Subgroups — #137×#10597010 #34061447

Order 19072.

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×#10597010	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#10597010	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	#23750618	C₂	1	1	yes	maximal, characteristic, complemented
9	#23750616	C₂	1	1	yes	maximal, characteristic
10	M₁₆×#10597010	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₁₄₉	C₂	1	1	yes	maximal, characteristic
12	D₁₄₉⋊M₁₆	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	1	yes	characteristic
17	Q₈×#10597010	C₂²	1	2	yes	—
18	Q₈×#10597010	C₂²	1	1	yes	characteristic
19	D₄×#10597010	—	2	1	no	—
20	D₄×#10597010	—	2	1	no	—
21	D₄×#10597010	C₂²	1	1	yes	characteristic
22	D₄×#10597010	C₂²	1	1	yes	characteristic
23	#18011990	—	2	1	no	—
24	#18011990	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#10597010	—	2	1	no	—
26	C₂×C₄⋊#10597010	C₂²	1	1	yes	characteristic
27	C₂×C₄×#10597010	—	2	1	no	—
28	C₂×C₄×#10597010	—	2	1	no	—
29	C₂×C₄×#10597010	C₂²	1	1	yes	characteristic
30	#18011980	C₂²	1	2	yes	—
31	#18011979	C₂²	1	2	yes	—
32	#18011974	C₂²	1	2	yes	—
33	#18011974	C₂²	1	2	yes	—
34	#18011974	C₂²	1	2	yes	—
35	#18011973	C₂²	1	2	yes	—
36	#18011965	C₂²	1	2	yes	—
37	C₈×#10597010	C₂²	1	2	yes	—
38	#41×D₁₄₉	C₂²	1	1	yes	characteristic
39	C₂×Q₈×D₁₄₉	C₂²	1	1	yes	characteristic
40	#18011944	C₄	1	1	yes	characteristic, complemented
41	#18011943	C₄	1	2	yes	complemented
42	#18011942	C₄	1	1	yes	characteristic, complemented
43	#18011941	C₄	1	1	yes	characteristic, complemented
44	#18011940	C₄	1	2	yes	complemented
45	Q₁₆×D₁₄₉	C₂²	1	2	yes	—
46	SD₁₆×D₁₄₉	C₂²	1	2	yes	—
47	M₁₆×D₁₄₉	C₂²	1	1	yes	characteristic
48	#137×C₁₄₉	C₄	1	1	yes	characteristic, complemented
49	C₂²×#10597010	—	2	1	no	—
50	C₂²×#10597010	—	2	1	no	—
51	C₂²×#10597010	—	2	1	no	—
52	C₂²×#10597010	D₄	1	1	yes	characteristic
53	C₂²×#10597010	D₄	1	1	yes	characteristic
54	#14558746	—	2	1	no	—
55	#14558745	—	2	1	no	—
56	D₁₄₉⋊C₈	C₂³	1	2	yes	—
57	C₄⋊#10597010	—	2	2	no	—
58	C₄⋊#10597010	—	2	2	no	—
59	C₄⋊#10597010	—	2	1	no	—
60	C₄⋊#10597010	—	2	1	no	—
61	C₄⋊#10597010	—	2	1	no	—
62	C₄⋊#10597010	C₂³	1	2	yes	—
63	C₄⋊#10597010	C₂³	1	1	yes	characteristic
64	C₄⋊#10597010	C₂³	1	1	yes	characteristic
65	C₂²⋊#10597010	—	2	1	no	—
66	C₂²⋊#10597010	—	2	1	no	—
67	C₂²⋊#10597010	—	2	1	no	—
68	C₄×#10597010	—	2	1	no	—
69	C₄×#10597010	—	2	1	no	—
70	C₄×#10597010	—	2	2	no	—
71	C₄×#10597010	—	2	2	no	—
72	C₄×#10597010	—	2	1	no	—
73	C₄×#10597010	D₄	1	1	yes	characteristic
74	C₄×#10597010	D₄	1	1	yes	characteristic
75	C₄×#10597010	C₂³	1	1	yes	characteristic
76	C₄×#10597010	C₂³	1	1	yes	characteristic
77	#14558737	C₂×C₄	1	1	yes	characteristic
78	#14558736	—	2	1	no	—
79	#14558736	C₂×C₄	1	1	yes	characteristic
80	#14558735	—	2	1	no	—
81	#14558735	C₂×C₄	1	1	yes	characteristic
82	C₂×Dic₂₉₈	—	2	1	no	—
83	C₂×Dic₂₉₈	C₂×C₄	1	1	yes	characteristic, complemented
84	Q₈×D₁₄₉	—	2	1	no	—
85	Q₈×D₁₄₉	—	2	1	no	—
86	Q₈×D₁₄₉	C₂×C₄	1	2	yes	complemented
87	Q₈×D₁₄₉	C₂³	1	2	yes	—
88	Q₈×D₁₄₉	C₂³	1	1	yes	characteristic
89	D₄×D₁₄₉	—	2	1	no	—
90	D₄×D₁₄₉	C₂³	1	1	yes	characteristic
91	C₂×C₄×D₁₄₉	—	2	1	no	—
92	C₂×C₄×D₁₄₉	—	2	1	no	—
93	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
94	Dic₅₉₆	C₂×C₄	1	2	yes	complemented
95	#14558726	C₂×C₄	1	2	yes	complemented
96	#14558726	C₂×C₄	1	2	yes	complemented
97	#14558725	C₂×C₄	1	2	yes	complemented
98	#14558724	C₂×C₄	1	2	yes	complemented
99	#14558723	C₂×C₄	1	2	yes	complemented
100	#14558720	C₂×C₄	1	1	yes	characteristic, complemented
101	#14558719	C₂×C₄	1	2	yes	complemented
102	C₈×D₁₄₉	C₂³	1	2	yes	—
103	#41×C₁₄₉	C₂×C₄	1	1	yes	characteristic
104	C₂×Q₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic, complemented
105	Q₁₆×C₁₄₉	C₂×C₄	1	2	yes	complemented
106	SD₁₆×C₁₄₉	C₂×C₄	1	2	yes	complemented
107	M₁₆×C₁₄₉	C₂×C₄	1	1	yes	characteristic, complemented
108	C₂×#10597010	—	2	2	no	—
109	C₂×#10597010	—	2	1	no	—
110	C₂×#10597010	—	2	1	no	—
111	C₂×#10597010	—	2	2	no	—
112	C₂×#10597010	—	4	1	no	—
113	C₂×#10597010	—	4	2	no	—
114	C₂×#10597010	—	4	2	no	—
115	C₂×#10597010	—	4	2	no	—
116	C₂×#10597010	C₂×D₄	1	1	yes	characteristic
117	C₂×#10597010	C₂×D₄	1	1	yes	characteristic
118	C₂×#10597010	C₂×D₄	1	1	yes	characteristic
119	C₂×#10597010	C₂×D₄	1	1	yes	characteristic
120	#11924753	—	2	2	no	—
121	C₂²×D₁₄₉	—	2	1	no	—
122	C₂²×D₁₄₉	C₂×D₄	1	1	yes	characteristic
123	Dic₂₉₈	—	2	1	no	—
124	Dic₂₉₈	—	2	2	no	—
125	Dic₂₉₈	—	2	2	no	—
126	Dic₂₉₈	—	2	1	no	—
127	Dic₂₉₈	C₂²×C₄	1	2	yes	—
128	Dic₂₉₈	C₂²×C₄	1	1	yes	characteristic
129	D₅₉₆	—	2	1	no	—
130	D₅₉₆	C₂²×C₄	1	1	yes	characteristic
131	C₂²⋊D₁₄₉	—	2	1	no	—
132	C₂²⋊D₁₄₉	—	2	1	no	—
133	C₂²⋊D₁₄₉	—	2	1	no	—
134	C₂×Dic₁₄₉	—	2	1	no	—
135	C₂×Dic₁₄₉	—	2	1	no	—
136	C₂×Dic₁₄₉	#41	1	1	yes	characteristic
137	C₄×D₁₄₉	—	2	2	no	—
138	C₄×D₁₄₉	—	2	1	no	—
139	C₄×D₁₄₉	—	2	2	no	—
140	C₄×D₁₄₉	—	2	1	no	—
141	C₄×D₁₄₉	—	2	1	no	—
142	C₄×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
143	C₄×D₁₄₉	C₂×D₄	1	1	yes	characteristic
144	C₄×D₁₄₉	#41	1	1	yes	characteristic
145	C₄×D₁₄₉	C₂⁴	1	1	yes	characteristic
146	#11924746	C₂²×C₄	1	2	yes	—
147	Q₈×C₁₄₉	—	2	1	no	—
148	Q₈×C₁₄₉	C₂²×C₄	1	2	yes	—
149	Q₈×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
150	D₄×C₁₄₉	—	2	1	no	—
151	D₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₄₉	—	2	1	no	—
153	C₂×C₄×C₁₄₉	—	2	1	no	—
154	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
155	C₈×C₁₄₉	C₂²×C₄	1	2	yes	—
156	C₂²×C₁₄₉	—	2	1	no	—
157	C₂²×C₁₄₉	C₄×D₄	1	1	yes	characteristic
158	C₂×D₁₄₉	—	2	1	no	—
159	C₂×D₁₄₉	—	2	1	no	—
160	C₂×D₁₄₉	—	2	1	no	—
161	C₂×D₁₄₉	—	4	1	no	—
162	C₂×D₁₄₉	—	4	1	no	—
163	C₂×D₁₄₉	C₄×D₄	1	1	yes	characteristic
164	C₂×D₁₄₉	C₂²×D₄	1	1	yes	characteristic
165	#10597010	—	2	1	no	—
166	#10597010	—	2	1	no	—
167	#10597010	—	2	1	no	—
168	#10597010	—	4	1	no	—
169	#10597010	—	4	1	no	—
170	#10597010	—	4	2	no	—
171	#10597010	#137	1	2	yes	direct product factor
172	C₄×C₁₄₉	—	2	1	no	—
173	C₄×C₁₄₉	—	2	2	no	—
174	C₄×C₁₄₉	C₄×D₄	1	1	yes	characteristic
175	C₄×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
176	Dic₁₄₉	—	2	1	no	—
177	Dic₁₄₉	—	2	2	no	—
178	Dic₁₄₉	C₄×D₄	1	1	yes	characteristic
179	Dic₁₄₉	C₂×#41	1	1	yes	characteristic
180	C₂×C₁₄₉	—	2	1	no	—
181	C₂×C₁₄₉	—	4	1	no	—
182	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
183	D₁₄₉	—	2	1	no	—
184	D₁₄₉	—	4	1	no	—
185	D₁₄₉	C₂×#137	1	1	yes	characteristic
186	D₁₄₉	#575	1	1	yes	characteristic
187	C₁₄₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
188	C₄×#137	—	149	1	no	maximal, Sylow
189	C₂×#137	—	149	1	no	—
190	C₄×#41	—	149	1	no	—
191	C₂×C₄×Q₈	—	149	1	no	—
192	#441	—	149	2	no	—
193	#440	—	149	2	no	—
194	C₄×Q₁₆	—	149	2	no	—
195	C₄×SD₁₆	—	149	2	no	—
196	#428	—	149	1	no	—
197	#419	—	149	1	no	—
198	#417	—	149	1	no	—
199	C₄×M₁₆	—	149	1	no	—
200	C₂×#41	—	149	1	no	—
201	C₂²×Q₈	—	149	1	no	—
202	#137	—	149	1	no	—
203	#137	—	149	1	no	—
204	#137	—	149	2	no	—
205	#137	—	149	2	no	—
206	#137	—	149	1	no	—
207	#137	#10597010	1	1	yes	characteristic, direct product factor
208	C₂×Q₁₆	—	149	2	no	—
209	C₂×SD₁₆	—	149	2	no	—
210	C₂×M₁₆	—	149	1	no	—
211	C₂×M₁₆	—	149	1	no	—
212	C₄×Q₈	—	298	1	no	—
213	C₄×Q₈	—	298	1	no	—
214	C₄×Q₈	—	149	2	no	—
215	C₄×Q₈	—	149	2	no	—
216	C₄×Q₈	—	149	1	no	—
217	C₄×Q₈	—	149	1	no	—
218	C₄×D₄	—	298	1	no	—
219	C₄×D₄	—	298	1	no	—
220	C₄×D₄	—	149	1	no	—
221	C₄×D₄	—	149	1	no	—
222	#117	—	298	1	no	—
223	#117	—	149	1	no	—
224	C₂×C₄⋊C₄	—	298	1	no	—
225	C₂×C₄⋊C₄	—	149	1	no	—
226	C₂×C₄²	—	298	1	no	—
227	C₂×C₄²	—	298	1	no	—
228	C₂×C₄²	—	149	1	no	—
229	#107	—	149	2	no	—
230	#106	—	149	2	no	—
231	#103	—	149	2	no	—
232	#103	—	149	2	no	—
233	#103	—	149	2	no	—
234	#102	—	149	2	no	—
235	#97	—	149	2	no	—
236	C₄×C₈	—	149	2	no	—
237	#41	—	298	1	no	—
238	#41	—	298	1	no	—
239	#41	—	149	1	no	—
240	#41	—	149	1	no	—
241	#41	—	149	1	no	—
242	#41	C₂×#10597010	1	1	yes	characteristic
243	C₂×Q₈	—	298	1	no	—
244	C₂×Q₈	—	298	1	no	—
245	C₂×Q₈	—	298	1	no	—
246	C₂×Q₈	—	149	2	no	—
247	C₂×Q₈	—	149	2	no	—
248	C₂×Q₈	—	149	1	no	—
249	C₂×Q₈	—	149	1	no	—
250	C₂×Q₈	C₂×#10597010	1	1	yes	characteristic, complemented
251	C₂×D₄	—	298	1	no	—
252	C₂×D₄	—	149	1	no	—
253	C₂²×C₄	—	298	1	no	—
254	C₂²×C₄	—	298	1	no	—
255	C₂²×C₄	—	298	1	no	—
256	C₂²×C₄	—	298	1	no	—
257	C₂²×C₄	—	298	1	no	—
258	C₂²×C₄	—	149	1	no	—
259	C₂²×C₄	—	149	1	no	—
260	C₂²×C₄	—	149	1	no	—
261	Q₁₆	—	149	2	no	—
262	Q₁₆	—	149	2	no	—
263	Q₁₆	—	149	2	no	—
264	Q₁₆	C₂×#10597010	1	2	yes	complemented
265	SD₁₆	—	149	2	no	—
266	SD₁₆	—	149	2	no	—
267	SD₁₆	—	149	2	no	—
268	SD₁₆	C₂×#10597010	1	2	yes	complemented
269	M₁₆	—	298	1	no	—
270	M₁₆	—	298	1	no	—
271	M₁₆	—	149	1	no	—
272	M₁₆	—	149	2	no	—
273	M₁₆	C₂×#10597010	1	1	yes	characteristic, complemented
274	C₂×C₈	—	149	2	no	—
275	C₂×C₈	—	149	2	no	—
276	C₄⋊C₄	—	298	2	no	—
277	C₄⋊C₄	—	298	2	no	—
278	C₄⋊C₄	—	298	1	no	—
279	C₄⋊C₄	—	298	1	no	—
280	C₄⋊C₄	—	298	1	no	—
281	C₄⋊C₄	—	149	2	no	—
282	C₄⋊C₄	—	149	1	no	—
283	C₄⋊C₄	—	149	1	no	—
284	C₂²⋊C₄	—	298	1	no	—
285	C₂²⋊C₄	—	298	1	no	—
286	C₂²⋊C₄	—	298	1	no	—
287	C₄²	—	298	1	no	—
288	C₄²	—	298	1	no	—
289	C₄²	—	298	1	no	—
290	C₄²	—	298	2	no	—
291	C₄²	—	298	2	no	—
292	C₄²	—	149	1	no	—
293	C₄²	—	149	1	no	—
294	C₄²	—	149	1	no	—
295	C₄²	—	149	1	no	—
296	C₂³	—	298	1	no	—
297	C₂³	—	149	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	298	2	no	—
300	Q₈	—	298	2	no	—
301	Q₈	—	298	1	no	—
302	Q₈	—	298	1	no	—
303	Q₈	—	149	2	no	—
304	Q₈	—	149	1	no	—
305	Q₈	C₂²×#10597010	1	2	yes	—
306	Q₈	C₂²×#10597010	1	1	yes	characteristic
307	D₄	—	2	1	no	—
308	D₄	—	298	1	no	—
309	D₄	—	298	1	no	—
310	D₄	—	298	1	no	—
311	D₄	—	298	1	no	—
312	D₄	—	149	1	no	—
313	D₄	C₂²×#10597010	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	298	1	no	—
317	C₂×C₄	—	298	2	no	—
318	C₂×C₄	—	298	2	no	—
319	C₂×C₄	—	298	1	no	—
320	C₂×C₄	—	298	1	no	—
321	C₂×C₄	—	298	1	no	—
322	C₂×C₄	—	298	2	no	—
323	C₂×C₄	—	298	1	no	—
324	C₂×C₄	—	298	1	no	—
325	C₂×C₄	—	298	1	no	—
326	C₂×C₄	—	298	2	no	—
327	C₂×C₄	—	596	1	no	—
328	C₂×C₄	—	596	2	no	—
329	C₂×C₄	—	596	2	no	—
330	C₂×C₄	—	596	2	no	—
331	C₂×C₄	—	149	1	no	—
332	C₂×C₄	—	149	1	no	—
333	C₂×C₄	—	149	1	no	—
334	C₂×C₄	—	149	1	no	—
335	C₂×C₄	—	149	1	no	—
336	C₂×C₄	—	149	1	no	—
337	C₂×C₄	—	149	1	no	—
338	C₂×C₄	—	149	1	no	—
339	C₂×C₄	—	149	1	no	—
340	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
341	C₈	—	298	2	no	—
342	C₈	—	149	2	no	—
343	C₈	C₂²×#10597010	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	298	1	no	—
346	C₂²	—	298	1	no	—
347	C₂²	—	298	1	no	—
348	C₂²	—	596	1	no	—
349	C₂²	—	596	1	no	—
350	C₂²	—	149	1	no	—
351	C₂²	—	149	1	no	—
352	C₂²	D₄×#10597010	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	298	1	no	—
356	C₄	—	298	1	no	—
357	C₄	—	298	1	no	—
358	C₄	—	298	1	no	—
359	C₄	—	298	2	no	—
360	C₄	—	596	1	no	—
361	C₄	—	596	2	no	—
362	C₄	—	596	1	no	—
363	C₄	—	149	2	no	—
364	C₄	—	149	1	no	—
365	C₄	—	149	1	no	—
366	C₄	D₄×#10597010	1	1	yes	characteristic
367	C₄	C₂³×#10597010	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	4	1	no	—
370	C₂	—	298	1	no	—
371	C₂	—	596	1	no	—
372	C₂	—	149	1	no	—
373	C₂	—	149	1	no	—
374	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	C₄×#137
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. C₂×Q₈×#10597010	C₂	semidirect
3. #23750665	C₂	semidirect
4. #23750664	C₂	semidirect
5. Q₁₆×#10597010	C₂	semidirect
6. SD₁₆×#10597010	C₂	semidirect
7. #23750636	C₂	semidirect
8. #23750618	C₂	semidirect
10. M₁₆×#10597010	C₂	semidirect
40. #18011944	C₄	semidirect
41. #18011943	C₄	semidirect
42. #18011942	C₄	semidirect
43. #18011941	C₄	semidirect
44. #18011940	C₄	semidirect
48. #137×C₁₄₉	C₄	semidirect
83. C₂×Dic₂₉₈	C₂×C₄	semidirect
86. Q₈×D₁₄₉	C₂×C₄	semidirect
94. Dic₅₉₆	C₂×C₄	semidirect
95. #14558726	C₂×C₄	semidirect
96. #14558726	C₂×C₄	semidirect
97. #14558725	C₂×C₄	semidirect
98. #14558724	C₂×C₄	semidirect
99. #14558723	C₂×C₄	semidirect
100. #14558720	C₂×C₄	semidirect
101. #14558719	C₂×C₄	semidirect
104. C₂×Q₈×C₁₄₉	C₂×C₄	semidirect
105. Q₁₆×C₁₄₉	C₂×C₄	semidirect
106. SD₁₆×C₁₄₉	C₂×C₄	semidirect
107. M₁₆×C₁₄₉	C₂×C₄	semidirect
171. #10597010	#137	direct (N × H)
187. C₁₄₉	C₄×#137	semidirect
207. #137	#10597010	direct (N × H)
250. C₂×Q₈	C₂×#10597010	semidirect
264. Q₁₆	C₂×#10597010	semidirect
268. SD₁₆	C₂×#10597010	semidirect
273. 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
#41×#10597010	2
C₂×Q₈×#10597010	2
#23750665	2
#23750664	2
Q₁₆×#10597010	2
SD₁₆×#10597010	2
#23750636	2
#23750618	2
#23750616	2
M₁₆×#10597010	2
#137×D₁₄₉	2
C₄×#137	149