Subgroups — #34058573

Order 19072.

992 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750352	C₂	1	4	yes	maximal, complemented
2	#23750342	C₂	1	4	yes	maximal, complemented
3	#23750311	C₂	1	4	yes	maximal, complemented
4	#23750307	C₂	1	4	yes	maximal, complemented
5	#23750238	C₂	1	1	yes	maximal, characteristic, complemented
6	#23750237	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750236	C₂	1	1	yes	maximal, characteristic, complemented
8	#23750198	C₂	1	2	yes	maximal
9	#23750181	C₂	1	1	yes	maximal, characteristic, complemented
10	#23750179	C₂	1	2	yes	maximal, complemented
11	#23750177	C₂	1	2	yes	maximal, complemented
12	C₄⋊Q₈×D₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#23750101	C₂	1	2	yes	maximal, complemented
14	#23750093	C₂	1	1	yes	maximal, characteristic
15	#541×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14558736	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₄₉	C₂²	1	2	yes	—
18	#18011911	C₂²	1	1	yes	characteristic
19	#18011910	C₂²	1	4	yes	—
20	#18011910	C₂²	1	2	yes	—
21	#18011910	C₂²	1	4	yes	—
22	#18011909	C₂²	1	2	yes	—
23	#18011906	C₂²	1	4	yes	—
24	#18011906	C₂²	1	4	yes	—
25	#18011904	C₂²	1	4	yes	—
26	#18011903	C₂²	1	1	yes	characteristic
27	#18011902	C₂²	1	4	yes	—
28	#18011901	C₂²	1	4	yes	—
29	#18011901	C₂²	1	4	yes	—
30	#18011901	C₂²	1	4	yes	—
31	#18011901	C₂²	1	4	yes	—
32	#18011897	C₂²	1	4	yes	—
33	#18011896	C₂²	1	4	yes	—
34	#18011893	C₂²	1	4	yes	—
35	#18011891	C₂²	1	2	yes	—
36	#18011890	C₂²	1	2	yes	—
37	#18011889	C₂²	1	4	yes	—
38	#18011889	C₂²	1	4	yes	—
39	#18011888	C₂²	1	4	yes	—
40	#18011887	C₂²	1	2	yes	—
41	Q₈×Dic₁₄₉	C₂²	1	2	yes	—
42	C₄×Dic₂₉₈	C₂²	1	2	yes	—
43	C₄×Dic₂₉₈	C₂²	1	4	yes	—
44	C₄×Dic₂₉₈	C₂²	1	4	yes	—
45	#18011876	C₂²	1	2	yes	—
46	#18011876	C₂²	1	4	yes	—
47	#18011876	C₂²	1	4	yes	—
48	#18011876	C₂²	1	4	yes	—
49	#18011876	C₂²	1	2	yes	—
50	#18011873	C₂²	1	2	yes	—
51	#18011871	C₂²	1	2	yes	—
52	#18011871	C₂²	1	2	yes	—
53	#18011869	C₂²	1	2	yes	—
54	#18011869	C₂²	1	4	yes	—
55	#18011868	C₂²	1	4	yes	—
56	#18011868	C₂²	1	1	yes	characteristic
57	#18011868	C₂²	1	4	yes	—
58	#18011868	C₂²	1	1	yes	characteristic
59	#18011867	C₂²	1	4	yes	—
60	C₄⋊C₄×D₁₄₉	C₂²	1	4	yes	—
61	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
62	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
63	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
64	C₄⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
65	#126×C₁₄₉	C₂²	1	4	yes	—
66	#125×C₁₄₉	C₂²	1	1	yes	characteristic
67	#125×C₁₄₉	C₂²	1	4	yes	—
68	C₂²⋊Q₈×C₁₄₉	C₂²	1	2	yes	—
69	C₄×Q₈×C₁₄₉	C₂²	1	2	yes	—
70	#117×C₁₄₉	C₂²	1	1	yes	characteristic
71	#14558736	—	2	4	no	—
72	C₂×Dic₂₉₈	C₂³	1	2	yes	—
73	C₂×Dic₂₉₈	C₂³	1	1	yes	characteristic
74	C₂×Dic₂₉₈	C₂³	1	4	yes	—
75	C₂×Dic₂₉₈	C₂³	1	4	yes	—
76	Q₈×D₁₄₉	—	2	8	no	—
77	C₂×D₅₉₆	C₂³	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₄₉	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₄₉	C₂³	1	1	yes	characteristic
80	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
81	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
82	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
83	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
84	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
85	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
87	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
88	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
89	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
90	#14558715	C₂³	1	4	yes	—
91	#14558715	C₂³	1	4	yes	—
92	#14558715	C₂³	1	4	yes	—
93	#14558715	C₂³	1	4	yes	—
94	#14558715	C₂³	1	4	yes	—
95	#14558715	C₂³	1	4	yes	—
96	#14558715	C₂³	1	4	yes	—
97	#14558715	C₂³	1	4	yes	—
98	#14558715	C₂³	1	2	yes	—
99	#14558715	C₂³	1	2	yes	—
100	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
101	#14558713	C₂³	1	2	yes	—
102	#14558713	C₂³	1	4	yes	—
103	#14558713	C₂³	1	4	yes	—
104	#14558713	C₂³	1	2	yes	—
105	#14558713	C₂³	1	4	yes	—
106	C₄×Dic₁₄₉	C₂³	1	4	yes	—
107	C₄×Dic₁₄₉	C₂³	1	4	yes	—
108	C₄×Dic₁₄₉	C₂³	1	4	yes	—
109	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
110	C₄×Dic₁₄₉	C₂³	1	2	yes	—
111	C₄×Dic₁₄₉	C₂³	1	4	yes	—
112	C₄×Dic₁₄₉	C₂³	1	2	yes	—
113	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
114	C₄×Dic₁₄₉	C₂³	1	2	yes	—
115	C₂×Q₈×C₁₄₉	C₂³	1	2	yes	—
116	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
117	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
118	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
119	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
120	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
121	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
122	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
123	C₂²⋊C₄×C₁₄₉	C₂³	1	2	yes	—
124	C₂²⋊C₄×C₁₄₉	C₂³	1	4	yes	—
125	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
126	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
127	C₄²×C₁₄₉	C₂³	1	4	yes	—
128	C₂²×D₁₄₉	C₂⁴	1	1	yes	characteristic
129	C₂²×D₁₄₉	C₂⁴	1	1	yes	characteristic
130	Dic₂₉₈	—	2	4	no	—
131	Dic₂₉₈	—	2	8	no	—
132	Dic₂₉₈	—	2	2	no	—
133	Dic₂₉₈	—	2	8	no	—
134	D₅₉₆	—	2	2	no	—
135	C₂²⋊D₁₄₉	—	2	2	no	—
136	C₂²⋊D₁₄₉	—	2	2	no	—
137	C₂×Dic₁₄₉	—	2	4	no	—
138	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
139	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
140	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
141	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
142	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
143	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
144	C₄×D₁₄₉	—	2	8	no	—
145	C₄×D₁₄₉	—	2	2	no	—
146	C₄×D₁₄₉	—	2	4	no	—
147	C₄×D₁₄₉	—	2	4	no	—
148	C₄×D₁₄₉	#41	1	4	yes	—
149	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
150	Q₈×C₁₄₉	—	2	4	no	—
151	C₂×C₄×C₁₄₉	—	2	2	no	—
152	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
153	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
154	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
155	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
156	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
157	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
158	C₂²×C₁₄₉	—	2	1	no	—
159	C₂²×C₁₄₉	—	2	2	no	—
160	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
161	C₂×D₁₄₉	—	2	2	no	—
162	C₂×D₁₄₉	—	2	1	no	—
163	C₂×D₁₄₉	—	2	2	no	—
164	C₂×D₁₄₉	C₂×#41	1	2	yes	—
165	C₄×C₁₄₉	—	2	1	no	—
166	C₄×C₁₄₉	—	2	4	no	—
167	C₄×C₁₄₉	—	2	2	no	—
168	C₄×C₁₄₉	—	2	2	no	—
169	C₄×C₁₄₉	—	2	4	no	—
170	C₄×C₁₄₉	C₂×#41	1	2	yes	—
171	Dic₁₄₉	—	2	1	no	—
172	Dic₁₄₉	—	2	4	no	—
173	Dic₁₄₉	—	2	2	no	—
174	Dic₁₄₉	—	2	4	no	—
175	Dic₁₄₉	C₂×#41	1	2	yes	—
176	Dic₁₄₉	2₋¹⁺⁴	1	4	yes	—
177	C₂×C₁₄₉	—	4	1	no	—
178	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
179	C₂×C₁₄₉	C₂×2₋¹⁺⁴	1	2	yes	—
180	D₁₄₉	—	2	2	no	—
181	D₁₄₉	—	4	1	no	—
182	C₁₄₉	#3509	1	1	yes	characteristic, complemented, Sylow
183	#3509	—	149	1	no	maximal, Sylow
184	#557	—	149	4	no	—
185	#556	—	149	4	no	—
186	#552	—	149	4	no	—
187	#552	—	149	4	no	—
188	#541	—	149	1	no	—
189	#541	—	149	1	no	—
190	#541	—	149	1	no	—
191	#541	D₁₄₉	1	1	yes	characteristic, complemented
192	#536	—	149	2	no	—
193	#533	—	149	1	no	—
194	#533	—	149	2	no	—
195	#533	—	149	2	no	—
196	C₂×C₄⋊Q₈	—	149	1	no	—
197	#519	—	149	2	no	—
198	C₄×#41	—	149	1	no	—
199	C₂×#41	—	149	1	no	—
200	C₂²×Q₈	—	149	2	no	—
201	C₄⋊Q₈	—	149	2	no	—
202	C₄⋊Q₈	—	149	4	no	—
203	C₄⋊Q₈	—	149	2	no	—
204	C₄⋊Q₈	—	149	4	no	—
205	C₄⋊Q₈	—	149	1	no	—
206	C₄⋊Q₈	C₂×D₁₄₉	1	1	yes	characteristic
207	#126	—	149	4	no	—
208	#126	—	149	4	no	—
209	#126	—	149	4	no	—
210	#126	C₂×D₁₄₉	1	4	yes	—
211	#125	—	149	4	no	—
212	#125	—	149	4	no	—
213	#125	—	149	4	no	—
214	#125	—	149	4	no	—
215	#125	—	149	4	no	—
216	#125	—	149	1	no	—
217	#125	C₂×D₁₄₉	1	1	yes	characteristic
218	#125	C₂×D₁₄₉	1	4	yes	—
219	#124	—	149	4	no	—
220	#123	—	149	4	no	—
221	#123	—	149	4	no	—
222	C₂²⋊Q₈	—	149	2	no	—
223	C₂²⋊Q₈	—	149	2	no	—
224	C₂²⋊Q₈	—	149	2	no	—
225	C₂²⋊Q₈	—	149	4	no	—
226	C₂²⋊Q₈	—	149	4	no	—
227	C₂²⋊Q₈	—	149	4	no	—
228	C₂²⋊Q₈	C₂×D₁₄₉	1	2	yes	—
229	C₄×Q₈	—	149	2	no	—
230	C₄×Q₈	—	149	2	no	—
231	C₄×Q₈	—	149	2	no	—
232	C₄×Q₈	—	149	4	no	—
233	C₄×Q₈	—	149	4	no	—
234	C₄×Q₈	—	149	4	no	—
235	C₄×Q₈	—	149	2	no	—
236	C₄×Q₈	—	149	4	no	—
237	C₄×Q₈	—	149	4	no	—
238	C₄×Q₈	C₂×D₁₄₉	1	2	yes	—
239	C₄×D₄	—	149	2	no	—
240	C₄×D₄	—	149	2	no	—
241	C₄×D₄	—	149	2	no	—
242	#117	—	149	2	no	—
243	#117	—	149	4	no	—
244	#117	—	149	4	no	—
245	#117	—	149	1	no	—
246	#117	—	149	1	no	—
247	#117	—	149	4	no	—
248	#117	—	149	4	no	—
249	#117	C₂×D₁₄₉	1	1	yes	characteristic
250	C₂×C₄⋊C₄	—	149	4	no	—
251	C₂×C₄²	—	149	1	no	—
252	C₂×C₄²	—	149	1	no	—
253	C₂×C₄²	—	149	1	no	—
254	#41	—	298	4	no	—
255	C₂×Q₈	—	298	8	no	—
256	C₂×Q₈	—	149	4	no	—
257	C₂×Q₈	—	149	2	no	—
258	C₂×Q₈	—	149	4	no	—
259	C₂×Q₈	—	149	1	no	—
260	C₂×Q₈	C₂²×D₁₄₉	1	2	yes	—
261	C₂×D₄	—	149	1	no	—
262	C₂×D₄	—	149	1	no	—
263	C₂×D₄	—	149	1	no	—
264	C₂²×C₄	—	149	2	no	—
265	C₂²×C₄	—	149	2	no	—
266	C₂²×C₄	—	149	2	no	—
267	C₂²×C₄	—	149	1	no	—
268	C₂²×C₄	—	149	4	no	—
269	C₂²×C₄	—	149	1	no	—
270	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
271	C₄⋊C₄	—	149	4	no	—
272	C₄⋊C₄	—	149	4	no	—
273	C₄⋊C₄	—	149	4	no	—
274	C₄⋊C₄	—	149	4	no	—
275	C₄⋊C₄	—	149	2	no	—
276	C₄⋊C₄	—	149	2	no	—
277	C₄⋊C₄	—	149	2	no	—
278	C₄⋊C₄	—	149	4	no	—
279	C₄⋊C₄	—	149	4	no	—
280	C₄⋊C₄	—	149	4	no	—
281	C₄⋊C₄	—	149	4	no	—
282	C₄⋊C₄	—	149	4	no	—
283	C₄⋊C₄	—	149	4	no	—
284	C₄⋊C₄	—	149	4	no	—
285	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
286	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
287	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
288	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
289	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
290	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
291	C₂²⋊C₄	—	149	4	no	—
292	C₂²⋊C₄	—	149	2	no	—
293	C₂²⋊C₄	—	149	4	no	—
294	C₂²⋊C₄	—	149	2	no	—
295	C₂²⋊C₄	—	149	4	no	—
296	C₂²⋊C₄	—	149	4	no	—
297	C₂²⋊C₄	C₂²×D₁₄₉	1	4	yes	—
298	C₂²⋊C₄	C₂²×D₁₄₉	1	2	yes	—
299	C₄²	—	149	1	no	—
300	C₄²	—	149	1	no	—
301	C₄²	—	149	2	no	—
302	C₄²	—	149	2	no	—
303	C₄²	—	149	2	no	—
304	C₄²	—	149	4	no	—
305	C₄²	—	149	4	no	—
306	C₄²	—	149	4	no	—
307	C₄²	—	149	4	no	—
308	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
309	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
310	C₄²	C₂²×D₁₄₉	1	4	yes	—
311	C₂³	—	149	1	no	—
312	C₂³	—	149	1	no	—
313	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
314	Q₈	—	2	4	no	—
315	Q₈	—	298	8	no	—
316	Q₈	—	298	8	no	—
317	Q₈	—	298	2	no	—
318	Q₈	—	298	4	no	—
319	D₄	—	298	2	no	—
320	D₄	—	298	2	no	—
321	D₄	—	298	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	298	8	no	—
324	C₂×C₄	—	298	4	no	—
325	C₂×C₄	—	298	4	no	—
326	C₂×C₄	—	298	4	no	—
327	C₂×C₄	—	298	2	no	—
328	C₂×C₄	—	149	2	no	—
329	C₂×C₄	—	149	4	no	—
330	C₂×C₄	—	149	4	no	—
331	C₂×C₄	—	149	1	no	—
332	C₂×C₄	—	149	1	no	—
333	C₂×C₄	—	149	2	no	—
334	C₂×C₄	—	149	4	no	—
335	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
336	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
337	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
338	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
339	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
340	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	298	2	no	—
344	C₂²	—	298	1	no	—
345	C₂²	—	298	2	no	—
346	C₂²	—	149	2	no	—
347	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	4	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	4	no	—
353	C₄	—	298	1	no	—
354	C₄	—	298	4	no	—
355	C₄	—	298	2	no	—
356	C₄	—	298	4	no	—
357	C₄	—	149	2	no	—
358	C₄	—	149	4	no	—
359	C₄	#41×D₁₄₉	1	2	yes	—
360	C₂	—	4	1	no	—
361	C₂	—	298	2	no	—
362	C₂	—	596	1	no	—
363	C₂	C₂×#41×D₁₄₉	1	1	yes	characteristic
364	C₂	2₋¹⁺⁴×D₁₄₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting	#541×C₁₄₉	C₂
prime radical	C₂×#14558736	C₂²
hypercenter	#541	D₁₄₉
hypocenter	C₁₄₉	#3509
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #541

Complements

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

N	H ≅ G/N	type
1. #23750352	C₂	semidirect
2. #23750342	C₂	semidirect
3. #23750311	C₂	semidirect
4. #23750307	C₂	semidirect
5. #23750238	C₂	semidirect
6. #23750237	C₂	semidirect
7. #23750236	C₂	semidirect
9. #23750181	C₂	semidirect
10. #23750179	C₂	semidirect
11. #23750177	C₂	semidirect
12. C₄⋊Q₈×D₁₄₉	C₂	semidirect
13. #23750101	C₂	semidirect
15. #541×C₁₄₉	C₂	semidirect
182. C₁₄₉	#3509	semidirect
191. #541	D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#23750352	2
#23750342	2
#23750311	2
#23750307	2
#23750238	2
#23750237	2
#23750236	2
#23750198	2
#23750181	2
#23750179	2
#23750177	2
C₄⋊Q₈×D₁₄₉	2
#23750101	2
#23750093	2
#541×C₁₄₉	2
#3509	149