Subgroups — #31789749

Order 17536.

412 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	C₄×#14555509	C₂	1	1	yes	maximal, characteristic
2	#22577532	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577531	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577479	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577478	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577475	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577474	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577471	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577455	C₂	1	1	yes	maximal, characteristic, complemented
10	#22577453	C₂	1	1	yes	maximal, characteristic, complemented
11	#22577453	C₂	1	1	yes	maximal, characteristic, complemented
12	#22577355	C₂	1	1	yes	maximal, characteristic, complemented
13	M₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555509	C₄	1	1	yes	characteristic
17	C₂×#14555494	C₂²	1	1	yes	characteristic
18	C₂×#14555493	C₂²	1	1	yes	characteristic
19	C₂×#14555493	C₂²	1	1	yes	characteristic
20	C₂²×#11764040	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
22	#17984053	C₄	1	1	yes	characteristic
23	D₄×Dic₁₃₇	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₃₇	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₃₇	C₂²	1	1	yes	characteristic
26	#17984048	C₄	1	1	yes	characteristic
27	#17984048	C₄	1	1	yes	characteristic
28	#17984046	C₄	1	1	yes	characteristic
29	#17984044	C₄	1	1	yes	characteristic
30	#17984044	C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
33	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
34	C₄⋊#11764040	C₂²	1	1	yes	characteristic
35	#17984017	C₂²	1	1	yes	characteristic
36	#17984017	C₂²	1	1	yes	characteristic
37	#17984017	C₂²	1	1	yes	characteristic
38	#17984017	C₂²	1	1	yes	characteristic
39	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
40	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
41	#17984000	C₂²	1	1	yes	characteristic
42	#17984000	C₂²	1	1	yes	characteristic
43	#17983999	C₂²	1	1	yes	characteristic
44	#17983998	C₂²	1	1	yes	characteristic
45	#17983998	C₂²	1	1	yes	characteristic
46	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
47	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
48	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
58	#97×C₁₃₇	C₂²	1	1	yes	characteristic
59	#14555509	—	2	4	no	—
60	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
70	#14555494	—	2	2	no	—
71	#14555493	—	2	2	no	—
72	#14555493	—	2	2	no	—
73	C₂×#11764040	—	2	2	no	—
74	C₂×#11764040	C₂³	1	1	yes	characteristic
75	C₂×#11764040	C₂³	1	1	yes	characteristic
76	C₂×#11764040	C₂³	1	1	yes	characteristic
77	C₂×#11764040	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
79	#14555489	C₂×C₄	1	1	yes	characteristic
80	#14555489	C₂×C₄	1	1	yes	characteristic
81	#14555489	C₂×C₄	1	1	yes	characteristic
82	#14555489	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
85	#14555487	C₂×C₄	1	1	yes	characteristic
86	#14555487	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
89	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
95	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
99	M₁₆×C₁₃₇	—	2	2	no	—
100	C₂×C₈×C₁₃₇	—	2	2	no	—
101	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
108	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
109	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
110	Dic₂₇₄	—	2	2	no	—
111	C₂²⋊D₁₃₇	—	2	2	no	—
112	C₂²⋊D₁₃₇	—	2	2	no	—
113	C₂×Dic₁₃₇	—	2	2	no	—
114	C₂×Dic₁₃₇	—	2	2	no	—
115	C₂×Dic₁₃₇	—	2	2	no	—
116	C₂×Dic₁₃₇	M₁₆	1	4	yes	complemented
117	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
124	C₄×D₁₃₇	—	2	1	no	—
125	C₄×D₁₃₇	—	2	1	no	—
126	C₄×D₁₃₇	—	2	2	no	—
127	C₄×D₁₃₇	—	2	2	no	—
128	#11764040	—	2	1	no	—
129	#11764040	—	2	1	no	—
130	#11764040	—	2	1	no	—
131	#11764040	#41	1	2	yes	complemented
132	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
133	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
134	D₄×C₁₃₇	—	2	2	no	—
135	C₂×C₄×C₁₃₇	—	2	1	no	—
136	C₂×C₄×C₁₃₇	—	2	1	no	—
137	C₂×C₄×C₁₃₇	—	2	1	no	—
138	C₂×C₄×C₁₃₇	—	2	1	no	—
139	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
144	C₈×C₁₃₇	—	2	1	no	—
145	C₈×C₁₃₇	—	2	1	no	—
146	C₈×C₁₃₇	—	2	1	no	—
147	C₈×C₁₃₇	#41	1	2	yes	complemented
148	C₂²×C₁₃₇	—	2	1	no	—
149	C₂²×C₁₃₇	—	2	1	no	—
150	C₂²×C₁₃₇	—	2	1	no	—
151	C₂²×C₁₃₇	—	2	1	no	—
152	C₂²×C₁₃₇	—	2	1	no	—
153	C₂²×C₁₃₇	C₂×M₁₆	1	2	yes	—
154	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₁₃₇	—	2	1	no	—
156	C₂×D₁₃₇	—	2	1	no	—
157	C₂×D₁₃₇	—	2	1	no	—
158	C₄×C₁₃₇	—	2	1	no	—
159	C₄×C₁₃₇	—	2	1	no	—
160	C₄×C₁₃₇	—	2	1	no	—
161	C₄×C₁₃₇	—	2	1	no	—
162	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
164	Dic₁₃₇	—	2	1	no	—
165	Dic₁₃₇	—	2	1	no	—
166	Dic₁₃₇	—	2	1	no	—
167	Dic₁₃₇	—	2	1	no	—
168	Dic₁₃₇	C₂×M₁₆	1	2	yes	—
169	Dic₁₃₇	C₂×M₁₆	1	2	yes	—
170	Dic₁₃₇	#131	1	2	yes	—
171	C₂×C₁₃₇	—	2	2	no	—
172	C₂×C₁₃₇	—	4	1	no	—
173	C₂×C₁₃₇	C₄×#41	1	1	yes	characteristic
174	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
175	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
176	D₁₃₇	—	4	1	no	—
177	C₁₃₇	#2965	1	1	yes	characteristic, complemented, Sylow
178	#2965	—	137	1	no	maximal, Sylow
179	C₄×#41	—	137	1	no	—
180	#446	—	137	1	no	—
181	#446	—	137	1	no	—
182	#436	—	137	1	no	—
183	#436	—	137	1	no	—
184	#435	—	137	1	no	—
185	#435	—	137	1	no	—
186	#435	—	137	1	no	—
187	#435	D₁₃₇	1	1	yes	characteristic, complemented
188	#431	—	137	1	no	—
189	#431	—	137	1	no	—
190	#431	—	137	1	no	—
191	C₄×M₁₆	—	137	1	no	—
192	C₄×M₁₆	—	137	1	no	—
193	C₂×C₄×C₈	—	137	1	no	—
194	C₂×#41	—	137	1	no	—
195	C₂×M₁₆	—	137	1	no	—
196	C₂×M₁₆	—	137	1	no	—
197	C₂×M₁₆	—	137	1	no	—
198	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
199	C₂²×C₈	—	137	1	no	—
200	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
201	C₄×Q₈	—	137	1	no	—
202	C₄×Q₈	—	137	1	no	—
203	C₄×D₄	—	137	1	no	—
204	C₄×D₄	—	137	1	no	—
205	C₄×D₄	—	137	1	no	—
206	C₄×D₄	—	137	1	no	—
207	C₄×D₄	—	137	1	no	—
208	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic
209	#117	—	137	1	no	—
210	#117	—	137	1	no	—
211	#117	—	137	1	no	—
212	C₂×C₄²	—	137	1	no	—
213	C₂×C₄²	—	137	1	no	—
214	C₂×C₄²	—	137	1	no	—
215	C₄⋊C₈	—	137	1	no	—
216	C₄⋊C₈	—	137	1	no	—
217	C₄⋊C₈	—	137	1	no	—
218	C₄⋊C₈	—	137	1	no	—
219	C₄⋊C₈	—	137	1	no	—
220	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
221	C₂²⋊C₈	—	137	1	no	—
222	C₂²⋊C₈	—	137	1	no	—
223	C₂²⋊C₈	—	137	1	no	—
224	C₂²⋊C₈	—	137	1	no	—
225	C₂²⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
226	C₂²⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
227	#97	—	137	1	no	—
228	#97	—	137	1	no	—
229	#97	—	137	1	no	—
230	#97	C₂×D₁₃₇	1	1	yes	characteristic
231	C₄×C₈	—	137	1	no	—
232	C₄×C₈	—	137	1	no	—
233	C₄×C₈	—	137	1	no	—
234	C₄×C₈	—	137	1	no	—
235	C₄×C₈	—	137	1	no	—
236	C₄×C₈	—	137	1	no	—
237	#41	—	274	4	no	—
238	C₂×Q₈	—	137	1	no	—
239	C₂×D₄	—	137	1	no	—
240	C₂×D₄	—	137	1	no	—
241	C₂×D₄	C₄×D₁₃₇	1	1	yes	characteristic
242	C₂²×C₄	—	137	1	no	—
243	C₂²×C₄	—	137	1	no	—
244	C₂²×C₄	—	137	1	no	—
245	C₂²×C₄	—	137	1	no	—
246	C₂²×C₄	—	137	1	no	—
247	C₂²×C₄	—	137	1	no	—
248	C₂²×C₄	—	137	1	no	—
249	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	274	2	no	—
253	M₁₆	—	274	2	no	—
254	M₁₆	—	274	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	274	2	no	—
257	C₂×C₈	—	137	1	no	—
258	C₂×C₈	—	137	1	no	—
259	C₂×C₈	—	137	1	no	—
260	C₂×C₈	—	137	1	no	—
261	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
265	C₄⋊C₄	—	137	1	no	—
266	C₄⋊C₄	—	137	1	no	—
267	C₄⋊C₄	—	137	1	no	—
268	C₄⋊C₄	—	137	1	no	—
269	C₄⋊C₄	—	137	1	no	—
270	C₄⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
271	C₂²⋊C₄	—	137	1	no	—
272	C₂²⋊C₄	—	137	1	no	—
273	C₂²⋊C₄	—	137	1	no	—
274	C₂²⋊C₄	—	137	1	no	—
275	C₂²⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
277	C₄²	—	137	1	no	—
278	C₄²	—	137	1	no	—
279	C₄²	—	137	1	no	—
280	C₄²	—	137	1	no	—
281	C₄²	—	137	1	no	—
282	C₄²	—	137	1	no	—
283	C₄²	—	137	1	no	—
284	C₄²	—	137	1	no	—
285	C₄²	—	137	1	no	—
286	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
287	C₂³	—	137	1	no	—
288	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
290	Q₈	—	274	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	274	2	no	—
293	D₄	—	274	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	274	2	no	—
299	C₂×C₄	—	274	2	no	—
300	C₂×C₄	—	274	2	no	—
301	C₂×C₄	—	274	2	no	—
302	C₂×C₄	—	274	2	no	—
303	C₂×C₄	—	274	1	no	—
304	C₂×C₄	—	274	1	no	—
305	C₂×C₄	—	137	1	no	—
306	C₂×C₄	—	137	1	no	—
307	C₂×C₄	—	137	1	no	—
308	C₂×C₄	—	137	1	no	—
309	C₂×C₄	—	137	4	no	—
310	C₂×C₄	—	137	1	no	—
311	C₂×C₄	—	137	1	no	—
312	C₂×C₄	—	137	1	no	—
313	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	274	1	no	—
322	C₈	—	274	1	no	—
323	C₈	—	274	1	no	—
324	C₈	—	137	2	no	—
325	C₈	#14555509	1	2	yes	complemented
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	—	274	1	no	—
332	C₂²	—	274	1	no	—
333	C₂²	—	274	1	no	—
334	C₂²	M₁₆×D₁₃₇	1	2	yes	—
335	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	274	1	no	—
341	C₄	—	274	1	no	—
342	C₄	—	274	1	no	—
343	C₄	—	274	1	no	—
344	C₄	—	137	2	no	—
345	C₄	—	137	2	no	—
346	C₄	—	137	2	no	—
347	C₄	#41×D₁₃₇	1	1	yes	characteristic
348	C₄	C₂×#14555509	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	548	1	no	—
352	C₂	C₄×#14555509	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₁₃₇	1	1	yes	characteristic
354	C₂	#131×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#435×C₁₃₇	C₂
prime radical	C₂×#14555509	C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#435	D₁₃₇
hypocenter	C₁₃₇	#2965
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #22577532	C₂	semidirect
3. #22577531	C₂	semidirect
4. #22577479	C₂	semidirect
5. #22577478	C₂	semidirect
6. #22577475	C₂	semidirect
7. #22577474	C₂	semidirect
8. #22577471	C₂	semidirect
9. #22577455	C₂	semidirect
10. #22577453	C₂	semidirect
11. #22577453	C₂	semidirect
12. #22577355	C₂	semidirect
13. M₁₆×Dic₁₃₇	C₂	semidirect
14. C₂×C₈×Dic₁₃₇	C₂	semidirect
15. #435×C₁₃₇	C₂	semidirect
116. C₂×Dic₁₃₇	M₁₆	semidirect
131. #11764040	#41	semidirect
147. C₈×C₁₃₇	#41	semidirect
177. C₁₃₇	#2965	semidirect
187. #435	D₁₃₇	semidirect
325. C₈	#14555509	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14555509	2
#22577532	2
#22577531	2
#22577479	2
#22577478	2
#22577475	2
#22577474	2
#22577471	2
#22577455	2
#22577453	2
#22577355	2
M₁₆×Dic₁₃₇	2
C₂×C₈×Dic₁₃₇	2
#435×C₁₃₇	2
#2965	137