Subgroups — #31790800

Order 17536.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577934	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577745	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577724	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577712	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577705	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577623	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577616	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577565	C₂	1	1	yes	maximal, characteristic, complemented
10	#22577495	C₂	1	1	yes	maximal, characteristic, complemented
11	#22577494	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	D₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#22577453	C₂	1	1	yes	maximal, characteristic, complemented
15	#449×C₁₃₇	C₂	1	1	yes	maximal, characteristic
16	C₂×#14555500	C₂²	1	1	yes	characteristic
17	C₂×#14555498	C₂²	1	1	yes	characteristic
18	#17984081	—	2	1	no	—
19	#17984079	C₂²	1	1	yes	characteristic
20	#17984075	C₂²	1	1	yes	characteristic
21	#17984069	—	2	1	no	—
22	#17984067	C₂²	1	1	yes	characteristic
23	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
25	#17984053	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₃₇	—	2	1	no	—
27	D₄×Dic₁₃₇	—	2	1	no	—
28	D₄×Dic₁₃₇	—	2	1	no	—
29	D₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
32	#17984047	—	2	1	no	—
33	#17984047	C₂²	1	1	yes	characteristic
34	#17984044	—	2	1	no	—
35	C₂×C₄×Dic₁₃₇	—	2	1	no	—
36	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
37	#17984022	C₂²	1	1	yes	characteristic
38	#17984021	C₂²	1	1	yes	characteristic
39	#17984020	C₂²	1	1	yes	characteristic
40	#17984019	C₂²	1	1	yes	characteristic
41	#17984017	C₂²	1	1	yes	characteristic
42	#17984017	C₂²	1	1	yes	characteristic
43	#17984013	C₂²	1	1	yes	characteristic
44	#17984013	C₂²	1	1	yes	characteristic
45	#17984012	C₂²	1	1	yes	characteristic
46	#17984009	C₂²	1	1	yes	characteristic
47	#17984009	C₂²	1	1	yes	characteristic
48	#17984009	C₂²	1	1	yes	characteristic
49	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₃₇	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
55	#121×C₁₃₇	C₂²	1	1	yes	characteristic
56	#103×C₁₃₇	C₂²	1	1	yes	characteristic
57	#102×C₁₃₇	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₃₇	—	2	1	no	—
61	C₂²×Dic₁₃₇	—	2	1	no	—
62	C₂²×Dic₁₃₇	—	2	1	no	—
63	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
64	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
65	#14555500	—	2	2	no	—
66	#14555498	—	2	2	no	—
67	C₂×#11764040	C₂³	1	1	yes	characteristic
68	C₂×#11764040	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₃₇	—	2	1	no	—
70	C₄⋊Dic₁₃₇	—	2	1	no	—
71	C₄⋊Dic₁₃₇	—	2	1	no	—
72	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
75	#14555489	—	2	1	no	—
76	#14555489	—	2	1	no	—
77	#14555489	—	2	1	no	—
78	C₂²⋊Dic₁₃₇	—	2	1	no	—
79	C₂²⋊Dic₁₃₇	—	2	1	no	—
80	C₂²⋊Dic₁₃₇	—	2	1	no	—
81	C₂²⋊Dic₁₃₇	—	2	1	no	—
82	C₂²⋊Dic₁₃₇	—	2	1	no	—
83	C₂²⋊Dic₁₃₇	—	2	1	no	—
84	C₂²⋊Dic₁₃₇	—	2	1	no	—
85	C₄×Dic₁₃₇	—	2	1	no	—
86	C₄×Dic₁₃₇	—	2	1	no	—
87	C₄×Dic₁₃₇	—	2	1	no	—
88	C₄×Dic₁₃₇	—	2	1	no	—
89	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
90	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
91	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
93	#41×C₁₃₇	—	2	2	no	—
94	#41×C₁₃₇	—	2	2	no	—
95	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₃₇	—	2	1	no	—
97	C₂×D₄×C₁₃₇	—	2	1	no	—
98	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₃₇	—	2	1	no	—
101	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
102	SD₁₆×C₁₃₇	—	2	2	no	—
103	D₈×C₁₃₇	—	2	2	no	—
104	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
108	Dic₂₇₄	—	2	1	no	—
109	Dic₂₇₄	—	2	1	no	—
110	C₂×Dic₁₃₇	—	2	2	no	—
111	C₂×Dic₁₃₇	—	2	1	no	—
112	C₂×Dic₁₃₇	—	2	1	no	—
113	C₂×Dic₁₃₇	—	2	1	no	—
114	C₂×Dic₁₃₇	—	2	1	no	—
115	C₂×Dic₁₃₇	—	4	1	no	—
116	C₂×Dic₁₃₇	—	4	2	no	—
117	C₂×Dic₁₃₇	—	4	2	no	—
118	C₂×Dic₁₃₇	—	4	2	no	—
119	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
123	#11764040	—	2	1	no	—
124	#11764040	—	2	1	no	—
125	C₂³×C₁₃₇	—	2	1	no	—
126	C₂³×C₁₃₇	—	2	1	no	—
127	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
128	Q₈×C₁₃₇	—	2	1	no	—
129	Q₈×C₁₃₇	#41	1	2	yes	—
130	D₄×C₁₃₇	—	2	4	no	—
131	D₄×C₁₃₇	—	2	1	no	—
132	D₄×C₁₃₇	—	2	1	no	—
133	D₄×C₁₃₇	—	2	1	no	—
134	D₄×C₁₃₇	—	4	2	no	—
135	D₄×C₁₃₇	#41	1	2	yes	—
136	C₂×C₄×C₁₃₇	—	2	4	no	—
137	C₂×C₄×C₁₃₇	—	2	1	no	—
138	C₂×C₄×C₁₃₇	—	2	1	no	—
139	C₂×C₄×C₁₃₇	—	2	1	no	—
140	C₂×C₄×C₁₃₇	—	2	1	no	—
141	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₃₇	—	2	1	no	—
144	C₈×C₁₃₇	—	2	1	no	—
145	C₂²×C₁₃₇	—	2	1	no	—
146	C₂²×C₁₃₇	—	2	2	no	—
147	C₂²×C₁₃₇	—	2	1	no	—
148	C₂²×C₁₃₇	—	2	1	no	—
149	C₂²×C₁₃₇	—	4	1	no	—
150	C₂²×C₁₃₇	—	4	1	no	—
151	C₂²×C₁₃₇	—	4	1	no	—
152	C₂²×C₁₃₇	—	4	1	no	—
153	C₂²×C₁₃₇	—	4	1	no	—
154	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₁₃₇	—	2	1	no	—
156	C₄×C₁₃₇	—	2	2	no	—
157	C₄×C₁₃₇	—	4	1	no	—
158	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
160	Dic₁₃₇	—	2	1	no	—
161	Dic₁₃₇	—	2	1	no	—
162	Dic₁₃₇	—	2	1	no	—
163	Dic₁₃₇	—	4	1	no	—
164	Dic₁₃₇	—	4	1	no	—
165	Dic₁₃₇	—	4	1	no	—
166	Dic₁₃₇	—	4	1	no	—
167	Dic₁₃₇	#135	1	2	yes	—
168	C₂×C₁₃₇	—	4	1	no	—
169	C₂×C₁₃₇	—	4	2	no	—
170	C₂×C₁₃₇	—	8	1	no	—
171	C₂×C₁₃₇	#525	1	1	yes	characteristic
172	C₂×C₁₃₇	C₂×#135	1	1	yes	characteristic
173	C₂×C₁₃₇	#575	1	1	yes	characteristic
174	C₁₃₇	#3102	1	1	yes	characteristic, complemented, Sylow
175	#3102	—	137	1	no	maximal, Sylow
176	#529	—	137	1	no	—
177	C₄×#41	—	137	1	no	—
178	#485	—	137	1	no	—
179	#482	—	137	1	no	—
180	#479	—	137	1	no	—
181	#478	—	137	1	no	—
182	#464	—	137	1	no	—
183	#463	—	137	1	no	—
184	#452	—	137	1	no	—
185	#449	D₁₃₇	1	1	yes	characteristic
186	C₄×Q₁₆	—	137	1	no	—
187	C₄×SD₁₆	—	137	1	no	—
188	C₄×SD₁₆	—	137	1	no	—
189	C₄×D₈	—	137	1	no	—
190	#431	—	137	1	no	—
191	C₂×#41	C₂×D₁₃₇	1	1	yes	characteristic
192	C₂×Q₁₆	—	137	1	no	—
193	C₂×SD₁₆	—	137	1	no	—
194	C₂×SD₁₆	C₂×D₁₃₇	1	1	yes	characteristic
195	C₂×D₈	C₂×D₁₃₇	1	1	yes	characteristic
196	#126	—	274	1	no	—
197	#125	—	137	1	no	—
198	#124	—	137	1	no	—
199	#123	—	274	1	no	—
200	C₂²⋊Q₈	—	137	1	no	—
201	#121	C₂×D₁₃₇	1	1	yes	characteristic
202	C₄×Q₈	—	137	1	no	—
203	C₄×Q₈	—	137	1	no	—
204	C₄×Q₈	—	137	1	no	—
205	C₄×D₄	—	274	1	no	—
206	C₄×D₄	—	274	1	no	—
207	C₄×D₄	—	274	1	no	—
208	C₄×D₄	—	137	1	no	—
209	C₄×D₄	—	137	1	no	—
210	C₄×D₄	—	137	1	no	—
211	#117	—	274	1	no	—
212	#117	—	274	1	no	—
213	#117	—	137	1	no	—
214	C₂×C₄²	—	274	1	no	—
215	C₂×C₄²	—	137	1	no	—
216	#107	—	137	1	no	—
217	#107	—	137	1	no	—
218	#106	—	137	1	no	—
219	#106	—	137	1	no	—
220	C₄⋊C₈	—	137	1	no	—
221	C₄⋊C₈	—	137	1	no	—
222	#103	—	137	1	no	—
223	#103	—	137	1	no	—
224	#103	—	137	1	no	—
225	#103	C₂×D₁₃₇	1	1	yes	characteristic
226	#102	—	137	1	no	—
227	#102	—	137	1	no	—
228	#102	—	137	1	no	—
229	#102	C₂×D₁₃₇	1	1	yes	characteristic
230	C₂²⋊C₈	—	137	1	no	—
231	C₂²⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
232	C₄×C₈	—	137	1	no	—
233	C₄×C₈	—	137	1	no	—
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	—	137	1	no	—
237	C₂×Q₈	C₂²×D₁₃₇	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
241	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	274	1	no	—
244	C₂²×C₄	—	274	1	no	—
245	C₂²×C₄	—	274	1	no	—
246	C₂²×C₄	—	137	1	no	—
247	C₂²×C₄	—	137	1	no	—
248	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
249	Q₁₆	—	274	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	274	2	no	—
252	D₈	—	2	2	no	—
253	C₂×C₈	—	137	1	no	—
254	C₂×C₈	—	137	1	no	—
255	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
257	C₄⋊C₄	—	274	1	no	—
258	C₄⋊C₄	—	274	1	no	—
259	C₄⋊C₄	—	274	1	no	—
260	C₄⋊C₄	—	274	1	no	—
261	C₄⋊C₄	—	274	1	no	—
262	C₄⋊C₄	—	274	1	no	—
263	C₄⋊C₄	—	137	1	no	—
264	C₄⋊C₄	—	137	1	no	—
265	C₄⋊C₄	—	137	1	no	—
266	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	274	1	no	—
269	C₂²⋊C₄	—	274	1	no	—
270	C₂²⋊C₄	—	274	1	no	—
271	C₂²⋊C₄	—	274	1	no	—
272	C₂²⋊C₄	—	274	1	no	—
273	C₂²⋊C₄	—	274	1	no	—
274	C₂²⋊C₄	—	274	1	no	—
275	C₄²	—	274	1	no	—
276	C₄²	—	274	1	no	—
277	C₄²	—	274	1	no	—
278	C₄²	—	274	1	no	—
279	C₄²	—	137	1	no	—
280	C₄²	—	137	1	no	—
281	C₄²	—	137	1	no	—
282	C₄²	—	137	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	D₄×D₁₃₇	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	274	1	no	—
288	Q₈	—	274	1	no	—
289	Q₈	#14555509	1	2	yes	—
290	D₄	—	2	4	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	4	2	no	—
295	D₄	#14555509	1	2	yes	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	274	2	no	—
302	C₂×C₄	—	274	1	no	—
303	C₂×C₄	—	274	1	no	—
304	C₂×C₄	—	274	1	no	—
305	C₂×C₄	—	274	1	no	—
306	C₂×C₄	—	548	1	no	—
307	C₂×C₄	—	548	2	no	—
308	C₂×C₄	—	548	2	no	—
309	C₂×C₄	—	548	2	no	—
310	C₂×C₄	—	137	1	no	—
311	C₂×C₄	—	137	1	no	—
312	C₂×C₄	—	137	1	no	—
313	C₂×C₄	—	137	1	no	—
314	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	274	1	no	—
319	C₈	—	274	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	274	1	no	—
334	C₄	—	274	1	no	—
335	C₄	—	274	1	no	—
336	C₄	—	548	1	no	—
337	C₄	—	548	1	no	—
338	C₄	—	548	1	no	—
339	C₄	—	548	1	no	—
340	C₄	—	137	2	no	—
341	C₄	C₂×#14555509	1	1	yes	characteristic
342	C₄	C₂×#14555509	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	4	2	no	—
345	C₂	—	8	1	no	—
346	C₂	#22577909	1	1	yes	characteristic
347	C₂	#135×D₁₃₇	1	1	yes	characteristic
348	C₂	#22578199	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting, prime radical	#449×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
hypercenter	#449	D₁₃₇
hypocenter	C₁₃₇	#3102
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #22577934	C₂	semidirect
2. #41×Dic₁₃₇	C₂	semidirect
3. #22577745	C₂	semidirect
4. #22577724	C₂	semidirect
5. #22577712	C₂	semidirect
6. #22577705	C₂	semidirect
7. #22577623	C₂	semidirect
8. #22577616	C₂	semidirect
9. #22577565	C₂	semidirect
10. #22577495	C₂	semidirect
11. #22577494	C₂	semidirect
12. SD₁₆×Dic₁₃₇	C₂	semidirect
13. D₈×Dic₁₃₇	C₂	semidirect
14. #22577453	C₂	semidirect
174. C₁₃₇	#3102	semidirect

Maximal subgroups

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

subgroup	index
#22577934	2
#41×Dic₁₃₇	2
#22577745	2
#22577724	2
#22577712	2
#22577705	2
#22577623	2
#22577616	2
#22577565	2
#22577495	2
#22577494	2
SD₁₆×Dic₁₃₇	2
D₈×Dic₁₃₇	2
#22577453	2
#449×C₁₃₇	2
#3102	137