Subgroups — #18011353

Order 4736.

506 conjugacy classes of subgroups.

The subgroup lattice has 372 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, complemented
2	#14558600	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558574	C₂	1	2	yes	maximal, complemented
4	#14558572	C₂	1	2	yes	maximal, complemented
5	SD₁₆×#3866	C₂	1	2	yes	maximal, complemented
6	D₈×#3866	C₂	1	2	yes	maximal, complemented
7	#14558544	C₂	1	1	yes	maximal, characteristic, complemented
8	#14558523	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11924598	C₂	1	1	yes	maximal, characteristic, complemented
10	#14558499	C₂	1	1	yes	maximal, characteristic, complemented
11	AGL(1,ℤ/8)×D₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×#3866	C₂²	1	2	yes	complemented
14	D₄×#3866	—	2	2	no	—
15	D₄×#3866	—	2	1	no	—
16	D₄×#3866	—	2	1	no	—
17	D₄×#3866	C₂²	1	2	yes	—
18	D₄×#3866	C₂²	1	2	yes	—
19	D₄×#3866	C₂²	1	2	yes	complemented
20	#11924615	—	2	1	no	—
21	#11924615	C₂²	1	1	yes	characteristic
22	#11924615	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#3866	—	2	1	no	—
24	C₂×C₄⋊#3866	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊#3866	—	2	2	no	—
26	#11924605	C₂²	1	2	yes	complemented
27	#11924604	C₂²	1	2	yes	complemented
28	#11924599	C₂²	1	1	yes	characteristic, complemented
29	#11924599	C₂²	1	1	yes	characteristic, complemented
30	#11924598	C₂²	1	2	yes	—
31	#11924598	C₂²	1	2	yes	—
32	#11924598	C₂²	1	1	yes	characteristic, complemented
33	#11924598	C₂²	1	1	yes	characteristic, complemented
34	#11924590	C₂²	1	2	yes	complemented
35	C₈×#3866	C₂²	1	2	yes	complemented
36	#41×D₃₇	C₂²	1	1	yes	characteristic
37	C₂×D₄×D₃₇	C₂²	1	1	yes	characteristic
38	#11924564	C₄	1	1	yes	characteristic, complemented
39	#11924563	C₄	1	2	yes	complemented
40	#11924562	C₄	1	1	yes	characteristic, complemented
41	#11924561	C₄	1	1	yes	characteristic, complemented
42	#11924560	C₄	1	2	yes	complemented
43	SD₁₆×D₃₇	C₂²	1	2	yes	—
44	D₈×D₃₇	C₂²	1	2	yes	—
45	M₁₆×D₃₇	C₂²	1	1	yes	characteristic
46	AGL(1,ℤ/8)×C₃₇	C₄	1	1	yes	characteristic, complemented
47	C₂³×D₃₇	—	2	1	no	—
48	C₂²×#3866	—	2	1	no	—
49	C₂²×#3866	—	2	2	no	—
50	C₂²×#3866	—	2	2	no	—
51	C₂²×#3866	—	2	2	no	—
52	#41×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
53	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
54	C₂×C₂²⋊D₃₇	—	2	1	no	—
55	#10596966	C₂×C₄	1	1	yes	characteristic, complemented
56	Q₈×D₃₇	C₂³	1	1	yes	characteristic
57	#10596964	—	2	1	no	—
58	#10596964	C₂×C₄	1	1	yes	characteristic, complemented
59	D₄×D₃₇	—	2	1	no	—
60	D₄×D₃₇	—	2	1	no	—
61	D₄×D₃₇	—	2	1	no	—
62	D₄×D₃₇	C₂×C₄	1	2	yes	complemented
63	D₄×D₃₇	C₂³	1	1	yes	characteristic
64	D₄×D₃₇	C₂³	1	2	yes	—
65	#10596962	—	2	1	no	—
66	#10596962	C₂×C₄	1	1	yes	characteristic, complemented
67	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic, complemented
68	C₂×C₄×D₃₇	—	2	1	no	—
69	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
70	C₂²⋊#3866	—	2	1	no	—
71	C₂²⋊#3866	—	2	2	no	—
72	C₂²⋊#3866	—	2	2	no	—
73	C₂²⋊#3866	—	2	2	no	—
74	C₂²⋊#3866	D₄	1	2	yes	—
75	C₂×#64438	—	2	1	no	—
76	C₄⋊#3866	—	2	2	no	—
77	C₄⋊#3866	—	2	1	no	—
78	C₄⋊#3866	D₄	1	1	yes	characteristic
79	C₄⋊#3866	D₄	1	1	yes	characteristic
80	C₄⋊#3866	C₂³	1	2	yes	—
81	C₄⋊#3866	C₂³	1	1	yes	characteristic
82	C₄⋊#3866	C₂³	1	1	yes	characteristic
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	SD₁₆×C₃₇	C₂×C₄	1	2	yes	complemented
90	D₈×C₃₇	C₂×C₄	1	2	yes	complemented
91	M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
92	#10596941	C₂×C₄	1	2	yes	complemented
93	#10596940	C₂×C₄	1	2	yes	complemented
94	#10596939	C₂×C₄	1	2	yes	complemented
95	#10596939	C₂×C₄	1	2	yes	complemented
96	#10596934	C₂×C₄	1	1	yes	characteristic, complemented
97	D₂₉₆	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	C₂³×C₃₇	—	2	1	no	—
102	C₂²×D₃₇	—	2	1	no	—
103	C₂²×D₃₇	—	2	1	no	—
104	C₂²×D₃₇	—	2	2	no	—
105	C₂²×D₃₇	—	2	2	no	—
106	C₂²×D₃₇	—	8	1	no	—
107	C₂²×D₃₇	—	8	1	no	—
108	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂×#3866	—	2	2	no	—
110	C₂×#3866	—	2	1	no	—
111	C₂×#3866	—	2	1	no	—
112	C₂×#3866	—	8	2	no	—
113	C₂×#3866	—	8	2	no	—
114	C₂×#3866	—	8	2	no	—
115	C₂×#3866	—	8	1	no	—
116	C₂×#3866	C₂×D₄	1	2	yes	—
117	C₂×#3866	C₂×D₄	1	2	yes	—
118	Q₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
119	D₄×C₃₇	—	2	1	no	—
120	D₄×C₃₇	—	2	1	no	—
121	D₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
122	D₄×C₃₇	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₃₇	—	2	1	no	—
124	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
125	C₂²⋊D₃₇	—	2	1	no	—
126	C₂²⋊D₃₇	—	2	1	no	—
127	C₂²⋊D₃₇	—	2	2	no	—
128	C₂²⋊D₃₇	—	2	1	no	—
129	C₂²⋊D₃₇	—	2	2	no	—
130	C₂×Dic₃₇	—	2	1	no	—
131	C₂×Dic₃₇	#41	1	1	yes	characteristic
132	D₁₄₈	—	2	1	no	—
133	D₁₄₈	—	2	1	no	—
134	D₁₄₈	C₂²×C₄	1	2	yes	—
135	D₁₄₈	C₂²×C₄	1	1	yes	characteristic
136	C₄×D₃₇	—	2	1	no	—
137	C₄×D₃₇	—	2	1	no	—
138	C₄×D₃₇	—	2	1	no	—
139	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
140	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
141	C₄×D₃₇	#41	1	1	yes	characteristic
142	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
143	Dic₇₄	—	2	1	no	—
144	Dic₇₄	C₂²×C₄	1	1	yes	characteristic
145	#64438	—	2	1	no	—
146	#64438	—	2	1	no	—
147	C₈×C₃₇	C₂²×C₄	1	2	yes	—
148	#64436	C₂²×C₄	1	2	yes	—
149	C₂²×C₃₇	—	2	1	no	—
150	C₂²×C₃₇	—	2	2	no	—
151	C₂²×C₃₇	—	8	1	no	—
152	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
153	C₂×D₃₇	—	2	1	no	—
154	C₂×D₃₇	—	2	1	no	—
155	C₂×D₃₇	—	2	1	no	—
156	C₂×D₃₇	—	2	2	no	—
157	C₂×D₃₇	—	4	1	no	—
158	C₂×D₃₇	—	4	2	no	—
159	C₂×D₃₇	—	4	2	no	—
160	C₂×D₃₇	—	4	1	no	—
161	C₂×D₃₇	—	8	2	no	—
162	C₂×D₃₇	—	8	1	no	—
163	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
164	C₂×D₃₇	C₂²×D₄	1	1	yes	characteristic
165	#3866	—	2	2	no	—
166	#3866	—	2	2	no	—
167	#3866	—	4	2	no	—
168	#3866	—	4	1	no	—
169	#3866	—	4	1	no	—
170	C₄×C₃₇	—	2	1	no	—
171	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
172	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
173	Dic₃₇	—	2	1	no	—
174	Dic₃₇	C₄×D₄	1	1	yes	characteristic
175	Dic₃₇	C₂×#41	1	1	yes	characteristic
176	#2946	—	37	1	no	maximal, Sylow
177	C₂×C₃₇	—	2	1	no	—
178	C₂×C₃₇	—	4	2	no	—
179	C₂×C₃₇	—	4	1	no	—
180	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
181	D₃₇	—	2	1	no	—
182	D₃₇	—	4	1	no	—
183	D₃₇	—	4	2	no	—
184	D₃₇	#576	1	1	yes	characteristic
185	D₃₇	#577	1	1	yes	characteristic
186	C₂×AGL(1,ℤ/8)	—	37	1	no	—
187	#520	—	37	1	no	—
188	#518	—	37	1	no	—
189	#442	—	37	2	no	—
190	#440	—	37	2	no	—
191	C₄×SD₁₆	—	37	2	no	—
192	C₄×D₈	—	37	2	no	—
193	#428	—	37	1	no	—
194	#416	—	37	1	no	—
195	C₂×#102	—	37	1	no	—
196	#405	—	37	1	no	—
197	C₃₇	#2946	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	37	1	no	—
199	C₂²×D₄	—	37	1	no	—
200	AGL(1,ℤ/8)	—	37	1	no	—
201	AGL(1,ℤ/8)	—	37	1	no	—
202	AGL(1,ℤ/8)	—	37	2	no	—
203	AGL(1,ℤ/8)	—	37	2	no	—
204	AGL(1,ℤ/8)	—	37	1	no	—
205	AGL(1,ℤ/8)	#3866	1	1	yes	characteristic, complemented
206	C₂×SD₁₆	—	37	2	no	—
207	C₂×D₈	—	37	2	no	—
208	C₂×M₁₆	—	37	1	no	—
209	C₂²×C₈	—	37	1	no	—
210	C₄×Q₈	—	37	2	no	—
211	C₄×D₄	—	37	2	no	—
212	C₄×D₄	—	37	2	no	—
213	C₄×D₄	—	37	2	no	—
214	C₄×D₄	—	74	2	no	—
215	C₄×D₄	—	74	1	no	—
216	C₄×D₄	—	74	1	no	—
217	#117	—	37	1	no	—
218	#117	—	37	1	no	—
219	#117	—	74	1	no	—
220	C₂×C₄⋊C₄	—	37	1	no	—
221	C₂×C₄⋊C₄	—	74	1	no	—
222	C₂×C₂²⋊C₄	—	74	2	no	—
223	#107	—	37	2	no	—
224	#106	—	37	2	no	—
225	#103	—	37	1	no	—
226	#103	—	37	1	no	—
227	#102	—	37	2	no	—
228	#102	—	37	2	no	—
229	#102	—	37	1	no	—
230	#102	—	37	1	no	—
231	#97	—	37	2	no	—
232	C₄×C₈	—	37	2	no	—
233	C₂⁴	—	74	1	no	—
234	#41	—	37	1	no	—
235	#41	—	37	1	no	—
236	#41	—	37	1	no	—
237	#41	—	74	1	no	—
238	#41	—	74	1	no	—
239	#41	C₂×#3866	1	1	yes	characteristic, complemented
240	C₂×Q₈	—	37	1	no	—
241	C₂×D₄	—	37	2	no	—
242	C₂×D₄	—	37	1	no	—
243	C₂×D₄	—	37	2	no	—
244	C₂×D₄	—	37	1	no	—
245	C₂×D₄	—	74	1	no	—
246	C₂×D₄	—	74	1	no	—
247	C₂×D₄	—	74	1	no	—
248	C₂×D₄	—	74	1	no	—
249	C₂×D₄	C₂×#3866	1	1	yes	characteristic, complemented
250	C₂²×C₄	—	37	1	no	—
251	C₂²×C₄	—	74	1	no	—
252	C₂²×C₄	—	74	2	no	—
253	C₂²×C₄	—	74	2	no	—
254	C₂²×C₄	—	74	2	no	—
255	C₂²×C₄	—	74	1	no	—
256	SD₁₆	—	37	2	no	—
257	SD₁₆	—	37	2	no	—
258	SD₁₆	—	37	2	no	—
259	SD₁₆	C₂×#3866	1	2	yes	complemented
260	D₈	—	37	2	no	—
261	D₈	—	37	2	no	—
262	D₈	—	37	2	no	—
263	D₈	C₂×#3866	1	2	yes	complemented
264	M₁₆	—	37	1	no	—
265	M₁₆	—	37	2	no	—
266	M₁₆	C₂×#3866	1	1	yes	characteristic, complemented
267	C₂×C₈	—	37	2	no	—
268	C₂×C₈	—	37	1	no	—
269	C₂×C₈	—	37	1	no	—
270	C₂×C₈	—	74	1	no	—
271	C₂×C₈	—	74	1	no	—
272	C₄⋊C₄	—	37	1	no	—
273	C₄⋊C₄	—	37	1	no	—
274	C₄⋊C₄	—	37	2	no	—
275	C₄⋊C₄	—	37	1	no	—
276	C₄⋊C₄	—	37	1	no	—
277	C₄⋊C₄	—	74	1	no	—
278	C₄⋊C₄	—	74	2	no	—
279	C₂²⋊C₄	—	37	2	no	—
280	C₂²⋊C₄	—	74	2	no	—
281	C₂²⋊C₄	—	74	2	no	—
282	C₂²⋊C₄	—	74	2	no	—
283	C₂²⋊C₄	—	74	1	no	—
284	C₄²	—	37	2	no	—
285	C₄²	—	74	1	no	—
286	C₄²	—	74	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	37	1	no	—
289	C₂³	—	296	1	no	—
290	C₂³	—	296	1	no	—
291	C₂³	—	74	2	no	—
292	C₂³	—	74	1	no	—
293	C₂³	—	74	2	no	—
294	C₂³	—	74	1	no	—
295	Q₈	—	37	1	no	—
296	Q₈	—	74	1	no	—
297	Q₈	C₂²×#3866	1	1	yes	characteristic
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	37	1	no	—
301	D₄	—	37	2	no	—
302	D₄	—	74	1	no	—
303	D₄	—	74	2	no	—
304	D₄	—	74	2	no	—
305	D₄	—	74	1	no	—
306	D₄	—	74	1	no	—
307	D₄	—	74	1	no	—
308	D₄	—	74	1	no	—
309	D₄	C₂²×#3866	1	1	yes	characteristic
310	D₄	C₂²×#3866	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	37	1	no	—
313	C₂×C₄	—	37	1	no	—
314	C₂×C₄	—	37	1	no	—
315	C₂×C₄	—	37	1	no	—
316	C₂×C₄	—	37	1	no	—
317	C₂×C₄	—	37	2	no	—
318	C₂×C₄	—	37	2	no	—
319	C₂×C₄	—	296	2	no	—
320	C₂×C₄	—	296	2	no	—
321	C₂×C₄	—	296	2	no	—
322	C₂×C₄	—	296	1	no	—
323	C₂×C₄	—	74	1	no	—
324	C₂×C₄	—	74	1	no	—
325	C₂×C₄	—	74	1	no	—
326	C₂×C₄	—	74	1	no	—
327	C₂×C₄	—	74	1	no	—
328	C₂×C₄	—	74	1	no	—
329	C₂×C₄	—	74	2	no	—
330	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
331	C₈	—	37	2	no	—
332	C₈	—	74	1	no	—
333	C₈	—	74	1	no	—
334	C₈	C₂²×#3866	1	2	yes	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	37	1	no	—
339	C₂²	—	37	1	no	—
340	C₂²	—	296	1	no	—
341	C₂²	—	296	2	no	—
342	C₂²	—	74	1	no	—
343	C₂²	—	74	1	no	—
344	C₂²	—	74	1	no	—
345	C₂²	—	74	2	no	—
346	C₂²	—	148	1	no	—
347	C₂²	—	148	2	no	—
348	C₂²	—	148	2	no	—
349	C₂²	—	148	1	no	—
350	C₂²	D₄×#3866	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	37	1	no	—
353	C₄	—	37	1	no	—
354	C₄	—	74	2	no	—
355	C₄	—	74	2	no	—
356	C₄	—	74	1	no	—
357	C₄	—	148	1	no	—
358	C₄	—	148	2	no	—
359	C₄	—	148	1	no	—
360	C₄	D₄×#3866	1	1	yes	characteristic
361	C₄	C₂³×#3866	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	4	2	no	—
364	C₂	—	4	1	no	—
365	C₂	—	37	1	no	—
366	C₂	—	37	1	no	—
367	C₂	—	74	1	no	—
368	C₂	—	148	2	no	—
369	C₂	—	148	1	no	—
370	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	AGL(1,ℤ/8)×C₃₇	C₄
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	AGL(1,ℤ/8)×D₃₇	C₂
cosocle	C₄×D₃₇	C₂⁴
hypercenter	AGL(1,ℤ/8)	#3866
hypocenter	C₃₇	#2946
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #14558602	C₂	semidirect
2. #14558600	C₂	semidirect
3. #14558574	C₂	semidirect
4. #14558572	C₂	semidirect
5. SD₁₆×#3866	C₂	semidirect
6. D₈×#3866	C₂	semidirect
7. #14558544	C₂	semidirect
8. #14558523	C₂	semidirect
9. C₂×#11924598	C₂	semidirect
10. #14558499	C₂	semidirect
13. Q₈×#3866	C₂²	semidirect
19. D₄×#3866	C₂²	semidirect
26. #11924605	C₂²	semidirect
27. #11924604	C₂²	semidirect
28. #11924599	C₂²	semidirect
29. #11924599	C₂²	semidirect
32. #11924598	C₂²	semidirect
33. #11924598	C₂²	semidirect
34. #11924590	C₂²	semidirect
35. C₈×#3866	C₂²	semidirect
38. #11924564	C₄	semidirect
39. #11924563	C₄	semidirect
40. #11924562	C₄	semidirect
41. #11924561	C₄	semidirect
42. #11924560	C₄	semidirect
46. AGL(1,ℤ/8)×C₃₇	C₄	semidirect
52. #41×C₃₇	C₂×C₄	semidirect
53. C₂×D₄×C₃₇	C₂×C₄	semidirect
55. #10596966	C₂×C₄	semidirect
58. #10596964	C₂×C₄	semidirect
62. D₄×D₃₇	C₂×C₄	semidirect
66. #10596962	C₂×C₄	semidirect
67. C₂×D₁₄₈	C₂×C₄	semidirect
89. SD₁₆×C₃₇	C₂×C₄	semidirect
90. D₈×C₃₇	C₂×C₄	semidirect
91. M₁₆×C₃₇	C₂×C₄	semidirect
92. #10596941	C₂×C₄	semidirect
93. #10596940	C₂×C₄	semidirect
94. #10596939	C₂×C₄	semidirect
95. #10596939	C₂×C₄	semidirect
96. #10596934	C₂×C₄	semidirect
97. D₂₉₆	C₂×C₄	semidirect
98. #10596930	C₂×C₄	semidirect
99. #10596929	C₂×C₄	semidirect
197. C₃₇	#2946	semidirect
205. AGL(1,ℤ/8)	#3866	semidirect
239. #41	C₂×#3866	semidirect
249. C₂×D₄	C₂×#3866	semidirect
259. SD₁₆	C₂×#3866	semidirect
263. D₈	C₂×#3866	semidirect
266. 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
#14558600	2
#14558574	2
#14558572	2
SD₁₆×#3866	2
D₈×#3866	2
#14558544	2
#14558523	2
C₂×#11924598	2
#14558499	2
AGL(1,ℤ/8)×D₃₇	2
#2946	37