Subgroups — #28348148

Order 13696.

526 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₈₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20293273	C₂	1	2	yes	maximal, complemented
4	#20293011	C₂	1	1	yes	maximal, characteristic, complemented
5	#20292868	C₂	1	1	yes	maximal, characteristic, complemented
6	#20292863	C₂	1	1	yes	maximal, characteristic, complemented
7	#20292843	C₂	1	2	yes	maximal, complemented
8	#20292780	C₂	1	2	yes	maximal, complemented
9	#20292774	C₂	1	2	yes	maximal, complemented
10	C₄×D₈₅₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133818	C₂²	1	2	yes	complemented
13	C₂²×D₄₂₈	C₂²	1	2	yes	—
14	#16812131	—	2	4	no	—
15	C₂×Dic₄₂₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×#13133807	C₂²	1	2	yes	complemented
17	C₂×#13133807	C₂²	1	2	yes	complemented
18	C₂×D₈₅₆	—	2	4	no	—
19	C₂×D₈₅₆	C₂²	1	1	yes	characteristic
20	C₂×D₈₅₆	C₂²	1	2	yes	—
21	C₂×D₈₅₆	C₂²	1	1	yes	characteristic
22	#16812097	C₂²	1	2	yes	complemented
23	#16812094	—	2	2	no	—
24	#16812088	C₂²	1	2	yes	—
25	#16812083	C₂²	1	2	yes	complemented
26	#16812082	—	2	2	no	—
27	#16812077	—	2	2	no	—
28	C₄×D₄₂₈	C₂²	1	2	yes	complemented
29	C₂²⋊C₄×D₁₀₇	—	2	2	no	—
30	#16812043	C₂²	1	1	yes	characteristic, complemented
31	#16812034	C₂²	1	2	yes	complemented
32	#16812031	C₂²	1	2	yes	complemented
33	#16812031	C₂²	1	2	yes	complemented
34	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
36	C₄×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
38	C₂²⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₁₀₇	—	2	2	no	—
42	#13133818	—	2	4	no	—
43	#13133818	—	4	2	no	—
44	C₂×Dic₂₁₄	C₂³	1	2	yes	—
45	C₂×D₄₂₈	—	2	4	no	—
46	C₂×D₄₂₈	—	2	2	no	—
47	C₂×D₄₂₈	—	4	2	no	—
48	C₂×D₄₂₈	C₂³	1	2	yes	—
49	C₂×D₄₂₈	C₂³	1	2	yes	—
50	C₂×C₂²⋊D₁₀₇	—	2	2	no	—
51	C₂×C₂²⋊D₁₀₇	—	2	2	no	—
52	C₂×C₄×D₁₀₇	—	2	2	no	—
53	C₂×C₄×D₁₀₇	—	2	2	no	—
54	Dic₄₂₈	—	2	2	no	—
55	#13133807	—	2	4	no	—
56	#13133807	—	4	2	no	—
57	D₈₅₆	—	2	4	no	—
58	D₈₅₆	—	2	2	no	—
59	D₈₅₆	D₄	1	4	yes	complemented
60	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
61	Dic₁₀₇⋊C₄	—	2	2	no	—
62	C₂²⋊Dic₁₀₇	—	2	2	no	—
63	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
64	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
65	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
66	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
67	C₂×D₄×C₁₀₇	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₀₇	—	2	2	no	—
71	C₂×C₈×C₁₀₇	—	2	2	no	—
72	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₇	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₀₇	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₀₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
80	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₁₀₇	—	2	2	no	—
83	C₂²×D₁₀₇	—	2	2	no	—
84	C₂²×D₁₀₇	—	4	4	no	—
85	C₂²×D₁₀₇	—	4	4	no	—
86	C₂²×D₁₀₇	—	8	2	no	—
87	D₄×C₁₀₇	—	2	2	no	—
88	C₂×C₄×C₁₀₇	—	2	1	no	—
89	C₂×C₄×C₁₀₇	—	2	1	no	—
90	C₂×C₄×C₁₀₇	—	2	1	no	—
91	C₂×C₄×C₁₀₇	—	2	1	no	—
92	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₀₇	—	4	4	no	—
98	C₂²⋊D₁₀₇	—	8	2	no	—
99	C₂×Dic₁₀₇	—	2	2	no	—
100	C₂×Dic₁₀₇	—	2	2	no	—
101	D₄₂₈	—	2	2	no	—
102	D₄₂₈	—	2	2	no	—
103	D₄₂₈	—	2	2	no	—
104	D₄₂₈	—	4	4	no	—
105	D₄₂₈	C₂×D₄	1	4	yes	—
106	C₄×D₁₀₇	—	4	4	no	—
107	C₄×D₁₀₇	—	8	2	no	—
108	Dic₂₁₄	—	2	2	no	—
109	Dic₂₁₄	—	2	2	no	—
110	C₈×C₁₀₇	—	2	1	no	—
111	C₈×C₁₀₇	—	2	1	no	—
112	C₈×C₁₀₇	—	2	1	no	—
113	C₈×C₁₀₇	C₂×D₄	1	2	yes	—
114	C₂²×C₁₀₇	—	2	1	no	—
115	C₂²×C₁₀₇	—	2	1	no	—
116	C₂²×C₁₀₇	—	2	1	no	—
117	C₂²×C₁₀₇	—	2	1	no	—
118	C₂²×C₁₀₇	—	2	1	no	—
119	C₂²×C₁₀₇	#135	1	2	yes	complemented
120	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₀₇	—	2	4	no	—
122	C₂×D₁₀₇	—	4	2	no	—
123	C₂×D₁₀₇	—	4	2	no	—
124	C₂×D₁₀₇	—	4	2	no	—
125	C₂×D₁₀₇	—	4	2	no	—
126	C₂×D₁₀₇	—	4	2	no	—
127	C₂×D₁₀₇	—	8	8	no	—
128	C₄×C₁₀₇	—	2	1	no	—
129	C₄×C₁₀₇	—	2	1	no	—
130	C₄×C₁₀₇	—	2	1	no	—
131	C₄×C₁₀₇	—	2	1	no	—
132	C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
134	Dic₁₀₇	—	4	2	no	—
135	Dic₁₀₇	—	4	2	no	—
136	C₂×C₁₀₇	—	2	2	no	—
137	C₂×C₁₀₇	—	4	1	no	—
138	C₂×C₁₀₇	D₄²	1	1	yes	characteristic
139	C₂×C₁₀₇	C₂×#135	1	1	yes	characteristic
140	C₂×C₁₀₇	#576	1	1	yes	characteristic
141	D₁₀₇	—	4	4	no	—
142	D₁₀₇	—	8	2	no	—
143	#3280	—	107	1	no	maximal, Sylow
144	C₁₀₇	#3280	1	1	yes	characteristic, complemented, Sylow
145	C₂×#135	—	107	1	no	—
146	C₂²×D₈	—	107	1	no	—
147	#546	—	107	2	no	—
148	#495	—	107	1	no	—
149	#467	—	107	1	no	—
150	#466	—	107	1	no	—
151	#463	—	107	2	no	—
152	#450	—	107	2	no	—
153	#449	—	107	2	no	—
154	C₄×D₈	—	107	1	no	—
155	C₈×D₄	D₁₀₇	1	1	yes	characteristic, complemented
156	C₂×#41	—	107	2	no	—
157	C₂²×D₄	—	107	2	no	—
158	#135	—	214	4	no	—
159	C₂×Q₁₆	—	107	1	no	—
160	C₂×SD₁₆	—	107	2	no	—
161	C₂×SD₁₆	—	107	2	no	—
162	C₂×D₈	—	107	1	no	—
163	C₂×D₈	—	107	2	no	—
164	C₂×D₈	—	107	1	no	—
165	C₂×D₈	—	214	4	no	—
166	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
168	#124	—	107	2	no	—
169	#123	—	214	2	no	—
170	C₂²⋊Q₈	—	107	2	no	—
171	#121	—	107	2	no	—
172	#121	—	214	2	no	—
173	#120	—	214	2	no	—
174	C₄×D₄	—	107	2	no	—
175	C₄×D₄	C₂×D₁₀₇	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	—	214	2	no	—
177	#107	—	107	1	no	—
178	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
179	#103	—	107	2	no	—
180	#102	—	107	2	no	—
181	#102	—	107	2	no	—
182	C₂²⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic
183	C₂²⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
184	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
185	C₂⁴	—	214	2	no	—
186	#41	—	428	2	no	—
187	#41	—	214	4	no	—
188	C₂×Q₈	—	107	2	no	—
189	C₂×D₄	—	107	2	no	—
190	C₂×D₄	—	107	2	no	—
191	C₂×D₄	—	428	2	no	—
192	C₂×D₄	—	214	4	no	—
193	C₂×D₄	—	214	2	no	—
194	C₂×D₄	—	214	2	no	—
195	C₂×D₄	—	214	2	no	—
196	C₂×D₄	D₄₂₈	1	1	yes	characteristic
197	C₂²×C₄	—	214	2	no	—
198	C₂²×C₄	—	214	2	no	—
199	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
201	Q₁₆	—	214	2	no	—
202	SD₁₆	—	428	2	no	—
203	SD₁₆	—	214	4	no	—
204	D₈	—	107	4	no	—
205	D₈	—	214	4	no	—
206	D₈	—	214	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
213	C₄⋊C₄	—	107	2	no	—
214	C₄⋊C₄	—	214	2	no	—
215	C₄⋊C₄	D₄₂₈	1	1	yes	characteristic
216	C₂²⋊C₄	—	214	2	no	—
217	C₂²⋊C₄	—	214	2	no	—
218	C₂²⋊C₄	—	214	2	no	—
219	C₂²⋊C₄	—	214	2	no	—
220	C₂²⋊C₄	—	214	2	no	—
221	C₂²⋊C₄	D₄₂₈	1	1	yes	characteristic
222	C₂²⋊C₄	D₄₂₈	1	1	yes	characteristic
223	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
224	C₂³	—	856	2	no	—
225	C₂³	—	428	4	no	—
226	C₂³	—	428	4	no	—
227	C₂³	—	214	2	no	—
228	C₂³	—	214	2	no	—
229	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
230	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
231	Q₈	—	214	2	no	—
232	Q₈	—	214	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	856	2	no	—
235	D₄	—	107	4	no	—
236	D₄	—	428	4	no	—
237	D₄	—	428	4	no	—
238	D₄	—	214	2	no	—
239	D₄	—	214	2	no	—
240	D₄	—	214	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	856	2	no	—
246	C₂×C₄	—	428	4	no	—
247	C₂×C₄	—	214	2	no	—
248	C₂×C₄	—	214	2	no	—
249	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	D₄×D₁₀₇	1	2	yes	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	856	8	no	—
264	C₂²	—	428	2	no	—
265	C₂²	—	428	2	no	—
266	C₂²	—	428	2	no	—
267	C₂²	—	428	2	no	—
268	C₂²	—	428	2	no	—
269	C₂²	—	214	4	no	—
270	C₂²	#16812131	1	2	yes	complemented
271	C₂²	C₂²×D₄₂₈	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	428	2	no	—
277	C₄	—	428	2	no	—
278	C₄	C₂×D₄×D₁₀₇	1	1	yes	characteristic
279	C₄	#16812155	1	1	yes	characteristic
280	C₂	—	2	2	no	—
281	C₂	—	4	1	no	—
282	C₂	—	856	2	no	—
283	C₂	—	428	4	no	—
284	C₂	D₄×D₄₂₈	1	1	yes	characteristic
285	C₂	C₂×#16812131	1	1	yes	characteristic
286	C₂	#20293432	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₂₈
commutator, cosocle	C₂×C₄×C₁₀₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₈×D₄×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
hypercenter	C₈×D₄	D₁₀₇
hypocenter	C₁₀₇	#3280
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16812131	C₂	semidirect
2. C₂²×D₈₅₆	C₂	semidirect
3. #20293273	C₂	semidirect
4. #20293011	C₂	semidirect
5. #20292868	C₂	semidirect
6. #20292863	C₂	semidirect
7. #20292843	C₂	semidirect
8. #20292780	C₂	semidirect
9. #20292774	C₂	semidirect
10. C₄×D₈₅₆	C₂	semidirect
11. C₈×D₄×C₁₀₇	C₂	semidirect
12. C₂×#13133818	C₂²	semidirect
15. C₂×Dic₄₂₈	C₂²	semidirect
16. C₂×#13133807	C₂²	semidirect
17. C₂×#13133807	C₂²	semidirect
22. #16812097	C₂²	semidirect
25. #16812083	C₂²	semidirect
28. C₄×D₄₂₈	C₂²	semidirect
30. #16812043	C₂²	semidirect
31. #16812034	C₂²	semidirect
32. #16812031	C₂²	semidirect
33. #16812031	C₂²	semidirect
35. C₂²×C₈×C₁₀₇	C₂²	semidirect
37. C₄⋊C₈×C₁₀₇	C₂²	semidirect
39. C₂²⋊C₈×C₁₀₇	C₂²	semidirect
40. C₄×C₈×C₁₀₇	C₂²	semidirect
59. D₈₅₆	D₄	semidirect
119. C₂²×C₁₀₇	#135	semidirect
144. C₁₀₇	#3280	semidirect
155. C₈×D₄	D₁₀₇	semidirect
167. C₂²×C₈	C₂×D₁₀₇	semidirect
178. C₄⋊C₈	C₂×D₁₀₇	semidirect
183. C₂²⋊C₈	C₂×D₁₀₇	semidirect
184. C₄×C₈	C₂×D₁₀₇	semidirect
270. C₂²	#16812131	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812131	2
C₂²×D₈₅₆	2
#20293273	2
#20293011	2
#20292868	2
#20292863	2
#20292843	2
#20292780	2
#20292774	2
C₄×D₈₅₆	2
C₈×D₄×C₁₀₇	2
#3280	107