Subgroups — C₄×#11924567 #18003415

Order 4736.

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×#11924567	C₂	1	1	yes	maximal, characteristic
2	C₄×#10596962	C₂	1	1	yes	maximal, characteristic
3	#14557638	C₂	1	2	yes	maximal, complemented
4	#14557631	C₂	1	2	yes	maximal, complemented
5	C₄×#10596942	C₂	1	2	yes	maximal, complemented
6	C₄×#10596941	C₂	1	2	yes	maximal, complemented
7	#14557573	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557542	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557532	C₂	1	1	yes	maximal, characteristic
10	C₄×#10596934	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	1	yes	characteristic
13	#11924567	C₄	1	8	yes	direct product factor
14	C₂×#10596942	C₂²	1	2	yes	—
15	C₂×#10596941	C₂²	1	2	yes	—
16	C₂×#10596934	C₂²	1	1	yes	characteristic
17	C₂×#10596934	C₂²	1	1	yes	characteristic
18	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
19	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
20	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
21	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₃₇	—	2	1	no	—
23	C₄×C₂²⋊D₃₇	—	2	1	no	—
24	#11924495	C₂²	1	1	yes	characteristic
25	#11924493	—	2	1	no	—
26	C₄²×D₃₇	—	2	1	no	—
27	#11924469	C₂²	1	2	yes	—
28	#11924467	C₂²	1	2	yes	—
29	#11924460	C₂²	1	2	yes	—
30	#11924460	C₂²	1	2	yes	—
31	#11924459	C₂²	1	2	yes	—
32	#11924456	C₂²	1	2	yes	—
33	#11924447	C₂²	1	2	yes	—
34	C₄×#64436	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₃₇	—	2	1	no	—
37	C₄×Q₈×C₃₇	—	2	1	no	—
38	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
39	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
40	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₃₇	—	2	1	no	—
43	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
44	C₂×Q₈×C₃₇	—	2	1	no	—
45	C₂×Q₈×C₃₇	—	2	2	no	—
46	C₂×Q₈×C₃₇	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₃₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₃₇	—	2	1	no	—
49	C₂²×C₄×C₃₇	—	2	1	no	—
50	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₃₇	—	2	1	no	—
54	#10596962	—	2	2	no	—
55	#10596962	C₂×C₄	1	4	yes	—
56	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₃₇	—	2	1	no	—
58	C₂×C₄×D₃₇	—	2	1	no	—
59	C₂×C₄×D₃₇	—	2	1	no	—
60	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₃₇	—	2	2	no	—
62	C₄⋊C₄×C₃₇	—	2	2	no	—
63	C₄⋊C₄×C₃₇	—	2	1	no	—
64	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
65	C₄²×C₃₇	—	2	2	no	—
66	C₄²×C₃₇	—	2	2	no	—
67	C₄²×C₃₇	D₄	1	1	yes	characteristic
68	C₄²×C₃₇	D₄	1	1	yes	characteristic
69	C₄²×C₃₇	C₂³	1	1	yes	characteristic
70	C₄²×C₃₇	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₃₇	—	2	1	no	—
72	#10596942	C₂×C₄	1	8	yes	complemented
73	#10596941	C₂×C₄	1	8	yes	complemented
74	#10596938	—	2	1	no	—
75	#10596938	—	2	1	no	—
76	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
78	#10596936	—	2	1	no	—
79	#10596936	—	2	1	no	—
80	C₄×Dic₃₇	—	2	1	no	—
81	C₄×Dic₃₇	—	2	1	no	—
82	C₄×Dic₃₇	—	2	1	no	—
83	#10596934	—	2	2	no	—
84	#10596934	C₂×C₄	1	4	yes	complemented
85	C₂×#64436	C₂³	1	2	yes	—
86	C₂×#64436	C₂³	1	2	yes	—
87	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₃₇	—	2	1	no	—
89	Q₈×C₃₇	—	2	4	no	—
90	Q₈×C₃₇	—	2	2	no	—
91	Q₈×C₃₇	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₃₇	—	2	4	no	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	—	2	2	no	—
95	C₂×C₄×C₃₇	—	2	2	no	—
96	C₂×C₄×C₃₇	—	4	1	no	—
97	C₂×C₄×C₃₇	—	4	2	no	—
98	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₇	#41	1	2	yes	—
105	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₃₇	—	2	4	no	—
107	C₂×Dic₃₇	—	2	1	no	—
108	C₂×Dic₃₇	—	2	1	no	—
109	C₂×Dic₃₇	—	2	1	no	—
110	D₁₄₈	—	2	1	no	—
111	D₁₄₈	C₂²×C₄	1	2	yes	—
112	C₄×D₃₇	—	2	4	no	—
113	C₄×D₃₇	—	4	2	no	—
114	C₄×D₃₇	—	4	2	no	—
115	Dic₇₄	—	2	1	no	—
116	Dic₇₄	C₂²×C₄	1	2	yes	—
117	#64436	—	2	2	no	—
118	#64436	C₂²×C₄	1	4	yes	—
119	C₂²×C₃₇	—	2	1	no	—
120	C₂²×C₃₇	—	2	1	no	—
121	C₂²×C₃₇	C₄×D₄	1	2	yes	—
122	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₃₇	—	2	2	no	—
124	C₂×D₃₇	—	4	1	no	—
125	C₂×D₃₇	—	4	1	no	—
126	C₄×C₃₇	—	2	4	no	—
127	C₄×C₃₇	—	2	1	no	—
128	C₄×C₃₇	—	2	1	no	—
129	C₄×C₃₇	—	2	1	no	—
130	C₄×C₃₇	—	4	2	no	—
131	C₄×C₃₇	C₄×D₄	1	2	yes	—
132	C₄×C₃₇	#137	1	2	yes	complemented
133	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
135	Dic₃₇	—	2	2	no	—
136	Dic₃₇	—	4	1	no	—
137	Dic₃₇	—	4	1	no	—
138	C₄×#137	—	37	1	no	maximal, Sylow
139	C₂×C₃₇	—	2	2	no	—
140	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₃₇	C₂×#137	1	1	yes	characteristic
142	C₂×C₃₇	#575	1	1	yes	characteristic
143	D₃₇	—	4	2	no	—
144	C₂×#137	—	37	1	no	—
145	C₄×#41	—	37	1	no	—
146	C₂×C₄×Q₈	D₃₇	1	1	yes	characteristic, complemented
147	#441	—	37	2	no	—
148	#440	—	37	2	no	—
149	C₄×Q₁₆	—	37	2	no	—
150	C₄×SD₁₆	—	37	2	no	—
151	#428	—	37	1	no	—
152	#419	—	37	1	no	—
153	#417	—	37	1	no	—
154	C₄×M₁₆	—	37	1	no	—
155	C₃₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	37	1	no	—
157	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
158	#137	—	37	8	no	—
159	C₂×Q₁₆	—	37	2	no	—
160	C₂×SD₁₆	—	37	2	no	—
161	C₂×M₁₆	—	37	1	no	—
162	C₂×M₁₆	—	37	1	no	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	—	2	1	no	—
165	C₄×Q₈	—	37	1	no	—
166	C₄×Q₈	—	37	1	no	—
167	C₄×Q₈	C₂×D₃₇	1	2	yes	—
168	C₄×Q₈	C₂×D₃₇	1	2	yes	—
169	C₄×D₄	—	37	1	no	—
170	C₄×D₄	—	37	1	no	—
171	C₄×D₄	—	74	1	no	—
172	C₄×D₄	—	74	1	no	—
173	#117	—	37	1	no	—
174	#117	—	74	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	74	1	no	—
179	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
180	#107	—	37	2	no	—
181	#106	—	37	2	no	—
182	#103	—	37	2	no	—
183	#103	—	37	2	no	—
184	#103	—	37	2	no	—
185	#102	—	37	2	no	—
186	#97	—	37	2	no	—
187	C₄×C₈	—	37	2	no	—
188	#41	—	37	4	no	—
189	#41	—	74	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	37	1	no	—
193	C₂×Q₈	C₄×D₃₇	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₃₇	1	2	yes	—
195	C₂×D₄	—	37	1	no	—
196	C₂×D₄	—	74	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	74	1	no	—
200	C₂²×C₄	—	74	1	no	—
201	C₂²×C₄	—	74	1	no	—
202	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
205	Q₁₆	—	37	8	no	—
206	SD₁₆	—	37	8	no	—
207	M₁₆	—	37	4	no	—
208	M₁₆	—	74	2	no	—
209	C₂×C₈	—	37	2	no	—
210	C₂×C₈	—	37	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	37	1	no	—
215	C₄⋊C₄	—	37	1	no	—
216	C₄⋊C₄	—	74	1	no	—
217	C₄⋊C₄	—	74	1	no	—
218	C₄⋊C₄	C₂²×D₃₇	1	2	yes	—
219	C₂²⋊C₄	—	74	1	no	—
220	C₂²⋊C₄	—	74	1	no	—
221	C₂²⋊C₄	—	74	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	74	1	no	—
225	C₄²	—	74	1	no	—
226	C₄²	—	74	1	no	—
227	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
228	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
229	C₄²	C₂²×D₃₇	1	1	yes	characteristic
230	C₄²	C₂²×D₃₇	1	1	yes	characteristic
231	C₂³	—	74	1	no	—
232	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	37	2	no	—
236	Q₈	—	74	1	no	—
237	Q₈	C₂×C₄×D₃₇	1	4	yes	—
238	D₄	—	37	2	no	—
239	D₄	—	74	4	no	—
240	D₄	—	74	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	—	74	1	no	—
248	C₂×C₄	—	74	4	no	—
249	C₂×C₄	—	74	1	no	—
250	C₂×C₄	—	74	1	no	—
251	C₂×C₄	—	148	2	no	—
252	C₂×C₄	—	148	2	no	—
253	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
254	C₂×C₄	#10596962	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
261	C₈	—	37	4	no	—
262	C₈	—	74	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	74	2	no	—
266	C₂²	—	148	1	no	—
267	C₂²	—	148	1	no	—
268	C₂²	C₄×C₂²⋊D₃₇	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₃₇	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	—	74	2	no	—
276	C₄	—	148	1	no	—
277	C₄	—	148	1	no	—
278	C₄	C₄×C₂²⋊D₃₇	1	2	yes	—
279	C₄	#11924567	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
281	C₄	C₂×#10596962	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	148	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₃₇	1	1	yes	characteristic
285	C₂	C₂×#11924567	1	1	yes	characteristic
286	C₂	#14558331	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×C₄×Q₈×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#10596962	C₂²
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₃₇
hypocenter	C₃₇	C₄×#137
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
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. #14557638	C₂	semidirect
4. #14557631	C₂	semidirect
5. C₄×#10596942	C₂	semidirect
6. C₄×#10596941	C₂	semidirect
7. #14557573	C₂	semidirect
8. #14557542	C₂	semidirect
10. C₄×#10596934	C₂	semidirect
11. C₂×C₄×Q₈×C₃₇	C₂	semidirect
13. #11924567	C₄	direct (N × H)
46. C₂×Q₈×C₃₇	C₂×C₄	semidirect
72. #10596942	C₂×C₄	semidirect
73. #10596941	C₂×C₄	semidirect
84. #10596934	C₂×C₄	semidirect
132. C₄×C₃₇	#137	semidirect
146. C₂×C₄×Q₈	D₃₇	semidirect
155. C₃₇	C₄×#137	semidirect
193. C₂×Q₈	C₄×D₃₇	semidirect
279. C₄	#11924567	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×#11924567	2
C₄×#10596962	2
#14557638	2
#14557631	2
C₄×#10596942	2
C₄×#10596941	2
#14557573	2
#14557542	2
#14557532	2
C₄×#10596934	2
C₂×C₄×Q₈×C₃₇	2
C₄×#137	37

Subgroups — C4×#11924567 #18003415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11924567 #18003415