Subgroups — C₄×#17984121 #31789612

Order 17536.

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×#17984121	C₂	1	1	yes	maximal, characteristic
2	C₄×#14555510	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577511	C₂	1	2	yes	maximal, complemented
5	#22577505	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₅₄₈	C₂	1	2	yes	maximal, complemented
7	C₄×#14555499	C₂	1	2	yes	maximal, complemented
8	#22577445	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577413	C₂	1	1	yes	maximal, characteristic, complemented
10	#22577405	C₂	1	1	yes	maximal, characteristic
11	C₄×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₇₄	C₂²	1	1	yes	characteristic
14	#17984121	C₄	1	8	yes	direct product factor
15	C₂×Dic₅₄₈	C₂²	1	2	yes	—
16	C₂×#14555499	C₂²	1	2	yes	—
17	C₄×Dic₂₇₄	—	2	1	no	—
18	C₄×Dic₂₇₄	—	2	1	no	—
19	C₄×Dic₂₇₄	C₂²	1	2	yes	—
20	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₇₄	C₂²	1	2	yes	—
22	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
23	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
24	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₃₇	—	2	1	no	—
26	C₄×C₂²⋊D₁₃₇	—	2	1	no	—
27	#17984047	C₂²	1	1	yes	characteristic
28	#17984045	—	2	1	no	—
29	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×#14555489	—	2	1	no	—
31	C₂×C₄×Dic₁₃₇	—	2	1	no	—
32	C₄²×D₁₃₇	—	2	1	no	—
33	#17984022	C₂²	1	2	yes	—
34	#17984020	C₂²	1	2	yes	—
35	#17984013	C₂²	1	2	yes	—
36	#17984013	C₂²	1	2	yes	—
37	#17984013	C₂²	1	2	yes	—
38	#17984010	C₂²	1	2	yes	—
39	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
42	#97×C₁₃₇	C₂²	1	2	yes	—
43	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
44	#14555510	—	2	2	no	—
45	#14555510	C₂×C₄	1	4	yes	—
46	C₂×Dic₂₇₄	—	2	1	no	—
47	C₂×Dic₂₇₄	—	2	2	no	—
48	C₂×Dic₂₇₄	C₂×C₄	1	4	yes	complemented
49	C₂×Dic₂₇₄	C₂³	1	2	yes	—
50	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
51	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
53	C₂²×Dic₁₃₇	—	2	1	no	—
54	C₂²×Dic₁₃₇	—	2	1	no	—
55	C₂×C₄×D₁₃₇	—	2	1	no	—
56	C₂×C₄×D₁₃₇	—	2	1	no	—
57	C₂×C₄×D₁₃₇	—	2	1	no	—
58	Dic₅₄₈	C₂×C₄	1	8	yes	complemented
59	#14555499	C₂×C₄	1	8	yes	complemented
60	C₄⋊Dic₁₃₇	—	2	1	no	—
61	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
62	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
63	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
64	#14555489	—	2	2	no	—
65	#14555489	—	2	2	no	—
66	#14555489	—	2	1	no	—
67	#14555489	—	2	1	no	—
68	C₂²⋊Dic₁₃₇	—	2	1	no	—
69	#14555487	—	2	1	no	—
70	#14555487	—	2	1	no	—
71	C₄×Dic₁₃₇	—	2	1	no	—
72	C₄×Dic₁₃₇	—	2	1	no	—
73	C₄×Dic₁₃₇	—	2	1	no	—
74	C₄×Dic₁₃₇	—	2	2	no	—
75	C₄×Dic₁₃₇	—	2	2	no	—
76	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	M₁₆×C₁₃₇	—	2	2	no	—
80	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
81	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
82	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
83	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
84	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
85	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
86	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
87	C₂²×D₁₃₇	—	2	1	no	—
88	Dic₂₇₄	—	2	4	no	—
89	Dic₂₇₄	—	2	2	no	—
90	Dic₂₇₄	—	2	1	no	—
91	Dic₂₇₄	C₂²×C₄	1	4	yes	—
92	Dic₂₇₄	C₂²×C₄	1	2	yes	—
93	D₅₄₈	—	2	1	no	—
94	D₅₄₈	C₂²×C₄	1	2	yes	—
95	C₂²⋊D₁₃₇	—	2	4	no	—
96	C₂×Dic₁₃₇	—	2	4	no	—
97	C₂×Dic₁₃₇	—	2	1	no	—
98	C₂×Dic₁₃₇	—	2	2	no	—
99	C₂×Dic₁₃₇	—	2	1	no	—
100	C₂×Dic₁₃₇	—	2	1	no	—
101	C₂×Dic₁₃₇	—	2	2	no	—
102	C₂×Dic₁₃₇	—	4	2	no	—
103	C₄×D₁₃₇	—	2	4	no	—
104	C₄×D₁₃₇	—	4	2	no	—
105	C₄×D₁₃₇	—	4	2	no	—
106	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	4	1	no	—
109	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	#41	1	2	yes	—
116	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
117	C₈×C₁₃₇	—	2	2	no	—
118	C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
119	C₂²×C₁₃₇	—	2	1	no	—
120	C₂²×C₁₃₇	—	2	1	no	—
121	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
122	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₃₇	—	2	2	no	—
124	C₂×D₁₃₇	—	4	1	no	—
125	C₂×D₁₃₇	—	4	1	no	—
126	C₄×C₁₃₇	—	2	1	no	—
127	C₄×C₁₃₇	—	2	1	no	—
128	C₄×C₁₃₇	—	2	1	no	—
129	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
130	C₄×C₁₃₇	#137	1	2	yes	complemented
131	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
133	Dic₁₃₇	—	2	2	no	—
134	Dic₁₃₇	—	2	4	no	—
135	Dic₁₃₇	—	4	1	no	—
136	Dic₁₃₇	—	4	1	no	—
137	Dic₁₃₇	—	4	2	no	—
138	C₂×C₁₃₇	—	2	2	no	—
139	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₃₇	C₂×#137	1	1	yes	characteristic
141	C₂×C₁₃₇	#575	1	1	yes	characteristic
142	D₁₃₇	—	4	2	no	—
143	C₁₃₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
144	C₄×#137	—	137	1	no	maximal, Sylow
145	C₂×#137	—	137	1	no	—
146	C₄×#41	—	137	1	no	—
147	C₂×C₄×Q₈	—	137	1	no	—
148	#441	—	137	2	no	—
149	#440	—	137	2	no	—
150	C₄×Q₁₆	—	137	2	no	—
151	C₄×SD₁₆	—	137	2	no	—
152	#428	—	137	1	no	—
153	#419	—	137	1	no	—
154	#417	—	137	1	no	—
155	C₄×M₁₆	D₁₃₇	1	1	yes	characteristic, complemented
156	C₂×#41	—	137	1	no	—
157	C₂²×Q₈	—	137	1	no	—
158	#137	—	137	8	no	—
159	C₂×Q₁₆	—	137	2	no	—
160	C₂×SD₁₆	—	137	2	no	—
161	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
162	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
163	C₄×Q₈	—	274	1	no	—
164	C₄×Q₈	—	274	1	no	—
165	C₄×Q₈	—	137	2	no	—
166	C₄×Q₈	—	137	2	no	—
167	C₄×Q₈	—	137	1	no	—
168	C₄×Q₈	—	137	1	no	—
169	C₄×D₄	—	274	1	no	—
170	C₄×D₄	—	274	1	no	—
171	C₄×D₄	—	137	1	no	—
172	C₄×D₄	—	137	1	no	—
173	#117	—	274	1	no	—
174	#117	—	137	1	no	—
175	C₂×C₄⋊C₄	—	274	1	no	—
176	C₂×C₄⋊C₄	—	137	1	no	—
177	C₂×C₄²	—	274	1	no	—
178	C₂×C₄²	—	274	1	no	—
179	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
180	#107	—	137	2	no	—
181	#106	—	137	2	no	—
182	#103	—	137	2	no	—
183	#103	—	137	2	no	—
184	#103	—	137	2	no	—
185	#102	—	137	2	no	—
186	#97	C₂×D₁₃₇	1	2	yes	—
187	C₄×C₈	C₂×D₁₃₇	1	2	yes	—
188	#41	—	274	2	no	—
189	#41	—	137	4	no	—
190	C₂×Q₈	—	274	1	no	—
191	C₂×Q₈	—	274	2	no	—
192	C₂×Q₈	—	137	4	no	—
193	C₂×Q₈	—	137	2	no	—
194	C₂×Q₈	—	137	1	no	—
195	C₂×D₄	—	274	1	no	—
196	C₂×D₄	—	137	1	no	—
197	C₂²×C₄	—	274	1	no	—
198	C₂²×C₄	—	274	1	no	—
199	C₂²×C₄	—	274	1	no	—
200	C₂²×C₄	—	274	1	no	—
201	C₂²×C₄	—	274	1	no	—
202	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
203	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
205	Q₁₆	—	137	8	no	—
206	SD₁₆	—	137	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	C₄×D₁₃₇	1	4	yes	complemented
209	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
210	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
211	C₄⋊C₄	—	274	2	no	—
212	C₄⋊C₄	—	274	2	no	—
213	C₄⋊C₄	—	274	1	no	—
214	C₄⋊C₄	—	274	1	no	—
215	C₄⋊C₄	—	274	1	no	—
216	C₄⋊C₄	—	137	2	no	—
217	C₄⋊C₄	—	137	1	no	—
218	C₄⋊C₄	—	137	1	no	—
219	C₂²⋊C₄	—	274	1	no	—
220	C₂²⋊C₄	—	274	1	no	—
221	C₂²⋊C₄	—	274	1	no	—
222	C₄²	—	274	1	no	—
223	C₄²	—	274	1	no	—
224	C₄²	—	274	2	no	—
225	C₄²	—	274	1	no	—
226	C₄²	—	274	2	no	—
227	C₄²	D₅₄₈	1	1	yes	characteristic
228	C₄²	D₅₄₈	1	1	yes	characteristic
229	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
230	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
231	C₂³	—	274	1	no	—
232	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
233	Q₈	—	274	4	no	—
234	Q₈	—	274	2	no	—
235	Q₈	—	274	1	no	—
236	Q₈	—	137	4	no	—
237	Q₈	—	137	2	no	—
238	D₄	—	274	4	no	—
239	D₄	—	274	1	no	—
240	D₄	—	137	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	274	4	no	—
244	C₂×C₄	—	274	1	no	—
245	C₂×C₄	—	274	2	no	—
246	C₂×C₄	—	274	4	no	—
247	C₂×C₄	—	274	1	no	—
248	C₂×C₄	—	274	1	no	—
249	C₂×C₄	—	274	2	no	—
250	C₂×C₄	—	548	2	no	—
251	C₂×C₄	—	548	2	no	—
252	C₂×C₄	—	548	2	no	—
253	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
254	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
255	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
256	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
257	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
258	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
259	C₂×C₄	#14555510	1	2	yes	—
260	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	C₂×C₄×D₁₃₇	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	274	2	no	—
266	C₂²	—	548	1	no	—
267	C₂²	—	548	1	no	—
268	C₂²	C₄×D₅₄₈	1	2	yes	—
269	C₂²	C₂²×D₅₄₈	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	274	2	no	—
274	C₄	—	274	4	no	—
275	C₄	—	548	1	no	—
276	C₄	—	548	1	no	—
277	C₄	—	548	2	no	—
278	C₄	C₄×D₅₄₈	1	2	yes	—
279	C₄	#17984121	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
281	C₄	C₂×#14555510	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	548	2	no	—
284	C₂	C₂×C₄×D₅₄₈	1	1	yes	characteristic
285	C₂	C₂×#17984121	1	1	yes	characteristic
286	C₂	#22578203	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₄₈
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₄×M₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#14555510	C₂²
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₄×M₁₆	D₁₃₇
hypocenter	C₁₃₇	C₄×#137
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
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×Dic₂₇₄	C₂	semidirect
4. #22577511	C₂	semidirect
5. #22577505	C₂	semidirect
6. C₄×Dic₅₄₈	C₂	semidirect
7. C₄×#14555499	C₂	semidirect
8. #22577445	C₂	semidirect
9. #22577413	C₂	semidirect
11. C₄×M₁₆×C₁₃₇	C₂	semidirect
14. #17984121	C₄	direct (N × H)
48. C₂×Dic₂₇₄	C₂×C₄	semidirect
58. Dic₅₄₈	C₂×C₄	semidirect
59. #14555499	C₂×C₄	semidirect
80. M₁₆×C₁₃₇	C₂×C₄	semidirect
130. C₄×C₁₃₇	#137	semidirect
143. C₁₃₇	C₄×#137	semidirect
155. C₄×M₁₆	D₁₃₇	semidirect
208. M₁₆	C₄×D₁₃₇	semidirect
279. C₄	#17984121	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×#17984121	2
C₄×#14555510	2
C₂×C₄×Dic₂₇₄	2
#22577511	2
#22577505	2
C₄×Dic₅₄₈	2
C₄×#14555499	2
#22577445	2
#22577413	2
#22577405	2
C₄×M₁₆×C₁₃₇	2
C₄×#137	137

Subgroups — C4×#17984121 #31789612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984121 #31789612