Subgroups — #137×#90673 #28407330

Order 14464.

484 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#90673	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#90673	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299638	C₂	1	2	yes	maximal, complemented
4	#20299637	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#90673	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#90673	C₂	1	2	yes	maximal, complemented
7	#20299609	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299591	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299589	C₂	1	1	yes	maximal, characteristic
10	M₁₆×#90673	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₁₁₃	C₂	1	1	yes	maximal, characteristic
12	D₁₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
13	Q₈×#90673	—	2	1	no	—
14	Q₈×#90673	—	2	1	no	—
15	Q₈×#90673	C₂²	1	2	yes	—
16	Q₈×#90673	C₂²	1	1	yes	characteristic
17	Q₈×#90673	C₂²	1	2	yes	—
18	Q₈×#90673	C₂²	1	1	yes	characteristic
19	D₄×#90673	—	2	1	no	—
20	D₄×#90673	—	2	1	no	—
21	D₄×#90673	C₂²	1	1	yes	characteristic
22	D₄×#90673	C₂²	1	1	yes	characteristic
23	#16813701	—	2	1	no	—
24	#16813701	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#90673	—	2	1	no	—
26	C₂×C₄⋊#90673	C₂²	1	1	yes	characteristic
27	C₂×C₄×#90673	—	2	1	no	—
28	C₂×C₄×#90673	—	2	1	no	—
29	C₂×C₄×#90673	C₂²	1	1	yes	characteristic
30	#16813691	C₂²	1	2	yes	—
31	#16813690	C₂²	1	2	yes	—
32	#16813685	C₂²	1	2	yes	—
33	#16813685	C₂²	1	2	yes	—
34	#16813685	C₂²	1	2	yes	—
35	#16813684	C₂²	1	2	yes	—
36	#16813676	C₂²	1	2	yes	—
37	C₈×#90673	C₂²	1	2	yes	—
38	#41×D₁₁₃	C₂²	1	1	yes	characteristic
39	C₂×Q₈×D₁₁₃	C₂²	1	1	yes	characteristic
40	#16813655	C₄	1	1	yes	characteristic, complemented
41	#16813654	C₄	1	2	yes	complemented
42	#16813653	C₄	1	1	yes	characteristic, complemented
43	#16813652	C₄	1	1	yes	characteristic, complemented
44	#16813651	C₄	1	2	yes	complemented
45	Q₁₆×D₁₁₃	C₂²	1	2	yes	—
46	SD₁₆×D₁₁₃	C₂²	1	2	yes	—
47	M₁₆×D₁₁₃	C₂²	1	1	yes	characteristic
48	#137×C₁₁₃	C₄	1	1	yes	characteristic, complemented
49	C₂²×#90673	—	2	1	no	—
50	C₂²×#90673	—	2	1	no	—
51	C₂²×#90673	—	2	1	no	—
52	C₂²×#90673	D₄	1	1	yes	characteristic
53	C₂²×#90673	D₄	1	1	yes	characteristic
54	#14268657	—	2	1	no	—
55	#14268656	—	2	1	no	—
56	#14268655	C₂³	1	2	yes	—
57	C₄⋊#90673	—	2	2	no	—
58	C₄⋊#90673	—	2	2	no	—
59	C₄⋊#90673	—	2	1	no	—
60	C₄⋊#90673	—	2	1	no	—
61	C₄⋊#90673	—	2	1	no	—
62	C₄⋊#90673	C₂³	1	2	yes	—
63	C₄⋊#90673	C₂³	1	1	yes	characteristic
64	C₄⋊#90673	C₂³	1	1	yes	characteristic
65	C₂²⋊#90673	—	2	1	no	—
66	C₂²⋊#90673	—	2	1	no	—
67	C₂²⋊#90673	—	2	1	no	—
68	C₄×#90673	—	2	1	no	—
69	C₄×#90673	—	2	1	no	—
70	C₄×#90673	—	2	2	no	—
71	C₄×#90673	—	2	2	no	—
72	C₄×#90673	—	2	1	no	—
73	C₄×#90673	D₄	1	1	yes	characteristic
74	C₄×#90673	D₄	1	1	yes	characteristic
75	C₄×#90673	C₂³	1	1	yes	characteristic
76	C₄×#90673	C₂³	1	1	yes	characteristic
77	#14268648	C₂×C₄	1	1	yes	characteristic
78	#14268647	—	2	1	no	—
79	#14268647	C₂×C₄	1	1	yes	characteristic
80	#14268646	—	2	1	no	—
81	#14268646	C₂×C₄	1	1	yes	characteristic
82	C₂×Dic₂₂₆	—	2	1	no	—
83	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic, complemented
84	Q₈×D₁₁₃	—	2	1	no	—
85	Q₈×D₁₁₃	—	2	1	no	—
86	Q₈×D₁₁₃	C₂×C₄	1	2	yes	complemented
87	Q₈×D₁₁₃	C₂³	1	2	yes	—
88	Q₈×D₁₁₃	C₂³	1	1	yes	characteristic
89	D₄×D₁₁₃	—	2	1	no	—
90	D₄×D₁₁₃	C₂³	1	1	yes	characteristic
91	C₂×C₄×D₁₁₃	—	2	1	no	—
92	C₂×C₄×D₁₁₃	—	2	1	no	—
93	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
94	Dic₄₅₂	C₂×C₄	1	2	yes	complemented
95	#14268637	C₂×C₄	1	2	yes	complemented
96	#14268637	C₂×C₄	1	2	yes	complemented
97	#14268636	C₂×C₄	1	2	yes	complemented
98	#14268635	C₂×C₄	1	2	yes	complemented
99	#14268634	C₂×C₄	1	2	yes	complemented
100	#14268631	C₂×C₄	1	1	yes	characteristic, complemented
101	#14268630	C₂×C₄	1	2	yes	complemented
102	C₈×D₁₁₃	C₂³	1	2	yes	—
103	#41×C₁₁₃	C₂×C₄	1	1	yes	characteristic
104	C₂×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
105	Q₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
106	SD₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
107	M₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
108	C₂²×D₁₁₃	—	2	1	no	—
109	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×#90673	—	2	2	no	—
111	C₂×#90673	—	2	1	no	—
112	C₂×#90673	—	2	1	no	—
113	C₂×#90673	—	2	2	no	—
114	C₂×#90673	—	4	1	no	—
115	C₂×#90673	—	4	2	no	—
116	C₂×#90673	—	4	2	no	—
117	C₂×#90673	—	4	2	no	—
118	C₂×#90673	C₂×D₄	1	1	yes	characteristic
119	C₂×#90673	C₂×D₄	1	1	yes	characteristic
120	C₂×#90673	C₂×D₄	1	1	yes	characteristic
121	C₂×#90673	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₁₁₃	—	2	1	no	—
123	Q₈×C₁₁₃	C₂²×C₄	1	2	yes	—
124	Q₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
125	D₄×C₁₁₃	—	2	1	no	—
126	D₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁₃	—	2	1	no	—
128	C₂×C₄×C₁₁₃	—	2	1	no	—
129	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²⋊D₁₁₃	—	2	1	no	—
131	C₂²⋊D₁₁₃	—	2	1	no	—
132	C₂²⋊D₁₁₃	—	2	1	no	—
133	C₂×Dic₁₁₃	—	2	1	no	—
134	C₂×Dic₁₁₃	—	2	1	no	—
135	C₂×Dic₁₁₃	#41	1	1	yes	characteristic
136	D₄₅₂	—	2	1	no	—
137	D₄₅₂	C₂²×C₄	1	1	yes	characteristic
138	C₄×D₁₁₃	—	2	2	no	—
139	C₄×D₁₁₃	—	2	1	no	—
140	C₄×D₁₁₃	—	2	2	no	—
141	C₄×D₁₁₃	—	2	1	no	—
142	C₄×D₁₁₃	—	2	1	no	—
143	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
144	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
145	C₄×D₁₁₃	#41	1	1	yes	characteristic
146	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
147	Dic₂₂₆	—	2	1	no	—
148	Dic₂₂₆	—	2	2	no	—
149	Dic₂₂₆	—	2	2	no	—
150	Dic₂₂₆	—	2	1	no	—
151	Dic₂₂₆	C₂²×C₄	1	2	yes	—
152	Dic₂₂₆	C₂²×C₄	1	1	yes	characteristic
153	#11744753	—	2	2	no	—
154	C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
155	#11744751	C₂²×C₄	1	2	yes	—
156	C₂²×C₁₁₃	—	2	1	no	—
157	C₂²×C₁₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×D₁₁₃	—	2	1	no	—
159	C₂×D₁₁₃	—	2	1	no	—
160	C₂×D₁₁₃	—	2	1	no	—
161	C₂×D₁₁₃	—	4	1	no	—
162	C₂×D₁₁₃	—	4	1	no	—
163	C₂×D₁₁₃	C₄×D₄	1	1	yes	characteristic
164	C₂×D₁₁₃	C₂²×D₄	1	1	yes	characteristic
165	#90673	—	2	1	no	—
166	#90673	—	2	1	no	—
167	#90673	—	2	1	no	—
168	#90673	—	4	1	no	—
169	#90673	—	4	1	no	—
170	#90673	—	4	2	no	—
171	#90673	#137	1	2	yes	direct product factor
172	C₄×C₁₁₃	—	2	2	no	—
173	C₄×C₁₁₃	—	2	1	no	—
174	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
175	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
176	Dic₁₁₃	—	2	2	no	—
177	Dic₁₁₃	—	2	1	no	—
178	Dic₁₁₃	C₄×D₄	1	1	yes	characteristic
179	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
180	C₂×C₁₁₃	—	2	1	no	—
181	C₂×C₁₁₃	—	4	1	no	—
182	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
183	D₁₁₃	—	2	1	no	—
184	D₁₁₃	—	4	1	no	—
185	D₁₁₃	C₂×#137	1	1	yes	characteristic
186	D₁₁₃	#575	1	1	yes	characteristic
187	C₄×#137	—	113	1	no	maximal, Sylow
188	C₁₁₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
189	C₂×#137	—	113	1	no	—
190	C₄×#41	—	113	1	no	—
191	C₂×C₄×Q₈	—	113	1	no	—
192	#441	—	113	2	no	—
193	#440	—	113	2	no	—
194	C₄×Q₁₆	—	113	2	no	—
195	C₄×SD₁₆	—	113	2	no	—
196	#428	—	113	1	no	—
197	#419	—	113	1	no	—
198	#417	—	113	1	no	—
199	C₄×M₁₆	—	113	1	no	—
200	C₂×#41	—	113	1	no	—
201	C₂²×Q₈	—	113	1	no	—
202	#137	—	113	1	no	—
203	#137	—	113	1	no	—
204	#137	—	113	2	no	—
205	#137	—	113	2	no	—
206	#137	—	113	1	no	—
207	#137	#90673	1	1	yes	characteristic, direct product factor
208	C₂×Q₁₆	—	113	2	no	—
209	C₂×SD₁₆	—	113	2	no	—
210	C₂×M₁₆	—	113	1	no	—
211	C₂×M₁₆	—	113	1	no	—
212	C₄×Q₈	—	113	2	no	—
213	C₄×Q₈	—	113	2	no	—
214	C₄×Q₈	—	113	1	no	—
215	C₄×Q₈	—	113	1	no	—
216	C₄×Q₈	—	226	1	no	—
217	C₄×Q₈	—	226	1	no	—
218	C₄×D₄	—	113	1	no	—
219	C₄×D₄	—	113	1	no	—
220	C₄×D₄	—	226	1	no	—
221	C₄×D₄	—	226	1	no	—
222	#117	—	113	1	no	—
223	#117	—	226	1	no	—
224	C₂×C₄⋊C₄	—	113	1	no	—
225	C₂×C₄⋊C₄	—	226	1	no	—
226	C₂×C₄²	—	113	1	no	—
227	C₂×C₄²	—	226	1	no	—
228	C₂×C₄²	—	226	1	no	—
229	#107	—	113	2	no	—
230	#106	—	113	2	no	—
231	#103	—	113	2	no	—
232	#103	—	113	2	no	—
233	#103	—	113	2	no	—
234	#102	—	113	2	no	—
235	#97	—	113	2	no	—
236	C₄×C₈	—	113	2	no	—
237	#41	—	113	1	no	—
238	#41	—	113	1	no	—
239	#41	—	113	1	no	—
240	#41	—	226	1	no	—
241	#41	—	226	1	no	—
242	#41	C₂×#90673	1	1	yes	characteristic
243	C₂×Q₈	—	113	2	no	—
244	C₂×Q₈	—	113	1	no	—
245	C₂×Q₈	—	113	2	no	—
246	C₂×Q₈	—	113	1	no	—
247	C₂×Q₈	—	226	1	no	—
248	C₂×Q₈	—	226	1	no	—
249	C₂×Q₈	—	226	1	no	—
250	C₂×Q₈	C₂×#90673	1	1	yes	characteristic, complemented
251	C₂×D₄	—	113	1	no	—
252	C₂×D₄	—	226	1	no	—
253	C₂²×C₄	—	113	1	no	—
254	C₂²×C₄	—	113	1	no	—
255	C₂²×C₄	—	113	1	no	—
256	C₂²×C₄	—	226	1	no	—
257	C₂²×C₄	—	226	1	no	—
258	C₂²×C₄	—	226	1	no	—
259	C₂²×C₄	—	226	1	no	—
260	C₂²×C₄	—	226	1	no	—
261	Q₁₆	—	113	2	no	—
262	Q₁₆	—	113	2	no	—
263	Q₁₆	—	113	2	no	—
264	Q₁₆	C₂×#90673	1	2	yes	complemented
265	SD₁₆	—	113	2	no	—
266	SD₁₆	—	113	2	no	—
267	SD₁₆	—	113	2	no	—
268	SD₁₆	C₂×#90673	1	2	yes	complemented
269	M₁₆	—	113	1	no	—
270	M₁₆	—	113	2	no	—
271	M₁₆	—	226	1	no	—
272	M₁₆	—	226	1	no	—
273	M₁₆	C₂×#90673	1	1	yes	characteristic, complemented
274	C₂×C₈	—	113	2	no	—
275	C₂×C₈	—	113	2	no	—
276	C₄⋊C₄	—	113	2	no	—
277	C₄⋊C₄	—	113	1	no	—
278	C₄⋊C₄	—	113	1	no	—
279	C₄⋊C₄	—	226	1	no	—
280	C₄⋊C₄	—	226	2	no	—
281	C₄⋊C₄	—	226	2	no	—
282	C₄⋊C₄	—	226	1	no	—
283	C₄⋊C₄	—	226	1	no	—
284	C₂²⋊C₄	—	226	1	no	—
285	C₂²⋊C₄	—	226	1	no	—
286	C₂²⋊C₄	—	226	1	no	—
287	C₄²	—	113	1	no	—
288	C₄²	—	113	1	no	—
289	C₄²	—	113	1	no	—
290	C₄²	—	113	1	no	—
291	C₄²	—	226	1	no	—
292	C₄²	—	226	1	no	—
293	C₄²	—	226	1	no	—
294	C₄²	—	226	2	no	—
295	C₄²	—	226	2	no	—
296	C₂³	—	113	1	no	—
297	C₂³	—	226	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	113	2	no	—
300	Q₈	—	113	1	no	—
301	Q₈	—	226	2	no	—
302	Q₈	—	226	2	no	—
303	Q₈	—	226	1	no	—
304	Q₈	—	226	1	no	—
305	Q₈	C₂²×#90673	1	2	yes	—
306	Q₈	C₂²×#90673	1	1	yes	characteristic
307	D₄	—	2	1	no	—
308	D₄	—	113	1	no	—
309	D₄	—	226	1	no	—
310	D₄	—	226	1	no	—
311	D₄	—	226	1	no	—
312	D₄	—	226	1	no	—
313	D₄	C₂²×#90673	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	113	1	no	—
317	C₂×C₄	—	113	1	no	—
318	C₂×C₄	—	113	1	no	—
319	C₂×C₄	—	113	1	no	—
320	C₂×C₄	—	113	1	no	—
321	C₂×C₄	—	113	1	no	—
322	C₂×C₄	—	113	1	no	—
323	C₂×C₄	—	113	1	no	—
324	C₂×C₄	—	113	1	no	—
325	C₂×C₄	—	452	2	no	—
326	C₂×C₄	—	452	2	no	—
327	C₂×C₄	—	452	2	no	—
328	C₂×C₄	—	452	1	no	—
329	C₂×C₄	—	226	1	no	—
330	C₂×C₄	—	226	1	no	—
331	C₂×C₄	—	226	2	no	—
332	C₂×C₄	—	226	2	no	—
333	C₂×C₄	—	226	1	no	—
334	C₂×C₄	—	226	2	no	—
335	C₂×C₄	—	226	1	no	—
336	C₂×C₄	—	226	1	no	—
337	C₂×C₄	—	226	2	no	—
338	C₂×C₄	—	226	1	no	—
339	C₂×C₄	—	226	1	no	—
340	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
341	C₈	—	113	2	no	—
342	C₈	—	226	2	no	—
343	C₈	C₂²×#90673	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	113	1	no	—
346	C₂²	—	113	1	no	—
347	C₂²	—	452	1	no	—
348	C₂²	—	452	1	no	—
349	C₂²	—	226	1	no	—
350	C₂²	—	226	1	no	—
351	C₂²	—	226	1	no	—
352	C₂²	D₄×#90673	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	113	2	no	—
356	C₄	—	113	1	no	—
357	C₄	—	113	1	no	—
358	C₄	—	452	1	no	—
359	C₄	—	452	2	no	—
360	C₄	—	452	1	no	—
361	C₄	—	226	1	no	—
362	C₄	—	226	1	no	—
363	C₄	—	226	1	no	—
364	C₄	—	226	1	no	—
365	C₄	—	226	2	no	—
366	C₄	D₄×#90673	1	1	yes	characteristic
367	C₄	C₂³×#90673	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	4	1	no	—
370	C₂	—	113	1	no	—
371	C₂	—	113	1	no	—
372	C₂	—	452	1	no	—
373	C₂	—	226	1	no	—
374	C₂	C₂×D₄×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#90673
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₄	C₂³×#90673
Fitting	#137×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₂×C₄×D₄
prime radical	#41×D₁₁₃	C₂²
cosocle	C₄×D₁₁₃	C₂⁴
hypercenter	#137	#90673
hypocenter	C₁₁₃	C₄×#137
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×#90673	C₂	semidirect
3. #20299638	C₂	semidirect
4. #20299637	C₂	semidirect
5. Q₁₆×#90673	C₂	semidirect
6. SD₁₆×#90673	C₂	semidirect
7. #20299609	C₂	semidirect
8. #20299591	C₂	semidirect
10. M₁₆×#90673	C₂	semidirect
40. #16813655	C₄	semidirect
41. #16813654	C₄	semidirect
42. #16813653	C₄	semidirect
43. #16813652	C₄	semidirect
44. #16813651	C₄	semidirect
48. #137×C₁₁₃	C₄	semidirect
83. C₂×Dic₂₂₆	C₂×C₄	semidirect
86. Q₈×D₁₁₃	C₂×C₄	semidirect
94. Dic₄₅₂	C₂×C₄	semidirect
95. #14268637	C₂×C₄	semidirect
96. #14268637	C₂×C₄	semidirect
97. #14268636	C₂×C₄	semidirect
98. #14268635	C₂×C₄	semidirect
99. #14268634	C₂×C₄	semidirect
100. #14268631	C₂×C₄	semidirect
101. #14268630	C₂×C₄	semidirect
104. C₂×Q₈×C₁₁₃	C₂×C₄	semidirect
105. Q₁₆×C₁₁₃	C₂×C₄	semidirect
106. SD₁₆×C₁₁₃	C₂×C₄	semidirect
107. M₁₆×C₁₁₃	C₂×C₄	semidirect
171. #90673	#137	direct (N × H)
188. C₁₁₃	C₄×#137	semidirect
207. #137	#90673	direct (N × H)
250. C₂×Q₈	C₂×#90673	semidirect
264. Q₁₆	C₂×#90673	semidirect
268. SD₁₆	C₂×#90673	semidirect
273. M₁₆	C₂×#90673	semidirect

Maximal subgroups

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

subgroup	index
#41×#90673	2
C₂×Q₈×#90673	2
#20299638	2
#20299637	2
Q₁₆×#90673	2
SD₁₆×#90673	2
#20299609	2
#20299591	2
#20299589	2
M₁₆×#90673	2
#137×D₁₁₃	2
C₄×#137	113