Subgroups — C₄×#11924565 #18003410

Order 4736.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924565	C₂	1	1	yes	maximal, characteristic
2	C₄×#10596964	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557639	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557637	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557634	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557630	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10596942	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#10596940	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557572	C₂	1	1	yes	maximal, characteristic, complemented
12	#14557540	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557531	C₂	1	1	yes	maximal, characteristic
14	C₄×#10596929	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10596964	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
18	#11924565	C₄	1	8	yes	direct product factor
19	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic
20	C₂×#10596942	C₂²	1	1	yes	characteristic
21	C₂×#10596940	C₂²	1	1	yes	characteristic
22	C₂×#10596929	C₂²	1	1	yes	characteristic
23	C₂×#10596929	C₂²	1	1	yes	characteristic
24	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₇₄	—	2	1	no	—
26	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
27	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
28	#11924501	—	2	1	no	—
29	#11924501	C₂²	1	1	yes	characteristic
30	#11924501	C₂²	1	1	yes	characteristic
31	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₃₇	—	2	1	no	—
33	#11924496	—	2	1	no	—
34	#11924494	C₂²	1	1	yes	characteristic
35	#11924492	—	2	1	no	—
36	C₄⋊C₄×D₃₇	—	2	1	no	—
37	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₃₇	—	2	1	no	—
39	C₄²×D₃₇	—	2	1	no	—
40	C₄²×D₃₇	C₂²	1	1	yes	characteristic
41	#11924470	C₂²	1	1	yes	characteristic
42	#11924469	C₂²	1	1	yes	characteristic
43	#11924467	C₂²	1	1	yes	characteristic
44	#11924461	C₂²	1	1	yes	characteristic
45	#11924461	C₂²	1	1	yes	characteristic
46	#11924460	C₂²	1	1	yes	characteristic
47	#11924459	C₂²	1	1	yes	characteristic
48	#11924459	C₂²	1	1	yes	characteristic
49	#11924457	C₂²	1	1	yes	characteristic
50	#11924446	C₂²	1	1	yes	characteristic
51	#11924446	C₂²	1	1	yes	characteristic
52	C₄×#64436	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
56	#106×C₃₇	C₂²	1	1	yes	characteristic
57	#103×C₃₇	C₂²	1	1	yes	characteristic
58	#102×C₃₇	C₂²	1	1	yes	characteristic
59	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	—	2	1	no	—
63	C₂×C₂²⋊D₃₇	—	2	1	no	—
64	C₂²×Dic₃₇	—	2	1	no	—
65	C₂²×Dic₃₇	—	2	1	no	—
66	Q₈×D₃₇	—	2	2	no	—
67	Q₈×D₃₇	C₂×C₄	1	4	yes	complemented
68	#10596964	—	2	2	no	—
69	#10596964	C₂×C₄	1	4	yes	—
70	C₂×C₄×D₃₇	—	2	1	no	—
71	C₂×C₄×D₃₇	—	2	1	no	—
72	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
73	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
74	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
75	C₂×Dic₇₄	—	2	1	no	—
76	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
77	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
78	SD₁₆×C₃₇	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₃₇	—	2	1	no	—
82	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₃₇	—	2	1	no	—
85	C₄²×C₃₇	—	2	1	no	—
86	C₄²×C₃₇	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₃₇	—	2	1	no	—
88	#10596942	C₂×C₄	1	4	yes	complemented
89	#10596940	C₂×C₄	1	4	yes	complemented
90	#10596938	—	2	1	no	—
91	C₄⋊Dic₃₇	—	2	1	no	—
92	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
94	#10596936	—	2	1	no	—
95	#10596936	—	2	1	no	—
96	#10596936	—	2	1	no	—
97	#10596936	—	2	1	no	—
98	#10596936	—	2	1	no	—
99	C₄×Dic₃₇	—	2	1	no	—
100	C₄×Dic₃₇	—	2	1	no	—
101	C₄×Dic₃₇	—	2	1	no	—
102	C₄×Dic₃₇	—	2	1	no	—
103	C₄×Dic₃₇	—	2	1	no	—
104	C₄×Dic₃₇	—	2	1	no	—
105	C₄×Dic₃₇	D₄	1	1	yes	characteristic
106	C₄×Dic₃₇	D₄	1	1	yes	characteristic
107	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
108	C₂×#64436	C₂³	1	1	yes	characteristic
109	C₂×#64436	C₂³	1	1	yes	characteristic
110	Dic₁₄₈	C₂×C₄	1	4	yes	complemented
111	#10596929	—	2	2	no	—
112	#10596929	C₂×C₄	1	4	yes	complemented
113	C₂³×C₃₇	—	2	1	no	—
114	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₃₇	—	2	1	no	—
116	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
117	D₄×C₃₇	—	2	1	no	—
118	D₄×C₃₇	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₃₇	—	2	1	no	—
120	C₂×C₄×C₃₇	—	2	1	no	—
121	C₂×C₄×C₃₇	—	2	1	no	—
122	C₂×C₄×C₃₇	—	4	2	no	—
123	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₃₇	—	2	4	no	—
127	C₂×Dic₃₇	—	2	1	no	—
128	C₂×Dic₃₇	—	2	1	no	—
129	C₂×Dic₃₇	—	2	4	no	—
130	C₂×Dic₃₇	—	2	1	no	—
131	C₂×Dic₃₇	—	2	1	no	—
132	C₂×Dic₃₇	—	4	2	no	—
133	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
136	C₄×D₃₇	—	2	4	no	—
137	C₄×D₃₇	—	2	2	no	—
138	C₄×D₃₇	—	4	1	no	—
139	C₄×D₃₇	—	4	2	no	—
140	C₄×D₃₇	C₂²×C₄	1	2	yes	—
141	C₄×D₃₇	#41	1	2	yes	—
142	Dic₇₄	—	2	1	no	—
143	Dic₇₄	—	2	1	no	—
144	Dic₇₄	—	2	4	no	—
145	Dic₇₄	C₂²×C₄	1	2	yes	—
146	Dic₇₄	C₂²×C₄	1	2	yes	—
147	C₈×C₃₇	—	2	1	no	—
148	C₈×C₃₇	C₂²×C₄	1	2	yes	—
149	#64436	—	2	1	no	—
150	#64436	C₂²×C₄	1	2	yes	—
151	C₂²×C₃₇	—	2	2	no	—
152	C₂²×C₃₇	—	4	1	no	—
153	C₂²×C₃₇	—	4	1	no	—
154	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₃₇	—	2	1	no	—
156	C₂×D₃₇	—	2	1	no	—
157	C₂×D₃₇	C₄×D₄	1	2	yes	—
158	C₄×C₃₇	—	2	2	no	—
159	C₄×C₃₇	—	2	1	no	—
160	C₄×C₃₇	—	4	1	no	—
161	C₄×C₃₇	—	4	1	no	—
162	C₄×C₃₇	#137	1	2	yes	complemented
163	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
164	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
165	Dic₃₇	—	2	2	no	—
166	Dic₃₇	—	2	2	no	—
167	Dic₃₇	—	2	1	no	—
168	Dic₃₇	—	2	1	no	—
169	Dic₃₇	—	4	1	no	—
170	Dic₃₇	—	4	1	no	—
171	Dic₃₇	C₄×D₄	1	2	yes	—
172	C₄×#137	—	37	1	no	maximal, Sylow
173	C₂×C₃₇	—	4	2	no	—
174	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×C₃₇	C₂×#137	1	1	yes	characteristic
176	C₂×C₃₇	#575	1	1	yes	characteristic
177	D₃₇	—	2	2	no	—
178	C₂×#137	—	37	1	no	—
179	C₄×#41	—	37	1	no	—
180	C₂×C₄×Q₈	—	37	1	no	—
181	#441	—	37	1	no	—
182	#441	—	37	1	no	—
183	#440	—	37	1	no	—
184	#440	—	37	1	no	—
185	C₄×Q₁₆	—	37	1	no	—
186	C₄×Q₁₆	—	37	1	no	—
187	C₄×SD₁₆	—	37	1	no	—
188	C₄×SD₁₆	D₃₇	1	1	yes	characteristic, complemented
189	#428	—	37	1	no	—
190	#419	—	37	1	no	—
191	#417	—	37	1	no	—
192	C₄×M₁₆	—	37	1	no	—
193	C₃₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	—	37	1	no	—
195	C₂²×Q₈	—	37	1	no	—
196	#137	—	37	8	no	—
197	C₂×Q₁₆	—	37	1	no	—
198	C₂×Q₁₆	—	37	1	no	—
199	C₂×SD₁₆	—	37	1	no	—
200	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
201	C₂×M₁₆	—	37	1	no	—
202	C₂×M₁₆	—	37	1	no	—
203	C₄×Q₈	—	37	1	no	—
204	C₄×Q₈	—	37	1	no	—
205	C₄×Q₈	—	37	1	no	—
206	C₄×Q₈	—	37	1	no	—
207	C₄×Q₈	—	37	1	no	—
208	C₄×Q₈	—	74	1	no	—
209	C₄×Q₈	—	74	1	no	—
210	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
211	C₄×D₄	—	37	1	no	—
212	C₄×D₄	—	74	1	no	—
213	C₄×D₄	—	74	1	no	—
214	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
215	#117	—	37	1	no	—
216	#117	—	74	1	no	—
217	C₂×C₄⋊C₄	—	37	1	no	—
218	C₂×C₄⋊C₄	—	74	1	no	—
219	C₂×C₄²	—	37	1	no	—
220	C₂×C₄²	—	74	1	no	—
221	C₂×C₄²	—	74	1	no	—
222	#107	—	37	1	no	—
223	#107	—	37	1	no	—
224	#106	—	37	1	no	—
225	#106	C₂×D₃₇	1	1	yes	characteristic
226	#103	—	37	1	no	—
227	#103	—	37	1	no	—
228	#103	—	37	1	no	—
229	#103	—	37	1	no	—
230	#103	—	37	1	no	—
231	#103	C₂×D₃₇	1	1	yes	characteristic
232	#102	—	37	1	no	—
233	#102	C₂×D₃₇	1	1	yes	characteristic
234	#97	—	37	1	no	—
235	#97	—	37	1	no	—
236	C₄×C₈	—	37	1	no	—
237	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
238	#41	—	37	4	no	—
239	#41	—	74	2	no	—
240	C₂×Q₈	—	37	4	no	—
241	C₂×Q₈	—	37	1	no	—
242	C₂×Q₈	—	37	1	no	—
243	C₂×Q₈	—	74	1	no	—
244	C₂×Q₈	—	74	2	no	—
245	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
246	C₂×D₄	—	74	1	no	—
247	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	37	1	no	—
250	C₂²×C₄	—	37	1	no	—
251	C₂²×C₄	—	37	1	no	—
252	C₂²×C₄	—	74	1	no	—
253	C₂²×C₄	—	74	1	no	—
254	C₂²×C₄	—	74	1	no	—
255	C₂²×C₄	—	74	1	no	—
256	Q₁₆	—	37	4	no	—
257	Q₁₆	—	37	4	no	—
258	SD₁₆	—	37	4	no	—
259	SD₁₆	C₄×D₃₇	1	4	yes	complemented
260	M₁₆	—	37	4	no	—
261	M₁₆	—	74	2	no	—
262	C₂×C₈	—	37	1	no	—
263	C₂×C₈	—	37	1	no	—
264	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	37	1	no	—
268	C₄⋊C₄	—	37	1	no	—
269	C₄⋊C₄	—	74	1	no	—
270	C₄⋊C₄	—	74	1	no	—
271	C₄⋊C₄	—	74	1	no	—
272	C₄⋊C₄	—	74	1	no	—
273	C₄⋊C₄	—	74	1	no	—
274	C₄⋊C₄	—	74	1	no	—
275	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	74	1	no	—
279	C₂²⋊C₄	—	74	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	37	1	no	—
282	C₄²	—	37	1	no	—
283	C₄²	—	37	1	no	—
284	C₄²	—	74	1	no	—
285	C₄²	—	74	1	no	—
286	C₄²	—	74	1	no	—
287	C₄²	—	74	1	no	—
288	C₄²	—	74	1	no	—
289	C₄²	—	74	1	no	—
290	C₄²	C₂²×D₃₇	1	1	yes	characteristic
291	C₂³	—	2	1	no	—
292	C₂³	—	37	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	37	2	no	—
295	Q₈	—	37	2	no	—
296	Q₈	—	74	4	no	—
297	Q₈	—	74	1	no	—
298	Q₈	—	74	1	no	—
299	Q₈	C₂×C₄×D₃₇	1	2	yes	—
300	D₄	—	2	1	no	—
301	D₄	—	74	4	no	—
302	D₄	C₂×C₄×D₃₇	1	2	yes	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	37	1	no	—
308	C₂×C₄	—	37	2	no	—
309	C₂×C₄	—	37	2	no	—
310	C₂×C₄	—	37	1	no	—
311	C₂×C₄	—	37	1	no	—
312	C₂×C₄	—	74	4	no	—
313	C₂×C₄	—	74	4	no	—
314	C₂×C₄	—	74	1	no	—
315	C₂×C₄	—	74	1	no	—
316	C₂×C₄	—	74	2	no	—
317	C₂×C₄	—	74	1	no	—
318	C₂×C₄	—	74	1	no	—
319	C₂×C₄	—	148	1	no	—
320	C₂×C₄	—	148	2	no	—
321	C₂×C₄	—	148	2	no	—
322	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
323	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	37	2	no	—
327	C₈	—	74	1	no	—
328	C₈	C₂×C₄×D₃₇	1	2	yes	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	37	2	no	—
333	C₂²	—	74	1	no	—
334	C₂²	—	74	1	no	—
335	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	37	2	no	—
341	C₄	—	74	2	no	—
342	C₄	—	74	2	no	—
343	C₄	—	74	1	no	—
344	C₄	—	74	1	no	—
345	C₄	—	148	1	no	—
346	C₄	—	148	1	no	—
347	C₄	#11924565	1	2	yes	direct product factor
348	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
349	C₄	#41×D₃₇	1	1	yes	characteristic
350	C₂	—	4	2	no	—
351	C₂	—	74	2	no	—
352	C₂	C₄×D₄×D₃₇	1	1	yes	characteristic
353	C₂	C₂×#11924565	1	1	yes	characteristic
354	C₂	#14558330	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₄×SD₁₆×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#10596964	C₂²
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₄×SD₁₆	D₃₇
hypocenter	C₃₇	C₄×#137
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Q₈×D₃₇	C₂	semidirect
4. #14557639	C₂	semidirect
5. #14557637	C₂	semidirect
6. #14557634	C₂	semidirect
7. #14557630	C₂	semidirect
8. C₄×Dic₁₄₈	C₂	semidirect
9. C₄×#10596942	C₂	semidirect
10. C₄×#10596940	C₂	semidirect
11. #14557572	C₂	semidirect
12. #14557540	C₂	semidirect
14. C₄×#10596929	C₂	semidirect
15. C₄×SD₁₆×C₃₇	C₂	semidirect
18. #11924565	C₄	direct (N × H)
67. Q₈×D₃₇	C₂×C₄	semidirect
78. SD₁₆×C₃₇	C₂×C₄	semidirect
88. #10596942	C₂×C₄	semidirect
89. #10596940	C₂×C₄	semidirect
110. Dic₁₄₈	C₂×C₄	semidirect
112. #10596929	C₂×C₄	semidirect
162. C₄×C₃₇	#137	semidirect
188. C₄×SD₁₆	D₃₇	semidirect
193. C₃₇	C₄×#137	semidirect
259. SD₁₆	C₄×D₃₇	semidirect
347. C₄	#11924565	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11924565	2
C₄×#10596964	2
C₄×Q₈×D₃₇	2
#14557639	2
#14557637	2
#14557634	2
#14557630	2
C₄×Dic₁₄₈	2
C₄×#10596942	2
C₄×#10596940	2
#14557572	2
#14557540	2
#14557531	2
C₄×#10596929	2
C₄×SD₁₆×C₃₇	2
C₄×#137	37

Subgroups — C4×#11924565 #18003410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11924565 #18003410