Subgroups — #18011319

Order 4736.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924623	C₂	1	2	yes	maximal, complemented
2	#14558568	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558567	C₂	1	1	yes	maximal, characteristic, complemented
4	#14558566	C₂	1	2	yes	maximal, complemented
5	#14558561	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558560	C₂	1	2	yes	maximal, complemented
7	D₄×#64438	C₂	1	1	yes	maximal, characteristic, complemented
8	#14558536	C₂	1	1	yes	maximal, characteristic, complemented
9	#14558506	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#64438	C₂	1	1	yes	maximal, characteristic, complemented
11	#14558078	C₂	1	1	yes	maximal, characteristic
12	#11924623	—	2	8	no	—
13	C₂×#10596953	C₂²	1	2	yes	—
14	C₂×#10596957	C₂²	1	2	yes	—
15	C₂×#10596957	C₂²	1	2	yes	—
16	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
17	C₂²×#64438	C₂²	1	1	yes	characteristic
18	C₂²×#64438	C₂²	1	2	yes	—
19	C₂²×#64438	C₂²	1	1	yes	characteristic
20	#11924603	C₂²	1	2	yes	—
21	C₄⋊#64438	C₂²	1	1	yes	characteristic
22	C₄⋊#64438	C₂²	1	1	yes	characteristic
23	#11924593	C₂²	1	2	yes	—
24	#11924593	C₂²	1	2	yes	—
25	C₂²⋊#64438	C₂²	1	2	yes	—
26	C₂²⋊#64438	C₂²	1	2	yes	—
27	C₄×#64438	C₂²	1	1	yes	characteristic
28	C₄×#64438	C₂²	1	1	yes	characteristic
29	C₄×#64438	C₂²	1	1	yes	characteristic
30	C₄×#64438	C₂²	1	1	yes	characteristic
31	C₂×#10596964	C₂²	1	2	yes	—
32	#11924535	C₄	1	1	yes	characteristic
33	#11924532	C₄	1	1	yes	characteristic
34	#11924522	C₄	1	2	yes	—
35	#11924510	C₄	1	1	yes	characteristic
36	#11924508	C₄	1	2	yes	—
37	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
38	#11924498	C₂²	1	1	yes	characteristic
39	#11924496	C₂²	1	2	yes	—
40	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
41	#121×C₃₇	C₄	1	1	yes	characteristic
42	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₃₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
48	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₃₇	C₂³	1	2	yes	—
50	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
51	#10596964	—	2	4	no	—
52	#10596964	—	2	4	no	—
53	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₃₇	C₂³	1	2	yes	—
55	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
56	#10596957	—	2	4	no	—
57	#10596957	—	2	4	no	—
58	C₂×#64438	—	2	2	no	—
59	C₂×#64438	—	2	4	no	—
60	C₂×#64438	D₄	1	4	yes	complemented
61	C₂×#64438	C₂³	1	1	yes	characteristic
62	C₂×#64438	C₂³	1	1	yes	characteristic
63	C₂×#64438	C₂³	1	2	yes	—
64	C₂×#64438	C₂³	1	2	yes	—
65	C₂×#64438	C₂³	1	1	yes	characteristic
66	C₂×#64438	C₂³	1	1	yes	characteristic
67	#10596953	—	2	4	no	—
68	D₃₇⋊C₈	—	2	4	no	—
69	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
70	C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
71	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
72	#10596938	C₂×C₄	1	2	yes	—
73	#10596938	C₂×C₄	1	2	yes	—
74	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
75	#10596936	C₂×C₄	1	2	yes	—
76	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
79	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
80	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
81	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
83	D₄×C₃₇	—	2	4	no	—
84	D₄×C₃₇	—	2	2	no	—
85	C₂×C₄×C₃₇	—	2	2	no	—
86	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²⋊D₃₇	—	2	4	no	—
90	C₂²⋊D₃₇	—	2	4	no	—
91	C₂²⋊D₃₇	—	2	4	no	—
92	C₂×Dic₃₇	—	2	2	no	—
93	C₂×Dic₃₇	—	2	2	no	—
94	C₂×Dic₃₇	—	2	2	no	—
95	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
97	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
100	C₂×Dic₃₇	#41	1	2	yes	—
101	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
102	D₁₄₈	—	2	2	no	—
103	C₄×D₃₇	—	2	2	no	—
104	C₄×D₃₇	—	2	2	no	—
105	Dic₇₄	—	2	4	no	—
106	#64438	—	2	2	no	—
107	#64438	—	2	2	no	—
108	#64438	—	2	1	no	—
109	#64438	—	2	1	no	—
110	#64438	C₂×D₄	1	2	yes	—
111	#64438	C₂×D₄	1	2	yes	—
112	C₂²×C₃₇	—	2	1	no	—
113	C₂²×C₃₇	—	2	1	no	—
114	C₂²×C₃₇	—	2	2	no	—
115	C₂²×C₃₇	—	2	2	no	—
116	C₂²×C₃₇	—	2	2	no	—
117	C₂²×C₃₇	C₄×D₄	1	2	yes	—
118	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₃₇	—	2	2	no	—
120	C₂×D₃₇	—	2	2	no	—
121	C₂×D₃₇	—	2	2	no	—
122	C₄×C₃₇	—	2	2	no	—
123	C₄×C₃₇	—	2	1	no	—
124	C₄×C₃₇	#131	1	2	yes	complemented
125	Dic₃₇	—	2	2	no	—
126	Dic₃₇	—	2	1	no	—
127	Dic₃₇	C₄×D₄	1	2	yes	—
128	Dic₃₇	#131	1	2	yes	complemented
129	Dic₃₇	C₂²×D₄	1	1	yes	characteristic
130	Dic₃₇	C₂×#41	1	1	yes	characteristic
131	#2931	—	37	1	no	maximal, Sylow
132	C₂×C₃₇	—	2	2	no	—
133	C₂×C₃₇	—	4	2	no	—
134	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
136	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
137	D₃₇	—	4	2	no	—
138	C₂×#131	—	37	2	no	—
139	#532	—	37	1	no	—
140	#436	—	37	2	no	—
141	#436	—	37	1	no	—
142	#436	—	37	1	no	—
143	C₈×D₄	—	37	1	no	—
144	C₈×D₄	—	37	2	no	—
145	C₈×D₄	—	37	1	no	—
146	#423	—	37	1	no	—
147	#408	—	37	2	no	—
148	C₂×C₄×C₈	—	37	1	no	—
149	C₃₇	#2931	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	—	37	2	no	—
151	#131	—	74	8	no	—
152	C₂×M₁₆	—	37	2	no	—
153	C₂×M₁₆	—	37	2	no	—
154	C₂×M₁₆	—	37	2	no	—
155	C₂²×C₈	—	37	1	no	—
156	C₂²×C₈	—	37	2	no	—
157	C₂²×C₈	—	37	2	no	—
158	C₂²×C₈	—	37	1	no	—
159	C₄⋊Q₈	—	37	1	no	—
160	#127	—	37	1	no	—
161	#124	—	37	2	no	—
162	#121	—	37	2	no	—
163	#121	—	37	1	no	—
164	#121	#3866	1	1	yes	characteristic
165	C₄×D₄	—	37	1	no	—
166	C₄×D₄	—	37	1	no	—
167	C₄×D₄	—	37	2	no	—
168	C₂×C₄²	—	37	1	no	—
169	C₄⋊C₈	—	37	1	no	—
170	C₄⋊C₈	—	37	1	no	—
171	C₄⋊C₈	—	37	2	no	—
172	C₂²⋊C₈	—	37	2	no	—
173	C₂²⋊C₈	—	37	2	no	—
174	C₂²⋊C₈	—	37	2	no	—
175	C₂²⋊C₈	—	37	2	no	—
176	C₄×C₈	—	37	1	no	—
177	C₄×C₈	—	37	1	no	—
178	C₄×C₈	—	37	1	no	—
179	C₄×C₈	—	37	1	no	—
180	#41	—	74	4	no	—
181	#41	—	74	4	no	—
182	C₂×Q₈	—	37	2	no	—
183	C₂×D₄	—	37	2	no	—
184	C₂×D₄	—	37	2	no	—
185	C₂×D₄	—	37	2	no	—
186	C₂×D₄	—	37	1	no	—
187	C₂×D₄	C₂×#3866	1	1	yes	characteristic
188	C₂×D₄	C₂×#3866	1	2	yes	—
189	C₂²×C₄	—	37	1	no	—
190	C₂²×C₄	—	37	2	no	—
191	C₂²×C₄	—	37	1	no	—
192	C₂²×C₄	—	37	2	no	—
193	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
194	M₁₆	—	74	4	no	—
195	M₁₆	—	74	4	no	—
196	M₁₆	—	74	4	no	—
197	C₂×C₈	—	37	4	no	—
198	C₂×C₈	—	37	1	no	—
199	C₂×C₈	—	37	1	no	—
200	C₂×C₈	—	37	1	no	—
201	C₂×C₈	—	37	1	no	—
202	C₂×C₈	—	37	2	no	—
203	C₂×C₈	—	37	2	no	—
204	C₂×C₈	—	74	4	no	—
205	C₂×C₈	—	74	4	no	—
206	C₂×C₈	—	74	2	no	—
207	C₄⋊C₄	—	37	2	no	—
208	C₄⋊C₄	—	37	1	no	—
209	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
210	C₂²⋊C₄	—	37	2	no	—
211	C₂²⋊C₄	—	37	2	no	—
212	C₂²⋊C₄	—	37	2	no	—
213	C₂²⋊C₄	C₂×#3866	1	2	yes	—
214	C₄²	—	37	1	no	—
215	C₄²	—	37	1	no	—
216	C₄²	—	37	1	no	—
217	C₄²	—	37	1	no	—
218	C₂³	—	37	2	no	—
219	C₂³	C₂²×#3866	1	2	yes	—
220	C₂³	C₂²×#3866	1	1	yes	characteristic
221	Q₈	—	74	4	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	D₄	—	74	4	no	—
225	D₄	—	74	4	no	—
226	D₄	—	74	4	no	—
227	D₄	—	74	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	37	2	no	—
230	C₂×C₄	—	37	2	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	2	no	—
236	C₂×C₄	—	74	2	no	—
237	C₂×C₄	—	74	2	no	—
238	C₂×C₄	—	74	2	no	—
239	C₂×C₄	—	74	2	no	—
240	C₂×C₄	—	74	2	no	—
241	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
242	C₂×C₄	C₂²×#3866	1	2	yes	—
243	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
244	C₈	—	37	2	no	—
245	C₈	—	37	2	no	—
246	C₈	—	74	2	no	—
247	C₈	—	74	2	no	—
248	C₈	—	74	1	no	—
249	C₈	—	74	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	74	2	no	—
256	C₂²	—	74	2	no	—
257	C₂²	—	74	2	no	—
258	C₂²	D₄×#3866	1	2	yes	—
259	C₂²	C₂³×#3866	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	37	1	no	—
263	C₄	—	37	1	no	—
264	C₄	—	37	2	no	—
265	C₄	—	37	2	no	—
266	C₄	—	74	2	no	—
267	C₄	—	74	1	no	—
268	C₄	#11924623	1	2	yes	complemented
269	C₂	—	2	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	148	2	no	—
272	C₂	C₂×D₄×#3866	1	1	yes	characteristic
273	C₂	C₂×#11924623	1	1	yes	characteristic
274	C₂	D₃₇⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	#121×C₃₇	C₄
prime radical	#14558078	C₂
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	#121	#3866
hypocenter	C₃₇	#2931
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂×#11924623	C₂	semidirect
2. #14558568	C₂	semidirect
3. #14558567	C₂	semidirect
4. #14558566	C₂	semidirect
5. #14558561	C₂	semidirect
6. #14558560	C₂	semidirect
7. D₄×#64438	C₂	semidirect
8. #14558536	C₂	semidirect
9. #14558506	C₂	semidirect
10. C₂×C₄×#64438	C₂	semidirect
60. C₂×#64438	D₄	semidirect
124. C₄×C₃₇	#131	semidirect
128. Dic₃₇	#131	semidirect
149. C₃₇	#2931	semidirect
268. C₄	#11924623	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924623	2
#14558568	2
#14558567	2
#14558566	2
#14558561	2
#14558560	2
D₄×#64438	2
#14558536	2
#14558506	2
C₂×C₄×#64438	2
#14558078	2
#2931	37