Subgroups — #28345308

Order 13696.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₁₀₇	C₂	1	1	yes	maximal, characteristic
2	#20293099	C₂	1	2	yes	maximal, complemented
3	#20292737	C₂	1	2	yes	maximal, complemented
4	#20292736	C₂	1	2	yes	maximal, complemented
5	C₄×D₈₅₆	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#20292709	C₂	1	1	yes	maximal, characteristic, complemented
8	#20292664	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292630	C₂	1	2	yes	maximal, complemented
10	#20292581	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₀₇	C₂²	1	2	yes	—
13	C₂×D₈₅₆	C₂²	1	1	yes	characteristic
14	C₂×#13133803	C₂²	1	2	yes	—
15	D₈×D₁₀₇	C₄	1	8	yes	complemented
16	C₂×#13133801	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₇	C₂²	1	1	yes	characteristic
18	C₄×D₄₂₈	C₂²	1	2	yes	complemented
19	D₄×Dic₁₀₇	C₂²	1	2	yes	complemented
20	#16812071	C₂²	1	2	yes	complemented
21	C₄×C₂²⋊D₁₀₇	—	2	2	no	—
22	(C₂²⋊D₁₀₇)⋊C₄	—	2	2	no	—
23	#16812067	C₂²	1	2	yes	—
24	#16812066	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
26	C₂²⋊C₄×D₁₀₇	—	2	2	no	—
27	#16812043	C₂²	1	1	yes	characteristic, complemented
28	#16812040	C₂²	1	2	yes	complemented
29	#16812031	C₂²	1	2	yes	complemented
30	#16812030	C₂²	1	2	yes	complemented
31	#16812029	C₂²	1	2	yes	complemented
32	#16812020	C₂²	1	1	yes	characteristic, complemented
33	#16812019	C₂²	1	1	yes	characteristic, complemented
34	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₈×C₁₀₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₀₇	C₂²	1	2	yes	complemented
37	#107×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
38	#102×C₁₀₇	C₂²	1	2	yes	complemented
39	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₁₀₇	—	2	2	no	—
41	C₂×D₄₂₈	C₂³	1	2	yes	—
42	C₂×C₂²⋊D₁₀₇	—	2	2	no	—
43	D₄×D₁₀₇	—	2	4	no	—
44	D₄×D₁₀₇	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₁₀₇	—	2	2	no	—
46	C₂×C₄×D₁₀₇	—	2	2	no	—
47	C₂×C₄×D₁₀₇	—	2	2	no	—
48	C₂×C₄×D₁₀₇	C₂³	1	1	yes	characteristic
49	D₈₅₆	C₂×C₄	1	4	yes	complemented
50	#13133803	C₂×C₄	1	8	yes	complemented
51	#13133801	—	2	2	no	—
52	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₈×D₁₀₇	C₂×C₄	1	4	yes	complemented
55	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
56	Dic₁₀₇⋊C₄	D₄	1	1	yes	characteristic
57	Dic₁₀₇⋊C₄	D₄	1	1	yes	characteristic
58	C₂²⋊Dic₁₀₇	—	2	2	no	—
59	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
60	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
61	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
62	(C₂×D₁₀₇)⋊C₄	D₄	1	1	yes	characteristic
63	(C₂×D₁₀₇)⋊C₄	D₄	1	1	yes	characteristic
64	C₄×Dic₁₀₇	—	2	2	no	—
65	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₁₀₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₇	—	2	2	no	—
68	D₈×C₁₀₇	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₇	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
73	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
74	C₂³×C₁₀₇	—	2	2	no	—
75	C₂²×D₁₀₇	—	2	2	no	—
76	C₂²×D₁₀₇	—	2	8	no	—
77	C₂²×D₁₀₇	—	8	2	no	—
78	C₂²×D₁₀₇	—	8	2	no	—
79	C₂²×D₁₀₇	C₂×D₄	1	1	yes	characteristic
80	D₄×C₁₀₇	—	2	2	no	—
81	D₄×C₁₀₇	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₀₇	—	2	2	no	—
83	C₂×C₄×C₁₀₇	—	8	2	no	—
84	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₁₀₇	—	2	8	no	—
88	C₂×Dic₁₀₇	—	2	2	no	—
89	C₂×Dic₁₀₇	—	8	2	no	—
90	C₂×Dic₁₀₇	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₁₀₇	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₁₀₇	C₂×D₄	1	1	yes	characteristic
93	D₄₂₈	—	2	2	no	—
94	D₄₂₈	C₂²×C₄	1	4	yes	—
95	C₄×D₁₀₇	—	8	2	no	—
96	C₄×D₁₀₇	—	8	2	no	—
97	C₄×D₁₀₇	C₂²×C₄	1	2	yes	—
98	C₄×D₁₀₇	#41	1	2	yes	—
99	C₈×C₁₀₇	—	2	1	no	—
100	C₈×C₁₀₇	C₂²×C₄	1	2	yes	—
101	C₁₀₇⋊C₈	—	2	1	no	—
102	C₁₀₇⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂²×C₁₀₇	—	2	4	no	—
104	C₂²×C₁₀₇	—	4	2	no	—
105	C₂²×C₁₀₇	—	4	2	no	—
106	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₁₀₇	—	2	1	no	—
108	C₂×D₁₀₇	—	2	1	no	—
109	C₂×D₁₀₇	—	2	4	no	—
110	C₂×D₁₀₇	—	4	2	no	—
111	C₂×D₁₀₇	—	4	2	no	—
112	C₂×D₁₀₇	—	8	8	no	—
113	C₂×D₁₀₇	C₄×D₄	1	2	yes	—
114	C₄×C₁₀₇	—	2	1	no	—
115	C₄×C₁₀₇	—	2	1	no	—
116	C₄×C₁₀₇	—	4	2	no	—
117	C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
119	Dic₁₀₇	—	2	1	no	—
120	Dic₁₀₇	—	2	1	no	—
121	Dic₁₀₇	—	4	2	no	—
122	Dic₁₀₇	C₄×D₄	1	2	yes	—
123	C₂×C₁₀₇	—	4	4	no	—
124	C₂×C₁₀₇	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂×C₁₀₇	#576	1	1	yes	characteristic
126	C₂×C₁₀₇	#576	1	1	yes	characteristic
127	D₁₀₇	—	2	2	no	—
128	D₁₀₇	—	4	4	no	—
129	#2948	—	107	1	no	maximal, Sylow
130	C₁₀₇	#2948	1	1	yes	characteristic, complemented, Sylow
131	C₂²×D₈	—	107	1	no	—
132	#518	—	107	2	no	—
133	#442	—	107	2	no	—
134	#442	—	107	2	no	—
135	C₄×D₈	—	107	1	no	—
136	C₄×D₈	—	107	1	no	—
137	C₄×D₈	—	107	1	no	—
138	C₄×D₈	D₁₀₇	1	1	yes	characteristic, complemented
139	#427	—	107	1	no	—
140	#418	—	107	2	no	—
141	#405	—	107	1	no	—
142	C₂²×D₄	—	107	2	no	—
143	C₂×D₈	—	107	2	no	—
144	C₂×D₈	—	107	8	no	—
145	C₂×D₈	—	107	1	no	—
146	C₂×D₈	C₂×D₁₀₇	1	1	yes	characteristic
147	C₂×M₁₆	—	107	1	no	—
148	C₂²×C₈	—	107	1	no	—
149	C₄×D₄	—	107	2	no	—
150	C₄×D₄	—	107	2	no	—
151	C₄×D₄	—	107	2	no	—
152	C₄×D₄	—	214	2	no	—
153	C₄×D₄	—	214	2	no	—
154	C₄×D₄	C₂×D₁₀₇	1	2	yes	complemented
155	#117	—	107	1	no	—
156	#117	—	107	2	no	—
157	C₂×C₂²⋊C₄	—	214	2	no	—
158	C₂×C₂²⋊C₄	—	214	2	no	—
159	#107	—	107	1	no	—
160	#107	C₂×D₁₀₇	1	1	yes	characteristic, complemented
161	#106	—	107	2	no	—
162	#102	—	107	2	no	—
163	#102	—	107	2	no	—
164	#102	—	107	2	no	—
165	#102	C₂×D₁₀₇	1	2	yes	complemented
166	#97	—	107	1	no	—
167	#97	—	107	1	no	—
168	C₄×C₈	—	107	1	no	—
169	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
170	C₂⁴	—	214	2	no	—
171	C₂×D₄	—	107	8	no	—
172	C₂×D₄	—	107	2	no	—
173	C₂×D₄	—	214	4	no	—
174	C₂×D₄	—	214	2	no	—
175	C₂×D₄	C₂²×D₁₀₇	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	107	1	no	—
178	C₂²×C₄	—	214	2	no	—
179	C₂²×C₄	—	214	2	no	—
180	C₂²×C₄	—	214	2	no	—
181	D₈	—	107	8	no	—
182	D₈	—	107	4	no	—
183	D₈	C₄×D₁₀₇	1	4	yes	complemented
184	M₁₆	—	214	2	no	—
185	C₂×C₈	—	107	1	no	—
186	C₂×C₈	—	107	4	no	—
187	C₂×C₈	—	107	1	no	—
188	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
190	C₄⋊C₄	—	107	1	no	—
191	C₄⋊C₄	—	107	1	no	—
192	C₄⋊C₄	—	107	2	no	—
193	C₄⋊C₄	C₂²×D₁₀₇	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	107	1	no	—
196	C₂²⋊C₄	—	107	1	no	—
197	C₂²⋊C₄	—	214	2	no	—
198	C₂²⋊C₄	—	214	2	no	—
199	C₂²⋊C₄	—	214	2	no	—
200	C₂²⋊C₄	—	214	2	no	—
201	C₄²	—	107	1	no	—
202	C₄²	—	214	2	no	—
203	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	—	856	2	no	—
206	C₂³	—	856	2	no	—
207	C₂³	—	107	1	no	—
208	C₂³	—	214	8	no	—
209	C₂³	—	214	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	107	4	no	—
212	D₄	—	214	8	no	—
213	D₄	—	214	2	no	—
214	D₄	C₂×C₄×D₁₀₇	1	4	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	856	2	no	—
218	C₂×C₄	—	856	2	no	—
219	C₂×C₄	—	856	2	no	—
220	C₂×C₄	—	107	1	no	—
221	C₂×C₄	—	107	2	no	—
222	C₂×C₄	—	107	2	no	—
223	C₂×C₄	—	107	1	no	—
224	C₂×C₄	—	107	1	no	—
225	C₂×C₄	—	214	2	no	—
226	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
227	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
228	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	107	2	no	—
231	C₈	—	214	1	no	—
232	C₈	C₂×C₄×D₁₀₇	1	2	yes	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	856	8	no	—
237	C₂²	—	107	2	no	—
238	C₂²	—	428	2	no	—
239	C₂²	—	428	2	no	—
240	C₂²	—	214	4	no	—
241	C₂²	—	214	1	no	—
242	C₂²	—	214	1	no	—
243	C₂²	C₂×D₄×D₁₀₇	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	107	2	no	—
248	C₄	—	428	2	no	—
249	C₄	—	214	1	no	—
250	C₄	—	214	1	no	—
251	C₄	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
252	C₄	#41×D₁₀₇	1	1	yes	characteristic
253	C₂	—	4	4	no	—
254	C₂	—	428	4	no	—
255	C₂	—	214	2	no	—
256	C₂	C₄×D₄×D₁₀₇	1	1	yes	characteristic
257	C₂	#20293428	1	1	yes	characteristic
258	C₂	#20293429	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₇
commutator	C₄×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₄×D₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	C₂×D₈×D₁₀₇	C₂
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₄×D₈	D₁₀₇
hypocenter	C₁₀₇	#2948
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #20293099	C₂	semidirect
3. #20292737	C₂	semidirect
4. #20292736	C₂	semidirect
5. C₄×D₈₅₆	C₂	semidirect
6. D₈×Dic₁₀₇	C₂	semidirect
7. #20292709	C₂	semidirect
8. #20292664	C₂	semidirect
9. #20292630	C₂	semidirect
10. #20292581	C₂	semidirect
11. C₄×D₈×C₁₀₇	C₂	semidirect
15. D₈×D₁₀₇	C₄	semidirect
18. C₄×D₄₂₈	C₂²	semidirect
19. D₄×Dic₁₀₇	C₂²	semidirect
20. #16812071	C₂²	semidirect
27. #16812043	C₂²	semidirect
28. #16812040	C₂²	semidirect
29. #16812031	C₂²	semidirect
30. #16812030	C₂²	semidirect
31. #16812029	C₂²	semidirect
32. #16812020	C₂²	semidirect
33. #16812019	C₂²	semidirect
34. C₈×Dic₁₀₇	C₂²	semidirect
36. C₄×D₄×C₁₀₇	C₂²	semidirect
37. #107×C₁₀₇	C₂²	semidirect
38. #102×C₁₀₇	C₂²	semidirect
39. C₄×C₈×C₁₀₇	C₂²	semidirect
44. D₄×D₁₀₇	C₂×C₄	semidirect
49. D₈₅₆	C₂×C₄	semidirect
50. #13133803	C₂×C₄	semidirect
54. C₈×D₁₀₇	C₂×C₄	semidirect
68. D₈×C₁₀₇	C₂×C₄	semidirect
130. C₁₀₇	#2948	semidirect
138. C₄×D₈	D₁₀₇	semidirect
154. C₄×D₄	C₂×D₁₀₇	semidirect
160. #107	C₂×D₁₀₇	semidirect
165. #102	C₂×D₁₀₇	semidirect
169. C₄×C₈	C₂×D₁₀₇	semidirect
183. D₈	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×D₈×D₁₀₇	2
#20293099	2
#20292737	2
#20292736	2
C₄×D₈₅₆	2
D₈×Dic₁₀₇	2
#20292709	2
#20292664	2
#20292630	2
#20292581	2
C₄×D₈×C₁₀₇	2
#2948	107