Subgroups — #18010306

Order 4736.

388 conjugacy classes of subgroups.

The subgroup lattice has 280 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	#14558415	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×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	C₄	1	2	yes	complemented
11	C₂×D₃₇⋊C₈	—	2	1	no	—
12	#11924615	—	2	1	no	—
13	#11924615	—	2	1	no	—
14	#11924615	C₄	1	2	yes	—
15	#11924615	C₂²	1	2	yes	—
16	C₂×C₄⋊#3866	—	2	1	no	—
17	C₂×C₂²⋊#3866	C₄	1	2	yes	—
18	C₂×C₄×#3866	—	2	1	no	—
19	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
20	C₂×#10596934	C₄	1	1	yes	characteristic, complemented
21	C₂×#10596929	C₂²	1	2	yes	—
22	M₁₆×D₃₇	—	2	1	no	—
23	M₁₆×D₃₇	—	2	1	no	—
24	M₁₆×D₃₇	—	2	1	no	—
25	M₁₆×D₃₇	—	2	1	no	—
26	C₂×C₈×D₃₇	C₄	1	2	yes	complemented
27	C₂×M₁₆×C₃₇	C₄	1	1	yes	characteristic, complemented
28	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×#3866	—	2	2	no	—
30	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
36	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
37	C₂×C₄×D₃₇	D₄	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	Q₈	1	1	yes	characteristic
42	C₂×C₄×D₃₇	Q₈	1	1	yes	characteristic
43	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
44	C₂²⋊#3866	—	2	2	no	—
45	C₂²⋊#3866	C₂×C₄	1	4	yes	—
46	#10596957	—	2	2	no	—
47	#10596957	C₂×C₄	1	4	yes	complemented
48	C₂×#64438	—	2	2	no	—
49	C₄⋊#3866	—	2	2	no	—
50	C₄⋊#3866	—	2	2	no	—
51	C₄×#3866	—	2	2	no	—
52	C₄×#3866	—	2	2	no	—
53	#10596953	—	2	2	no	—
54	#10596953	—	2	2	no	—
55	D₃₇⋊C₈	—	2	2	no	—
56	D₃₇⋊C₈	—	2	2	no	—
57	M₁₆×C₃₇	—	2	1	no	—
58	M₁₆×C₃₇	—	2	1	no	—
59	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	complemented
60	#10596934	—	2	1	no	—
61	#10596934	—	2	1	no	—
62	C₂×#64436	C₂×C₄	1	2	yes	complemented
63	#10596929	—	2	2	no	—
64	#10596929	—	2	2	no	—
65	C₈×D₃₇	—	2	2	no	—
66	C₈×D₃₇	—	2	2	no	—
67	C₂³×C₃₇	C₄²	1	1	yes	characteristic
68	C₂²×D₃₇	—	4	2	no	—
69	C₂²×D₃₇	C₄²	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	C₂²×C₄	1	1	yes	characteristic
75	C₂×#3866	—	2	4	no	—
76	C₂×#3866	—	4	4	no	—
77	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
78	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₃₇	C₄²	1	1	yes	characteristic
84	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
96	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
97	C₄×D₃₇	C₄⋊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	C₂×D₄	1	1	yes	characteristic
102	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
103	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
104	C₄×D₃₇	C₂×Q₈	1	1	yes	characteristic
105	#64438	—	2	4	no	—
106	C₈×C₃₇	—	2	2	no	—
107	#64436	—	2	2	no	—
108	C₂²×C₃₇	—	4	1	no	—
109	C₂²×C₃₇	#95	1	1	yes	characteristic
110	C₂²×C₃₇	#95	1	1	yes	characteristic
111	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
112	C₂×D₃₇	—	2	2	no	—
113	C₂×D₃₇	—	2	2	no	—
114	C₂×D₃₇	—	2	2	no	—
115	C₂×D₃₇	—	4	1	no	—
116	C₂×D₃₇	—	4	1	no	—
117	C₂×D₃₇	—	4	1	no	—
118	C₂×D₃₇	#95	1	1	yes	characteristic
119	C₂×D₃₇	#95	1	1	yes	characteristic
120	C₂×D₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×D₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	#3866	—	4	4	no	—
123	C₄×C₃₇	#95	1	1	yes	characteristic
124	C₄×C₃₇	#95	1	1	yes	characteristic
125	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	Dic₃₇	#95	1	1	yes	characteristic
128	Dic₃₇	#95	1	1	yes	characteristic
129	Dic₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	Dic₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×#344	—	37	1	no	maximal, Sylow
132	C₂×C₃₇	—	2	1	no	—
133	C₂×C₃₇	—	2	1	no	—
134	C₂×C₃₇	—	2	1	no	—
135	C₂×C₃₇	C₂×#95	1	1	yes	characteristic
136	D₃₇	—	2	1	no	—
137	D₃₇	—	2	1	no	—
138	D₃₇	—	2	1	no	—
139	D₃₇	#344	1	2	yes	complemented
140	C₂²×M₁₆	—	37	1	no	—
141	C₂²×M₁₆	—	37	1	no	—
142	C₂×#117	—	37	1	no	—
143	#344	—	37	4	no	—
144	C₃₇	C₂×#344	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	—	37	1	no	—
146	C₂×M₁₆	—	37	1	no	—
147	C₂×M₁₆	—	37	2	no	—
148	C₂×M₁₆	—	37	2	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	C₂×M₁₆	—	74	1	no	—
156	C₂×M₁₆	—	74	1	no	—
157	C₂×M₁₆	—	74	1	no	—
158	C₂×M₁₆	#3866	1	1	yes	characteristic, complemented
159	C₂²×C₈	—	37	2	no	—
160	C₂²×C₈	—	74	1	no	—
161	#117	—	37	2	no	—
162	#117	—	37	2	no	—
163	#117	—	74	1	no	—
164	#117	—	74	1	no	—
165	C₂×C₄⋊C₄	—	74	1	no	—
166	C₂×C₂²⋊C₄	—	37	2	no	—
167	C₂×C₄²	—	74	1	no	—
168	C₂⁴	—	37	1	no	—
169	C₂²×C₄	—	37	1	no	—
170	C₂²×C₄	—	37	1	no	—
171	C₂²×C₄	—	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	—	74	2	no	—
183	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
184	M₁₆	—	2	1	no	—
185	M₁₆	—	2	1	no	—
186	M₁₆	—	37	4	no	—
187	M₁₆	—	74	2	no	—
188	M₁₆	—	74	2	no	—
189	M₁₆	—	74	2	no	—
190	M₁₆	—	74	2	no	—
191	M₁₆	—	74	2	no	—
192	M₁₆	—	74	1	no	—
193	M₁₆	—	74	1	no	—
194	C₂×C₈	—	37	2	no	—
195	C₂×C₈	—	74	2	no	—
196	C₂×C₈	—	74	2	no	—
197	C₂×C₈	—	74	2	no	—
198	C₂×C₈	—	74	2	no	—
199	C₂×C₈	—	74	2	no	—
200	C₂×C₈	C₂×#3866	1	2	yes	complemented
201	C₄⋊C₄	—	74	2	no	—
202	C₄⋊C₄	—	74	2	no	—
203	C₂²⋊C₄	—	37	4	no	—
204	C₂²⋊C₄	—	74	2	no	—
205	C₄²	—	74	2	no	—
206	C₄²	—	74	2	no	—
207	C₂³	—	37	1	no	—
208	C₂³	—	37	1	no	—
209	C₂³	—	37	1	no	—
210	C₂³	—	37	1	no	—
211	C₂³	—	37	1	no	—
212	C₂³	—	37	1	no	—
213	C₂³	—	148	2	no	—
214	C₂³	C₄×#3866	1	1	yes	characteristic
215	C₂×C₄	—	37	1	no	—
216	C₂×C₄	—	37	1	no	—
217	C₂×C₄	—	37	1	no	—
218	C₂×C₄	—	37	1	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	—	74	4	no	—
238	C₂×C₄	—	148	4	no	—
239	C₂×C₄	C₄×#3866	1	1	yes	characteristic
240	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
241	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
242	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
244	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	74	4	no	—
247	C₈	—	74	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	37	1	no	—
250	C₂²	—	37	1	no	—
251	C₂²	—	37	1	no	—
252	C₂²	—	37	1	no	—
253	C₂²	—	74	2	no	—
254	C₂²	—	74	2	no	—
255	C₂²	—	74	2	no	—
256	C₂²	—	148	1	no	—
257	C₂²	—	148	1	no	—
258	C₂²	—	148	1	no	—
259	C₂²	(C₂×#3866)⋊C₄	1	1	yes	characteristic
260	C₂²	(C₂×#3866)⋊C₄	1	1	yes	characteristic
261	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
262	C₄	—	37	1	no	—
263	C₄	—	37	1	no	—
264	C₄	—	37	1	no	—
265	C₄	—	37	1	no	—
266	C₄	—	148	4	no	—
267	C₄	(C₂×#3866)⋊C₄	1	1	yes	characteristic
268	C₄	(C₂×#3866)⋊C₄	1	1	yes	characteristic
269	C₄	C₂×C₂²⋊#3866	1	1	yes	characteristic
270	C₄	C₂×C₄⋊#3866	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	37	2	no	—
275	C₂	—	74	1	no	—
276	C₂	—	74	1	no	—
277	C₂	—	74	1	no	—
278	C₂	C₂×(C₂×#3866)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#3866
commutator	C₂²×C₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂×M₁₆×C₃₇	C₄
socle	C₂×C₃₇	C₂×#95
prime radical	C₂³×D₃₇	C₂×C₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₂×M₁₆	#3866
hypocenter	C₃₇	C₂×#344
LCS	G ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14558415	C₂	semidirect
8. D₃₇⋊M₁₆	C₄	semidirect
10. C₂×#10596957	C₄	semidirect
20. C₂×#10596934	C₄	semidirect
26. C₂×C₈×D₃₇	C₄	semidirect
27. C₂×M₁₆×C₃₇	C₄	semidirect
47. #10596957	C₂×C₄	semidirect
59. C₂×C₈×C₃₇	C₂×C₄	semidirect
62. C₂×#64436	C₂×C₄	semidirect
139. D₃₇	#344	semidirect
144. C₃₇	C₂×#344	semidirect
158. C₂×M₁₆	#3866	semidirect
200. C₂×C₈	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
#14558415	2
C₂×M₁₆×D₃₇	2
C₂×#344	37