Subgroups — C₄×#16812729 #28371849

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×#16812729	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295131	C₂	1	2	yes	maximal, complemented
4	#20295126	C₂	1	2	yes	maximal, complemented
5	C₄×#13182019	C₂	1	2	yes	maximal, complemented
6	C₄×#13182017	C₂	1	2	yes	maximal, complemented
7	#20295069	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295034	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295027	C₂	1	1	yes	maximal, characteristic
10	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
13	#16812729	C₄	1	8	yes	direct product factor
14	C₂×#13182019	C₂²	1	2	yes	—
15	C₂×#13182017	C₂²	1	2	yes	—
16	C₂×#13182013	C₂²	1	1	yes	characteristic
17	C₂×#13182013	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₁₈	—	2	1	no	—
19	C₄×Dic₂₁₈	—	2	1	no	—
20	C₄×Dic₂₁₈	C₂²	1	2	yes	—
21	C₄×Dic₂₁₈	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×#13182008	—	2	1	no	—
24	C₂×C₄×Dic₁₀₉	—	2	1	no	—
25	#16812632	C₂²	1	2	yes	—
26	#16812630	C₂²	1	2	yes	—
27	#16812623	C₂²	1	2	yes	—
28	#16812622	C₂²	1	2	yes	—
29	#16812622	C₂²	1	2	yes	—
30	#16812620	C₂²	1	2	yes	—
31	#16812610	C₂²	1	2	yes	—
32	C₄×#11724784	C₂²	1	2	yes	—
33	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₀₉	—	2	1	no	—
37	C₄×D₄×C₁₀₉	—	2	1	no	—
38	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
40	#117×C₁₀₉	—	2	1	no	—
41	#117×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₁₀₉	—	2	1	no	—
43	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
44	C₂×Dic₂₁₈	—	2	1	no	—
45	C₂×Dic₂₁₈	—	2	2	no	—
46	C₂×Dic₂₁₈	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₂₁₈	C₂³	1	2	yes	—
48	C₂²×Dic₁₀₉	—	2	1	no	—
49	C₂²×Dic₁₀₉	—	2	1	no	—
50	#13182019	C₂×C₄	1	8	yes	complemented
51	#13182017	C₂×C₄	1	8	yes	complemented
52	#13182013	—	2	2	no	—
53	#13182013	C₂×C₄	1	4	yes	complemented
54	C₂×#11724784	C₂³	1	2	yes	—
55	C₂×#11724784	C₂³	1	2	yes	—
56	C₄⋊Dic₁₀₉	—	2	1	no	—
57	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
58	#13182008	—	2	2	no	—
59	#13182008	—	2	2	no	—
60	C₄×Dic₁₀₉	—	2	2	no	—
61	C₄×Dic₁₀₉	—	2	2	no	—
62	#41×C₁₀₉	—	2	2	no	—
63	#41×C₁₀₉	C₂×C₄	1	4	yes	—
64	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₀₉	—	2	1	no	—
66	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	—	2	1	no	—
68	C₂²×C₄×C₁₀₉	—	2	1	no	—
69	C₂²×C₄×C₁₀₉	—	2	1	no	—
70	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₉	—	2	1	no	—
74	C₄⋊C₄×C₁₀₉	—	2	1	no	—
75	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
78	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
79	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
80	C₄²×C₁₀₉	—	2	1	no	—
81	C₄²×C₁₀₉	—	2	1	no	—
82	C₄²×C₁₀₉	—	2	1	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	—	2	1	no	—
88	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₁₀₉	—	2	1	no	—
90	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
91	D₄×C₁₀₉	—	2	4	no	—
92	D₄×C₁₀₉	—	2	1	no	—
93	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	4	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₄×C₁₀₉	—	4	1	no	—
100	C₂×C₄×C₁₀₉	—	4	2	no	—
101	C₂×C₄×C₁₀₉	—	4	2	no	—
102	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	#41	1	2	yes	—
109	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
110	C₂×Dic₁₀₉	—	2	4	no	—
111	C₂×Dic₁₀₉	—	2	2	no	—
112	C₂×Dic₁₀₉	—	2	2	no	—
113	C₂×Dic₁₀₉	—	4	2	no	—
114	Dic₂₁₈	—	2	4	no	—
115	Dic₂₁₈	—	2	2	no	—
116	Dic₂₁₈	C₂²×C₄	1	4	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	—	2	2	no	—
122	C₂²×C₁₀₉	—	4	1	no	—
123	C₂²×C₁₀₉	—	4	1	no	—
124	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
125	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₁₀₉	—	2	2	no	—
127	C₄×C₁₀₉	—	2	1	no	—
128	C₄×C₁₀₉	—	2	1	no	—
129	C₄×C₁₀₉	—	2	1	no	—
130	C₄×C₁₀₉	—	4	1	no	—
131	C₄×C₁₀₉	—	4	1	no	—
132	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
133	C₄×C₁₀₉	#137	1	2	yes	complemented
134	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
136	Dic₁₀₉	—	2	4	no	—
137	Dic₁₀₉	—	4	2	no	—
138	C₂×C₁₀₉	—	2	2	no	—
139	C₂×C₁₀₉	—	4	2	no	—
140	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
142	C₂×C₁₀₉	#575	1	1	yes	characteristic
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	D₁₀₉	1	1	yes	characteristic
147	C₂×C₄×Q₈	—	109	1	no	—
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	C₂×D₁₀₉	1	1	yes	characteristic
157	C₂²×Q₈	—	109	1	no	—
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	—	109	2	no	—
164	C₄×Q₈	—	109	2	no	—
165	C₄×Q₈	—	218	1	no	—
166	C₄×Q₈	—	218	1	no	—
167	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
168	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
172	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	C₂×D₁₀₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	109	1	no	—
176	C₂×C₄⋊C₄	—	218	1	no	—
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	—	2	2	no	—
189	#41	C₄×D₁₀₉	1	4	yes	—
190	C₂×Q₈	—	109	4	no	—
191	C₂×Q₈	—	109	2	no	—
192	C₂×Q₈	—	218	1	no	—
193	C₂×Q₈	—	218	2	no	—
194	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	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	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	109	2	no	—
214	C₄⋊C₄	—	218	2	no	—
215	C₄⋊C₄	—	218	2	no	—
216	C₄⋊C₄	—	218	1	no	—
217	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	218	2	no	—
226	C₄²	—	218	2	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	—	2	1	no	—
232	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
233	Q₈	—	2	1	no	—
234	Q₈	—	109	4	no	—
235	Q₈	—	218	4	no	—
236	Q₈	—	218	2	no	—
237	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	1	no	—
240	D₄	C₂×C₄×D₁₀₉	1	2	yes	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	436	2	no	—
250	C₂×C₄	—	218	2	no	—
251	C₂×C₄	—	218	4	no	—
252	C₂×C₄	—	218	2	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	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	C₄×C₂²⋊D₁₀₉	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	436	2	no	—
277	C₄	—	218	4	no	—
278	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
279	C₄	#16812729	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	—	4	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
285	C₂	C₂×#16812729	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	C₄×#41×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#41×C₁₀₉	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₄×#41	D₁₀₉
hypocenter	C₁₀₉	C₄×#137
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₂₁₈	C₂	semidirect
3. #20295131	C₂	semidirect
4. #20295126	C₂	semidirect
5. C₄×#13182019	C₂	semidirect
6. C₄×#13182017	C₂	semidirect
7. #20295069	C₂	semidirect
8. #20295034	C₂	semidirect
10. C₄×#13182013	C₂	semidirect
13. #16812729	C₄	direct (N × H)
46. C₂×Dic₂₁₈	C₂×C₄	semidirect
50. #13182019	C₂×C₄	semidirect
51. #13182017	C₂×C₄	semidirect
53. #13182013	C₂×C₄	semidirect
133. C₄×C₁₀₉	#137	semidirect
144. C₁₀₉	C₄×#137	semidirect
279. C₄	#16812729	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×#16812729	2
C₂×C₄×Dic₂₁₈	2
#20295131	2
#20295126	2
C₄×#13182019	2
C₄×#13182017	2
#20295069	2
#20295034	2
#20295027	2
C₄×#13182013	2
C₄×#41×C₁₀₉	2
C₄×#137	109

Subgroups — C4×#16812729 #28371849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812729 #28371849