Subgroups — C₄×#16812730 #28371851

Order 13952.

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×#16812730	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182029	C₂	1	1	yes	maximal, characteristic
3	#20295131	C₂	1	2	yes	maximal, complemented
4	#20295124	C₂	1	2	yes	maximal, complemented
5	C₄×#13182019	C₂	1	2	yes	maximal, complemented
6	C₄×#13182016	C₂	1	2	yes	maximal, complemented
7	#20295066	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295035	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295025	C₂	1	1	yes	maximal, characteristic
10	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	#16812730	C₄	1	8	yes	direct product factor
14	C₂×#13182019	C₂²	1	2	yes	—
15	C₂×#13182016	C₂²	1	2	yes	—
16	C₂×#13182013	C₂²	1	1	yes	characteristic
17	C₂×#13182013	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
20	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
21	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
23	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
24	#16812658	C₂²	1	1	yes	characteristic
25	#16812656	—	2	1	no	—
26	C₄²×D₁₀₉	—	2	1	no	—
27	#16812632	C₂²	1	2	yes	—
28	#16812630	C₂²	1	2	yes	—
29	#16812623	C₂²	1	2	yes	—
30	#16812623	C₂²	1	2	yes	—
31	#16812622	C₂²	1	2	yes	—
32	#16812619	C₂²	1	2	yes	—
33	#16812610	C₂²	1	2	yes	—
34	C₄×#11724784	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₀₉	—	2	1	no	—
37	C₄×Q₈×C₁₀₉	—	2	1	no	—
38	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
39	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₁₀₉	—	2	1	no	—
43	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
44	#13182029	—	2	2	no	—
45	#13182029	C₂×C₄	1	4	yes	—
46	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
47	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
49	C₂×C₄×D₁₀₉	—	2	1	no	—
50	C₂×C₄×D₁₀₉	—	2	1	no	—
51	C₂×C₄×D₁₀₉	—	2	1	no	—
52	#13182019	C₂×C₄	1	8	yes	complemented
53	#13182016	C₂×C₄	1	8	yes	complemented
54	#13182013	—	2	2	no	—
55	#13182013	C₂×C₄	1	4	yes	complemented
56	C₂×#11724784	C₂³	1	2	yes	—
57	C₂×#11724784	C₂³	1	2	yes	—
58	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
60	#13182008	—	2	1	no	—
61	#13182008	—	2	1	no	—
62	C₂²⋊Dic₁₀₉	—	2	1	no	—
63	#13182006	—	2	1	no	—
64	#13182006	—	2	1	no	—
65	C₄×Dic₁₀₉	—	2	1	no	—
66	C₄×Dic₁₀₉	—	2	1	no	—
67	C₄×Dic₁₀₉	—	2	1	no	—
68	C₂×Q₈×C₁₀₉	—	2	1	no	—
69	C₂×Q₈×C₁₀₉	—	2	2	no	—
70	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
71	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
72	C₂²×C₄×C₁₀₉	—	2	1	no	—
73	C₂²×C₄×C₁₀₉	—	2	1	no	—
74	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₉	—	2	2	no	—
78	C₄⋊C₄×C₁₀₉	—	2	2	no	—
79	C₄⋊C₄×C₁₀₉	—	2	1	no	—
80	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
81	C₄²×C₁₀₉	—	2	2	no	—
82	C₄²×C₁₀₉	—	2	2	no	—
83	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
84	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
85	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₁₀₉	—	2	1	no	—
89	Q₈×C₁₀₉	—	2	4	no	—
90	Q₈×C₁₀₉	—	2	2	no	—
91	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₁₀₉	—	2	4	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	4	1	no	—
97	C₂×C₄×C₁₀₉	—	4	2	no	—
98	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	#41	1	2	yes	—
105	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₁₀₉	—	2	4	no	—
107	C₂×Dic₁₀₉	—	2	1	no	—
108	C₂×Dic₁₀₉	—	2	1	no	—
109	C₂×Dic₁₀₉	—	2	1	no	—
110	D₄₃₆	—	2	1	no	—
111	D₄₃₆	C₂²×C₄	1	2	yes	—
112	C₄×D₁₀₉	—	2	4	no	—
113	C₄×D₁₀₉	—	4	2	no	—
114	C₄×D₁₀₉	—	4	2	no	—
115	Dic₂₁₈	—	2	1	no	—
116	Dic₂₁₈	C₂²×C₄	1	2	yes	—
117	#11724784	—	2	2	no	—
118	#11724784	C₂²×C₄	1	4	yes	—
119	C₂²×C₁₀₉	—	2	1	no	—
120	C₂²×C₁₀₉	—	2	1	no	—
121	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
122	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₀₉	—	2	2	no	—
124	C₂×D₁₀₉	—	4	1	no	—
125	C₂×D₁₀₉	—	4	1	no	—
126	C₄×C₁₀₉	—	2	4	no	—
127	C₄×C₁₀₉	—	2	1	no	—
128	C₄×C₁₀₉	—	2	1	no	—
129	C₄×C₁₀₉	—	2	1	no	—
130	C₄×C₁₀₉	—	4	2	no	—
131	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
132	C₄×C₁₀₉	#137	1	2	yes	complemented
133	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
135	Dic₁₀₉	—	2	2	no	—
136	Dic₁₀₉	—	4	1	no	—
137	Dic₁₀₉	—	4	1	no	—
138	C₂×C₁₀₉	—	2	2	no	—
139	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
141	C₂×C₁₀₉	#575	1	1	yes	characteristic
142	D₁₀₉	—	4	2	no	—
143	C₄×#137	—	109	1	no	maximal, Sylow
144	C₁₀₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
145	C₂×#137	—	109	1	no	—
146	C₄×#41	—	109	1	no	—
147	C₂×C₄×Q₈	D₁₀₉	1	1	yes	characteristic, complemented
148	#441	—	109	2	no	—
149	#440	—	109	2	no	—
150	C₄×Q₁₆	—	109	2	no	—
151	C₄×SD₁₆	—	109	2	no	—
152	#428	—	109	1	no	—
153	#419	—	109	1	no	—
154	#417	—	109	1	no	—
155	C₄×M₁₆	—	109	1	no	—
156	C₂×#41	—	109	1	no	—
157	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
158	#137	—	109	8	no	—
159	C₂×Q₁₆	—	109	2	no	—
160	C₂×SD₁₆	—	109	2	no	—
161	C₂×M₁₆	—	109	1	no	—
162	C₂×M₁₆	—	109	1	no	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	—	2	1	no	—
165	C₄×Q₈	—	109	1	no	—
166	C₄×Q₈	—	109	1	no	—
167	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
168	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
169	C₄×D₄	—	109	1	no	—
170	C₄×D₄	—	109	1	no	—
171	C₄×D₄	—	218	1	no	—
172	C₄×D₄	—	218	1	no	—
173	#117	—	109	1	no	—
174	#117	—	218	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	218	1	no	—
179	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
180	#107	—	109	2	no	—
181	#106	—	109	2	no	—
182	#103	—	109	2	no	—
183	#103	—	109	2	no	—
184	#103	—	109	2	no	—
185	#102	—	109	2	no	—
186	#97	—	109	2	no	—
187	C₄×C₈	—	109	2	no	—
188	#41	—	109	4	no	—
189	#41	—	218	2	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	109	1	no	—
193	C₂×Q₈	C₄×D₁₀₉	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
195	C₂×D₄	—	109	1	no	—
196	C₂×D₄	—	218	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	218	1	no	—
200	C₂²×C₄	—	218	1	no	—
201	C₂²×C₄	—	218	1	no	—
202	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
205	Q₁₆	—	109	8	no	—
206	SD₁₆	—	109	8	no	—
207	M₁₆	—	109	4	no	—
208	M₁₆	—	218	2	no	—
209	C₂×C₈	—	109	2	no	—
210	C₂×C₈	—	109	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	109	1	no	—
215	C₄⋊C₄	—	109	1	no	—
216	C₄⋊C₄	—	218	1	no	—
217	C₄⋊C₄	—	218	1	no	—
218	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
219	C₂²⋊C₄	—	218	1	no	—
220	C₂²⋊C₄	—	218	1	no	—
221	C₂²⋊C₄	—	218	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	218	1	no	—
225	C₄²	—	218	1	no	—
226	C₄²	—	218	1	no	—
227	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
228	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
229	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
230	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
231	C₂³	—	218	1	no	—
232	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	109	2	no	—
236	Q₈	—	218	1	no	—
237	Q₈	C₂×C₄×D₁₀₉	1	4	yes	—
238	D₄	—	109	2	no	—
239	D₄	—	218	4	no	—
240	D₄	—	218	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	436	2	no	—
248	C₂×C₄	—	436	2	no	—
249	C₂×C₄	—	218	1	no	—
250	C₂×C₄	—	218	1	no	—
251	C₂×C₄	—	218	1	no	—
252	C₂×C₄	—	218	4	no	—
253	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
254	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
259	C₂×C₄	#13182029	1	2	yes	—
260	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
261	C₈	—	109	4	no	—
262	C₈	—	218	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	436	1	no	—
266	C₂²	—	436	1	no	—
267	C₂²	—	218	2	no	—
268	C₂²	C₄×C₂²⋊D₁₀₉	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	436	1	no	—
276	C₄	—	436	1	no	—
277	C₄	—	218	2	no	—
278	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
279	C₄	#16812730	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
281	C₄	C₂×#13182029	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	436	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
285	C₂	C₂×#16812730	1	1	yes	characteristic
286	C₂	#20295824	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄×Q₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182029	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₀₉
hypocenter	C₁₀₉	C₄×#137
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #20295131	C₂	semidirect
4. #20295124	C₂	semidirect
5. C₄×#13182019	C₂	semidirect
6. C₄×#13182016	C₂	semidirect
7. #20295066	C₂	semidirect
8. #20295035	C₂	semidirect
10. C₄×#13182013	C₂	semidirect
11. C₂×C₄×Q₈×C₁₀₉	C₂	semidirect
13. #16812730	C₄	direct (N × H)
52. #13182019	C₂×C₄	semidirect
53. #13182016	C₂×C₄	semidirect
55. #13182013	C₂×C₄	semidirect
70. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
132. C₄×C₁₀₉	#137	semidirect
144. C₁₀₉	C₄×#137	semidirect
147. C₂×C₄×Q₈	D₁₀₉	semidirect
193. C₂×Q₈	C₄×D₁₀₉	semidirect
279. C₄	#16812730	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×#16812730	2
C₄×#13182029	2
#20295131	2
#20295124	2
C₄×#13182019	2
C₄×#13182016	2
#20295066	2
#20295035	2
#20295025	2
C₄×#13182013	2
C₂×C₄×Q₈×C₁₀₉	2
C₄×#137	109

Subgroups — C4×#16812730 #28371851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812730 #28371851