₈	1	2	yes	—
269	C₄	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
270	C₄	C₂×#13133818	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	428	4	no	—
273	C₂	—	428	2	no	—
274	C₂	C₂×C₄×D₄₂₈	1	1	yes	characteristic
275	C₂	#20293432	1	1	yes	characteristic
276	C₂	#20293439	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₂₈
commutator	C₄×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	#405×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	C₂×#16812137	C₂
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#405	D₁₀₇
hypocenter	C₁₀₇	#2946
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20293107	C₂	semidirect
3. #20293101	C₂	semidirect
4. #20292740	C₂	semidirect
5. #20292729	C₂	semidirect
6. C₄×#13133807	C₂	semidirect
7. C₄×D₈₅₆	C₂	semidirect
8. #20292669	C₂	semidirect
9. #20292624	C₂	semidirect
10. C₂×#16812031	C₂	semidirect
11. #405×C₁₀₇	C₂	semidirect
14. #16812137	C₄	semidirect
17. C₄×Dic₂₁₄	C₂²	semidirect
20. C₄×D₄₂₈	C₂²	semidirect
29. #16812043	C₂²	semidirect
30. #16812041	C₂²	semidirect
31. #16812034	C₂²	semidirect
32. #16812034	C₂²	semidirect
35. #16812031	C₂²	semidirect
36. #16812031	C₂²	semidirect
40. #97×C₁₀₇	C₂²	semidirect
41. C₄×C₈×C₁₀₇	C₂²	semidirect
44. #13133818	C₂×C₄	semidirect
48. C₂×D₄₂₈	C₂×C₄	semidirect
57. #13133807	C₂×C₄	semidirect
58. D₈₅₆	C₂×C₄	semidirect
71. M₁₆×C₁₀₇	C₂×C₄	semidirect
139. C₁₀₇	#2946	semidirect
150. #405	D₁₀₇	semidirect
179. #97	C₂×D₁₀₇	semidirect
180. C₄×C₈	C₂×D₁₀₇	semidirect
199. M₁₆	C₄×D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812137	2
#20293107	2
#20293101	2
#20292740	2
#20292729	2
C₄×#13133807	2
C₄×D₈₅₆	2
#20292669	2
#20292624	2
C₂×#16812031	2
#405×C₁₀₇	2
#2946	107