Subgroups — #18010266

Order 4736.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924615	C₂	1	1	yes	maximal, characteristic
3	#14558400	C₂	1	4	yes	maximal, complemented
4	#117×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596953	—	2	1	no	—
6	D₃₇⋊M₁₆	—	2	1	no	—
7	D₃₇⋊M₁₆	—	2	1	no	—
8	D₃₇⋊M₁₆	C₄	1	2	yes	complemented
9	D₃₇⋊M₁₆	C₂²	1	2	yes	—
10	C₂×#10596957	—	2	1	no	—
11	C₂×D₃₇⋊C₈	C₄	1	2	yes	complemented
12	#11924615	—	2	1	no	—
13	#11924615	—	2	1	no	—
14	#11924615	C₄	1	2	yes	complemented
15	#11924615	C₂²	1	2	yes	—
16	C₂×C₄⋊#3866	—	2	1	no	—
17	C₂×C₂²⋊#3866	—	2	1	no	—
18	C₂×C₄×#3866	C₄	1	2	yes	complemented
19	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
20	#11924495	C₄	1	1	yes	characteristic, complemented
21	#11924494	—	2	1	no	—
22	#11924493	C₂²	1	2	yes	—
23	#11924492	—	2	1	no	—
24	C₄⋊C₄×D₃₇	—	2	1	no	—
25	C₂²⋊C₄×D₃₇	—	2	1	no	—
26	C₄²×D₃₇	C₄	1	2	yes	complemented
27	#117×C₃₇	C₄	1	1	yes	characteristic, complemented
28	C₂³×D₃₇	D₄	1	1	yes	characteristic
29	C₂²×#3866	—	2	2	no	—
30	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
32	C₂×C₄×D₃₇	—	2	2	no	—
33	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
39	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
40	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
41	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
42	C₂×C₄×D₃₇	Q₈	1	1	yes	characteristic
43	C₂×C₄×D₃₇	Q₈	1	1	yes	characteristic
44	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
45	C₂²⋊#3866	—	2	2	no	—
46	C₂²⋊#3866	—	2	2	no	—
47	#10596957	—	2	2	no	—
48	#10596957	—	2	2	no	—
49	C₂×#64438	—	2	2	no	—
50	C₄⋊#3866	—	2	2	no	—
51	C₄⋊#3866	—	2	2	no	—
52	C₄×#3866	—	2	2	no	—
53	C₄×#3866	C₂×C₄	1	4	yes	complemented
54	#10596953	—	2	2	no	—
55	#10596953	—	2	2	no	—
56	D₃₇⋊C₈	—	2	2	no	—
57	D₃₇⋊C₈	C₂×C₄	1	4	yes	complemented
58	C₄⋊C₄×C₃₇	—	2	1	no	—
59	C₂²⋊C₄×C₃₇	—	2	1	no	—
60	C₄²×C₃₇	C₂×C₄	1	2	yes	complemented
61	C₂²⋊Dic₃₇	—	2	1	no	—
62	#10596938	—	2	2	no	—
63	C₄⋊Dic₃₇	—	2	1	no	—
64	#10596936	—	2	2	no	—
65	C₄×Dic₃₇	—	2	2	no	—
66	C₄×Dic₃₇	C₂×C₄	1	2	yes	complemented
67	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×D₃₇	—	4	2	no	—
69	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₃₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×D₃₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂×#3866	—	2	4	no	—
76	C₂×#3866	—	4	4	no	—
77	C₂×C₄×C₃₇	—	2	2	no	—
78	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
79	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
80	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₃₇	—	2	2	no	—
85	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
91	C₄×D₃₇	—	4	4	no	—
92	C₄×D₃₇	C₄²	1	1	yes	characteristic
93	C₄×D₃₇	C₄²	1	1	yes	characteristic
94	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
100	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
102	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
103	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
106	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
107	C₄×D₃₇	C₂×Q₈	1	1	yes	characteristic
108	#64438	—	2	4	no	—
109	C₂²×C₃₇	—	4	1	no	—
110	C₂²×C₃₇	#95	1	1	yes	characteristic
111	C₂²×C₃₇	#95	1	1	yes	characteristic
112	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×D₃₇	—	2	2	no	—
114	C₂×D₃₇	—	2	2	no	—
115	C₂×D₃₇	—	2	2	no	—
116	C₂×D₃₇	—	4	1	no	—
117	C₂×D₃₇	—	4	1	no	—
118	C₂×D₃₇	—	4	1	no	—
119	C₂×D₃₇	#95	1	1	yes	characteristic
120	C₂×D₃₇	#95	1	1	yes	characteristic
121	C₂×D₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×D₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	#3866	—	4	4	no	—
124	C₄×C₃₇	—	4	2	no	—
125	C₄×C₃₇	#95	1	1	yes	characteristic
126	C₄×C₃₇	#95	1	1	yes	characteristic
127	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
128	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Dic₃₇	—	4	2	no	—
130	Dic₃₇	#95	1	1	yes	characteristic
131	Dic₃₇	#95	1	1	yes	characteristic
132	Dic₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	Dic₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×H(3,ℤ/4)	—	37	1	no	maximal, Sylow
135	C₂×C₃₇	—	2	1	no	—
136	C₂×C₃₇	—	2	1	no	—
137	C₂×C₃₇	—	2	1	no	—
138	C₂×C₃₇	C₂×#95	1	1	yes	characteristic
139	D₃₇	—	2	1	no	—
140	D₃₇	—	2	1	no	—
141	D₃₇	—	2	1	no	—
142	D₃₇	H(3,ℤ/4)	1	2	yes	complemented
143	C₂²×M₁₆	—	37	1	no	—
144	C₂×#117	—	37	1	no	—
145	C₂×#117	—	37	1	no	—
146	H(3,ℤ/4)	—	37	4	no	—
147	C₃₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	—	37	1	no	—
149	C₂×M₁₆	—	37	2	no	—
150	C₂×M₁₆	—	37	2	no	—
151	C₂×M₁₆	—	74	1	no	—
152	C₂×M₁₆	—	74	1	no	—
153	C₂×M₁₆	—	74	1	no	—
154	C₂×M₁₆	—	74	1	no	—
155	C₂²×C₈	—	37	2	no	—
156	#117	—	37	2	no	—
157	#117	—	37	2	no	—
158	#117	—	37	1	no	—
159	#117	—	37	2	no	—
160	#117	—	74	1	no	—
161	#117	—	74	1	no	—
162	#117	—	74	1	no	—
163	#117	—	74	1	no	—
164	#117	#3866	1	1	yes	characteristic, complemented
165	C₂×C₄⋊C₄	—	74	1	no	—
166	C₂×C₄⋊C₄	—	74	1	no	—
167	C₂×C₂²⋊C₄	—	74	1	no	—
168	C₂×C₂²⋊C₄	—	74	1	no	—
169	C₂×C₄²	—	37	2	no	—
170	C₂×C₄²	—	37	2	no	—
171	C₂⁴	—	37	1	no	—
172	C₂²×C₄	—	37	1	no	—
173	C₂²×C₄	—	37	1	no	—
174	C₂²×C₄	—	37	1	no	—
175	C₂²×C₄	—	37	1	no	—
176	C₂²×C₄	—	37	1	no	—
177	C₂²×C₄	—	37	1	no	—
178	C₂²×C₄	—	37	1	no	—
179	C₂²×C₄	—	37	1	no	—
180	C₂²×C₄	—	37	1	no	—
181	C₂²×C₄	—	37	1	no	—
182	C₂²×C₄	—	37	1	no	—
183	C₂²×C₄	—	37	1	no	—
184	C₂²×C₄	—	37	1	no	—
185	C₂²×C₄	—	74	2	no	—
186	C₂²×C₄	—	74	2	no	—
187	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
188	M₁₆	—	74	2	no	—
189	M₁₆	—	74	2	no	—
190	M₁₆	—	74	2	no	—
191	M₁₆	—	74	2	no	—
192	C₂×C₈	—	37	4	no	—
193	C₂×C₈	—	74	2	no	—
194	C₂×C₈	—	74	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	74	2	no	—
197	C₄⋊C₄	—	74	2	no	—
198	C₄⋊C₄	—	74	2	no	—
199	C₄⋊C₄	—	74	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	74	2	no	—
202	C₂²⋊C₄	—	74	2	no	—
203	C₂²⋊C₄	—	74	2	no	—
204	C₂²⋊C₄	—	74	1	no	—
205	C₄²	—	37	4	no	—
206	C₄²	—	37	2	no	—
207	C₄²	—	74	2	no	—
208	C₄²	—	74	2	no	—
209	C₄²	C₂×#3866	1	2	yes	complemented
210	C₂³	—	37	1	no	—
211	C₂³	—	37	1	no	—
212	C₂³	—	37	1	no	—
213	C₂³	—	37	1	no	—
214	C₂³	—	37	1	no	—
215	C₂³	—	37	1	no	—
216	C₂³	—	148	2	no	—
217	C₂³	C₂²⋊#3866	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	37	1	no	—
220	C₂×C₄	—	37	1	no	—
221	C₂×C₄	—	37	1	no	—
222	C₂×C₄	—	37	1	no	—
223	C₂×C₄	—	37	1	no	—
224	C₂×C₄	—	37	1	no	—
225	C₂×C₄	—	37	1	no	—
226	C₂×C₄	—	37	1	no	—
227	C₂×C₄	—	37	1	no	—
228	C₂×C₄	—	37	1	no	—
229	C₂×C₄	—	37	1	no	—
230	C₂×C₄	—	37	1	no	—
231	C₂×C₄	—	37	1	no	—
232	C₂×C₄	—	37	1	no	—
233	C₂×C₄	—	37	1	no	—
234	C₂×C₄	—	37	1	no	—
235	C₂×C₄	—	37	1	no	—
236	C₂×C₄	—	37	1	no	—
237	C₂×C₄	—	37	1	no	—
238	C₂×C₄	—	37	1	no	—
239	C₂×C₄	—	37	1	no	—
240	C₂×C₄	—	37	1	no	—
241	C₂×C₄	—	74	4	no	—
242	C₂×C₄	—	74	2	no	—
243	C₂×C₄	—	148	4	no	—
244	C₂×C₄	—	148	4	no	—
245	C₂×C₄	C₄×#3866	1	1	yes	characteristic
246	C₂×C₄	C₄×#3866	1	1	yes	characteristic
247	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
248	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
249	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
250	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
251	C₈	—	74	4	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	37	1	no	—
254	C₂²	—	37	1	no	—
255	C₂²	—	37	1	no	—
256	C₂²	—	37	1	no	—
257	C₂²	—	74	2	no	—
258	C₂²	—	74	2	no	—
259	C₂²	—	74	2	no	—
260	C₂²	—	148	1	no	—
261	C₂²	—	148	1	no	—
262	C₂²	—	148	1	no	—
263	C₂²	(C₂×#3866)⋊C₄	1	1	yes	characteristic
264	C₂²	(C₂×#3866)⋊C₄	1	1	yes	characteristic
265	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
266	C₄	—	4	2	no	—
267	C₄	—	37	1	no	—
268	C₄	—	37	1	no	—
269	C₄	—	37	1	no	—
270	C₄	—	37	1	no	—
271	C₄	—	148	2	no	—
272	C₄	—	148	4	no	—
273	C₄	(C₂×#3866)⋊C₄	1	1	yes	characteristic
274	C₄	(C₂×#3866)⋊C₄	1	1	yes	characteristic
275	C₄	C₂×C₄×#3866	1	1	yes	characteristic
276	C₄	C₂×C₄⋊#3866	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	37	2	no	—
281	C₂	—	74	1	no	—
282	C₂	—	74	1	no	—
283	C₂	—	74	1	no	—
284	C₂	C₂×(C₂×#3866)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄×#3866
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#3866
Fitting	#117×C₃₇	C₄
socle	C₂×C₃₇	C₂×#95
prime radical	C₂³×D₃₇	D₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#117	#3866
hypocenter	C₃₇	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #14558400	C₂	semidirect
8. D₃₇⋊M₁₆	C₄	semidirect
11. C₂×D₃₇⋊C₈	C₄	semidirect
14. #11924615	C₄	semidirect
18. C₂×C₄×#3866	C₄	semidirect
20. #11924495	C₄	semidirect
26. C₄²×D₃₇	C₄	semidirect
27. #117×C₃₇	C₄	semidirect
53. C₄×#3866	C₂×C₄	semidirect
57. D₃₇⋊C₈	C₂×C₄	semidirect
60. C₄²×C₃₇	C₂×C₄	semidirect
66. C₄×Dic₃₇	C₂×C₄	semidirect
142. D₃₇	H(3,ℤ/4)	semidirect
147. C₃₇	C₂×H(3,ℤ/4)	semidirect
164. #117	#3866	semidirect
209. C₄²	C₂×#3866	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₃₇⋊M₁₆	2
C₂×#11924615	2
#14558400	2
#117×D₃₇	2
C₂×H(3,ℤ/4)	37