Subgroups — #18011541

Order 4736.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558584	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#64438	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	#14558565	C₂	1	2	yes	maximal, complemented
6	#14558561	C₂	1	1	yes	maximal, characteristic, complemented
7	#14558558	C₂	1	2	yes	maximal, complemented
8	#14558549	C₂	1	1	yes	maximal, characteristic, complemented
9	#14558540	C₂	1	2	yes	maximal, complemented
10	#14558536	C₂	1	1	yes	maximal, characteristic, complemented
11	#14558069	C₂	1	1	yes	maximal, characteristic
12	C₂×#10596953	C₂²	1	2	yes	—
13	C₂×#10596957	C₂²	1	2	yes	—
14	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
15	#11924603	C₂²	1	2	yes	—
16	#11924603	C₂²	1	2	yes	—
17	C₄⋊#64438	C₂²	1	1	yes	characteristic
18	C₄⋊#64438	C₂²	1	1	yes	characteristic
19	C₄⋊#64438	C₂²	1	1	yes	characteristic
20	C₄⋊#64438	C₂²	1	2	yes	—
21	C₄⋊#64438	C₂²	1	1	yes	characteristic
22	#11924593	C₂²	1	2	yes	—
23	#11924593	C₂²	1	2	yes	—
24	C₂²⋊#64438	C₂²	1	1	yes	characteristic
25	C₂²⋊#64438	C₂²	1	1	yes	characteristic
26	#11924591	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	2	yes	—
30	#11924531	C₄	1	2	yes	—
31	#11924527	C₄	1	1	yes	characteristic
32	#11924525	C₄	1	1	yes	characteristic
33	#11924520	C₄	1	2	yes	—
34	#11924510	C₄	1	1	yes	characteristic
35	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
36	#11924498	C₂²	1	1	yes	characteristic
37	#11924496	C₂²	1	2	yes	—
38	#11924494	C₂²	1	2	yes	—
39	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₃₇	C₄	1	1	yes	characteristic
41	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
44	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
46	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₃₇	C₂³	1	2	yes	—
48	#10596957	—	4	2	no	—
49	C₂×#64438	C₂³	1	1	yes	characteristic
50	C₂×#64438	C₂³	1	1	yes	characteristic
51	C₂×#64438	C₂³	1	1	yes	characteristic
52	C₂×#64438	C₂³	1	1	yes	characteristic
53	C₂×#64438	C₂³	1	2	yes	—
54	C₂×#64438	C₂³	1	2	yes	—
55	#10596953	—	4	2	no	—
56	D₃₇⋊C₈	—	4	2	no	—
57	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
60	#10596938	C₂×C₄	1	2	yes	—
61	#10596938	C₂×C₄	1	2	yes	—
62	#10596938	C₂×C₄	1	2	yes	—
63	#10596938	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
66	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
67	#10596936	C₂×C₄	1	2	yes	—
68	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
71	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
72	C₄×Dic₃₇	C₂³	1	2	yes	—
73	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
75	Q₈×C₃₇	—	2	2	no	—
76	C₂×C₄×C₃₇	—	2	2	no	—
77	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊D₃₇	—	4	2	no	—
82	C₂×Dic₃₇	—	2	1	no	—
83	C₂×Dic₃₇	—	2	1	no	—
84	C₂×Dic₃₇	—	4	1	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	C₂²×C₄	1	2	yes	—
89	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
90	D₁₄₈	—	2	2	no	—
91	C₄×D₃₇	—	2	2	no	—
92	C₄×D₃₇	—	2	2	no	—
93	#64438	—	2	1	no	—
94	#64438	—	2	2	no	—
95	#64438	—	2	1	no	—
96	#64438	—	2	2	no	—
97	#64438	—	2	1	no	—
98	#64438	—	2	1	no	—
99	C₂²×C₃₇	—	2	1	no	—
100	C₂²×C₃₇	—	2	1	no	—
101	C₂²×C₃₇	—	2	1	no	—
102	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₃₇	—	2	2	no	—
104	C₂×D₃₇	—	2	2	no	—
105	C₂×D₃₇	—	2	2	no	—
106	C₄×C₃₇	—	2	2	no	—
107	C₄×C₃₇	—	2	1	no	—
108	C₄×C₃₇	—	2	2	no	—
109	C₄×C₃₇	#131	1	2	yes	—
110	Dic₃₇	—	2	1	no	—
111	Dic₃₇	—	2	1	no	—
112	Dic₃₇	—	2	1	no	—
113	Dic₃₇	—	2	2	no	—
114	Dic₃₇	H(5,𝔽₂)	1	1	yes	characteristic
115	Dic₃₇	2₋¹⁺⁴	1	1	yes	characteristic
116	#2993	—	37	1	no	maximal, Sylow
117	C₂×C₃₇	—	4	1	no	—
118	C₂×C₃₇	#520	1	1	yes	characteristic
119	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
120	C₂×C₃₇	#568	1	1	yes	characteristic
121	D₃₇	—	4	2	no	—
122	#529	—	37	1	no	—
123	#446	—	37	1	no	—
124	C₈×Q₈	—	37	1	no	—
125	#436	—	37	2	no	—
126	#436	—	37	1	no	—
127	#435	—	37	2	no	—
128	C₈×D₄	—	37	1	no	—
129	#433	—	37	2	no	—
130	#431	—	37	1	no	—
131	#424	—	37	2	no	—
132	#423	—	37	1	no	—
133	C₃₇	#2993	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	37	2	no	—
135	C₂×M₁₆	—	37	2	no	—
136	C₂²×C₈	—	37	2	no	—
137	#126	—	37	2	no	—
138	#125	—	37	1	no	—
139	#124	—	37	1	no	—
140	#123	—	37	2	no	—
141	C₂²⋊Q₈	#3866	1	1	yes	characteristic
142	#121	—	37	1	no	—
143	C₄×Q₈	—	37	1	no	—
144	C₄×D₄	—	37	1	no	—
145	C₄×D₄	—	37	2	no	—
146	#117	—	37	2	no	—
147	C₂×C₄²	—	37	1	no	—
148	C₄⋊C₈	—	37	1	no	—
149	C₄⋊C₈	—	37	1	no	—
150	C₄⋊C₈	—	37	1	no	—
151	C₄⋊C₈	—	37	2	no	—
152	C₄⋊C₈	—	37	1	no	—
153	C₄⋊C₈	—	37	2	no	—
154	C₄⋊C₈	—	37	2	no	—
155	C₂²⋊C₈	—	37	1	no	—
156	C₂²⋊C₈	—	37	1	no	—
157	C₂²⋊C₈	—	37	2	no	—
158	C₂²⋊C₈	—	37	2	no	—
159	#97	—	37	2	no	—
160	C₄×C₈	—	37	1	no	—
161	C₄×C₈	—	37	1	no	—
162	C₄×C₈	—	37	2	no	—
163	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
164	C₂×D₄	—	37	2	no	—
165	C₂×D₄	—	37	1	no	—
166	C₂²×C₄	—	37	1	no	—
167	C₂²×C₄	—	37	1	no	—
168	C₂²×C₄	—	37	2	no	—
169	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
170	M₁₆	—	148	2	no	—
171	M₁₆	—	148	2	no	—
172	C₂×C₈	—	37	1	no	—
173	C₂×C₈	—	37	1	no	—
174	C₂×C₈	—	37	1	no	—
175	C₂×C₈	—	37	1	no	—
176	C₂×C₈	—	37	2	no	—
177	C₂×C₈	—	37	2	no	—
178	C₂×C₈	—	148	2	no	—
179	C₄⋊C₄	—	37	2	no	—
180	C₄⋊C₄	—	37	2	no	—
181	C₄⋊C₄	—	37	1	no	—
182	C₄⋊C₄	—	37	2	no	—
183	C₄⋊C₄	C₂×#3866	1	2	yes	—
184	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
185	C₂²⋊C₄	—	37	2	no	—
186	C₂²⋊C₄	—	37	2	no	—
187	C₂²⋊C₄	—	37	2	no	—
188	C₂²⋊C₄	—	37	2	no	—
189	C₂²⋊C₄	C₂×#3866	1	2	yes	—
190	C₄²	—	37	1	no	—
191	C₄²	—	37	1	no	—
192	C₄²	—	37	1	no	—
193	C₄²	—	37	2	no	—
194	C₄²	—	37	1	no	—
195	C₂³	—	37	2	no	—
196	C₂³	C₂²×#3866	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	74	2	no	—
199	D₄	—	148	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	37	1	no	—
202	C₂×C₄	—	37	1	no	—
203	C₂×C₄	—	37	1	no	—
204	C₂×C₄	—	37	1	no	—
205	C₂×C₄	—	37	2	no	—
206	C₂×C₄	—	74	2	no	—
207	C₂×C₄	—	74	2	no	—
208	C₂×C₄	—	74	1	no	—
209	C₂×C₄	—	74	1	no	—
210	C₂×C₄	—	148	1	no	—
211	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
212	C₂×C₄	C₂²×#3866	1	2	yes	—
213	C₂×C₄	C₂²×#3866	1	2	yes	—
214	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
215	C₈	—	74	1	no	—
216	C₈	—	74	2	no	—
217	C₈	—	74	1	no	—
218	C₈	—	74	2	no	—
219	C₈	—	74	1	no	—
220	C₈	—	74	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	74	2	no	—
225	C₂²	—	74	2	no	—
226	C₂²	—	74	2	no	—
227	C₂²	C₂³×#3866	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	37	1	no	—
232	C₄	—	37	1	no	—
233	C₄	—	74	1	no	—
234	C₄	—	74	1	no	—
235	C₄	—	74	1	no	—
236	C₄	—	74	2	no	—
237	C₄	#11924623	1	2	yes	—
238	C₂	—	4	1	no	—
239	C₂	—	148	2	no	—
240	C₂	#14558602	1	1	yes	characteristic
241	C₂	C₂×#11924623	1	1	yes	characteristic
242	C₂	#14558613	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₂²⋊Q₈×C₃₇	C₄
prime radical	#11924510	C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₂²⋊Q₈	#3866
hypocenter	C₃₇	#2993
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14558584	C₂	semidirect
2. Q₈×#64438	C₂	semidirect
3. #14558567	C₂	semidirect
4. #14558566	C₂	semidirect
5. #14558565	C₂	semidirect
6. #14558561	C₂	semidirect
7. #14558558	C₂	semidirect
8. #14558549	C₂	semidirect
9. #14558540	C₂	semidirect
10. #14558536	C₂	semidirect
133. C₃₇	#2993	semidirect

Maximal subgroups

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

subgroup	index
#14558584	2
Q₈×#64438	2
#14558567	2
#14558566	2
#14558565	2
#14558561	2
#14558558	2
#14558549	2
#14558540	2
#14558536	2
#14558069	2
#2993	37