Subgroups — #18011282

Order 4736.

384 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558558	C₂	1	2	yes	maximal, complemented
2	#14558557	C₂	1	2	yes	maximal, complemented
3	#14558556	C₂	1	1	yes	maximal, characteristic, complemented
4	#14558555	C₂	1	1	yes	maximal, characteristic, complemented
5	#14558554	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558553	C₂	1	1	yes	maximal, characteristic, complemented
7	#14558539	C₂	1	2	yes	maximal, complemented
8	#14558506	C₂	1	2	yes	maximal
9	C₄×#10596957	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11924591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#10596964	C₂	1	1	yes	maximal, characteristic
12	C₂×#10596957	C₂²	1	2	yes	—
13	C₂²×#64438	C₂²	1	2	yes	—
14	#11924603	C₂²	1	2	yes	—
15	#11924603	C₂²	1	2	yes	—
16	C₄⋊#64438	C₂²	1	2	yes	—
17	C₄⋊#64438	C₂²	1	2	yes	—
18	#11924593	C₂²	1	2	yes	—
19	#11924593	C₂²	1	2	yes	—
20	C₂²⋊#64438	C₂²	1	2	yes	—
21	C₂²⋊#64438	C₂²	1	2	yes	—
22	#11924591	C₂²	1	1	yes	characteristic
23	#11924591	C₂²	1	1	yes	characteristic
24	#11924591	C₂²	1	2	yes	—
25	#11924591	C₂²	1	1	yes	characteristic
26	#11924591	C₂²	1	1	yes	characteristic
27	C₄×#64438	C₂²	1	2	yes	—
28	C₂×#10596964	C₂²	1	1	yes	characteristic
29	C₄×Dic₇₄	C₄	1	1	yes	characteristic
30	#11924501	C₄	1	1	yes	characteristic
31	D₄×Dic₃₇	C₄	1	1	yes	characteristic
32	C₄×C₂²⋊D₃₇	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₃₇	C₄	1	1	yes	characteristic
34	#11924496	C₄	1	1	yes	characteristic
35	#11924496	C₄	1	1	yes	characteristic
36	#11924494	C₂²	1	1	yes	characteristic
37	#11924492	C₂²	1	1	yes	characteristic
38	#11924492	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
41	C₄²×D₃₇	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₃₇	C₄	1	1	yes	characteristic
43	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
51	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
52	#10596964	—	2	4	no	—
53	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
56	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
57	#10596957	—	2	4	no	—
58	C₂×#64438	—	2	4	no	—
59	C₂×#64438	C₂³	1	2	yes	—
60	C₂×#64438	C₂³	1	2	yes	—
61	C₂×#64438	C₂³	1	2	yes	—
62	C₂×#64438	C₂³	1	2	yes	—
63	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
69	#10596938	C₂×C₄	1	1	yes	characteristic
70	#10596938	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
72	#10596936	C₂×C₄	1	1	yes	characteristic
73	#10596936	C₂×C₄	1	1	yes	characteristic
74	#10596936	C₂×C₄	1	1	yes	characteristic
75	#10596936	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
82	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
83	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
84	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
85	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
88	D₄×C₃₇	—	2	2	no	—
89	C₂×C₄×C₃₇	—	4	1	no	—
90	C₂×C₄×C₃₇	—	4	1	no	—
91	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊D₃₇	—	2	2	no	—
97	C₂²⋊D₃₇	—	2	2	no	—
98	C₂×Dic₃₇	—	2	2	no	—
99	C₂×Dic₃₇	—	4	1	no	—
100	C₂×Dic₃₇	—	4	1	no	—
101	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₃₇	#41	1	4	yes	—
108	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
109	C₄×D₃₇	—	2	2	no	—
110	C₄×D₃₇	—	4	1	no	—
111	C₄×D₃₇	—	4	1	no	—
112	Dic₇₄	—	2	2	no	—
113	#64438	—	2	2	no	—
114	#64438	—	2	2	no	—
115	#64438	—	2	2	no	—
116	#64438	—	2	2	no	—
117	C₂²×C₃₇	—	2	1	no	—
118	C₂²×C₃₇	—	2	1	no	—
119	C₂²×C₃₇	—	2	1	no	—
120	C₂²×C₃₇	—	2	1	no	—
121	C₂²×C₃₇	—	2	1	no	—
122	C₂²×C₃₇	#117	1	2	yes	—
123	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₃₇	—	2	1	no	—
125	C₂×D₃₇	—	2	1	no	—
126	C₂×D₃₇	—	2	1	no	—
127	C₄×C₃₇	—	2	1	no	—
128	C₄×C₃₇	—	2	1	no	—
129	C₄×C₃₇	—	2	1	no	—
130	C₄×C₃₇	—	2	1	no	—
131	C₄×C₃₇	—	2	1	no	—
132	Dic₃₇	—	2	1	no	—
133	Dic₃₇	—	2	1	no	—
134	Dic₃₇	—	2	1	no	—
135	Dic₃₇	—	2	1	no	—
136	Dic₃₇	—	2	1	no	—
137	Dic₃₇	#117	1	2	yes	—
138	Dic₃₇	C₂×#41	1	2	yes	—
139	#2923	—	37	1	no	maximal, Sylow
140	C₂×C₃₇	—	2	2	no	—
141	C₂×C₃₇	—	4	1	no	—
142	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
143	C₂×C₃₇	#568	1	1	yes	characteristic
144	C₂×C₃₇	#568	1	1	yes	characteristic
145	D₃₇	—	4	1	no	—
146	C₄×#41	—	37	1	no	—
147	#433	—	37	2	no	—
148	#433	—	37	2	no	—
149	#432	—	37	1	no	—
150	#432	—	37	1	no	—
151	#432	—	37	1	no	—
152	#432	—	37	1	no	—
153	#424	—	37	2	no	—
154	#408	—	37	2	no	—
155	C₄×M₁₆	—	37	1	no	—
156	C₂×#97	—	37	1	no	—
157	C₃₇	#2923	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	37	1	no	—
159	C₂×M₁₆	—	37	2	no	—
160	C₂²×C₈	—	37	2	no	—
161	C₄×Q₈	—	37	1	no	—
162	C₄×Q₈	—	37	1	no	—
163	C₄×D₄	—	37	1	no	—
164	C₄×D₄	—	37	1	no	—
165	C₄×D₄	—	37	1	no	—
166	C₄×D₄	—	37	1	no	—
167	C₄×D₄	—	37	1	no	—
168	C₄×D₄	#3866	1	1	yes	characteristic
169	#117	—	37	1	no	—
170	#117	—	37	1	no	—
171	#117	—	37	1	no	—
172	C₂×C₄²	—	37	1	no	—
173	C₂×C₄²	—	37	1	no	—
174	C₂×C₄²	—	37	1	no	—
175	C₄⋊C₈	—	37	2	no	—
176	C₄⋊C₈	—	37	2	no	—
177	C₄⋊C₈	—	37	2	no	—
178	C₄⋊C₈	—	37	2	no	—
179	C₂²⋊C₈	—	37	2	no	—
180	C₂²⋊C₈	—	37	2	no	—
181	C₂²⋊C₈	—	37	2	no	—
182	C₂²⋊C₈	—	37	2	no	—
183	#97	—	37	1	no	—
184	#97	—	37	1	no	—
185	#97	—	37	2	no	—
186	#97	—	37	1	no	—
187	#97	—	37	1	no	—
188	C₄×C₈	—	37	2	no	—
189	#41	—	74	4	no	—
190	C₂×Q₈	—	37	1	no	—
191	C₂×D₄	—	37	1	no	—
192	C₂×D₄	—	37	1	no	—
193	C₂×D₄	C₂×#3866	1	1	yes	characteristic
194	C₂²×C₄	—	37	1	no	—
195	C₂²×C₄	—	37	1	no	—
196	C₂²×C₄	—	37	1	no	—
197	C₂²×C₄	—	37	1	no	—
198	C₂²×C₄	—	37	1	no	—
199	C₂²×C₄	—	37	1	no	—
200	C₂²×C₄	—	37	1	no	—
201	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
202	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
203	M₁₆	—	74	4	no	—
204	C₂×C₈	—	37	2	no	—
205	C₂×C₈	—	37	2	no	—
206	C₂×C₈	—	37	2	no	—
207	C₂×C₈	—	37	2	no	—
208	C₂×C₈	—	74	4	no	—
209	C₄⋊C₄	—	37	1	no	—
210	C₄⋊C₄	—	37	1	no	—
211	C₄⋊C₄	—	37	1	no	—
212	C₄⋊C₄	—	37	1	no	—
213	C₄⋊C₄	—	37	1	no	—
214	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
215	C₂²⋊C₄	—	37	1	no	—
216	C₂²⋊C₄	—	37	1	no	—
217	C₂²⋊C₄	—	37	1	no	—
218	C₂²⋊C₄	—	37	1	no	—
219	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
220	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
221	C₄²	—	37	1	no	—
222	C₄²	—	37	1	no	—
223	C₄²	—	37	1	no	—
224	C₄²	—	37	1	no	—
225	C₄²	—	37	1	no	—
226	C₄²	—	37	1	no	—
227	C₄²	—	37	1	no	—
228	C₄²	—	37	1	no	—
229	C₄²	—	37	1	no	—
230	C₄²	C₂×#3866	1	1	yes	characteristic
231	C₂³	—	37	1	no	—
232	C₂³	C₂²×#3866	1	1	yes	characteristic
233	C₂³	C₂²×#3866	1	1	yes	characteristic
234	Q₈	—	74	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	74	2	no	—
237	D₄	—	74	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	37	4	no	—
241	C₂×C₄	—	37	1	no	—
242	C₂×C₄	—	37	1	no	—
243	C₂×C₄	—	37	1	no	—
244	C₂×C₄	—	37	1	no	—
245	C₂×C₄	—	37	1	no	—
246	C₂×C₄	—	37	1	no	—
247	C₂×C₄	—	37	1	no	—
248	C₂×C₄	—	74	2	no	—
249	C₂×C₄	—	74	2	no	—
250	C₂×C₄	—	148	1	no	—
251	C₂×C₄	—	148	1	no	—
252	C₂×C₄	—	148	1	no	—
253	C₂×C₄	—	148	1	no	—
254	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
255	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
256	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
257	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
258	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
259	C₈	—	74	2	no	—
260	C₈	—	74	2	no	—
261	C₈	—	74	2	no	—
262	C₈	—	74	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	74	1	no	—
269	C₂²	—	74	1	no	—
270	C₂²	—	74	1	no	—
271	C₂²	#11924615	1	2	yes	—
272	C₂²	C₂³×#3866	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	37	2	no	—
279	C₄	—	37	2	no	—
280	C₄	—	74	1	no	—
281	C₄	—	74	1	no	—
282	C₄	—	74	1	no	—
283	C₄	—	74	1	no	—
284	C₄	—	74	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	4	1	no	—
287	C₂	—	148	1	no	—
288	C₂	C₂×#11924615	1	1	yes	characteristic
289	C₂	#14558611	1	1	yes	characteristic
290	C₂	#14558612	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄×D₄×C₃₇	C₄
prime radical	C₂×#10596964	C₂²
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄×D₄	#3866
hypocenter	C₃₇	#2923
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #14558558	C₂	semidirect
2. #14558557	C₂	semidirect
3. #14558556	C₂	semidirect
4. #14558555	C₂	semidirect
5. #14558554	C₂	semidirect
6. #14558553	C₂	semidirect
7. #14558539	C₂	semidirect
9. C₄×#10596957	C₂	semidirect
10. C₂×#11924591	C₂	semidirect
157. C₃₇	#2923	semidirect

Maximal subgroups

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

subgroup	index
#14558558	2
#14558557	2
#14558556	2
#14558555	2
#14558554	2
#14558553	2
#14558539	2
#14558506	2
C₄×#10596957	2
C₂×#11924591	2
C₄×#10596964	2
#2923	37