Subgroups — #18004051

Order 4736.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924556	C₂	1	2	yes	maximal, complemented
2	#14558083	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557755	C₂	1	2	yes	maximal, complemented
4	#14557748	C₂	1	2	yes	maximal, complemented
5	#14557743	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557738	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557733	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557728	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊#64436	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557527	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	2	yes	—
13	#11924556	—	2	8	no	—
14	C₂×#10596942	C₂²	1	2	yes	—
15	C₂×#10596940	C₂²	1	2	yes	—
16	C₂×#10596941	C₂²	1	2	yes	—
17	C₂×#10596939	C₂²	1	2	yes	—
18	C₂²×#64436	C₂²	1	2	yes	—
19	#11924536	C₂²	1	1	yes	characteristic
20	#11924533	C₂²	1	1	yes	characteristic
21	#11924524	C₂²	1	2	yes	—
22	#11924510	—	2	1	no	—
23	#11924510	—	2	1	no	—
24	C₄×D₁₄₈	—	2	1	no	—
25	C₄×D₁₄₈	—	2	1	no	—
26	C₄⋊#64436	C₂²	1	1	yes	characteristic
27	C₄⋊#64436	C₂²	1	1	yes	characteristic
28	C₄⋊#64436	C₂²	1	2	yes	—
29	#11924459	C₂²	1	2	yes	—
30	#11924459	C₂²	1	2	yes	—
31	#11924456	C₂²	1	2	yes	—
32	#11924456	C₂²	1	2	yes	—
33	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₃₇	—	2	1	no	—
37	C₄×D₄×C₃₇	—	2	1	no	—
38	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
40	#117×C₃₇	—	2	1	no	—
41	#117×C₃₇	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₃₇	—	2	1	no	—
43	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
44	#41×C₃₇	—	2	2	no	—
45	#41×C₃₇	D₄	1	4	yes	complemented
46	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₃₇	—	2	1	no	—
48	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₃₇	—	2	1	no	—
50	C₂²×C₄×C₃₇	—	2	1	no	—
51	C₂²×C₄×C₃₇	—	2	1	no	—
52	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₃₇	—	2	2	no	—
56	#10596962	—	2	4	no	—
57	#10596962	—	2	4	no	—
58	C₂×D₁₄₈	—	2	1	no	—
59	C₂×D₁₄₈	—	2	1	no	—
60	C₂×D₁₄₈	C₂³	1	2	yes	—
61	C₂×C₄×D₃₇	—	2	2	no	—
62	C₂×Dic₇₄	C₂³	1	2	yes	—
63	C₄⋊C₄×C₃₇	—	2	1	no	—
64	C₄⋊C₄×C₃₇	—	2	1	no	—
65	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₃₇	—	2	1	no	—
68	C₂²⋊C₄×C₃₇	—	2	1	no	—
69	C₂²⋊C₄×C₃₇	—	2	1	no	—
70	C₄²×C₃₇	—	2	1	no	—
71	C₄²×C₃₇	—	2	1	no	—
72	C₄²×C₃₇	—	2	1	no	—
73	C₄²×C₃₇	D₄	1	1	yes	characteristic
74	C₄²×C₃₇	D₄	1	1	yes	characteristic
75	C₄²×C₃₇	C₂³	1	1	yes	characteristic
76	C₄²×C₃₇	C₂³	1	1	yes	characteristic
77	#10596942	—	2	4	no	—
78	#10596941	—	2	4	no	—
79	#10596940	—	2	4	no	—
80	#10596939	—	2	4	no	—
81	#10596938	—	2	2	no	—
82	#10596938	—	2	2	no	—
83	C₄⋊Dic₃₇	—	2	1	no	—
84	C₄⋊Dic₃₇	—	2	1	no	—
85	C₂×#64436	—	2	4	no	—
86	C₂×#64436	C₂³	1	2	yes	—
87	C₂×#64436	C₂³	1	2	yes	—
88	C₂³×C₃₇	—	2	1	no	—
89	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₃₇	—	2	2	no	—
91	Q₈×C₃₇	—	2	1	no	—
92	Q₈×C₃₇	C₂×D₄	1	2	yes	—
93	D₄×C₃₇	—	2	4	no	—
94	D₄×C₃₇	—	2	1	no	—
95	D₄×C₃₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₃₇	—	2	1	no	—
97	C₂×C₄×C₃₇	—	2	4	no	—
98	C₂×C₄×C₃₇	—	2	2	no	—
99	C₂×C₄×C₃₇	—	2	1	no	—
100	C₂×C₄×C₃₇	—	2	1	no	—
101	C₂×C₄×C₃₇	—	4	1	no	—
102	C₂×C₄×C₃₇	—	4	2	no	—
103	C₂×C₄×C₃₇	—	4	2	no	—
104	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₇	#41	1	2	yes	—
111	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₃₇	—	4	4	no	—
113	C₂×Dic₃₇	—	2	2	no	—
114	D₁₄₈	—	2	2	no	—
115	D₁₄₈	—	2	2	no	—
116	D₁₄₈	—	4	2	no	—
117	D₁₄₈	—	4	2	no	—
118	C₄×D₃₇	—	4	4	no	—
119	Dic₇₄	—	2	2	no	—
120	Dic₇₄	—	2	2	no	—
121	#64436	—	2	2	no	—
122	#64436	—	2	2	no	—
123	C₂²×C₃₇	—	2	1	no	—
124	C₂²×C₃₇	—	2	1	no	—
125	C₂²×C₃₇	—	2	2	no	—
126	C₂²×C₃₇	—	4	1	no	—
127	C₂²×C₃₇	—	4	1	no	—
128	C₂²×C₃₇	#121	1	2	yes	—
129	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×D₃₇	—	4	2	no	—
131	C₂×D₃₇	—	4	2	no	—
132	C₂×D₃₇	—	4	2	no	—
133	C₄×C₃₇	—	2	2	no	—
134	C₄×C₃₇	—	2	1	no	—
135	C₄×C₃₇	—	2	1	no	—
136	C₄×C₃₇	—	2	1	no	—
137	C₄×C₃₇	—	4	1	no	—
138	C₄×C₃₇	—	4	1	no	—
139	C₄×C₃₇	#121	1	2	yes	—
140	C₄×C₃₇	#135	1	2	yes	complemented
141	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
143	Dic₃₇	—	4	2	no	—
144	#3035	—	37	1	no	maximal, Sylow
145	C₂×C₃₇	—	2	2	no	—
146	C₂×C₃₇	—	4	2	no	—
147	C₂×C₃₇	C₂×#121	1	1	yes	characteristic
148	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
149	C₂×C₃₇	#575	1	1	yes	characteristic
150	D₃₇	—	8	2	no	—
151	C₂×#135	—	37	2	no	—
152	#532	—	37	1	no	—
153	C₄×#41	D₃₇	1	1	yes	characteristic, complemented
154	#464	—	37	2	no	—
155	#463	—	37	2	no	—
156	#462	—	37	1	no	—
157	#461	—	37	1	no	—
158	#460	—	37	1	no	—
159	#459	—	37	1	no	—
160	C₂×C₄⋊C₈	—	37	1	no	—
161	#416	—	37	2	no	—
162	C₃₇	#3035	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	37	2	no	—
164	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
165	#135	—	74	8	no	—
166	C₂×Q₁₆	—	37	2	no	—
167	C₂×SD₁₆	—	37	2	no	—
168	C₂×SD₁₆	—	37	2	no	—
169	C₂×D₈	—	37	2	no	—
170	C₂²×C₈	—	37	2	no	—
171	C₄⋊Q₈	—	37	1	no	—
172	#127	—	37	1	no	—
173	#124	—	37	2	no	—
174	#121	—	74	1	no	—
175	#121	—	74	1	no	—
176	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
177	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	74	1	no	—
181	C₄×D₄	—	74	1	no	—
182	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
183	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
184	#117	—	2	1	no	—
185	#117	C₂×D₃₇	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
188	C₄⋊C₈	—	37	1	no	—
189	C₄⋊C₈	—	37	1	no	—
190	C₄⋊C₈	—	37	2	no	—
191	#103	—	37	2	no	—
192	#103	—	37	2	no	—
193	#102	—	37	2	no	—
194	#102	—	37	2	no	—
195	#41	—	2	2	no	—
196	#41	—	74	4	no	—
197	#41	—	74	4	no	—
198	#41	D₁₄₈	1	4	yes	complemented
199	C₂×Q₈	—	37	2	no	—
200	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	37	2	no	—
203	C₂×D₄	—	74	2	no	—
204	C₂×D₄	—	74	1	no	—
205	C₂×D₄	—	74	1	no	—
206	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	74	2	no	—
211	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
214	Q₁₆	—	74	4	no	—
215	SD₁₆	—	74	4	no	—
216	SD₁₆	—	74	4	no	—
217	D₈	—	74	4	no	—
218	C₂×C₈	—	37	2	no	—
219	C₂×C₈	—	37	2	no	—
220	C₂×C₈	—	74	4	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	74	1	no	—
224	C₄⋊C₄	—	74	1	no	—
225	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
226	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	74	2	no	—
231	C₂²⋊C₄	—	74	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
236	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
237	C₄²	C₂²×D₃₇	1	1	yes	characteristic
238	C₄²	C₂²×D₃₇	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	74	2	no	—
241	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	—	74	2	no	—
244	Q₈	—	74	2	no	—
245	Q₈	C₂×D₁₄₈	1	2	yes	—
246	D₄	—	2	4	no	—
247	D₄	—	2	1	no	—
248	D₄	—	74	2	no	—
249	D₄	—	74	2	no	—
250	D₄	—	148	2	no	—
251	D₄	—	148	4	no	—
252	D₄	—	148	2	no	—
253	D₄	C₂×D₁₄₈	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	74	2	no	—
263	C₂×C₄	—	148	4	no	—
264	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
265	C₂×C₄	#10596962	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
272	C₈	—	74	2	no	—
273	C₈	—	74	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	148	2	no	—
280	C₂²	—	148	2	no	—
281	C₂²	—	148	2	no	—
282	C₂²	#11924510	1	2	yes	—
283	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	148	2	no	—
291	C₄	#11924510	1	2	yes	—
292	C₄	#11924556	1	2	yes	complemented
293	C₄	C₂²×D₁₄₈	1	1	yes	characteristic
294	C₄	C₂×#10596962	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	2	no	—
297	C₂	—	296	2	no	—
298	C₂	C₂×#11924510	1	1	yes	characteristic
299	C₂	C₂×#11924556	1	1	yes	characteristic
300	C₂	#14558331	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₄×#41×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	C₄×#41	D₃₇
hypocenter	C₃₇	#3035
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#11924556	C₂	semidirect
2. #14558083	C₂	semidirect
3. #14557755	C₂	semidirect
4. #14557748	C₂	semidirect
5. #14557743	C₂	semidirect
6. #14557738	C₂	semidirect
7. #14557733	C₂	semidirect
8. #14557728	C₂	semidirect
9. C₂×C₄⋊#64436	C₂	semidirect
10. #14557527	C₂	semidirect
11. C₄×#41×C₃₇	C₂	semidirect
45. #41×C₃₇	D₄	semidirect
140. C₄×C₃₇	#135	semidirect
153. C₄×#41	D₃₇	semidirect
162. C₃₇	#3035	semidirect
198. #41	D₁₄₈	semidirect
292. C₄	#11924556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924556	2
#14558083	2
#14557755	2
#14557748	2
#14557743	2
#14557738	2
#14557733	2
#14557728	2
C₂×C₄⋊#64436	2
#14557527	2
C₄×#41×C₃₇	2
#3035	37