Subgroups — #18011334

Order 4736.

544 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10596957	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558566	C₂	1	2	yes	maximal, complemented
4	#14558565	C₂	1	2	yes	maximal, complemented
5	#14558562	C₂	1	2	yes	maximal, complemented
6	#14558560	C₂	1	2	yes	maximal, complemented
7	#14558536	C₂	1	1	yes	maximal, characteristic, complemented
8	#14558505	C₂	1	1	yes	maximal, characteristic, complemented
9	#14558503	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#10596957	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#11924496	C₂	1	1	yes	maximal, characteristic
12	C₂×#10596953	C₂²	1	2	yes	complemented
13	D₃₇⋊M₁₆	—	2	4	no	—
14	C₂×#10596957	—	2	4	no	—
15	C₂×#10596957	C₂²	1	1	yes	characteristic
16	C₂×#10596957	C₂²	1	1	yes	characteristic
17	C₂×#10596957	C₂²	1	1	yes	characteristic
18	C₂×#10596957	C₂²	1	1	yes	characteristic
19	C₂×#10596957	C₂²	1	2	yes	complemented
20	C₂×D₃₇⋊C₈	C₂²	1	2	yes	complemented
21	C₂²×#64438	C₂²	1	2	yes	complemented
22	#11924603	C₂²	1	2	yes	complemented
23	C₄⋊#64438	C₂²	1	2	yes	complemented
24	#11924593	C₂²	1	2	yes	complemented
25	#11924593	C₂²	1	2	yes	complemented
26	C₂²⋊#64438	C₂²	1	2	yes	complemented
27	C₂²⋊#64438	C₂²	1	2	yes	complemented
28	#11924591	C₂²	1	2	yes	complemented
29	C₄×#64438	C₂²	1	2	yes	complemented
30	C₂²×C₂²⋊D₃₇	C₄	1	1	yes	characteristic
31	C₂³×Dic₃₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
33	#11924496	C₄	1	1	yes	characteristic
34	#11924496	C₄	1	1	yes	characteristic
35	#11924496	C₄	1	1	yes	characteristic
36	#11924496	C₄	1	1	yes	characteristic
37	#11924496	C₂²	1	1	yes	characteristic
38	#11924496	C₂²	1	1	yes	characteristic
39	#11924496	C₂²	1	1	yes	characteristic
40	#11924496	C₂²	1	1	yes	characteristic
41	C₂×#10596936	C₄	1	1	yes	characteristic
42	C₂×#10596938	C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₃₇	C₄	1	1	yes	characteristic
45	C₂⁴×C₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊D₃₇	—	2	4	no	—
50	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₃₇	—	2	2	no	—
55	C₂²×Dic₃₇	—	2	2	no	—
56	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
61	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
62	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₃₇	—	2	2	no	—
64	C₂×C₄×D₃₇	—	2	2	no	—
65	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
69	#10596957	—	2	2	no	—
70	#10596957	—	2	2	no	—
71	#10596957	—	2	2	no	—
72	#10596957	—	2	4	no	—
73	#10596957	D₄	1	4	yes	complemented
74	C₂×#64438	—	2	4	no	—
75	C₂×#64438	C₂³	1	2	yes	—
76	C₂×#64438	C₂³	1	2	yes	—
77	C₂×#64438	C₂³	1	2	yes	—
78	C₂×#64438	C₂³	1	2	yes	—
79	#10596953	—	2	4	no	—
80	D₃₇⋊C₈	—	2	4	no	—
81	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
85	#10596938	C₂×C₄	1	1	yes	characteristic
86	#10596938	C₂×C₄	1	1	yes	characteristic
87	#10596938	C₂×C₄	1	1	yes	characteristic
88	#10596938	C₂×C₄	1	1	yes	characteristic
89	#10596936	C₂×C₄	1	1	yes	characteristic
90	#10596936	C₂×C₄	1	1	yes	characteristic
91	#10596936	C₂×C₄	1	1	yes	characteristic
92	#10596936	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
96	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
97	C₂³×C₃₇	—	2	2	no	—
98	C₂³×C₃₇	—	2	2	no	—
99	C₂³×C₃₇	—	4	1	no	—
100	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×D₃₇	—	2	2	no	—
104	C₂²×D₃₇	—	2	2	no	—
105	C₂²×D₃₇	—	4	1	no	—
106	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₇	—	2	2	no	—
109	C₂×C₄×C₃₇	—	4	1	no	—
110	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²⋊D₃₇	—	2	2	no	—
115	C₂²⋊D₃₇	—	2	2	no	—
116	C₂²⋊D₃₇	—	2	2	no	—
117	C₂²⋊D₃₇	M₁₆	1	4	yes	complemented
118	C₂×Dic₃₇	—	2	2	no	—
119	C₂×Dic₃₇	—	2	2	no	—
120	C₂×Dic₃₇	—	2	2	no	—
121	C₂×Dic₃₇	—	2	2	no	—
122	C₂×Dic₃₇	—	2	1	no	—
123	C₂×Dic₃₇	—	2	1	no	—
124	C₂×Dic₃₇	—	2	1	no	—
125	C₂×Dic₃₇	—	2	1	no	—
126	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
132	C₂×Dic₃₇	#41	1	2	yes	—
133	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
134	C₄×D₃₇	—	2	2	no	—
135	C₄×D₃₇	—	2	2	no	—
136	C₄×D₃₇	—	2	1	no	—
137	C₄×D₃₇	—	2	1	no	—
138	C₄×D₃₇	—	2	1	no	—
139	C₄×D₃₇	—	2	1	no	—
140	#64438	—	2	2	no	—
141	#64438	—	2	2	no	—
142	#64438	—	2	2	no	—
143	#64438	C₂×D₄	1	4	yes	complemented
144	C₂²×C₃₇	—	2	1	no	—
145	C₂²×C₃₇	—	2	1	no	—
146	C₂²×C₃₇	—	2	1	no	—
147	C₂²×C₃₇	—	2	1	no	—
148	C₂²×C₃₇	—	2	1	no	—
149	C₂²×C₃₇	—	2	1	no	—
150	C₂²×C₃₇	—	2	1	no	—
151	C₂²×C₃₇	—	4	4	no	—
152	C₂²×C₃₇	C₄×D₄	1	2	yes	—
153	C₂²×C₃₇	C₂×M₁₆	1	2	yes	complemented
154	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₃₇	—	2	1	no	—
156	C₂×D₃₇	—	2	1	no	—
157	C₂×D₃₇	—	2	1	no	—
158	C₂×D₃₇	—	2	1	no	—
159	C₂×D₃₇	—	2	1	no	—
160	C₂×D₃₇	—	4	4	no	—
161	C₂×D₃₇	C₂×M₁₆	1	2	yes	complemented
162	C₄×C₃₇	—	2	1	no	—
163	C₄×C₃₇	—	2	1	no	—
164	C₄×C₃₇	—	2	1	no	—
165	C₄×C₃₇	—	2	1	no	—
166	Dic₃₇	—	2	1	no	—
167	Dic₃₇	—	2	1	no	—
168	Dic₃₇	—	2	1	no	—
169	Dic₃₇	C₄×D₄	1	2	yes	—
170	Dic₃₇	C₂×M₁₆	1	2	yes	complemented
171	Dic₃₇	C₂²×D₄	1	1	yes	characteristic
172	Dic₃₇	C₂×#41	1	1	yes	characteristic
173	D₄×M₁₆	—	37	1	no	maximal, Sylow
174	C₂×C₃₇	—	2	2	no	—
175	C₂×C₃₇	—	2	2	no	—
176	C₂×C₃₇	—	4	1	no	—
177	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
179	C₂×C₃₇	#568	1	1	yes	characteristic
180	D₃₇	—	2	2	no	—
181	D₃₇	—	4	1	no	—
182	C₂²×M₁₆	—	37	1	no	—
183	C₂²×M₁₆	—	37	1	no	—
184	C₂×C₄×D₄	—	37	1	no	—
185	#436	—	37	2	no	—
186	#435	—	37	2	no	—
187	#435	—	37	2	no	—
188	C₈×D₄	—	37	2	no	—
189	#423	—	37	1	no	—
190	#407	—	37	1	no	—
191	#407	—	37	1	no	—
192	C₄×M₁₆	—	37	1	no	—
193	C₃₇	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
194	C₂²×D₄	—	37	1	no	—
195	C₂³×C₄	—	37	1	no	—
196	C₂³×C₄	—	37	1	no	—
197	C₂×M₁₆	—	37	1	no	—
198	C₂×M₁₆	—	37	1	no	—
199	C₂×M₁₆	—	37	1	no	—
200	C₂×M₁₆	—	37	2	no	—
201	C₂×M₁₆	—	37	2	no	—
202	C₂×M₁₆	—	37	1	no	—
203	C₂×M₁₆	—	74	4	no	—
204	C₂×M₁₆	—	74	4	no	—
205	C₂²×C₈	—	37	2	no	—
206	C₂²×C₈	—	37	2	no	—
207	C₄×D₄	—	37	1	no	—
208	C₄×D₄	—	37	1	no	—
209	C₄×D₄	—	37	1	no	—
210	C₄×D₄	—	37	1	no	—
211	C₄×D₄	—	37	1	no	—
212	C₄×D₄	—	37	1	no	—
213	C₄×D₄	—	37	1	no	—
214	C₄×D₄	—	37	1	no	—
215	C₂×C₄⋊C₄	—	37	1	no	—
216	C₂×C₂²⋊C₄	—	37	1	no	—
217	C₂×C₂²⋊C₄	#3866	1	1	yes	characteristic
218	C₂×C₄²	—	37	1	no	—
219	C₄⋊C₈	—	37	2	no	—
220	C₄⋊C₈	—	37	2	no	—
221	C₂²⋊C₈	—	37	2	no	—
222	C₂²⋊C₈	—	37	2	no	—
223	C₂²⋊C₈	—	37	2	no	—
224	C₂²⋊C₈	—	37	2	no	—
225	#97	—	37	2	no	—
226	C₄×C₈	—	37	2	no	—
227	C₂⁴	—	37	1	no	—
228	C₂⁴	C₂×#3866	1	1	yes	characteristic
229	C₂×D₄	—	37	1	no	—
230	C₂×D₄	—	37	1	no	—
231	C₂×D₄	—	37	1	no	—
232	C₂×D₄	—	37	1	no	—
233	C₂×D₄	—	74	4	no	—
234	C₂²×C₄	—	37	1	no	—
235	C₂²×C₄	—	37	1	no	—
236	C₂²×C₄	—	37	1	no	—
237	C₂²×C₄	—	37	1	no	—
238	C₂²×C₄	—	37	1	no	—
239	C₂²×C₄	—	37	1	no	—
240	C₂²×C₄	—	37	1	no	—
241	C₂²×C₄	—	37	1	no	—
242	C₂²×C₄	—	37	1	no	—
243	C₂²×C₄	—	37	1	no	—
244	C₂²×C₄	—	37	1	no	—
245	C₂²×C₄	—	74	2	no	—
246	C₂²×C₄	—	74	2	no	—
247	C₂²×C₄	—	74	2	no	—
248	C₂²×C₄	—	74	2	no	—
249	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
250	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
251	M₁₆	—	37	4	no	—
252	M₁₆	—	74	4	no	—
253	M₁₆	—	74	4	no	—
254	M₁₆	—	74	2	no	—
255	M₁₆	—	74	2	no	—
256	M₁₆	—	74	2	no	—
257	C₂×C₈	—	37	2	no	—
258	C₂×C₈	—	37	2	no	—
259	C₂×C₈	—	37	2	no	—
260	C₂×C₈	—	37	2	no	—
261	C₂×C₈	—	74	4	no	—
262	C₂×C₈	—	74	4	no	—
263	C₄⋊C₄	—	37	1	no	—
264	C₄⋊C₄	—	37	1	no	—
265	C₄⋊C₄	—	37	1	no	—
266	C₄⋊C₄	—	37	1	no	—
267	C₂²⋊C₄	—	37	1	no	—
268	C₂²⋊C₄	—	37	1	no	—
269	C₂²⋊C₄	—	37	1	no	—
270	C₂²⋊C₄	—	37	1	no	—
271	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
272	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
275	C₄²	—	37	1	no	—
276	C₄²	—	37	1	no	—
277	C₄²	—	37	1	no	—
278	C₄²	—	37	1	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	37	1	no	—
283	C₂³	—	37	1	no	—
284	C₂³	—	74	2	no	—
285	C₂³	—	74	2	no	—
286	C₂³	—	148	1	no	—
287	C₂³	C₂²×#3866	1	1	yes	characteristic
288	C₂³	C₂²×#3866	1	1	yes	characteristic
289	C₂³	C₂²×#3866	1	1	yes	characteristic
290	D₄	—	37	4	no	—
291	D₄	—	74	2	no	—
292	D₄	—	74	2	no	—
293	D₄	—	74	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	37	2	no	—
297	C₂×C₄	—	37	2	no	—
298	C₂×C₄	—	37	1	no	—
299	C₂×C₄	—	37	1	no	—
300	C₂×C₄	—	37	1	no	—
301	C₂×C₄	—	37	1	no	—
302	C₂×C₄	—	37	1	no	—
303	C₂×C₄	—	37	1	no	—
304	C₂×C₄	—	74	1	no	—
305	C₂×C₄	—	74	1	no	—
306	C₂×C₄	—	74	1	no	—
307	C₂×C₄	—	74	1	no	—
308	C₂×C₄	—	74	2	no	—
309	C₂×C₄	—	74	2	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	—	74	2	no	—
316	C₂×C₄	—	74	2	no	—
317	C₂×C₄	—	74	2	no	—
318	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
319	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
320	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
321	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
322	C₈	—	37	4	no	—
323	C₈	—	74	2	no	—
324	C₈	—	74	2	no	—
325	C₈	—	74	2	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	4	no	—
334	C₂²	—	37	2	no	—
335	C₂²	—	74	1	no	—
336	C₂²	—	74	1	no	—
337	C₂²	—	74	1	no	—
338	C₂²	—	74	1	no	—
339	C₂²	—	74	1	no	—
340	C₂²	—	148	4	no	—
341	C₂²	D₄×#3866	1	2	yes	—
342	C₂²	D₃₇⋊M₁₆	1	2	yes	complemented
343	C₂²	C₂³×#3866	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	37	1	no	—
349	C₄	—	37	1	no	—
350	C₄	—	37	2	no	—
351	C₄	—	37	2	no	—
352	C₄	—	74	1	no	—
353	C₄	—	74	1	no	—
354	C₄	—	74	1	no	—
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	4	1	no	—
358	C₂	—	74	2	no	—
359	C₂	—	148	1	no	—
360	C₂	C₂×D₄×#3866	1	1	yes	characteristic
361	C₂	C₂×D₃₇⋊M₁₆	1	1	yes	characteristic
362	C₂	#14558611	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₂×C₂²⋊C₄×C₃₇	C₄
prime radical	C₂²×C₂²⋊D₃₇	C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#3866
hypocenter	C₃₇	D₄×M₁₆
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₃₇⋊M₁₆	C₂	semidirect
2. C₂²×#10596957	C₂	semidirect
3. #14558566	C₂	semidirect
4. #14558565	C₂	semidirect
5. #14558562	C₂	semidirect
6. #14558560	C₂	semidirect
7. #14558536	C₂	semidirect
8. #14558505	C₂	semidirect
9. #14558503	C₂	semidirect
10. C₄×#10596957	C₂	semidirect
12. C₂×#10596953	C₂²	semidirect
19. C₂×#10596957	C₂²	semidirect
20. C₂×D₃₇⋊C₈	C₂²	semidirect
21. C₂²×#64438	C₂²	semidirect
22. #11924603	C₂²	semidirect
23. C₄⋊#64438	C₂²	semidirect
24. #11924593	C₂²	semidirect
25. #11924593	C₂²	semidirect
26. C₂²⋊#64438	C₂²	semidirect
27. C₂²⋊#64438	C₂²	semidirect
28. #11924591	C₂²	semidirect
29. C₄×#64438	C₂²	semidirect
73. #10596957	D₄	semidirect
117. C₂²⋊D₃₇	M₁₆	semidirect
143. #64438	C₂×D₄	semidirect
153. C₂²×C₃₇	C₂×M₁₆	semidirect
161. C₂×D₃₇	C₂×M₁₆	semidirect
170. Dic₃₇	C₂×M₁₆	semidirect
193. C₃₇	D₄×M₁₆	semidirect
342. C₂²	D₃₇⋊M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₇⋊M₁₆	2
C₂²×#10596957	2
#14558566	2
#14558565	2
#14558562	2
#14558560	2
#14558536	2
#14558505	2
#14558503	2
C₄×#10596957	2
C₂×#11924496	2
D₄×M₁₆	37