Subgroups — #31797743

Order 17536.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578455	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578453	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578438	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578436	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578435	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578435	C₂	1	1	yes	maximal, characteristic, complemented
7	#22578434	C₂	1	1	yes	maximal, characteristic, complemented
8	#22578432	C₂	1	1	yes	maximal, characteristic, complemented
9	#22578428	C₂	1	1	yes	maximal, characteristic, complemented
10	#22578427	C₂	1	1	yes	maximal, characteristic, complemented
11	#22578424	C₂	1	1	yes	maximal, characteristic, complemented
12	#22578410	C₂	1	1	yes	maximal, characteristic, complemented
13	#22578409	C₂	1	1	yes	maximal, characteristic, complemented
14	#22578406	C₂	1	1	yes	maximal, characteristic, complemented
15	#22577930	C₂	1	1	yes	maximal, characteristic
16	C₂×#14555520	C₂²	1	1	yes	characteristic
17	C₂×#14555519	C₂²	1	1	yes	characteristic
18	C₂×#14555519	C₂²	1	1	yes	characteristic
19	C₂×#14555519	C₂²	1	1	yes	characteristic
20	C₂×#14555518	C₂²	1	1	yes	characteristic
21	C₂²×#11764047	C₂²	1	1	yes	characteristic
22	#17984155	C₂²	1	1	yes	characteristic
23	#17984155	C₂²	1	1	yes	characteristic
24	#17984155	C₂²	1	1	yes	characteristic
25	#17984155	C₂²	1	1	yes	characteristic
26	C₄⋊#11764047	C₂²	1	1	yes	characteristic
27	C₄⋊#11764047	C₂²	1	1	yes	characteristic
28	C₄⋊#11764047	C₂²	1	1	yes	characteristic
29	C₄⋊#11764047	C₂²	1	1	yes	characteristic
30	C₄⋊#11764047	C₂²	1	1	yes	characteristic
31	C₄⋊#11764047	C₂²	1	1	yes	characteristic
32	#17984145	C₂²	1	1	yes	characteristic
33	#17984145	C₂²	1	1	yes	characteristic
34	C₂²⋊#11764047	C₂²	1	1	yes	characteristic
35	C₂²⋊#11764047	C₂²	1	1	yes	characteristic
36	C₂²⋊#11764047	C₂²	1	1	yes	characteristic
37	C₂²⋊#11764047	C₂²	1	1	yes	characteristic
38	#17984143	C₂²	1	1	yes	characteristic
39	#17984143	C₂²	1	1	yes	characteristic
40	#17984143	C₂²	1	1	yes	characteristic
41	#17984143	C₂²	1	1	yes	characteristic
42	C₄×#11764047	C₂²	1	1	yes	characteristic
43	C₄×#11764047	C₂²	1	1	yes	characteristic
44	#17984083	C₄	1	1	yes	characteristic
45	#17984081	C₄	1	1	yes	characteristic
46	#17984078	C₄	1	1	yes	characteristic
47	#17984074	C₄	1	1	yes	characteristic
48	#17984071	C₄	1	1	yes	characteristic
49	#17984065	C₄	1	1	yes	characteristic
50	#17984060	C₄	1	1	yes	characteristic
51	#17984053	C₂²	1	1	yes	characteristic
52	D₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
53	#17984048	C₂²	1	1	yes	characteristic
54	#17984048	C₂²	1	1	yes	characteristic
55	#17984046	C₂²	1	1	yes	characteristic
56	#17984044	C₂²	1	1	yes	characteristic
57	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
58	#123×C₁₃₇	C₄	1	1	yes	characteristic
59	#14555520	—	4	1	no	—
60	#14555519	—	4	1	no	—
61	#14555519	—	4	1	no	—
62	#14555519	—	4	1	no	—
63	#14555518	—	4	1	no	—
64	C₂×#11764047	—	4	1	no	—
65	C₂×#11764047	C₂³	1	1	yes	characteristic
66	C₂×#11764047	C₂³	1	1	yes	characteristic
67	C₂×#11764047	C₂³	1	1	yes	characteristic
68	C₂×#11764047	C₂³	1	1	yes	characteristic
69	C₂×#11764047	C₂³	1	1	yes	characteristic
70	C₂×#11764047	C₂³	1	1	yes	characteristic
71	C₂×#11764047	C₂³	1	1	yes	characteristic
72	C₂×#11764047	C₂³	1	1	yes	characteristic
73	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
77	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
78	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
82	#14555489	C₂×C₄	1	1	yes	characteristic
83	#14555489	C₂×C₄	1	1	yes	characteristic
84	#14555489	C₂×C₄	1	1	yes	characteristic
85	#14555489	C₂×C₄	1	1	yes	characteristic
86	#14555489	C₂×C₄	1	1	yes	characteristic
87	#14555489	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
90	C₂²⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
91	#14555487	C₂×C₄	1	1	yes	characteristic
92	#14555487	C₂×C₄	1	1	yes	characteristic
93	#14555487	C₂×C₄	1	1	yes	characteristic
94	#14555487	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
98	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
99	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊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	#11764047	—	2	1	no	—
109	#11764047	—	2	1	no	—
110	#11764047	—	2	1	no	—
111	#11764047	—	2	1	no	—
112	#11764047	—	2	1	no	—
113	#11764047	—	2	1	no	—
114	#11764047	—	2	1	no	—
115	#11764047	—	2	1	no	—
116	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
117	Dic₂₇₄	—	4	1	no	—
118	C₂²⋊D₁₃₇	—	4	1	no	—
119	C₂²⋊D₁₃₇	—	4	1	no	—
120	C₂×Dic₁₃₇	—	2	1	no	—
121	C₂×Dic₁₃₇	—	2	1	no	—
122	C₂×Dic₁₃₇	—	2	1	no	—
123	C₂×Dic₁₃₇	—	2	1	no	—
124	C₂×Dic₁₃₇	—	4	1	no	—
125	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
132	C₄×D₁₃₇	—	2	1	no	—
133	C₄×D₁₃₇	—	2	1	no	—
134	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
135	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
136	D₄×C₁₃₇	—	4	1	no	—
137	C₂×C₄×C₁₃₇	—	4	1	no	—
138	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₁₃₇	—	2	1	no	—
144	C₂²×C₁₃₇	—	2	1	no	—
145	C₂²×C₁₃₇	—	2	1	no	—
146	C₂²×C₁₃₇	—	2	1	no	—
147	C₂²×C₁₃₇	—	2	1	no	—
148	C₂²×C₁₃₇	—	2	1	no	—
149	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×D₁₃₇	—	2	1	no	—
151	C₂×D₁₃₇	—	2	1	no	—
152	C₂×D₁₃₇	—	2	1	no	—
153	C₄×C₁₃₇	—	2	1	no	—
154	C₄×C₁₃₇	—	2	1	no	—
155	C₄×C₁₃₇	—	2	1	no	—
156	C₄×C₁₃₇	—	2	1	no	—
157	C₄×C₁₃₇	—	2	1	no	—
158	Dic₁₃₇	—	2	1	no	—
159	Dic₁₃₇	—	2	1	no	—
160	Dic₁₃₇	—	2	1	no	—
161	Dic₁₃₇	—	2	1	no	—
162	Dic₁₃₇	—	2	1	no	—
163	Dic₁₃₇	—	2	1	no	—
164	Dic₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂×C₁₃₇	—	4	1	no	—
167	C₂×C₁₃₇	—	4	1	no	—
168	C₂×C₁₃₇	#520	1	1	yes	characteristic
169	C₂×C₁₃₇	#568	1	1	yes	characteristic
170	C₂×C₁₃₇	#568	1	1	yes	characteristic
171	D₁₃₇	—	4	1	no	—
172	C₁₃₇	#2995	1	1	yes	characteristic, complemented, Sylow
173	#2995	—	137	1	no	maximal, Sylow
174	#529	—	137	1	no	—
175	#446	—	137	1	no	—
176	#446	—	137	1	no	—
177	#436	—	137	1	no	—
178	#436	—	137	1	no	—
179	#435	—	137	1	no	—
180	#435	—	137	1	no	—
181	#435	—	137	1	no	—
182	#435	—	137	1	no	—
183	#433	—	137	1	no	—
184	#433	—	137	1	no	—
185	#432	—	137	1	no	—
186	#424	—	137	1	no	—
187	#424	—	137	1	no	—
188	#423	—	137	1	no	—
189	C₂×M₁₆	—	137	1	no	—
190	C₂×M₁₆	—	137	1	no	—
191	C₂×M₁₆	—	137	1	no	—
192	C₂×M₁₆	—	137	1	no	—
193	C₂²×C₈	—	137	1	no	—
194	C₂²×C₈	—	137	1	no	—
195	#126	—	137	1	no	—
196	#126	—	137	1	no	—
197	#125	—	137	1	no	—
198	#124	—	137	1	no	—
199	#123	—	137	1	no	—
200	#123	#10587739	1	1	yes	characteristic
201	C₂²⋊Q₈	—	137	1	no	—
202	#121	—	137	1	no	—
203	C₄×Q₈	—	137	1	no	—
204	C₄×D₄	—	137	1	no	—
205	C₄×D₄	—	137	1	no	—
206	C₄×D₄	—	137	1	no	—
207	#117	—	137	1	no	—
208	#117	—	137	1	no	—
209	C₂×C₄²	—	137	1	no	—
210	C₄⋊C₈	—	137	1	no	—
211	C₄⋊C₈	—	137	1	no	—
212	C₄⋊C₈	—	137	1	no	—
213	C₄⋊C₈	—	137	1	no	—
214	C₄⋊C₈	—	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	C₂²⋊C₈	—	137	1	no	—
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	—	137	1	no	—
226	#97	—	137	1	no	—
227	#97	—	137	1	no	—
228	#97	—	137	1	no	—
229	#97	—	137	1	no	—
230	C₄×C₈	—	137	1	no	—
231	C₄×C₈	—	137	1	no	—
232	C₂×Q₈	—	137	1	no	—
233	C₂×D₄	—	137	1	no	—
234	C₂×D₄	—	137	1	no	—
235	C₂×D₄	C₂×#10587739	1	1	yes	characteristic
236	C₂²×C₄	—	137	1	no	—
237	C₂²×C₄	—	137	1	no	—
238	C₂²×C₄	—	137	1	no	—
239	C₂²×C₄	—	137	1	no	—
240	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
241	M₁₆	—	548	1	no	—
242	M₁₆	—	548	1	no	—
243	M₁₆	—	548	1	no	—
244	M₁₆	—	548	1	no	—
245	C₂×C₈	—	548	1	no	—
246	C₂×C₈	—	548	1	no	—
247	C₂×C₈	—	137	1	no	—
248	C₂×C₈	—	137	1	no	—
249	C₂×C₈	—	137	1	no	—
250	C₂×C₈	—	137	1	no	—
251	C₂×C₈	—	137	1	no	—
252	C₂×C₈	—	137	1	no	—
253	C₂×C₈	—	137	1	no	—
254	C₂×C₈	—	137	1	no	—
255	C₄⋊C₄	—	137	1	no	—
256	C₄⋊C₄	—	137	1	no	—
257	C₄⋊C₄	—	137	1	no	—
258	C₄⋊C₄	—	137	1	no	—
259	C₄⋊C₄	—	137	1	no	—
260	C₄⋊C₄	—	137	1	no	—
261	C₄⋊C₄	—	137	1	no	—
262	C₄⋊C₄	—	137	1	no	—
263	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
264	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
265	C₂²⋊C₄	—	137	1	no	—
266	C₂²⋊C₄	—	137	1	no	—
267	C₂²⋊C₄	—	137	1	no	—
268	C₂²⋊C₄	—	137	1	no	—
269	C₂²⋊C₄	—	137	1	no	—
270	C₂²⋊C₄	—	137	1	no	—
271	C₂²⋊C₄	—	137	1	no	—
272	C₂²⋊C₄	C₂×#10587739	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×#10587739	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×#10587739	1	1	yes	characteristic
275	C₄²	—	137	1	no	—
276	C₄²	—	137	1	no	—
277	C₄²	—	137	1	no	—
278	C₄²	—	137	1	no	—
279	C₄²	—	137	1	no	—
280	C₄²	—	137	1	no	—
281	C₂³	—	137	1	no	—
282	C₂³	C₂²×#10587739	1	1	yes	characteristic
283	C₂³	C₂²×#10587739	1	1	yes	characteristic
284	Q₈	—	548	1	no	—
285	D₄	—	4	1	no	—
286	D₄	—	548	1	no	—
287	D₄	—	548	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	274	1	no	—
290	C₂×C₄	—	274	1	no	—
291	C₂×C₄	—	274	1	no	—
292	C₂×C₄	—	274	1	no	—
293	C₂×C₄	—	274	1	no	—
294	C₂×C₄	—	274	1	no	—
295	C₂×C₄	—	548	1	no	—
296	C₂×C₄	—	137	1	no	—
297	C₂×C₄	—	137	1	no	—
298	C₂×C₄	—	137	1	no	—
299	C₂×C₄	—	137	1	no	—
300	C₂×C₄	—	137	1	no	—
301	C₂×C₄	—	137	1	no	—
302	C₂×C₄	—	137	1	no	—
303	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
304	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
305	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
306	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
307	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
308	C₈	—	274	1	no	—
309	C₈	—	274	1	no	—
310	C₈	—	274	1	no	—
311	C₈	—	274	1	no	—
312	C₈	—	274	1	no	—
313	C₈	—	274	1	no	—
314	C₈	—	274	1	no	—
315	C₈	—	274	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	274	1	no	—
323	C₂²	—	274	1	no	—
324	C₂²	—	274	1	no	—
325	C₂²	C₂³×#10587739	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	274	1	no	—
332	C₄	—	274	1	no	—
333	C₄	—	274	1	no	—
334	C₄	—	274	1	no	—
335	C₄	—	274	1	no	—
336	C₄	—	274	1	no	—
337	C₄	—	137	1	no	—
338	C₄	—	137	1	no	—
339	C₂	—	4	1	no	—
340	C₂	—	4	1	no	—
341	C₂	—	548	1	no	—
342	C₂	#22578472	1	1	yes	characteristic
343	C₂	#22578481	1	1	yes	characteristic
344	C₂	#22578482	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10587739
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#123×C₁₃₇	C₄
prime radical	#17984060	C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	#123	#10587739
hypocenter	C₁₃₇	#2995
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. #22578455	C₂	semidirect
2. #22578453	C₂	semidirect
3. #22578438	C₂	semidirect
4. #22578436	C₂	semidirect
5. #22578435	C₂	semidirect
6. #22578435	C₂	semidirect
7. #22578434	C₂	semidirect
8. #22578432	C₂	semidirect
9. #22578428	C₂	semidirect
10. #22578427	C₂	semidirect
11. #22578424	C₂	semidirect
12. #22578410	C₂	semidirect
13. #22578409	C₂	semidirect
14. #22578406	C₂	semidirect
172. C₁₃₇	#2995	semidirect

Maximal subgroups

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

subgroup	index
#22578455	2
#22578453	2
#22578438	2
#22578436	2
#22578435	2
#22578434	2
#22578432	2
#22578428	2
#22578427	2
#22578424	2
#22578410	2
#22578409	2
#22578406	2
#22577930	2
#2995	137