Subgroups — C₄×#16812142 #28345284

Order 13696.

484 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×#16812142	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133818	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₂₁₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292735	C₂	1	2	yes	maximal, complemented
5	#20292729	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₄₂₈	C₂	1	2	yes	maximal, complemented
7	C₄×#13133807	C₂	1	2	yes	maximal, complemented
8	#20292669	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292637	C₂	1	1	yes	maximal, characteristic, complemented
10	#20292629	C₂	1	1	yes	maximal, characteristic
11	C₄×M₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133818	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₁₄	C₂²	1	1	yes	characteristic
14	#16812142	C₄	1	8	yes	direct product factor
15	C₂×Dic₄₂₈	C₂²	1	2	yes	—
16	C₂×#13133807	C₂²	1	2	yes	—
17	C₄×Dic₂₁₄	—	2	1	no	—
18	C₄×Dic₂₁₄	—	2	1	no	—
19	C₄×Dic₂₁₄	C₂²	1	2	yes	—
20	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₁₄	C₂²	1	2	yes	—
22	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic
23	C₄×D₄₂₈	C₂²	1	1	yes	characteristic
24	C₄×D₄₂₈	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₀₇	—	2	1	no	—
26	C₄×C₂²⋊D₁₀₇	—	2	1	no	—
27	#16812068	C₂²	1	1	yes	characteristic
28	#16812066	—	2	1	no	—
29	C₂×C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
30	C₂×Dic₁₀₇⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₁₀₇	—	2	1	no	—
32	C₄²×D₁₀₇	—	2	1	no	—
33	#16812043	C₂²	1	2	yes	—
34	#16812041	C₂²	1	2	yes	—
35	#16812034	C₂²	1	2	yes	—
36	#16812034	C₂²	1	2	yes	—
37	#16812034	C₂²	1	2	yes	—
38	#16812031	C₂²	1	2	yes	—
39	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
42	#97×C₁₀₇	C₂²	1	2	yes	—
43	C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
44	#13133818	—	2	2	no	—
45	#13133818	C₂×C₄	1	4	yes	—
46	C₂×Dic₂₁₄	—	2	1	no	—
47	C₂×Dic₂₁₄	—	2	2	no	—
48	C₂×Dic₂₁₄	C₂×C₄	1	4	yes	complemented
49	C₂×Dic₂₁₄	C₂³	1	2	yes	—
50	C₂×Dic₂₁₄	C₂³	1	1	yes	characteristic
51	C₂×D₄₂₈	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₀₇	—	2	1	no	—
53	C₂²×Dic₁₀₇	—	2	1	no	—
54	C₂²×Dic₁₀₇	—	2	1	no	—
55	C₂×C₄×D₁₀₇	—	2	1	no	—
56	C₂×C₄×D₁₀₇	—	2	1	no	—
57	C₂×C₄×D₁₀₇	—	2	1	no	—
58	Dic₄₂₈	C₂×C₄	1	8	yes	complemented
59	#13133807	C₂×C₄	1	8	yes	complemented
60	C₄⋊Dic₁₀₇	—	2	1	no	—
61	C₄⋊Dic₁₀₇	C₂³	1	1	yes	characteristic
62	C₄⋊Dic₁₀₇	C₂³	1	1	yes	characteristic
63	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
64	Dic₁₀₇⋊C₄	—	2	2	no	—
65	Dic₁₀₇⋊C₄	—	2	2	no	—
66	Dic₁₀₇⋊C₄	—	2	1	no	—
67	Dic₁₀₇⋊C₄	—	2	1	no	—
68	C₂²⋊Dic₁₀₇	—	2	1	no	—
69	(C₂×D₁₀₇)⋊C₄	—	2	1	no	—
70	(C₂×D₁₀₇)⋊C₄	—	2	1	no	—
71	C₄×Dic₁₀₇	—	2	1	no	—
72	C₄×Dic₁₀₇	—	2	1	no	—
73	C₄×Dic₁₀₇	—	2	1	no	—
74	C₄×Dic₁₀₇	—	2	2	no	—
75	C₄×Dic₁₀₇	—	2	2	no	—
76	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
79	M₁₆×C₁₀₇	—	2	2	no	—
80	M₁₆×C₁₀₇	C₂×C₄	1	4	yes	complemented
81	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
82	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
83	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
84	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
85	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
86	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
87	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₁₀₇	—	2	1	no	—
89	C₂×C₄×C₁₀₇	—	2	2	no	—
90	C₂×C₄×C₁₀₇	—	4	1	no	—
91	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₇	#41	1	2	yes	—
98	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₁₀₇	—	2	4	no	—
100	C₂×Dic₁₀₇	—	2	4	no	—
101	C₂×Dic₁₀₇	—	2	1	no	—
102	C₂×Dic₁₀₇	—	2	2	no	—
103	C₂×Dic₁₀₇	—	2	1	no	—
104	C₂×Dic₁₀₇	—	2	1	no	—
105	C₂×Dic₁₀₇	—	2	2	no	—
106	C₂×Dic₁₀₇	—	4	2	no	—
107	D₄₂₈	—	2	1	no	—
108	D₄₂₈	C₂²×C₄	1	2	yes	—
109	C₄×D₁₀₇	—	2	4	no	—
110	C₄×D₁₀₇	—	4	2	no	—
111	C₄×D₁₀₇	—	4	2	no	—
112	Dic₂₁₄	—	2	4	no	—
113	Dic₂₁₄	—	2	2	no	—
114	Dic₂₁₄	—	2	1	no	—
115	Dic₂₁₄	C₂²×C₄	1	4	yes	—
116	Dic₂₁₄	C₂²×C₄	1	2	yes	—
117	C₈×C₁₀₇	—	2	2	no	—
118	C₈×C₁₀₇	C₂²×C₄	1	4	yes	—
119	C₂²×C₁₀₇	—	2	1	no	—
120	C₂²×C₁₀₇	—	2	1	no	—
121	C₂²×C₁₀₇	C₄×D₄	1	2	yes	—
122	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₀₇	—	2	2	no	—
124	C₂×D₁₀₇	—	4	1	no	—
125	C₂×D₁₀₇	—	4	1	no	—
126	C₄×C₁₀₇	—	2	1	no	—
127	C₄×C₁₀₇	—	2	1	no	—
128	C₄×C₁₀₇	—	2	1	no	—
129	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
130	C₄×C₁₀₇	#137	1	2	yes	complemented
131	C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
133	Dic₁₀₇	—	2	4	no	—
134	Dic₁₀₇	—	2	2	no	—
135	Dic₁₀₇	—	4	1	no	—
136	Dic₁₀₇	—	4	1	no	—
137	Dic₁₀₇	—	4	2	no	—
138	C₂×C₁₀₇	—	2	2	no	—
139	C₂×C₁₀₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₀₇	C₂×#137	1	1	yes	characteristic
141	C₂×C₁₀₇	#575	1	1	yes	characteristic
142	D₁₀₇	—	4	2	no	—
143	C₄×#137	—	107	1	no	maximal, Sylow
144	C₁₀₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
145	C₂×#137	—	107	1	no	—
146	C₄×#41	—	107	1	no	—
147	C₂×C₄×Q₈	—	107	1	no	—
148	#441	—	107	2	no	—
149	#440	—	107	2	no	—
150	C₄×Q₁₆	—	107	2	no	—
151	C₄×SD₁₆	—	107	2	no	—
152	#428	—	107	1	no	—
153	#419	—	107	1	no	—
154	#417	—	107	1	no	—
155	C₄×M₁₆	D₁₀₇	1	1	yes	characteristic, complemented
156	C₂×#41	—	107	1	no	—
157	C₂²×Q₈	—	107	1	no	—
158	#137	—	107	8	no	—
159	C₂×Q₁₆	—	107	2	no	—
160	C₂×SD₁₆	—	107	2	no	—
161	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
162	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
163	C₄×Q₈	—	107	2	no	—
164	C₄×Q₈	—	107	2	no	—
165	C₄×Q₈	—	107	1	no	—
166	C₄×Q₈	—	107	1	no	—
167	C₄×Q₈	—	214	1	no	—
168	C₄×Q₈	—	214	1	no	—
169	C₄×D₄	—	107	1	no	—
170	C₄×D₄	—	107	1	no	—
171	C₄×D₄	—	214	1	no	—
172	C₄×D₄	—	214	1	no	—
173	#117	—	107	1	no	—
174	#117	—	214	1	no	—
175	C₂×C₄⋊C₄	—	107	1	no	—
176	C₂×C₄⋊C₄	—	214	1	no	—
177	C₂×C₄²	—	214	1	no	—
178	C₂×C₄²	—	214	1	no	—
179	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
180	#107	—	107	2	no	—
181	#106	—	107	2	no	—
182	#103	—	107	2	no	—
183	#103	—	107	2	no	—
184	#103	—	107	2	no	—
185	#102	—	107	2	no	—
186	#97	C₂×D₁₀₇	1	2	yes	—
187	C₄×C₈	C₂×D₁₀₇	1	2	yes	—
188	#41	—	107	4	no	—
189	#41	—	214	2	no	—
190	C₂×Q₈	—	107	4	no	—
191	C₂×Q₈	—	107	2	no	—
192	C₂×Q₈	—	107	1	no	—
193	C₂×Q₈	—	214	1	no	—
194	C₂×Q₈	—	214	2	no	—
195	C₂×D₄	—	107	1	no	—
196	C₂×D₄	—	214	1	no	—
197	C₂²×C₄	—	214	1	no	—
198	C₂²×C₄	—	214	1	no	—
199	C₂²×C₄	—	214	1	no	—
200	C₂²×C₄	—	214	1	no	—
201	C₂²×C₄	—	214	1	no	—
202	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
203	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
205	Q₁₆	—	107	8	no	—
206	SD₁₆	—	107	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	C₄×D₁₀₇	1	4	yes	complemented
209	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
210	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
211	C₄⋊C₄	—	107	2	no	—
212	C₄⋊C₄	—	107	1	no	—
213	C₄⋊C₄	—	107	1	no	—
214	C₄⋊C₄	—	214	1	no	—
215	C₄⋊C₄	—	214	2	no	—
216	C₄⋊C₄	—	214	2	no	—
217	C₄⋊C₄	—	214	1	no	—
218	C₄⋊C₄	—	214	1	no	—
219	C₂²⋊C₄	—	214	1	no	—
220	C₂²⋊C₄	—	214	1	no	—
221	C₂²⋊C₄	—	214	1	no	—
222	C₄²	—	214	1	no	—
223	C₄²	—	214	1	no	—
224	C₄²	—	214	2	no	—
225	C₄²	—	214	1	no	—
226	C₄²	—	214	2	no	—
227	C₄²	D₄₂₈	1	1	yes	characteristic
228	C₄²	D₄₂₈	1	1	yes	characteristic
229	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
230	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
231	C₂³	—	214	1	no	—
232	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
233	Q₈	—	107	4	no	—
234	Q₈	—	107	2	no	—
235	Q₈	—	214	4	no	—
236	Q₈	—	214	2	no	—
237	Q₈	—	214	1	no	—
238	D₄	—	107	2	no	—
239	D₄	—	214	4	no	—
240	D₄	—	214	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	428	2	no	—
244	C₂×C₄	—	428	2	no	—
245	C₂×C₄	—	428	2	no	—
246	C₂×C₄	—	214	1	no	—
247	C₂×C₄	—	214	1	no	—
248	C₂×C₄	—	214	2	no	—
249	C₂×C₄	—	214	4	no	—
250	C₂×C₄	—	214	1	no	—
251	C₂×C₄	—	214	2	no	—
252	C₂×C₄	—	214	4	no	—
253	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
254	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
259	C₂×C₄	#13133818	1	2	yes	—
260	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	C₂×C₄×D₁₀₇	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	428	1	no	—
266	C₂²	—	428	1	no	—
267	C₂²	—	214	2	no	—
268	C₂²	C₄×D₄₂₈	1	2	yes	—
269	C₂²	C₂²×D₄₂₈	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	428	1	no	—
274	C₄	—	428	1	no	—
275	C₄	—	428	2	no	—
276	C₄	—	214	2	no	—
277	C₄	—	214	4	no	—
278	C₄	C₄×D₄₂₈	1	2	yes	—
279	C₄	#16812142	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
281	C₄	C₂×#13133818	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	428	2	no	—
284	C₂	C₂×C₄×D₄₂₈	1	1	yes	characteristic
285	C₂	C₂×#16812142	1	1	yes	characteristic
286	C₂	#20293427	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₈
commutator	C₄×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₄×M₁₆×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	C₂×#13133818	C₂²
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₄×M₁₆	D₁₀₇
hypocenter	C₁₀₇	C₄×#137
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₂₁₄	C₂	semidirect
4. #20292735	C₂	semidirect
5. #20292729	C₂	semidirect
6. C₄×Dic₄₂₈	C₂	semidirect
7. C₄×#13133807	C₂	semidirect
8. #20292669	C₂	semidirect
9. #20292637	C₂	semidirect
11. C₄×M₁₆×C₁₀₇	C₂	semidirect
14. #16812142	C₄	direct (N × H)
48. C₂×Dic₂₁₄	C₂×C₄	semidirect
58. Dic₄₂₈	C₂×C₄	semidirect
59. #13133807	C₂×C₄	semidirect
80. M₁₆×C₁₀₇	C₂×C₄	semidirect
130. C₄×C₁₀₇	#137	semidirect
144. C₁₀₇	C₄×#137	semidirect
155. C₄×M₁₆	D₁₀₇	semidirect
208. M₁₆	C₄×D₁₀₇	semidirect
279. C₄	#16812142	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×#16812142	2
C₄×#13133818	2
C₂×C₄×Dic₂₁₄	2
#20292735	2
#20292729	2
C₄×Dic₄₂₈	2
C₄×#13133807	2
#20292669	2
#20292637	2
#20292629	2
C₄×M₁₆×C₁₀₇	2
C₄×#137	107

Subgroups — C4×#16812142 #28345284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812142 #28345284