Subgroups — #28345362

Order 13696.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293425	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812115	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292747	C₂	1	4	yes	maximal, complemented
4	#20292742	C₂	1	4	yes	maximal, complemented
5	#20292675	C₂	1	1	yes	maximal, characteristic, complemented
6	#20292644	C₂	1	1	yes	maximal, characteristic, complemented
7	#20292643	C₂	1	1	yes	maximal, characteristic, complemented
8	#20292582	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133818	C₂²	1	1	yes	characteristic
11	#16812140	C₄	1	2	yes	complemented
12	#16812139	C₄	1	2	yes	complemented
13	#16812135	C₄	1	2	yes	complemented
14	#16812134	C₄	1	2	yes	complemented
15	#16812129	C₂²	1	4	yes	—
16	#16812115	—	2	1	no	—
17	#16812115	—	2	1	no	—
18	#16812115	C₂²	1	2	yes	complemented
19	#16812115	C₂²	1	2	yes	complemented
20	C₂×#13133802	C₂²	1	1	yes	characteristic
21	C₂×#13133802	C₂²	1	1	yes	characteristic
22	C₂×#13133801	—	2	1	no	—
23	C₂×C₈×D₁₀₇	—	2	1	no	—
24	#16812068	C₂²	1	1	yes	characteristic
25	#16812044	C₂²	1	4	yes	complemented
26	#16812036	C₂²	1	4	yes	complemented
27	#16812035	C₂²	1	4	yes	complemented
28	#16812019	C₂²	1	2	yes	complemented
29	C₈×Dic₁₀₇	C₂²	1	2	yes	complemented
30	C₂×#41×C₁₀₇	C₂²	1	1	yes	characteristic
31	#131×C₁₀₇	—	2	1	no	—
32	#131×C₁₀₇	—	2	1	no	—
33	#131×C₁₀₇	C₂²	1	2	yes	complemented
34	#131×C₁₀₇	C₂²	1	2	yes	complemented
35	C₂×M₁₆×C₁₀₇	—	2	1	no	—
36	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₀₇	—	2	1	no	—
38	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
39	#13133818	—	2	1	no	—
40	#13133818	—	2	1	no	—
41	#13133818	C₂×C₄	1	2	yes	—
42	#13133818	C₂³	1	2	yes	—
43	C₂×Dic₂₁₄	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄₂₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₀₇	—	2	1	no	—
46	C₂×C₄×D₁₀₇	—	2	1	no	—
47	#13133808	C₂×C₄	1	4	yes	—
48	#13133806	C₂×C₄	1	4	yes	—
49	#13133805	C₂×C₄	1	4	yes	—
50	#13133803	C₂×C₄	1	4	yes	—
51	#13133802	—	2	1	no	—
52	#13133802	C₂×C₄	1	2	yes	—
53	#13133802	C₂×C₄	1	2	yes	—
54	#13133802	C₂³	1	2	yes	—
55	#13133801	—	2	2	no	—
56	#13133801	—	2	2	no	—
57	C₂×C₁₀₇⋊C₈	—	2	1	no	—
58	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
59	C₈×D₁₀₇	—	2	2	no	—
60	C₈×D₁₀₇	—	2	2	no	—
61	C₄⋊Dic₁₀₇	—	2	1	no	—
62	C₂²⋊Dic₁₀₇	—	2	1	no	—
63	C₄×Dic₁₀₇	C₂³	1	2	yes	—
64	#41×C₁₀₇	—	2	1	no	—
65	#41×C₁₀₇	—	2	1	no	—
66	#41×C₁₀₇	C₂×C₄	1	2	yes	—
67	#41×C₁₀₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
69	C₂×D₄×C₁₀₇	—	2	1	no	—
70	C₂×D₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₇	—	2	1	no	—
72	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
73	M₁₆×C₁₀₇	—	2	1	no	—
74	M₁₆×C₁₀₇	—	2	2	no	—
75	M₁₆×C₁₀₇	—	2	2	no	—
76	M₁₆×C₁₀₇	C₂³	1	2	yes	—
77	C₂×C₈×C₁₀₇	—	2	1	no	—
78	C₂×C₈×C₁₀₇	—	2	2	no	—
79	C₂×C₈×C₁₀₇	—	2	2	no	—
80	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
85	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
86	C₂³×C₁₀₇	—	2	1	no	—
87	C₂³×C₁₀₇	#41	1	1	yes	characteristic
88	C₂²×D₁₀₇	—	2	1	no	—
89	Q₈×C₁₀₇	—	2	1	no	—
90	Q₈×C₁₀₇	C₂²×C₄	1	2	yes	—
91	D₄×C₁₀₇	—	2	2	no	—
92	D₄×C₁₀₇	—	2	2	no	—
93	D₄×C₁₀₇	—	2	1	no	—
94	D₄×C₁₀₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₀₇	—	2	1	no	—
96	C₂×C₄×C₁₀₇	—	2	2	no	—
97	C₂×C₄×C₁₀₇	—	2	2	no	—
98	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
103	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₁₀₇	—	2	2	no	—
105	C₂²⋊D₁₀₇	—	2	2	no	—
106	C₂×Dic₁₀₇	—	2	1	no	—
107	C₂×Dic₁₀₇	—	2	2	no	—
108	D₄₂₈	—	2	1	no	—
109	D₄₂₈	C₂²×C₄	1	2	yes	—
110	C₄×D₁₀₇	—	2	2	no	—
111	C₄×D₁₀₇	—	2	2	no	—
112	Dic₂₁₄	—	2	1	no	—
113	Dic₂₁₄	C₂²×C₄	1	2	yes	—
114	C₈×C₁₀₇	—	2	2	no	—
115	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
116	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
117	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
118	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
119	C₁₀₇⋊C₈	—	2	2	no	—
120	C₁₀₇⋊C₈	C₂²×C₄	1	4	yes	—
121	C₂²×C₁₀₇	—	2	2	no	—
122	C₂²×C₁₀₇	—	4	1	no	—
123	C₂²×C₁₀₇	—	4	2	no	—
124	C₂²×C₁₀₇	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₀₇	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₁₀₇	C₂×#41	1	1	yes	characteristic
127	C₂×D₁₀₇	—	2	2	no	—
128	C₂×D₁₀₇	—	4	2	no	—
129	C₄×C₁₀₇	—	2	2	no	—
130	C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
131	C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
134	Dic₁₀₇	—	2	2	no	—
135	Dic₁₀₇	—	4	2	no	—
136	C₂×C₁₀₇	—	2	1	no	—
137	C₂×C₁₀₇	—	2	1	no	—
138	C₂×C₁₀₇	—	2	1	no	—
139	C₂×C₁₀₇	—	4	2	no	—
140	C₂×C₁₀₇	C₂×C₄×D₄	1	1	yes	characteristic
141	D₁₀₇	—	4	2	no	—
142	#2955	—	107	1	no	maximal, Sylow
143	C₁₀₇	#2955	1	1	yes	characteristic, complemented, Sylow
144	#575	—	107	1	no	—
145	C₂×#131	—	107	1	no	—
146	C₂×#131	D₁₀₇	1	1	yes	characteristic, complemented
147	#444	—	107	4	no	—
148	#443	—	107	4	no	—
149	#430	—	107	1	no	—
150	#421	—	107	1	no	—
151	#421	—	107	1	no	—
152	#405	—	107	1	no	—
153	C₂×#41	—	107	1	no	—
154	C₂×#41	C₂×D₁₀₇	1	1	yes	characteristic
155	#137	—	107	2	no	—
156	#137	—	107	2	no	—
157	AGL(1,ℤ/8)	—	107	2	no	—
158	AGL(1,ℤ/8)	—	107	2	no	—
159	#135	—	107	4	no	—
160	#131	—	2	1	no	—
161	#131	—	2	1	no	—
162	#131	—	107	2	no	—
163	#131	—	107	2	no	—
164	#131	—	214	1	no	—
165	#131	—	214	1	no	—
166	#131	C₂×D₁₀₇	1	2	yes	complemented
167	#131	C₂×D₁₀₇	1	2	yes	complemented
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	107	1	no	—
170	C₂×M₁₆	—	107	1	no	—
171	C₂×M₁₆	—	214	1	no	—
172	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	214	1	no	—
175	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
176	#117	—	107	1	no	—
177	#108	—	107	4	no	—
178	#104	—	107	4	no	—
179	#104	—	107	4	no	—
180	#97	—	107	2	no	—
181	C₄×C₈	—	107	2	no	—
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	107	2	no	—
185	#41	—	107	2	no	—
186	#41	—	214	1	no	—
187	#41	—	214	1	no	—
188	#41	C₄×D₁₀₇	1	2	yes	—
189	#41	C₂²×D₁₀₇	1	2	yes	—
190	C₂×Q₈	—	107	1	no	—
191	C₂×Q₈	C₄×D₁₀₇	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	107	1	no	—
194	C₂×D₄	—	214	1	no	—
195	C₂×D₄	C₄×D₁₀₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	214	1	no	—
198	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
199	Q₁₆	—	107	4	no	—
200	SD₁₆	—	107	4	no	—
201	SD₁₆	—	107	4	no	—
202	D₈	—	107	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	107	2	no	—
207	M₁₆	—	107	2	no	—
208	M₁₆	—	107	2	no	—
209	M₁₆	—	214	1	no	—
210	M₁₆	—	214	2	no	—
211	M₁₆	—	214	2	no	—
212	M₁₆	C₂²×D₁₀₇	1	2	yes	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	107	2	no	—
217	C₂×C₈	—	214	1	no	—
218	C₂×C₈	—	214	2	no	—
219	C₂×C₈	—	214	2	no	—
220	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
222	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
226	C₄⋊C₄	—	214	1	no	—
227	C₂²⋊C₄	—	214	1	no	—
228	C₄²	—	107	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	214	1	no	—
231	C₂³	#13133818	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	107	2	no	—
234	Q₈	—	214	1	no	—
235	Q₈	C₂×C₄×D₁₀₇	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	107	2	no	—
240	D₄	—	214	2	no	—
241	D₄	—	214	2	no	—
242	D₄	—	214	1	no	—
243	D₄	C₂×C₄×D₁₀₇	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	214	2	no	—
248	C₂×C₄	—	214	1	no	—
249	C₂×C₄	—	214	2	no	—
250	C₂×C₄	—	214	2	no	—
251	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
255	C₂×C₄	#13133818	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	107	4	no	—
259	C₈	—	214	2	no	—
260	C₈	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
261	C₈	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
262	C₈	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
263	C₈	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	428	2	no	—
268	C₂²	—	214	2	no	—
269	C₂²	C₄×C₂²⋊D₁₀₇	1	1	yes	characteristic
270	C₂²	C₄×C₂²⋊D₁₀₇	1	1	yes	characteristic
271	C₂²	C₂×#13133818	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	428	2	no	—
274	C₄	—	214	2	no	—
275	C₄	C₄×C₂²⋊D₁₀₇	1	1	yes	characteristic
276	C₄	C₄×C₂²⋊D₁₀₇	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	428	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₁₀₇
commutator	C₄×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₂×#131×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂×C₄×D₄
prime radical	#20293425	C₂
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₂×#131	D₁₀₇
hypocenter	C₁₀₇	#2955
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#16812115	C₂	semidirect
3. #20292747	C₂	semidirect
4. #20292742	C₂	semidirect
5. #20292675	C₂	semidirect
6. #20292644	C₂	semidirect
7. #20292643	C₂	semidirect
8. #20292582	C₂	semidirect
9. C₂×#131×C₁₀₇	C₂	semidirect
11. #16812140	C₄	semidirect
12. #16812139	C₄	semidirect
13. #16812135	C₄	semidirect
14. #16812134	C₄	semidirect
18. #16812115	C₂²	semidirect
19. #16812115	C₂²	semidirect
25. #16812044	C₂²	semidirect
26. #16812036	C₂²	semidirect
27. #16812035	C₂²	semidirect
28. #16812019	C₂²	semidirect
29. C₈×Dic₁₀₇	C₂²	semidirect
33. #131×C₁₀₇	C₂²	semidirect
34. #131×C₁₀₇	C₂²	semidirect
143. C₁₀₇	#2955	semidirect
146. C₂×#131	D₁₀₇	semidirect
166. #131	C₂×D₁₀₇	semidirect
167. #131	C₂×D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293425	2
C₂×#16812115	2
#20292747	2
#20292742	2
#20292675	2
#20292644	2
#20292643	2
#20292582	2
C₂×#131×C₁₀₇	2
#2955	107