Subgroups — #18011354

Order 4736.

470 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558602	C₂	1	1	yes	maximal, characteristic
2	#14558601	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558573	C₂	1	2	yes	maximal, complemented
4	#14558572	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#3866	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#3866	C₂	1	2	yes	maximal, complemented
7	#14558544	C₂	1	1	yes	maximal, characteristic, complemented
8	#14558523	C₂	1	1	yes	maximal, characteristic
9	C₂×#11924599	C₂	1	1	yes	maximal, characteristic, complemented
10	#14558499	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×#3866	—	2	1	no	—
14	Q₈×#3866	—	2	1	no	—
15	Q₈×#3866	C₂²	1	2	yes	—
16	Q₈×#3866	C₂²	1	2	yes	—
17	Q₈×#3866	C₂²	1	2	yes	—
18	D₄×#3866	—	2	2	no	—
19	D₄×#3866	C₂²	1	2	yes	—
20	#11924615	—	2	2	no	—
21	#11924615	—	2	1	no	—
22	#11924615	C₂²	1	1	yes	characteristic
23	#11924615	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#3866	—	2	1	no	—
25	C₂×C₄⋊#3866	C₂²	1	1	yes	characteristic
26	#11924605	C₂²	1	2	yes	—
27	#11924604	C₂²	1	2	yes	—
28	#11924599	C₂²	1	2	yes	—
29	#11924599	C₂²	1	2	yes	—
30	#11924599	C₂²	1	1	yes	characteristic
31	#11924599	C₂²	1	1	yes	characteristic
32	#11924598	C₂²	1	1	yes	characteristic
33	#11924598	C₂²	1	1	yes	characteristic
34	#11924590	C₂²	1	2	yes	—
35	C₈×#3866	C₂²	1	2	yes	—
36	#41×D₃₇	C₂²	1	1	yes	characteristic
37	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
38	#11924569	C₄	1	1	yes	characteristic, complemented
39	#11924568	C₄	1	2	yes	complemented
40	#11924567	C₄	1	1	yes	characteristic, complemented
41	#11924566	C₄	1	1	yes	characteristic, complemented
42	#11924565	C₄	1	2	yes	complemented
43	Q₁₆×D₃₇	C₂²	1	2	yes	—
44	SD₁₆×D₃₇	C₂²	1	2	yes	—
45	M₁₆×D₃₇	C₂²	1	1	yes	characteristic
46	#137×C₃₇	C₄	1	1	yes	characteristic, complemented
47	C₂²×#3866	—	2	1	no	—
48	C₂²×#3866	—	2	2	no	—
49	#41×C₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
51	#10596966	C₂×C₄	1	1	yes	characteristic
52	Q₈×D₃₇	—	2	1	no	—
53	Q₈×D₃₇	—	2	1	no	—
54	Q₈×D₃₇	C₂×C₄	1	2	yes	complemented
55	Q₈×D₃₇	C₂³	1	2	yes	—
56	Q₈×D₃₇	C₂³	1	1	yes	characteristic
57	#10596964	—	2	1	no	—
58	#10596964	C₂×C₄	1	1	yes	characteristic
59	D₄×D₃₇	—	2	1	no	—
60	D₄×D₃₇	C₂³	1	1	yes	characteristic
61	#10596962	—	2	1	no	—
62	#10596962	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₃₇	—	2	1	no	—
64	C₂×C₄×D₃₇	—	2	1	no	—
65	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
66	C₂×Dic₇₄	—	2	1	no	—
67	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic, complemented
68	C₂²⋊#3866	—	2	1	no	—
69	C₂²⋊#3866	—	2	2	no	—
70	C₂²⋊#3866	D₄	1	2	yes	—
71	C₂×#64438	—	2	1	no	—
72	C₄⋊#3866	—	2	2	no	—
73	C₄⋊#3866	—	2	2	no	—
74	C₄⋊#3866	—	2	2	no	—
75	C₄⋊#3866	—	2	1	no	—
76	C₄⋊#3866	D₄	1	1	yes	characteristic
77	C₄⋊#3866	D₄	1	1	yes	characteristic
78	C₄⋊#3866	C₂³	1	1	yes	characteristic
79	C₄⋊#3866	C₂³	1	1	yes	characteristic
80	C₄⋊#3866	C₂³	1	2	yes	—
81	C₄×#3866	—	2	2	no	—
82	C₄×#3866	—	2	2	no	—
83	C₄×#3866	—	2	2	no	—
84	C₄×#3866	—	2	1	no	—
85	C₄×#3866	C₂³	1	2	yes	—
86	D₃₇⋊C₈	—	2	1	no	—
87	D₃₇⋊C₈	C₂³	1	1	yes	characteristic
88	D₃₇⋊C₈	C₂³	1	1	yes	characteristic
89	Q₁₆×C₃₇	C₂×C₄	1	2	yes	complemented
90	SD₁₆×C₃₇	C₂×C₄	1	2	yes	complemented
91	M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
92	#10596942	C₂×C₄	1	2	yes	complemented
93	#10596942	C₂×C₄	1	2	yes	complemented
94	#10596941	C₂×C₄	1	2	yes	complemented
95	#10596940	C₂×C₄	1	2	yes	complemented
96	#10596934	C₂×C₄	1	1	yes	characteristic, complemented
97	Dic₁₄₈	C₂×C₄	1	2	yes	complemented
98	#10596930	C₂×C₄	1	2	yes	complemented
99	#10596929	C₂×C₄	1	2	yes	complemented
100	C₈×D₃₇	C₂³	1	2	yes	—
101	C₂²×D₃₇	—	2	1	no	—
102	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×#3866	—	2	1	no	—
104	C₂×#3866	—	2	1	no	—
105	C₂×#3866	—	2	2	no	—
106	C₂×#3866	—	8	2	no	—
107	C₂×#3866	—	8	1	no	—
108	C₂×#3866	C₂×D₄	1	2	yes	—
109	C₂×#3866	C₂×D₄	1	2	yes	—
110	Q₈×C₃₇	—	2	1	no	—
111	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
112	Q₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
113	D₄×C₃₇	—	2	1	no	—
114	D₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₇	—	2	1	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²⋊D₃₇	—	2	1	no	—
119	C₂²⋊D₃₇	—	2	1	no	—
120	C₂²⋊D₃₇	—	2	1	no	—
121	C₂×Dic₃₇	—	2	1	no	—
122	C₂×Dic₃₇	—	2	1	no	—
123	C₂×Dic₃₇	#41	1	1	yes	characteristic
124	D₁₄₈	—	2	1	no	—
125	D₁₄₈	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₃₇	—	2	2	no	—
127	C₄×D₃₇	—	2	1	no	—
128	C₄×D₃₇	—	2	2	no	—
129	C₄×D₃₇	—	2	1	no	—
130	C₄×D₃₇	—	2	1	no	—
131	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
133	C₄×D₃₇	#41	1	1	yes	characteristic
134	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
135	Dic₇₄	—	2	1	no	—
136	Dic₇₄	—	2	2	no	—
137	Dic₇₄	—	2	2	no	—
138	Dic₇₄	—	2	1	no	—
139	Dic₇₄	C₂²×C₄	1	2	yes	—
140	Dic₇₄	C₂²×C₄	1	1	yes	characteristic
141	#64438	—	2	1	no	—
142	#64438	—	2	1	no	—
143	C₈×C₃₇	C₂²×C₄	1	2	yes	—
144	#64436	C₂²×C₄	1	2	yes	—
145	C₂²×C₃₇	—	2	1	no	—
146	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
147	C₂×D₃₇	—	2	1	no	—
148	C₂×D₃₇	—	2	1	no	—
149	C₂×D₃₇	—	2	1	no	—
150	C₂×D₃₇	—	4	1	no	—
151	C₂×D₃₇	—	4	1	no	—
152	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
153	C₂×D₃₇	C₂²×D₄	1	1	yes	characteristic
154	#3866	—	2	2	no	—
155	#3866	—	2	2	no	—
156	#3866	—	4	1	no	—
157	#3866	—	4	2	no	—
158	#3866	—	4	1	no	—
159	C₄×C₃₇	—	2	2	no	—
160	C₄×C₃₇	—	2	1	no	—
161	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
162	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
163	Dic₃₇	—	2	2	no	—
164	Dic₃₇	—	2	1	no	—
165	Dic₃₇	C₄×D₄	1	1	yes	characteristic
166	Dic₃₇	C₂×#41	1	1	yes	characteristic
167	#2947	—	37	1	no	maximal, Sylow
168	C₂×C₃₇	—	2	1	no	—
169	C₂×C₃₇	—	4	1	no	—
170	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
171	D₃₇	—	2	1	no	—
172	D₃₇	—	4	1	no	—
173	D₃₇	#577	1	1	yes	characteristic
174	D₃₇	#578	1	1	yes	characteristic
175	C₂×#137	—	37	1	no	—
176	#520	—	37	1	no	—
177	#519	—	37	1	no	—
178	#441	—	37	2	no	—
179	#440	—	37	2	no	—
180	C₄×Q₁₆	—	37	2	no	—
181	C₄×SD₁₆	—	37	2	no	—
182	#428	—	37	1	no	—
183	#416	—	37	1	no	—
184	C₂×#103	—	37	1	no	—
185	#405	—	37	1	no	—
186	C₃₇	#2947	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	37	1	no	—
188	C₂²×Q₈	—	37	1	no	—
189	#137	—	37	1	no	—
190	#137	—	37	1	no	—
191	#137	—	37	2	no	—
192	#137	—	37	2	no	—
193	#137	—	37	1	no	—
194	#137	#3866	1	1	yes	characteristic, complemented
195	C₂×Q₁₆	—	37	2	no	—
196	C₂×SD₁₆	—	37	2	no	—
197	C₂×M₁₆	—	37	1	no	—
198	C₂²×C₈	—	37	1	no	—
199	C₄×Q₈	—	37	2	no	—
200	C₄×Q₈	—	37	2	no	—
201	C₄×Q₈	—	37	2	no	—
202	C₄×Q₈	—	74	1	no	—
203	C₄×Q₈	—	74	1	no	—
204	C₄×D₄	—	37	2	no	—
205	C₄×D₄	—	74	2	no	—
206	#117	—	37	1	no	—
207	#117	—	37	1	no	—
208	#117	—	74	2	no	—
209	#117	—	74	1	no	—
210	C₂×C₄⋊C₄	—	37	1	no	—
211	C₂×C₄⋊C₄	—	74	1	no	—
212	#107	—	37	2	no	—
213	#106	—	37	2	no	—
214	#103	—	37	2	no	—
215	#103	—	37	2	no	—
216	#103	—	37	1	no	—
217	#103	—	37	1	no	—
218	#102	—	37	1	no	—
219	#102	—	37	1	no	—
220	#97	—	37	2	no	—
221	C₄×C₈	—	37	2	no	—
222	#41	—	37	1	no	—
223	#41	—	37	1	no	—
224	#41	—	37	1	no	—
225	#41	—	74	1	no	—
226	#41	—	74	1	no	—
227	#41	C₂×#3866	1	1	yes	characteristic
228	C₂×Q₈	—	37	2	no	—
229	C₂×Q₈	—	37	2	no	—
230	C₂×Q₈	—	37	1	no	—
231	C₂×Q₈	—	37	1	no	—
232	C₂×Q₈	—	74	1	no	—
233	C₂×Q₈	—	74	1	no	—
234	C₂×Q₈	—	74	1	no	—
235	C₂×Q₈	C₂×#3866	1	1	yes	characteristic, complemented
236	C₂×D₄	—	37	1	no	—
237	C₂×D₄	—	74	1	no	—
238	C₂²×C₄	—	37	1	no	—
239	C₂²×C₄	—	74	1	no	—
240	C₂²×C₄	—	74	1	no	—
241	C₂²×C₄	—	74	2	no	—
242	C₂²×C₄	—	74	1	no	—
243	Q₁₆	—	37	2	no	—
244	Q₁₆	—	37	2	no	—
245	Q₁₆	—	37	2	no	—
246	Q₁₆	C₂×#3866	1	2	yes	complemented
247	SD₁₆	—	37	2	no	—
248	SD₁₆	—	37	2	no	—
249	SD₁₆	—	37	2	no	—
250	SD₁₆	C₂×#3866	1	2	yes	complemented
251	M₁₆	—	37	1	no	—
252	M₁₆	—	37	2	no	—
253	M₁₆	C₂×#3866	1	1	yes	characteristic, complemented
254	C₂×C₈	—	37	2	no	—
255	C₂×C₈	—	37	1	no	—
256	C₂×C₈	—	37	1	no	—
257	C₂×C₈	—	74	1	no	—
258	C₂×C₈	—	74	1	no	—
259	C₄⋊C₄	—	37	1	no	—
260	C₄⋊C₄	—	37	1	no	—
261	C₄⋊C₄	—	37	1	no	—
262	C₄⋊C₄	—	37	1	no	—
263	C₄⋊C₄	—	37	2	no	—
264	C₄⋊C₄	—	74	1	no	—
265	C₄⋊C₄	—	74	2	no	—
266	C₄⋊C₄	—	74	2	no	—
267	C₄⋊C₄	—	74	2	no	—
268	C₂²⋊C₄	—	37	2	no	—
269	C₂²⋊C₄	—	74	2	no	—
270	C₂²⋊C₄	—	74	1	no	—
271	C₄²	—	37	2	no	—
272	C₄²	—	74	1	no	—
273	C₄²	—	74	2	no	—
274	C₄²	—	74	2	no	—
275	C₄²	—	74	2	no	—
276	C₂³	—	37	1	no	—
277	C₂³	—	74	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	37	2	no	—
280	Q₈	—	37	1	no	—
281	Q₈	—	74	2	no	—
282	Q₈	—	74	2	no	—
283	Q₈	—	74	1	no	—
284	Q₈	—	74	1	no	—
285	Q₈	C₂²×#3866	1	2	yes	—
286	Q₈	C₂²×#3866	1	1	yes	characteristic
287	D₄	—	2	1	no	—
288	D₄	—	37	1	no	—
289	D₄	—	74	1	no	—
290	D₄	—	74	1	no	—
291	D₄	—	74	1	no	—
292	D₄	—	74	1	no	—
293	D₄	C₂²×#3866	1	1	yes	characteristic
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	37	1	no	—
297	C₂×C₄	—	37	1	no	—
298	C₂×C₄	—	37	1	no	—
299	C₂×C₄	—	37	1	no	—
300	C₂×C₄	—	37	1	no	—
301	C₂×C₄	—	37	2	no	—
302	C₂×C₄	—	37	2	no	—
303	C₂×C₄	—	296	2	no	—
304	C₂×C₄	—	296	1	no	—
305	C₂×C₄	—	74	1	no	—
306	C₂×C₄	—	74	1	no	—
307	C₂×C₄	—	74	2	no	—
308	C₂×C₄	—	74	2	no	—
309	C₂×C₄	—	74	1	no	—
310	C₂×C₄	—	74	1	no	—
311	C₂×C₄	—	74	1	no	—
312	C₂×C₄	—	74	1	no	—
313	C₂×C₄	—	74	1	no	—
314	C₂×C₄	—	74	2	no	—
315	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
316	C₈	—	37	2	no	—
317	C₈	—	74	1	no	—
318	C₈	—	74	1	no	—
319	C₈	C₂²×#3866	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	37	1	no	—
322	C₂²	—	37	1	no	—
323	C₂²	—	74	1	no	—
324	C₂²	—	74	1	no	—
325	C₂²	—	74	1	no	—
326	C₂²	—	148	1	no	—
327	C₂²	—	148	1	no	—
328	C₂²	D₄×#3866	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	37	1	no	—
332	C₄	—	37	1	no	—
333	C₄	—	74	2	no	—
334	C₄	—	74	2	no	—
335	C₄	—	74	1	no	—
336	C₄	—	74	2	no	—
337	C₄	—	148	2	no	—
338	C₄	—	148	1	no	—
339	C₄	—	148	1	no	—
340	C₄	D₄×#3866	1	1	yes	characteristic
341	C₄	C₂³×#3866	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	4	1	no	—
344	C₂	—	37	1	no	—
345	C₂	—	37	1	no	—
346	C₂	—	74	1	no	—
347	C₂	—	148	1	no	—
348	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	#137×C₃₇	C₄
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	#41×D₃₇	C₂²
cosocle	C₄×D₃₇	C₂⁴
hypercenter	#137	#3866
hypocenter	C₃₇	#2947
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. #14558601	C₂	semidirect
3. #14558573	C₂	semidirect
4. #14558572	C₂	semidirect
5. Q₁₆×#3866	C₂	semidirect
6. SD₁₆×#3866	C₂	semidirect
7. #14558544	C₂	semidirect
9. C₂×#11924599	C₂	semidirect
10. #14558499	C₂	semidirect
38. #11924569	C₄	semidirect
39. #11924568	C₄	semidirect
40. #11924567	C₄	semidirect
41. #11924566	C₄	semidirect
42. #11924565	C₄	semidirect
46. #137×C₃₇	C₄	semidirect
50. C₂×Q₈×C₃₇	C₂×C₄	semidirect
54. Q₈×D₃₇	C₂×C₄	semidirect
67. C₂×Dic₇₄	C₂×C₄	semidirect
89. Q₁₆×C₃₇	C₂×C₄	semidirect
90. SD₁₆×C₃₇	C₂×C₄	semidirect
91. M₁₆×C₃₇	C₂×C₄	semidirect
92. #10596942	C₂×C₄	semidirect
93. #10596942	C₂×C₄	semidirect
94. #10596941	C₂×C₄	semidirect
95. #10596940	C₂×C₄	semidirect
96. #10596934	C₂×C₄	semidirect
97. Dic₁₄₈	C₂×C₄	semidirect
98. #10596930	C₂×C₄	semidirect
99. #10596929	C₂×C₄	semidirect
186. C₃₇	#2947	semidirect
194. #137	#3866	semidirect
235. C₂×Q₈	C₂×#3866	semidirect
246. Q₁₆	C₂×#3866	semidirect
250. SD₁₆	C₂×#3866	semidirect
253. M₁₆	C₂×#3866	semidirect

Maximal subgroups

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

subgroup	index
#14558602	2
#14558601	2
#14558573	2
#14558572	2
Q₁₆×#3866	2
SD₁₆×#3866	2
#14558544	2
#14558523	2
C₂×#11924599	2
#14558499	2
#137×D₃₇	2
#2947	37