Subgroups — #18011375

Order 4736.

452 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558612	C₂	1	2	yes	maximal, complemented
2	#14558580	C₂	1	2	yes	maximal, complemented
3	#14558579	C₂	1	2	yes	maximal, complemented
4	#14558576	C₂	1	4	yes	maximal, complemented
5	C₂×#11924606	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558530	C₂	1	2	yes	maximal, complemented
7	#14558496	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₃₇	C₂	1	1	yes	maximal, characteristic
9	#11924624	C₂²	1	4	yes	complemented
10	#11924623	—	2	4	no	—
11	#11924623	C₂²	1	4	yes	complemented
12	C₂×#10596953	—	2	2	no	—
13	C₂×#10596953	C₂²	1	2	yes	—
14	D₃₇⋊M₁₆	—	2	2	no	—
15	D₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
16	#11924615	C₂²	1	1	yes	characteristic
17	#11924606	C₂²	1	2	yes	complemented
18	#11924606	C₂²	1	2	yes	complemented
19	#11924601	C₂²	1	4	yes	complemented
20	#11924600	C₂²	1	4	yes	complemented
21	#11924590	C₂²	1	2	yes	complemented
22	C₈×#3866	C₂²	1	2	yes	complemented
23	#41×D₃₇	C₂²	1	2	yes	—
24	#11924559	C₂²	1	1	yes	characteristic
25	#11924558	C₄	1	1	yes	characteristic, complemented
26	#11924557	C₂²	1	2	yes	—
27	#11924556	C₄	1	2	yes	complemented
28	#11924555	C₂²	1	1	yes	characteristic
29	Q₁₆×D₃₇	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₃₇	C₄	1	2	yes	complemented
31	D₈×D₃₇	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
33	#135×C₃₇	C₄	1	1	yes	characteristic, complemented
34	#41×C₃₇	C₂×C₄	1	2	yes	—
35	#10596966	C₂³	1	2	yes	—
36	Q₈×D₃₇	C₂×C₄	1	2	yes	—
37	#10596964	—	2	2	no	—
38	#10596964	C₂³	1	2	yes	—
39	D₄×D₃₇	—	2	2	no	—
40	D₄×D₃₇	C₂×C₄	1	2	yes	—
41	#10596962	—	2	2	no	—
42	#10596962	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₃₇	—	2	2	no	—
44	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
45	C₂²⋊#3866	—	2	1	no	—
46	#10596957	—	2	4	no	—
47	#10596957	D₄	1	2	yes	—
48	C₂×#64438	—	2	2	no	—
49	C₂×#64438	—	2	4	no	—
50	C₄⋊#3866	—	2	1	no	—
51	C₄×#3866	C₂³	1	2	yes	—
52	#10596953	—	2	2	no	—
53	#10596953	—	2	4	no	—
54	#10596953	D₄	1	2	yes	—
55	#10596953	C₂³	1	4	yes	—
56	D₃₇⋊C₈	—	2	4	no	—
57	D₃₇⋊C₈	C₂³	1	2	yes	—
58	Q₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
59	SD₁₆×C₃₇	C₂×C₄	1	2	yes	—
60	D₈×C₃₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
62	#10596942	C₂×C₄	1	2	yes	—
63	#10596941	C₂×C₄	1	2	yes	—
64	#10596940	C₂×C₄	1	2	yes	—
65	#10596939	C₂×C₄	1	2	yes	—
66	C₂×#64436	C₂×C₄	1	1	yes	characteristic
67	Dic₁₄₈	C₂×C₄	1	1	yes	characteristic
68	D₂₉₆	C₂×C₄	1	1	yes	characteristic
69	#10596930	C₂×C₄	1	2	yes	—
70	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
71	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
72	C₈×D₃₇	C₂³	1	1	yes	characteristic
73	C₈×D₃₇	C₂³	1	1	yes	characteristic
74	C₂²×D₃₇	—	2	2	no	—
75	C₂²×D₃₇	#41	1	1	yes	characteristic
76	C₂×#3866	—	2	2	no	—
77	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
78	D₄×C₃₇	—	2	2	no	—
79	D₄×C₃₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₃₇	—	2	2	no	—
81	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊D₃₇	—	2	2	no	—
83	C₂²⋊D₃₇	—	2	2	no	—
84	C₂²⋊D₃₇	—	2	2	no	—
85	C₂×Dic₃₇	—	2	2	no	—
86	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
87	D₁₄₈	—	2	2	no	—
88	D₁₄₈	C₂²×C₄	1	2	yes	—
89	C₄×D₃₇	—	2	2	no	—
90	C₄×D₃₇	—	2	2	no	—
91	C₄×D₃₇	—	2	2	no	—
92	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
94	C₄×D₃₇	#41	1	1	yes	characteristic
95	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
96	Dic₇₄	—	2	2	no	—
97	Dic₇₄	C₂²×C₄	1	2	yes	—
98	#64438	—	2	4	no	—
99	#64438	C₂×D₄	1	4	yes	—
100	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
102	#64436	C₂²×C₄	1	1	yes	characteristic
103	#64436	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₃₇	—	2	2	no	—
105	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
106	C₂×D₃₇	—	2	2	no	—
107	C₂×D₃₇	—	4	1	no	—
108	C₂×D₃₇	—	8	2	no	—
109	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
110	C₂×D₃₇	C₂×#41	1	1	yes	characteristic
111	#3866	—	4	2	no	—
112	C₄×C₃₇	—	2	2	no	—
113	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
114	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
115	Dic₃₇	—	2	2	no	—
116	Dic₃₇	C₄×D₄	1	1	yes	characteristic
117	Dic₃₇	C₂²×D₄	1	1	yes	characteristic
118	#2957	—	37	1	no	maximal, Sylow
119	C₂×C₃₇	—	2	1	no	—
120	C₂×C₃₇	—	4	2	no	—
121	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
122	D₃₇	—	2	1	no	—
123	D₃₇	—	2	1	no	—
124	D₃₇	—	4	2	no	—
125	C₂×#135	—	37	1	no	—
126	#568	—	37	2	no	—
127	#444	—	37	2	no	—
128	#444	—	37	2	no	—
129	#443	—	37	4	no	—
130	C₂×#108	—	37	1	no	—
131	#421	—	37	2	no	—
132	#405	—	37	1	no	—
133	C₃₇	#2957	1	1	yes	characteristic, complemented, Sylow
134	C₂×#41	—	37	2	no	—
135	#135	—	37	2	no	—
136	#135	—	37	1	no	—
137	#135	—	37	2	no	—
138	#135	—	37	1	no	—
139	#135	—	37	1	no	—
140	#135	#3866	1	1	yes	characteristic, complemented
141	C₂×Q₁₆	—	37	1	no	—
142	C₂×SD₁₆	—	37	2	no	—
143	C₂×D₈	—	37	1	no	—
144	#131	—	37	4	no	—
145	#131	—	37	4	no	—
146	#131	—	74	4	no	—
147	C₂×M₁₆	—	37	1	no	—
148	C₂×M₁₆	—	37	2	no	—
149	C₂×M₁₆	—	74	2	no	—
150	C₂×M₁₆	—	74	2	no	—
151	C₂²×C₈	—	37	1	no	—
152	#117	—	37	1	no	—
153	#108	—	37	2	no	—
154	#108	—	37	2	no	—
155	#104	—	37	4	no	—
156	#104	—	37	4	no	—
157	#97	—	37	2	no	—
158	C₄×C₈	—	37	2	no	—
159	#41	—	37	2	no	—
160	#41	—	37	2	no	—
161	#41	—	37	2	no	—
162	#41	—	74	2	no	—
163	#41	—	74	2	no	—
164	#41	C₂×#3866	1	2	yes	—
165	C₂×Q₈	—	37	2	no	—
166	C₂×D₄	—	37	2	no	—
167	C₂×D₄	—	74	2	no	—
168	C₂²×C₄	—	37	1	no	—
169	C₂²×C₄	—	74	2	no	—
170	Q₁₆	—	37	2	no	—
171	Q₁₆	—	37	1	no	—
172	Q₁₆	C₂×#3866	1	1	yes	characteristic
173	SD₁₆	—	37	2	no	—
174	SD₁₆	—	37	2	no	—
175	SD₁₆	—	37	2	no	—
176	SD₁₆	C₂×#3866	1	2	yes	—
177	D₈	—	37	2	no	—
178	D₈	—	37	1	no	—
179	D₈	C₂×#3866	1	1	yes	characteristic
180	M₁₆	—	37	2	no	—
181	M₁₆	—	37	2	no	—
182	M₁₆	—	37	4	no	—
183	M₁₆	—	74	4	no	—
184	M₁₆	—	74	4	no	—
185	M₁₆	—	74	2	no	—
186	C₂×C₈	—	37	2	no	—
187	C₂×C₈	—	37	1	no	—
188	C₂×C₈	—	37	1	no	—
189	C₂×C₈	—	37	1	no	—
190	C₂×C₈	—	37	1	no	—
191	C₂×C₈	—	37	1	no	—
192	C₂×C₈	—	74	4	no	—
193	C₂×C₈	—	74	4	no	—
194	C₂×C₈	—	74	2	no	—
195	C₂×C₈	C₂×#3866	1	1	yes	characteristic
196	C₄⋊C₄	—	74	1	no	—
197	C₂²⋊C₄	—	74	1	no	—
198	C₄²	—	37	2	no	—
199	C₂³	—	37	1	no	—
200	C₂³	—	74	2	no	—
201	Q₈	—	37	2	no	—
202	Q₈	—	74	2	no	—
203	Q₈	C₂²×#3866	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	37	2	no	—
206	D₄	—	74	2	no	—
207	D₄	—	74	2	no	—
208	D₄	—	74	2	no	—
209	D₄	—	74	2	no	—
210	D₄	C₂²×#3866	1	2	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	37	1	no	—
213	C₂×C₄	—	37	1	no	—
214	C₂×C₄	—	37	1	no	—
215	C₂×C₄	—	37	1	no	—
216	C₂×C₄	—	37	1	no	—
217	C₂×C₄	—	74	2	no	—
218	C₂×C₄	—	74	2	no	—
219	C₂×C₄	—	74	2	no	—
220	C₂×C₄	—	74	2	no	—
221	C₂×C₄	—	74	2	no	—
222	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
223	C₈	—	37	4	no	—
224	C₈	—	37	1	no	—
225	C₈	—	37	1	no	—
226	C₈	—	74	4	no	—
227	C₈	C₂²×#3866	1	1	yes	characteristic
228	C₈	C₂²×#3866	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	37	1	no	—
231	C₂²	—	37	1	no	—
232	C₂²	—	296	2	no	—
233	C₂²	—	74	2	no	—
234	C₂²	—	148	1	no	—
235	C₂²	D₄×#3866	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	37	1	no	—
238	C₄	—	37	1	no	—
239	C₄	—	74	2	no	—
240	C₄	—	148	2	no	—
241	C₄	D₄×#3866	1	1	yes	characteristic
242	C₄	C₂³×#3866	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	—	74	1	no	—
246	C₂	—	74	1	no	—
247	C₂	—	148	2	no	—
248	C₂	C₂×D₄×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#3866
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₄	C₂³×#3866
Fitting	#135×C₃₇	C₄
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	#135×D₃₇	C₂
cosocle	C₄×D₃₇	C₂⁴
hypercenter	#135	#3866
hypocenter	C₃₇	#2957
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #14558612	C₂	semidirect
2. #14558580	C₂	semidirect
3. #14558579	C₂	semidirect
4. #14558576	C₂	semidirect
5. C₂×#11924606	C₂	semidirect
6. #14558530	C₂	semidirect
7. #14558496	C₂	semidirect
9. #11924624	C₂²	semidirect
11. #11924623	C₂²	semidirect
17. #11924606	C₂²	semidirect
18. #11924606	C₂²	semidirect
19. #11924601	C₂²	semidirect
20. #11924600	C₂²	semidirect
21. #11924590	C₂²	semidirect
22. C₈×#3866	C₂²	semidirect
25. #11924558	C₄	semidirect
27. #11924556	C₄	semidirect
29. Q₁₆×D₃₇	C₄	semidirect
30. SD₁₆×D₃₇	C₄	semidirect
31. D₈×D₃₇	C₄	semidirect
33. #135×C₃₇	C₄	semidirect
133. C₃₇	#2957	semidirect
140. #135	#3866	semidirect

Maximal subgroups

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

subgroup	index
#14558612	2
#14558580	2
#14558579	2
#14558576	2
C₂×#11924606	2
#14558530	2
#14558496	2
#135×D₃₇	2
#2957	37