Subgroups — C₄×#16812129 #28345231

Order 13696.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812129	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133818	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133808	C₂	1	2	yes	maximal, complemented
4	C₄×#13133805	C₂	1	2	yes	maximal, complemented
5	C₄×#13133806	C₂	1	2	yes	maximal, complemented
6	C₄×#13133803	C₂	1	2	yes	maximal, complemented
7	#20292663	C₂	1	1	yes	maximal, characteristic, complemented
8	#20292623	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292621	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133818	C₂²	1	1	yes	characteristic
13	#16812129	C₄	1	8	yes	direct product factor
14	C₂×#13133808	C₂²	1	1	yes	characteristic
15	C₂×#13133806	C₂²	1	1	yes	characteristic
16	C₂×#13133805	C₂²	1	1	yes	characteristic
17	C₂×#13133803	C₂²	1	1	yes	characteristic
18	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₁₄	C₂²	1	2	yes	—
21	C₄×D₄₂₈	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₁₀₇	—	2	2	no	—
23	#16812068	C₂²	1	1	yes	characteristic
24	#16812066	—	2	1	no	—
25	C₄²×D₁₀₇	—	2	1	no	—
26	#16812042	C₂²	1	2	yes	—
27	#16812040	C₂²	1	2	yes	—
28	#16812033	C₂²	1	2	yes	—
29	#16812032	C₂²	1	2	yes	—
30	#16812030	C₂²	1	2	yes	—
31	#16812029	C₂²	1	2	yes	—
32	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
33	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
34	C₂×#41×C₁₀₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₀₇	C₂²	1	2	yes	—
36	C₄×D₄×C₁₀₇	—	2	2	no	—
37	C₄×D₄×C₁₀₇	C₂²	1	2	yes	—
38	#117×C₁₀₇	—	2	1	no	—
39	#117×C₁₀₇	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₇	—	2	1	no	—
41	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
42	#13133818	—	2	2	no	—
43	#13133818	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₁₄	C₂³	1	1	yes	characteristic
45	C₂×D₄₂₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₇	—	2	1	no	—
47	C₂×C₄×D₁₀₇	—	2	1	no	—
48	C₂×C₄×D₁₀₇	—	2	2	no	—
49	#13133808	C₂×C₄	1	4	yes	complemented
50	#13133806	C₂×C₄	1	4	yes	complemented
51	#13133805	C₂×C₄	1	4	yes	complemented
52	#13133803	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₀₇⋊C₈	—	2	2	no	—
54	C₂×C₁₀₇⋊C₈	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
56	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
59	Dic₁₀₇⋊C₄	—	2	2	no	—
60	C₂²⋊Dic₁₀₇	—	2	1	no	—
61	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
62	C₄×Dic₁₀₇	—	2	1	no	—
63	C₄×Dic₁₀₇	—	2	2	no	—
64	#41×C₁₀₇	—	2	2	no	—
65	#41×C₁₀₇	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₀₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₀₇	—	2	1	no	—
68	C₂×D₄×C₁₀₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₇	—	2	1	no	—
70	C₂²×C₄×C₁₀₇	—	2	2	no	—
71	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
72	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₇	—	2	2	no	—
74	C₄⋊C₄×C₁₀₇	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
76	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
77	C₄²×C₁₀₇	—	2	1	no	—
78	C₄²×C₁₀₇	—	2	2	no	—
79	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
80	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
81	C₄²×C₁₀₇	C₂³	1	2	yes	—
82	C₂³×C₁₀₇	—	2	1	no	—
83	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₁₀₇	—	2	1	no	—
85	Q₈×C₁₀₇	—	2	1	no	—
86	Q₈×C₁₀₇	C₂²×C₄	1	2	yes	—
87	D₄×C₁₀₇	—	2	4	no	—
88	D₄×C₁₀₇	—	2	1	no	—
89	D₄×C₁₀₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₇	—	2	1	no	—
91	C₂×C₄×C₁₀₇	—	2	4	no	—
92	C₂×C₄×C₁₀₇	—	2	4	no	—
93	C₂×C₄×C₁₀₇	—	2	2	no	—
94	C₂×C₄×C₁₀₇	—	4	4	no	—
95	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₇	#41	1	2	yes	—
100	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₁₀₇	—	2	4	no	—
102	C₂×Dic₁₀₇	—	2	1	no	—
103	C₂×Dic₁₀₇	—	2	2	no	—
104	D₄₂₈	—	2	1	no	—
105	D₄₂₈	C₂²×C₄	1	2	yes	—
106	C₄×D₁₀₇	—	2	4	no	—
107	C₄×D₁₀₇	—	4	4	no	—
108	Dic₂₁₄	—	2	1	no	—
109	Dic₂₁₄	C₂²×C₄	1	2	yes	—
110	C₁₀₇⋊C₈	—	2	2	no	—
111	C₁₀₇⋊C₈	C₂²×C₄	1	2	yes	—
112	C₁₀₇⋊C₈	C₂²×C₄	1	2	yes	—
113	C₂²×C₁₀₇	—	2	2	no	—
114	C₂²×C₁₀₇	—	2	2	no	—
115	C₂²×C₁₀₇	—	4	2	no	—
116	C₂²×C₁₀₇	C₄×D₄	1	2	yes	—
117	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₀₇	—	2	2	no	—
119	C₂×D₁₀₇	—	4	2	no	—
120	C₄×C₁₀₇	—	2	2	no	—
121	C₄×C₁₀₇	—	2	2	no	—
122	C₄×C₁₀₇	—	4	2	no	—
123	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
124	C₄×C₁₀₇	#135	1	4	yes	complemented
125	C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
127	Dic₁₀₇	—	2	2	no	—
128	Dic₁₀₇	—	4	2	no	—
129	C₂×C₁₀₇	—	2	2	no	—
130	C₂×C₁₀₇	—	4	2	no	—
131	C₂×C₁₀₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₀₇	C₂×#135	1	2	yes	—
133	D₁₀₇	—	4	2	no	—
134	C₄×#135	—	107	1	no	maximal, Sylow
135	C₁₀₇	C₄×#135	1	1	yes	characteristic, complemented, Sylow
136	C₂×#135	—	107	1	no	—
137	C₄×#41	—	107	1	no	—
138	C₄×#41	D₁₀₇	1	1	yes	characteristic, complemented
139	C₄×Q₁₆	—	107	2	no	—
140	C₄×SD₁₆	—	107	2	no	—
141	C₄×SD₁₆	—	107	2	no	—
142	C₄×D₈	—	107	2	no	—
143	#427	—	107	1	no	—
144	#416	—	107	1	no	—
145	#416	—	107	1	no	—
146	C₂×C₄×C₈	—	107	1	no	—
147	C₂×#41	—	107	1	no	—
148	C₂×#41	C₂×D₁₀₇	1	1	yes	characteristic
149	#135	—	107	8	no	—
150	C₂×Q₁₆	—	107	1	no	—
151	C₂×SD₁₆	—	107	1	no	—
152	C₂×SD₁₆	—	107	1	no	—
153	C₂×D₈	—	107	1	no	—
154	C₂²×C₈	—	107	1	no	—
155	C₂²×C₈	—	107	1	no	—
156	C₄×Q₈	—	107	2	no	—
157	C₄×Q₈	C₂×D₁₀₇	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	107	2	no	—
160	C₄×D₄	—	214	2	no	—
161	C₄×D₄	C₂×D₁₀₇	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	107	1	no	—
164	#117	—	214	1	no	—
165	#117	C₂×D₁₀₇	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	214	1	no	—
168	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
169	#107	—	107	2	no	—
170	#106	—	107	2	no	—
171	#103	—	107	2	no	—
172	#103	—	107	2	no	—
173	#102	—	107	2	no	—
174	#102	—	107	2	no	—
175	C₄×C₈	—	107	2	no	—
176	C₄×C₈	—	107	2	no	—
177	#41	—	2	2	no	—
178	#41	—	107	4	no	—
179	#41	—	214	2	no	—
180	#41	C₄×D₁₀₇	1	4	yes	complemented
181	C₂×Q₈	—	107	1	no	—
182	C₂×Q₈	C₂²×D₁₀₇	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	107	1	no	—
185	C₂×D₄	—	214	1	no	—
186	C₂×D₄	C₂²×D₁₀₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	214	2	no	—
190	C₂²×C₄	—	214	1	no	—
191	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
193	Q₁₆	—	107	4	no	—
194	SD₁₆	—	107	4	no	—
195	SD₁₆	—	107	4	no	—
196	D₈	—	107	4	no	—
197	C₂×C₈	—	107	4	no	—
198	C₂×C₈	—	107	1	no	—
199	C₂×C₈	—	107	1	no	—
200	C₂×C₈	—	107	2	no	—
201	C₂×C₈	—	214	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	107	2	no	—
204	C₄⋊C₄	—	214	2	no	—
205	C₄⋊C₄	C₂²×D₁₀₇	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	214	1	no	—
209	C₂²⋊C₄	—	214	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	214	1	no	—
213	C₄²	—	214	2	no	—
214	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
215	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
216	C₄²	C₂²×D₁₀₇	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	214	1	no	—
219	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	107	2	no	—
222	Q₈	—	214	1	no	—
223	Q₈	C₂×C₄×D₁₀₇	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	107	2	no	—
227	D₄	—	214	4	no	—
228	D₄	—	214	1	no	—
229	D₄	C₂×C₄×D₁₀₇	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	428	4	no	—
236	C₂×C₄	—	214	2	no	—
237	C₂×C₄	—	214	1	no	—
238	C₂×C₄	—	214	4	no	—
239	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
243	C₂×C₄	#13133818	1	2	yes	—
244	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
245	C₈	—	107	2	no	—
246	C₈	—	107	2	no	—
247	C₈	—	214	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	428	2	no	—
252	C₂²	—	214	2	no	—
253	C₂²	C₄×C₂²⋊D₁₀₇	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	428	2	no	—
259	C₄	—	214	2	no	—
260	C₄	C₄×C₂²⋊D₁₀₇	1	2	yes	—
261	C₄	#16812129	1	4	yes	direct product factor
262	C₄	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
263	C₄	C₂×#13133818	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	428	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₁₀₇	1	1	yes	characteristic
268	C₂	C₂×#16812129	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊D₁₀₇
commutator	C₄×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₄×#41×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	C₂×#16812129	C₂
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₄×#41	D₁₀₇
hypocenter	C₁₀₇	C₄×#135
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#13133818	C₂	semidirect
3. C₄×#13133808	C₂	semidirect
4. C₄×#13133805	C₂	semidirect
5. C₄×#13133806	C₂	semidirect
6. C₄×#13133803	C₂	semidirect
7. #20292663	C₂	semidirect
8. #20292623	C₂	semidirect
9. #20292621	C₂	semidirect
10. C₂×C₄×C₁₀₇⋊C₈	C₂	semidirect
11. C₄×#41×C₁₀₇	C₂	semidirect
13. #16812129	C₄	direct (N × H)
43. #13133818	C₂×C₄	semidirect
49. #13133808	C₂×C₄	semidirect
50. #13133806	C₂×C₄	semidirect
51. #13133805	C₂×C₄	semidirect
52. #13133803	C₂×C₄	semidirect
54. C₂×C₁₀₇⋊C₈	C₂×C₄	semidirect
65. #41×C₁₀₇	C₂×C₄	semidirect
124. C₄×C₁₀₇	#135	semidirect
135. C₁₀₇	C₄×#135	semidirect
138. C₄×#41	D₁₀₇	semidirect
180. #41	C₄×D₁₀₇	semidirect
261. C₄	#16812129	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16812129	2
C₄×#13133818	2
C₄×#13133808	2
C₄×#13133805	2
C₄×#13133806	2
C₄×#13133803	2
#20292663	2
#20292623	2
#20292621	2
C₂×C₄×C₁₀₇⋊C₈	2
C₄×#41×C₁₀₇	2
C₄×#135	107

Subgroups — C4×#16812129 #28345231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812129 #28345231