Subgroups — C₄×#17984114 #31789597

Order 17536.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984114	C₂	1	1	yes	maximal, characteristic
2	C₄×#14555510	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577513	C₂	1	2	yes	maximal, complemented
4	#22577503	C₂	1	2	yes	maximal, complemented
5	C₄×#14555498	C₂	1	2	yes	maximal, complemented
6	C₄×#14555495	C₂	1	2	yes	maximal, complemented
7	#22577443	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577408	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577402	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14555494	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	1	yes	characteristic
13	#17984114	C₄	1	8	yes	direct product factor
14	C₂×#14555498	C₂²	1	2	yes	—
15	C₂×#14555495	C₂²	1	2	yes	—
16	C₂×#14555494	C₂²	1	1	yes	characteristic
17	C₂×#14555494	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic, complemented
19	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic, complemented
20	C₄×D₅₄₈	C₂²	1	1	yes	characteristic, complemented
21	C₄×D₅₄₈	C₂²	1	1	yes	characteristic, complemented
22	C₄×C₂²⋊D₁₃₇	—	2	1	no	—
23	C₄×C₂²⋊D₁₃₇	—	2	1	no	—
24	#17984047	C₂²	1	1	yes	characteristic
25	#17984045	—	2	1	no	—
26	C₄²×D₁₃₇	—	2	1	no	—
27	#17984021	C₂²	1	2	yes	complemented
28	#17984019	C₂²	1	2	yes	complemented
29	#17984011	C₂²	1	2	yes	complemented
30	#17984009	C₂²	1	2	yes	—
31	#17984009	C₂²	1	2	yes	—
32	#17984008	C₂²	1	2	yes	complemented
33	#17983999	C₂²	1	2	yes	complemented
34	C₄×#11764040	C₂²	1	2	yes	complemented
35	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
36	C₂³×C₄×C₁₃₇	—	2	1	no	—
37	C₄×D₄×C₁₃₇	—	2	1	no	—
38	C₄×D₄×C₁₃₇	—	2	1	no	—
39	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
40	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
43	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
44	#14555510	—	2	2	no	—
45	#14555510	C₂×C₄	1	4	yes	complemented
46	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
47	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
49	C₂×C₄×D₁₃₇	—	2	1	no	—
50	C₂×C₄×D₁₃₇	—	2	1	no	—
51	C₂×C₄×D₁₃₇	—	2	1	no	—
52	#14555498	C₂×C₄	1	8	yes	complemented
53	#14555495	C₂×C₄	1	8	yes	complemented
54	#14555494	—	2	2	no	—
55	#14555494	C₂×C₄	1	4	yes	complemented
56	C₂×#11764040	C₂³	1	2	yes	—
57	C₂×#11764040	C₂³	1	2	yes	—
58	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
60	#14555489	—	2	1	no	—
61	#14555489	—	2	1	no	—
62	C₂²⋊Dic₁₃₇	—	2	1	no	—
63	#14555487	—	2	1	no	—
64	#14555487	—	2	1	no	—
65	C₄×Dic₁₃₇	—	2	1	no	—
66	C₄×Dic₁₃₇	—	2	1	no	—
67	C₄×Dic₁₃₇	—	2	1	no	—
68	C₂⁴×C₁₃₇	—	2	1	no	—
69	C₂×D₄×C₁₃₇	—	2	1	no	—
70	C₂×D₄×C₁₃₇	—	2	2	no	—
71	C₂×D₄×C₁₃₇	C₂×C₄	1	4	yes	complemented
72	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₃₇	—	2	1	no	—
74	C₂²×C₄×C₁₃₇	—	2	2	no	—
75	C₂²×C₄×C₁₃₇	—	2	2	no	—
76	C₂²×C₄×C₁₃₇	—	4	4	no	—
77	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₃₇	—	2	1	no	—
81	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
82	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
83	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
84	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
85	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
86	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
87	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
88	C₂²×D₁₃₇	—	2	1	no	—
89	Dic₂₇₄	—	2	1	no	—
90	Dic₂₇₄	C₂²×C₄	1	2	yes	complemented
91	D₅₄₈	—	2	1	no	—
92	D₅₄₈	C₂²×C₄	1	2	yes	complemented
93	C₂²⋊D₁₃₇	—	2	4	no	—
94	C₂×Dic₁₃₇	—	2	1	no	—
95	C₂×Dic₁₃₇	—	2	1	no	—
96	C₂×Dic₁₃₇	—	2	1	no	—
97	C₄×D₁₃₇	—	2	4	no	—
98	C₄×D₁₃₇	—	4	2	no	—
99	C₄×D₁₃₇	—	4	2	no	—
100	#11764040	—	2	2	no	—
101	#11764040	C₂²×C₄	1	4	yes	complemented
102	C₂³×C₁₃₇	—	2	4	no	—
103	C₂³×C₁₃₇	—	2	2	no	—
104	C₂³×C₁₃₇	—	4	2	no	—
105	C₂³×C₁₃₇	—	4	2	no	—
106	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
107	D₄×C₁₃₇	—	2	4	no	—
108	D₄×C₁₃₇	—	2	2	no	—
109	D₄×C₁₃₇	C₂²×C₄	1	4	yes	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	—	2	2	no	—
112	C₂×C₄×C₁₃₇	—	4	1	no	—
113	C₂×C₄×C₁₃₇	—	4	4	no	—
114	C₂×C₄×C₁₃₇	—	4	2	no	—
115	C₂×C₄×C₁₃₇	—	4	4	no	—
116	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	#41	1	2	yes	—
123	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
124	C₂²×C₁₃₇	—	2	1	no	—
125	C₂²×C₁₃₇	—	2	1	no	—
126	C₂²×C₁₃₇	—	2	4	no	—
127	C₂²×C₁₃₇	—	4	8	no	—
128	C₂²×C₁₃₇	—	4	2	no	—
129	C₂²×C₁₃₇	—	4	2	no	—
130	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
131	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₃₇	—	2	2	no	—
133	C₂×D₁₃₇	—	4	1	no	—
134	C₂×D₁₃₇	—	4	1	no	—
135	C₄×C₁₃₇	—	2	1	no	—
136	C₄×C₁₃₇	—	2	1	no	—
137	C₄×C₁₃₇	—	2	1	no	—
138	C₄×C₁₃₇	—	4	2	no	—
139	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
140	C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	complemented
141	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
143	Dic₁₃₇	—	2	2	no	—
144	Dic₁₃₇	—	4	1	no	—
145	Dic₁₃₇	—	4	1	no	—
146	C₂×C₁₃₇	—	2	2	no	—
147	C₂×C₁₃₇	—	4	4	no	—
148	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₁₃₇	#575	1	1	yes	characteristic
151	D₁₃₇	—	4	2	no	—
152	C₁₃₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
153	C₄×AGL(1,ℤ/8)	—	137	1	no	maximal, Sylow
154	C₂×AGL(1,ℤ/8)	—	137	1	no	—
155	C₄×#41	—	137	1	no	—
156	C₂×C₄×D₄	D₁₃₇	1	1	yes	characteristic, complemented
157	#442	—	137	2	no	—
158	#440	—	137	2	no	—
159	C₄×SD₁₆	—	137	2	no	—
160	C₄×D₈	—	137	2	no	—
161	#428	—	137	1	no	—
162	#418	—	137	1	no	—
163	#417	—	137	1	no	—
164	C₄×M₁₆	—	137	1	no	—
165	C₂×#41	—	137	1	no	—
166	C₂²×D₄	C₂×D₁₃₇	1	1	yes	characteristic
167	C₂³×C₄	—	2	1	no	—
168	AGL(1,ℤ/8)	—	137	8	no	—
169	C₂×SD₁₆	—	137	2	no	—
170	C₂×D₈	—	137	2	no	—
171	C₂×M₁₆	—	137	1	no	—
172	C₂×M₁₆	—	137	1	no	—
173	C₄×Q₈	—	137	1	no	—
174	C₄×Q₈	—	137	1	no	—
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	274	1	no	—
178	C₄×D₄	—	274	1	no	—
179	C₄×D₄	—	137	1	no	—
180	C₄×D₄	—	137	1	no	—
181	C₄×D₄	C₂×D₁₃₇	1	2	yes	—
182	C₄×D₄	C₂×D₁₃₇	1	2	yes	—
183	#117	—	274	1	no	—
184	#117	—	137	1	no	—
185	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	2	1	no	—
187	C₂×C₄²	—	274	1	no	—
188	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
189	#107	—	137	2	no	—
190	#106	—	137	2	no	—
191	#103	—	137	2	no	—
192	#102	—	137	2	no	—
193	#102	—	137	2	no	—
194	#102	—	137	2	no	—
195	#97	—	137	2	no	—
196	C₄×C₈	—	137	2	no	—
197	C₂⁴	—	2	1	no	—
198	#41	—	274	2	no	—
199	#41	—	137	4	no	—
200	C₂×Q₈	—	137	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	274	1	no	—
204	C₂×D₄	—	137	1	no	—
205	C₂×D₄	C₄×D₁₃₇	1	4	yes	complemented
206	C₂×D₄	C₂²×D₁₃₇	1	2	yes	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	274	1	no	—
212	C₂²×C₄	—	274	1	no	—
213	C₂²×C₄	—	274	1	no	—
214	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
217	SD₁₆	—	137	8	no	—
218	D₈	—	137	8	no	—
219	M₁₆	—	274	2	no	—
220	M₁₆	—	137	4	no	—
221	C₂×C₈	—	137	2	no	—
222	C₂×C₈	—	137	2	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	274	1	no	—
225	C₄⋊C₄	—	274	1	no	—
226	C₄⋊C₄	—	137	1	no	—
227	C₄⋊C₄	—	137	1	no	—
228	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	274	1	no	—
232	C₂²⋊C₄	—	274	1	no	—
233	C₂²⋊C₄	—	274	1	no	—
234	C₄²	—	274	1	no	—
235	C₄²	—	274	1	no	—
236	C₄²	—	274	1	no	—
237	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
238	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
239	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
240	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
241	C₂³	—	2	4	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	274	1	no	—
246	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
247	Q₈	—	274	1	no	—
248	Q₈	—	137	2	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	2	no	—
251	D₄	—	274	4	no	—
252	D₄	—	274	1	no	—
253	D₄	—	137	2	no	—
254	D₄	C₂×C₄×D₁₃₇	1	4	yes	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	274	1	no	—
262	C₂×C₄	—	274	4	no	—
263	C₂×C₄	—	274	1	no	—
264	C₂×C₄	—	274	1	no	—
265	C₂×C₄	—	548	2	no	—
266	C₂×C₄	—	548	2	no	—
267	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
272	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
273	C₂×C₄	#14555510	1	2	yes	—
274	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
275	C₈	—	274	2	no	—
276	C₈	—	137	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	274	2	no	—
284	C₂²	—	548	1	no	—
285	C₂²	—	548	1	no	—
286	C₂²	C₄×C₂²⋊D₁₃₇	1	2	yes	—
287	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	2	no	—
292	C₄	—	274	2	no	—
293	C₄	—	548	1	no	—
294	C₄	—	548	1	no	—
295	C₄	C₄×C₂²⋊D₁₃₇	1	2	yes	—
296	C₄	#17984114	1	2	yes	direct product factor
297	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
298	C₄	C₂×#14555510	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	—	548	2	no	—
302	C₂	C₂×C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
303	C₂	C₂×#17984114	1	1	yes	characteristic
304	C₂	#22578201	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×C₄×D₄×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#17984114	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₃₇
hypocenter	C₁₃₇	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄×#14555510	C₂	semidirect
3. #22577513	C₂	semidirect
4. #22577503	C₂	semidirect
5. C₄×#14555498	C₂	semidirect
6. C₄×#14555495	C₂	semidirect
7. #22577443	C₂	semidirect
8. #22577408	C₂	semidirect
9. #22577402	C₂	semidirect
10. C₄×#14555494	C₂	semidirect
11. C₂×C₄×D₄×C₁₃₇	C₂	semidirect
13. #17984114	C₄	direct (N × H)
18. C₄×Dic₂₇₄	C₂²	semidirect
19. C₄×Dic₂₇₄	C₂²	semidirect
20. C₄×D₅₄₈	C₂²	semidirect
21. C₄×D₅₄₈	C₂²	semidirect
27. #17984021	C₂²	semidirect
28. #17984019	C₂²	semidirect
29. #17984011	C₂²	semidirect
32. #17984008	C₂²	semidirect
33. #17983999	C₂²	semidirect
34. C₄×#11764040	C₂²	semidirect
45. #14555510	C₂×C₄	semidirect
52. #14555498	C₂×C₄	semidirect
53. #14555495	C₂×C₄	semidirect
55. #14555494	C₂×C₄	semidirect
71. C₂×D₄×C₁₃₇	C₂×C₄	semidirect
90. Dic₂₇₄	C₂²×C₄	semidirect
92. D₅₄₈	C₂²×C₄	semidirect
101. #11764040	C₂²×C₄	semidirect
140. C₄×C₁₃₇	AGL(1,ℤ/8)	semidirect
152. C₁₃₇	C₄×AGL(1,ℤ/8)	semidirect
156. C₂×C₄×D₄	D₁₃₇	semidirect
205. C₂×D₄	C₄×D₁₃₇	semidirect
296. C₄	#17984114	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×#17984114	2
C₄×#14555510	2
#22577513	2
#22577503	2
C₄×#14555498	2
C₄×#14555495	2
#22577443	2
#22577408	2
#22577402	2
C₄×#14555494	2
C₂×C₄×D₄×C₁₃₇	2
C₄×AGL(1,ℤ/8)	137

Subgroups — C4×#17984114 #31789597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984114 #31789597