Subgroups — #34061449

Order 19072.

470 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	#23750694	C₂	1	1	yes	maximal, characteristic
2	#23750693	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750665	C₂	1	2	yes	maximal, complemented
4	#23750664	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#10597010	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#10597010	C₂	1	2	yes	maximal, complemented
7	#23750636	C₂	1	1	yes	maximal, characteristic, complemented
8	#23750615	C₂	1	1	yes	maximal, characteristic
9	C₂×#18011974	C₂	1	1	yes	maximal, characteristic, complemented
10	#23750591	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₁₄₉	C₂	1	1	yes	maximal, characteristic
12	C₂×D₁₄₉⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×#10597010	—	2	1	no	—
14	Q₈×#10597010	—	2	1	no	—
15	Q₈×#10597010	C₂²	1	2	yes	—
16	Q₈×#10597010	C₂²	1	2	yes	—
17	Q₈×#10597010	C₂²	1	2	yes	—
18	D₄×#10597010	—	2	2	no	—
19	D₄×#10597010	C₂²	1	2	yes	—
20	#18011990	—	2	1	no	—
21	#18011990	—	2	2	no	—
22	#18011990	C₂²	1	1	yes	characteristic
23	#18011990	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#10597010	—	2	1	no	—
25	C₂×C₄⋊#10597010	C₂²	1	1	yes	characteristic
26	#18011980	C₂²	1	2	yes	—
27	#18011979	C₂²	1	2	yes	—
28	#18011974	C₂²	1	2	yes	—
29	#18011974	C₂²	1	1	yes	characteristic
30	#18011974	C₂²	1	2	yes	—
31	#18011974	C₂²	1	1	yes	characteristic
32	#18011973	C₂²	1	1	yes	characteristic
33	#18011973	C₂²	1	1	yes	characteristic
34	#18011965	C₂²	1	2	yes	—
35	C₈×#10597010	C₂²	1	2	yes	—
36	#41×D₁₄₉	C₂²	1	1	yes	characteristic
37	C₂×Q₈×D₁₄₉	C₂²	1	1	yes	characteristic
38	#18011944	C₄	1	1	yes	characteristic, complemented
39	#18011943	C₄	1	2	yes	complemented
40	#18011942	C₄	1	1	yes	characteristic, complemented
41	#18011941	C₄	1	1	yes	characteristic, complemented
42	#18011940	C₄	1	2	yes	complemented
43	Q₁₆×D₁₄₉	C₂²	1	2	yes	—
44	SD₁₆×D₁₄₉	C₂²	1	2	yes	—
45	M₁₆×D₁₄₉	C₂²	1	1	yes	characteristic
46	#137×C₁₄₉	C₄	1	1	yes	characteristic, complemented
47	C₂²×#10597010	—	2	2	no	—
48	C₂²×#10597010	—	2	1	no	—
49	D₁₄₉⋊C₈	—	2	1	no	—
50	D₁₄₉⋊C₈	C₂³	1	1	yes	characteristic
51	D₁₄₉⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×#11924753	—	2	1	no	—
53	C₄⋊#10597010	—	2	2	no	—
54	C₄⋊#10597010	—	2	2	no	—
55	C₄⋊#10597010	—	2	2	no	—
56	C₄⋊#10597010	—	2	1	no	—
57	C₄⋊#10597010	D₄	1	1	yes	characteristic
58	C₄⋊#10597010	D₄	1	1	yes	characteristic
59	C₄⋊#10597010	C₂³	1	1	yes	characteristic
60	C₄⋊#10597010	C₂³	1	1	yes	characteristic
61	C₄⋊#10597010	C₂³	1	2	yes	—
62	C₂²⋊#10597010	—	2	2	no	—
63	C₂²⋊#10597010	—	2	1	no	—
64	C₂²⋊#10597010	D₄	1	2	yes	—
65	C₄×#10597010	—	2	2	no	—
66	C₄×#10597010	—	2	2	no	—
67	C₄×#10597010	—	2	2	no	—
68	C₄×#10597010	—	2	1	no	—
69	C₄×#10597010	C₂³	1	2	yes	—
70	#14558737	C₂×C₄	1	1	yes	characteristic
71	#14558736	—	2	1	no	—
72	#14558736	C₂×C₄	1	1	yes	characteristic
73	#14558735	—	2	1	no	—
74	#14558735	C₂×C₄	1	1	yes	characteristic
75	C₂×Dic₂₉₈	—	2	1	no	—
76	C₂×Dic₂₉₈	C₂×C₄	1	1	yes	characteristic, complemented
77	Q₈×D₁₄₉	—	2	1	no	—
78	Q₈×D₁₄₉	—	2	1	no	—
79	Q₈×D₁₄₉	C₂×C₄	1	2	yes	complemented
80	Q₈×D₁₄₉	C₂³	1	2	yes	—
81	Q₈×D₁₄₉	C₂³	1	1	yes	characteristic
82	D₄×D₁₄₉	—	2	1	no	—
83	D₄×D₁₄₉	C₂³	1	1	yes	characteristic
84	C₂×C₄×D₁₄₉	—	2	1	no	—
85	C₂×C₄×D₁₄₉	—	2	1	no	—
86	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
87	Dic₅₉₆	C₂×C₄	1	2	yes	complemented
88	#14558726	C₂×C₄	1	2	yes	complemented
89	#14558726	C₂×C₄	1	2	yes	complemented
90	#14558725	C₂×C₄	1	2	yes	complemented
91	#14558724	C₂×C₄	1	2	yes	complemented
92	#14558723	C₂×C₄	1	2	yes	complemented
93	#14558720	C₂×C₄	1	1	yes	characteristic, complemented
94	#14558719	C₂×C₄	1	2	yes	complemented
95	C₈×D₁₄₉	C₂³	1	2	yes	—
96	#41×C₁₄₉	C₂×C₄	1	1	yes	characteristic
97	C₂×Q₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic, complemented
98	Q₁₆×C₁₄₉	C₂×C₄	1	2	yes	complemented
99	SD₁₆×C₁₄₉	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₄₉	C₂×C₄	1	1	yes	characteristic, complemented
101	C₂×#10597010	—	2	1	no	—
102	C₂×#10597010	—	2	1	no	—
103	C₂×#10597010	—	2	2	no	—
104	C₂×#10597010	—	8	2	no	—
105	C₂×#10597010	—	8	1	no	—
106	C₂×#10597010	C₂×D₄	1	2	yes	—
107	C₂×#10597010	C₂×D₄	1	2	yes	—
108	#11924753	—	2	1	no	—
109	#11924753	—	2	1	no	—
110	C₂²×D₁₄₉	—	2	1	no	—
111	C₂²×D₁₄₉	C₂×D₄	1	1	yes	characteristic
112	Dic₂₉₈	—	2	1	no	—
113	Dic₂₉₈	—	2	2	no	—
114	Dic₂₉₈	—	2	2	no	—
115	Dic₂₉₈	—	2	1	no	—
116	Dic₂₉₈	C₂²×C₄	1	2	yes	—
117	Dic₂₉₈	C₂²×C₄	1	1	yes	characteristic
118	D₅₉₆	—	2	1	no	—
119	D₅₉₆	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊D₁₄₉	—	2	1	no	—
121	C₂²⋊D₁₄₉	—	2	1	no	—
122	C₂²⋊D₁₄₉	—	2	1	no	—
123	C₂×Dic₁₄₉	—	2	1	no	—
124	C₂×Dic₁₄₉	—	2	1	no	—
125	C₂×Dic₁₄₉	#41	1	1	yes	characteristic
126	C₄×D₁₄₉	—	2	2	no	—
127	C₄×D₁₄₉	—	2	1	no	—
128	C₄×D₁₄₉	—	2	2	no	—
129	C₄×D₁₄₉	—	2	1	no	—
130	C₄×D₁₄₉	—	2	1	no	—
131	C₄×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₁₄₉	C₂×D₄	1	1	yes	characteristic
133	C₄×D₁₄₉	#41	1	1	yes	characteristic
134	C₄×D₁₄₉	C₂⁴	1	1	yes	characteristic
135	#11924746	C₂²×C₄	1	2	yes	—
136	Q₈×C₁₄₉	—	2	1	no	—
137	Q₈×C₁₄₉	C₂²×C₄	1	2	yes	—
138	Q₈×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
139	D₄×C₁₄₉	—	2	1	no	—
140	D₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₄₉	—	2	1	no	—
142	C₂×C₄×C₁₄₉	—	2	1	no	—
143	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
144	C₈×C₁₄₉	C₂²×C₄	1	2	yes	—
145	C₂²×C₁₄₉	—	2	1	no	—
146	C₂²×C₁₄₉	C₄×D₄	1	1	yes	characteristic
147	C₂×D₁₄₉	—	2	1	no	—
148	C₂×D₁₄₉	—	2	1	no	—
149	C₂×D₁₄₉	—	2	1	no	—
150	C₂×D₁₄₉	—	4	1	no	—
151	C₂×D₁₄₉	—	4	1	no	—
152	C₂×D₁₄₉	C₄×D₄	1	1	yes	characteristic
153	C₂×D₁₄₉	C₂²×D₄	1	1	yes	characteristic
154	#10597010	—	2	2	no	—
155	#10597010	—	2	2	no	—
156	#10597010	—	4	1	no	—
157	#10597010	—	4	2	no	—
158	#10597010	—	4	1	no	—
159	C₄×C₁₄₉	—	2	1	no	—
160	C₄×C₁₄₉	—	2	2	no	—
161	C₄×C₁₄₉	C₄×D₄	1	1	yes	characteristic
162	C₄×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
163	Dic₁₄₉	—	2	1	no	—
164	Dic₁₄₉	—	2	2	no	—
165	Dic₁₄₉	C₄×D₄	1	1	yes	characteristic
166	Dic₁₄₉	C₂×#41	1	1	yes	characteristic
167	C₂×C₁₄₉	—	2	1	no	—
168	C₂×C₁₄₉	—	4	1	no	—
169	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
170	D₁₄₉	—	2	1	no	—
171	D₁₄₉	—	4	1	no	—
172	D₁₄₉	#577	1	1	yes	characteristic
173	D₁₄₉	#578	1	1	yes	characteristic
174	C₁₄₉	#2947	1	1	yes	characteristic, complemented, Sylow
175	#2947	—	149	1	no	maximal, Sylow
176	C₂×#137	—	149	1	no	—
177	#520	—	149	1	no	—
178	#519	—	149	1	no	—
179	#441	—	149	2	no	—
180	#440	—	149	2	no	—
181	C₄×Q₁₆	—	149	2	no	—
182	C₄×SD₁₆	—	149	2	no	—
183	#428	—	149	1	no	—
184	#416	—	149	1	no	—
185	C₂×#103	—	149	1	no	—
186	#405	—	149	1	no	—
187	C₂×#41	—	149	1	no	—
188	C₂²×Q₈	—	149	1	no	—
189	#137	—	149	1	no	—
190	#137	—	149	1	no	—
191	#137	—	149	2	no	—
192	#137	—	149	2	no	—
193	#137	—	149	1	no	—
194	#137	#10597010	1	1	yes	characteristic, complemented
195	C₂×Q₁₆	—	149	2	no	—
196	C₂×SD₁₆	—	149	2	no	—
197	C₂×M₁₆	—	149	1	no	—
198	C₂²×C₈	—	149	1	no	—
199	C₄×Q₈	—	298	1	no	—
200	C₄×Q₈	—	298	1	no	—
201	C₄×Q₈	—	149	2	no	—
202	C₄×Q₈	—	149	2	no	—
203	C₄×Q₈	—	149	2	no	—
204	C₄×D₄	—	298	2	no	—
205	C₄×D₄	—	149	2	no	—
206	#117	—	298	1	no	—
207	#117	—	298	2	no	—
208	#117	—	149	1	no	—
209	#117	—	149	1	no	—
210	C₂×C₄⋊C₄	—	298	1	no	—
211	C₂×C₄⋊C₄	—	149	1	no	—
212	#107	—	149	2	no	—
213	#106	—	149	2	no	—
214	#103	—	149	2	no	—
215	#103	—	149	2	no	—
216	#103	—	149	1	no	—
217	#103	—	149	1	no	—
218	#102	—	149	1	no	—
219	#102	—	149	1	no	—
220	#97	—	149	2	no	—
221	C₄×C₈	—	149	2	no	—
222	#41	—	298	1	no	—
223	#41	—	298	1	no	—
224	#41	—	149	1	no	—
225	#41	—	149	1	no	—
226	#41	—	149	1	no	—
227	#41	C₂×#10597010	1	1	yes	characteristic
228	C₂×Q₈	—	298	1	no	—
229	C₂×Q₈	—	298	1	no	—
230	C₂×Q₈	—	298	1	no	—
231	C₂×Q₈	—	149	2	no	—
232	C₂×Q₈	—	149	2	no	—
233	C₂×Q₈	—	149	1	no	—
234	C₂×Q₈	—	149	1	no	—
235	C₂×Q₈	C₂×#10597010	1	1	yes	characteristic, complemented
236	C₂×D₄	—	298	1	no	—
237	C₂×D₄	—	149	1	no	—
238	C₂²×C₄	—	298	1	no	—
239	C₂²×C₄	—	298	1	no	—
240	C₂²×C₄	—	298	1	no	—
241	C₂²×C₄	—	298	2	no	—
242	C₂²×C₄	—	149	1	no	—
243	Q₁₆	—	149	2	no	—
244	Q₁₆	—	149	2	no	—
245	Q₁₆	—	149	2	no	—
246	Q₁₆	C₂×#10597010	1	2	yes	complemented
247	SD₁₆	—	149	2	no	—
248	SD₁₆	—	149	2	no	—
249	SD₁₆	—	149	2	no	—
250	SD₁₆	C₂×#10597010	1	2	yes	complemented
251	M₁₆	—	149	1	no	—
252	M₁₆	—	149	2	no	—
253	M₁₆	C₂×#10597010	1	1	yes	characteristic, complemented
254	C₂×C₈	—	298	1	no	—
255	C₂×C₈	—	298	1	no	—
256	C₂×C₈	—	149	1	no	—
257	C₂×C₈	—	149	1	no	—
258	C₂×C₈	—	149	2	no	—
259	C₄⋊C₄	—	298	1	no	—
260	C₄⋊C₄	—	298	2	no	—
261	C₄⋊C₄	—	298	2	no	—
262	C₄⋊C₄	—	298	2	no	—
263	C₄⋊C₄	—	149	1	no	—
264	C₄⋊C₄	—	149	1	no	—
265	C₄⋊C₄	—	149	2	no	—
266	C₄⋊C₄	—	149	1	no	—
267	C₄⋊C₄	—	149	1	no	—
268	C₂²⋊C₄	—	298	1	no	—
269	C₂²⋊C₄	—	298	2	no	—
270	C₂²⋊C₄	—	149	2	no	—
271	C₄²	—	298	2	no	—
272	C₄²	—	298	2	no	—
273	C₄²	—	298	2	no	—
274	C₄²	—	298	1	no	—
275	C₄²	—	149	2	no	—
276	C₂³	—	298	1	no	—
277	C₂³	—	149	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	298	2	no	—
280	Q₈	—	298	2	no	—
281	Q₈	—	298	1	no	—
282	Q₈	—	298	1	no	—
283	Q₈	—	149	2	no	—
284	Q₈	—	149	1	no	—
285	Q₈	C₂²×#10597010	1	2	yes	—
286	Q₈	C₂²×#10597010	1	1	yes	characteristic
287	D₄	—	2	1	no	—
288	D₄	—	298	1	no	—
289	D₄	—	298	1	no	—
290	D₄	—	298	1	no	—
291	D₄	—	298	1	no	—
292	D₄	—	149	1	no	—
293	D₄	C₂²×#10597010	1	1	yes	characteristic
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	298	1	no	—
297	C₂×C₄	—	298	2	no	—
298	C₂×C₄	—	298	2	no	—
299	C₂×C₄	—	298	1	no	—
300	C₂×C₄	—	298	1	no	—
301	C₂×C₄	—	298	1	no	—
302	C₂×C₄	—	298	1	no	—
303	C₂×C₄	—	298	1	no	—
304	C₂×C₄	—	298	1	no	—
305	C₂×C₄	—	298	2	no	—
306	C₂×C₄	—	149	2	no	—
307	C₂×C₄	—	149	2	no	—
308	C₂×C₄	—	149	1	no	—
309	C₂×C₄	—	149	1	no	—
310	C₂×C₄	—	149	1	no	—
311	C₂×C₄	—	149	1	no	—
312	C₂×C₄	—	149	1	no	—
313	C₂×C₄	—	1192	2	no	—
314	C₂×C₄	—	1192	1	no	—
315	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
316	C₈	—	298	1	no	—
317	C₈	—	298	1	no	—
318	C₈	—	149	2	no	—
319	C₈	C₂²×#10597010	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	298	1	no	—
322	C₂²	—	298	1	no	—
323	C₂²	—	298	1	no	—
324	C₂²	—	596	1	no	—
325	C₂²	—	596	1	no	—
326	C₂²	—	149	1	no	—
327	C₂²	—	149	1	no	—
328	C₂²	D₄×#10597010	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	298	2	no	—
332	C₄	—	298	2	no	—
333	C₄	—	298	1	no	—
334	C₄	—	298	2	no	—
335	C₄	—	596	2	no	—
336	C₄	—	596	1	no	—
337	C₄	—	596	1	no	—
338	C₄	—	149	1	no	—
339	C₄	—	149	1	no	—
340	C₄	D₄×#10597010	1	1	yes	characteristic
341	C₄	C₂³×#10597010	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	4	1	no	—
344	C₂	—	298	1	no	—
345	C₂	—	596	1	no	—
346	C₂	—	149	1	no	—
347	C₂	—	149	1	no	—
348	C₂	C₂×D₄×#10597010	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10597010
commutator	C₄×C₁₄₉	C₂³×C₄
Frattini	C₄	C₂³×#10597010
Fitting	#137×C₁₄₉	C₄
socle	C₂×C₁₄₉	C₂×C₄×D₄
prime radical	#41×D₁₄₉	C₂²
cosocle	C₄×D₁₄₉	C₂⁴
hypercenter	#137	#10597010
hypocenter	C₁₄₉	#2947
LCS	G ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. #23750693	C₂	semidirect
3. #23750665	C₂	semidirect
4. #23750664	C₂	semidirect
5. Q₁₆×#10597010	C₂	semidirect
6. SD₁₆×#10597010	C₂	semidirect
7. #23750636	C₂	semidirect
9. C₂×#18011974	C₂	semidirect
10. #23750591	C₂	semidirect
38. #18011944	C₄	semidirect
39. #18011943	C₄	semidirect
40. #18011942	C₄	semidirect
41. #18011941	C₄	semidirect
42. #18011940	C₄	semidirect
46. #137×C₁₄₉	C₄	semidirect
76. C₂×Dic₂₉₈	C₂×C₄	semidirect
79. Q₈×D₁₄₉	C₂×C₄	semidirect
87. Dic₅₉₆	C₂×C₄	semidirect
88. #14558726	C₂×C₄	semidirect
89. #14558726	C₂×C₄	semidirect
90. #14558725	C₂×C₄	semidirect
91. #14558724	C₂×C₄	semidirect
92. #14558723	C₂×C₄	semidirect
93. #14558720	C₂×C₄	semidirect
94. #14558719	C₂×C₄	semidirect
97. C₂×Q₈×C₁₄₉	C₂×C₄	semidirect
98. Q₁₆×C₁₄₉	C₂×C₄	semidirect
99. SD₁₆×C₁₄₉	C₂×C₄	semidirect
100. M₁₆×C₁₄₉	C₂×C₄	semidirect
174. C₁₄₉	#2947	semidirect
194. #137	#10597010	semidirect
235. C₂×Q₈	C₂×#10597010	semidirect
246. Q₁₆	C₂×#10597010	semidirect
250. SD₁₆	C₂×#10597010	semidirect
253. M₁₆	C₂×#10597010	semidirect

Maximal subgroups

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

subgroup	index
#23750694	2
#23750693	2
#23750665	2
#23750664	2
Q₁₆×#10597010	2
SD₁₆×#10597010	2
#23750636	2
#23750615	2
C₂×#18011974	2
#23750591	2
#137×D₁₄₉	2
#2947	149