Subgroups — #34047705

Order 19072.

788 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18011885	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₅₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#18011871	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄⋊Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#23749527	C₂	1	1	yes	maximal, characteristic, complemented
6	#23749523	C₂	1	1	yes	maximal, characteristic, complemented
7	#23749517	C₂	1	2	yes	maximal, complemented
8	#23749502	C₂	1	2	yes	maximal, complemented
9	#23749499	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₉₆	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₄₉	C₂²	1	2	yes	—
14	C₂³×Dic₁₄₉	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₄₉	C₂²	1	2	yes	—
16	#18011885	C₄	1	8	yes	complemented
17	C₄×D₅₉₆	—	2	4	no	—
18	#18011871	—	2	8	no	—
19	C₂×C₄⋊Dic₁₄₉	—	2	4	no	—
20	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
24	C₂×#14558715	C₂²	1	2	yes	—
25	C₂×#14558713	C₂²	1	2	yes	—
26	C₂×#14558713	C₂²	1	2	yes	—
27	C₂×#14558713	C₂²	1	2	yes	—
28	C₂×#14558713	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
30	#18011818	C₂²	1	2	yes	—
31	#18011818	C₂²	1	2	yes	—
32	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
37	#95×C₁₄₉	C₂²	1	1	yes	characteristic
38	#95×C₁₄₉	C₂²	1	1	yes	characteristic
39	C₂³×D₁₄₉	C₂³	1	2	yes	—
40	C₂×D₅₉₆	—	2	2	no	—
41	C₂×D₅₉₆	—	2	2	no	—
42	C₂×D₅₉₆	C₂×C₄	1	4	yes	complemented
43	C₂×C₂²⋊D₁₄₉	—	2	4	no	—
44	C₂×C₂²⋊D₁₄₉	—	2	4	no	—
45	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	8	yes	complemented
46	C₂²×Dic₁₄₉	—	2	4	no	—
47	C₂²×Dic₁₄₉	—	2	4	no	—
48	C₂²×Dic₁₄₉	—	4	2	no	—
49	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
50	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
51	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
52	C₂×C₄×D₁₄₉	—	2	4	no	—
53	C₂×C₄×D₁₄₉	—	2	4	no	—
54	C₂×C₄×D₁₄₉	—	4	2	no	—
55	C₄⋊Dic₁₄₉	—	2	4	no	—
56	C₄⋊Dic₁₄₉	—	2	2	no	—
57	C₄⋊Dic₁₄₉	C₂×C₄	1	4	yes	complemented
58	C₄⋊Dic₁₄₉	D₄	1	4	yes	complemented
59	#14558715	—	2	4	no	—
60	#14558713	—	2	4	no	—
61	#14558713	—	2	4	no	—
62	#14558713	—	4	2	no	—
63	#14558713	C₂×C₄	1	8	yes	complemented
64	C₄×Dic₁₄₉	—	2	4	no	—
65	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₄₉	—	2	2	no	—
67	C₂²×C₄×C₁₄₉	—	2	2	no	—
68	C₂²×C₄×C₁₄₉	C₂×C₄	1	4	yes	complemented
69	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
76	C₂²⋊C₄×C₁₄₉	D₄	1	4	yes	complemented
77	C₄²×C₁₄₉	—	2	2	no	—
78	C₄²×C₁₄₉	—	2	2	no	—
79	C₂²×D₁₄₉	—	2	2	no	—
80	C₂²×D₁₄₉	—	2	2	no	—
81	C₂²×D₁₄₉	—	2	2	no	—
82	C₂²×D₁₄₉	—	2	2	no	—
83	C₂²×D₁₄₉	—	2	2	no	—
84	C₂²×D₁₄₉	—	2	2	no	—
85	C₂²×D₁₄₉	C₂²×C₄	1	4	yes	—
86	D₅₉₆	—	2	8	no	—
87	C₂²⋊D₁₄₉	—	2	16	no	—
88	C₂×Dic₁₄₉	—	2	2	no	—
89	C₂×Dic₁₄₉	—	2	2	no	—
90	C₂×Dic₁₄₉	—	2	2	no	—
91	C₂×Dic₁₄₉	—	2	2	no	—
92	C₂×Dic₁₄₉	—	2	2	no	—
93	C₂×Dic₁₄₉	—	2	2	no	—
94	C₂×Dic₁₄₉	—	2	2	no	—
95	C₂×Dic₁₄₉	—	4	8	no	—
96	C₂×Dic₁₄₉	C₂²×C₄	1	4	yes	—
97	C₂×Dic₁₄₉	C₂×D₄	1	4	yes	—
98	C₄×D₁₄₉	—	4	8	no	—
99	C₂³×C₁₄₉	—	2	1	no	—
100	C₂³×C₁₄₉	—	2	1	no	—
101	C₂³×C₁₄₉	—	2	1	no	—
102	C₂³×C₁₄₉	—	2	1	no	—
103	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
104	C₂³×C₁₄₉	C₂×D₄	1	2	yes	—
105	C₂³×C₁₄₉	#41	1	2	yes	—
106	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
107	C₂×C₄×C₁₄₉	—	2	1	no	—
108	C₂×C₄×C₁₄₉	—	2	1	no	—
109	C₂×C₄×C₁₄₉	—	2	1	no	—
110	C₂×C₄×C₁₄₉	—	2	1	no	—
111	C₂×C₄×C₁₄₉	—	2	1	no	—
112	C₂×C₄×C₁₄₉	—	2	1	no	—
113	C₂×C₄×C₁₄₉	—	2	8	no	—
114	C₂×C₄×C₁₄₉	—	2	1	no	—
115	C₂×C₄×C₁₄₉	—	2	1	no	—
116	C₂×C₄×C₁₄₉	—	2	1	no	—
117	C₂×C₄×C₁₄₉	—	2	1	no	—
118	C₂×C₄×C₁₄₉	—	2	1	no	—
119	C₂×C₄×C₁₄₉	—	2	1	no	—
120	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
121	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
122	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₁₄₉	#41	1	2	yes	—
126	C₂²×C₁₄₉	—	2	2	no	—
127	C₂²×C₁₄₉	—	2	2	no	—
128	C₂²×C₁₄₉	—	2	2	no	—
129	C₂²×C₁₄₉	—	2	2	no	—
130	C₂²×C₁₄₉	—	2	2	no	—
131	C₂²×C₁₄₉	—	2	2	no	—
132	C₂²×C₁₄₉	C₄×D₄	1	4	yes	complemented
133	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
134	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
135	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
136	C₂²×C₁₄₉	C₂×#41	1	1	yes	characteristic
137	C₂²×C₁₄₉	C₂×#41	1	1	yes	characteristic
138	C₂²×C₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂²×C₁₄₉	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×D₁₄₉	—	2	4	no	—
141	C₂×D₁₄₉	—	2	4	no	—
142	C₂×D₁₄₉	—	2	4	no	—
143	C₂×D₁₄₉	—	4	2	no	—
144	C₂×D₁₄₉	—	4	2	no	—
145	C₂×D₁₄₉	—	4	2	no	—
146	C₂×D₁₄₉	—	4	2	no	—
147	C₄×C₁₄₉	—	2	2	no	—
148	C₄×C₁₄₉	—	2	2	no	—
149	C₄×C₁₄₉	—	2	2	no	—
150	C₄×C₁₄₉	—	4	1	no	—
151	C₄×C₁₄₉	—	4	1	no	—
152	C₄×C₁₄₉	C₄×D₄	1	4	yes	complemented
153	Dic₁₄₉	—	2	4	no	—
154	Dic₁₄₉	—	2	4	no	—
155	Dic₁₄₉	—	4	2	no	—
156	C₂×C₁₄₉	—	2	4	no	—
157	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂×C₁₄₉	#520	1	1	yes	characteristic
160	C₂×C₁₄₉	D₄²	1	1	yes	characteristic
161	C₂×C₁₄₉	#547	1	1	yes	characteristic
162	C₂×C₁₄₉	#548	1	1	yes	characteristic
163	C₂×C₁₄₉	#553	1	1	yes	characteristic
164	D₁₄₉	—	4	4	no	—
165	C₁₄₉	#2361	1	1	yes	characteristic, complemented, Sylow
166	#2361	—	149	1	no	maximal, Sylow
167	C₂×#121	—	149	1	no	—
168	C₂×C₄×D₄	—	149	2	no	—
169	C₂×C₄×D₄	—	149	1	no	—
170	C₂²×C₄⋊C₄	—	149	1	no	—
171	#390	—	149	1	no	—
172	#389	—	149	1	no	—
173	#388	—	149	2	no	—
174	#386	—	149	2	no	—
175	#385	—	149	2	no	—
176	C₄×C₄⋊C₄	—	149	1	no	—
177	C₄×C₂²⋊C₄	D₁₄₉	1	1	yes	characteristic, complemented
178	C₂²×D₄	—	149	2	no	—
179	C₂²×D₄	—	149	1	no	—
180	C₂³×C₄	—	149	2	no	—
181	C₂³×C₄	—	149	2	no	—
182	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
183	#121	—	149	8	no	—
184	C₄×D₄	—	298	4	no	—
185	C₄×D₄	—	298	8	no	—
186	C₂×C₄⋊C₄	—	298	4	no	—
187	C₂×C₄⋊C₄	—	149	2	no	—
188	C₂×C₄⋊C₄	—	149	1	no	—
189	C₂×C₄⋊C₄	—	149	2	no	—
190	C₂×C₄⋊C₄	—	149	1	no	—
191	C₂×C₄⋊C₄	—	149	1	no	—
192	C₂×C₂²⋊C₄	—	149	2	no	—
193	C₂×C₂²⋊C₄	—	149	2	no	—
194	C₂×C₂²⋊C₄	—	149	2	no	—
195	C₂×C₂²⋊C₄	—	149	2	no	—
196	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
198	C₂×C₄²	—	149	2	no	—
199	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
200	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
201	#95	—	149	2	no	—
202	#95	—	149	2	no	—
203	#95	C₂×D₁₄₉	1	1	yes	characteristic
204	#95	C₂×D₁₄₉	1	1	yes	characteristic
205	C₂⁴	—	149	2	no	—
206	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
207	C₂×D₄	—	298	4	no	—
208	C₂×D₄	—	298	4	no	—
209	C₂×D₄	—	298	2	no	—
210	C₂×D₄	—	298	2	no	—
211	C₂×D₄	—	149	8	no	—
212	C₂×D₄	—	149	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	298	4	no	—
216	C₂²×C₄	—	298	4	no	—
217	C₂²×C₄	—	298	4	no	—
218	C₂²×C₄	—	298	4	no	—
219	C₂²×C₄	—	596	2	no	—
220	C₂²×C₄	—	596	2	no	—
221	C₂²×C₄	—	149	2	no	—
222	C₂²×C₄	—	149	2	no	—
223	C₂²×C₄	—	149	2	no	—
224	C₂²×C₄	C₄×D₁₄₉	1	4	yes	complemented
225	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
228	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
231	C₄⋊C₄	—	298	4	no	—
232	C₄⋊C₄	—	298	2	no	—
233	C₄⋊C₄	—	298	4	no	—
234	C₄⋊C₄	—	149	4	no	—
235	C₄⋊C₄	—	149	4	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	298	4	no	—
238	C₂²⋊C₄	—	298	4	no	—
239	C₂²⋊C₄	—	596	2	no	—
240	C₂²⋊C₄	—	149	8	no	—
241	C₂²⋊C₄	D₅₉₆	1	4	yes	complemented
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	298	4	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	298	2	no	—
250	C₂³	—	298	2	no	—
251	C₂³	—	298	2	no	—
252	C₂³	—	298	2	no	—
253	C₂³	—	298	2	no	—
254	C₂³	—	298	2	no	—
255	C₂³	—	149	4	no	—
256	C₂³	C₂×C₄×D₁₄₉	1	2	yes	—
257	C₂³	C₂×D₅₉₆	1	2	yes	—
258	C₂³	#14558736	1	2	yes	—
259	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
260	D₄	—	298	16	no	—
261	D₄	—	298	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	298	2	no	—
276	C₂×C₄	—	298	2	no	—
277	C₂×C₄	—	298	2	no	—
278	C₂×C₄	—	298	2	no	—
279	C₂×C₄	—	298	2	no	—
280	C₂×C₄	—	298	2	no	—
281	C₂×C₄	—	298	2	no	—
282	C₂×C₄	—	596	8	no	—
283	C₂×C₄	—	596	8	no	—
284	C₂×C₄	—	149	4	no	—
285	C₂×C₄	—	149	4	no	—
286	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
287	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
288	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
289	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
290	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
291	C₂×C₄	#14558735	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	298	4	no	—
299	C₂²	—	298	4	no	—
300	C₂²	—	298	4	no	—
301	C₂²	—	596	2	no	—
302	C₂²	—	596	2	no	—
303	C₂²	—	596	2	no	—
304	C₂²	—	596	2	no	—
305	C₂²	C₄×D₅₉₆	1	4	yes	complemented
306	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
307	C₂²	C₂×D₄×D₁₄₉	1	1	yes	characteristic
308	C₂²	C₂²×D₅₉₆	1	1	yes	characteristic
309	C₂²	C₂×#14558735	1	1	yes	characteristic
310	C₂²	C₂×#14558736	1	1	yes	characteristic
311	C₂²	#18011957	1	1	yes	characteristic
312	C₂²	#18011958	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	298	4	no	—
319	C₄	—	298	4	no	—
320	C₄	—	596	2	no	—
321	C₄	#18011871	1	4	yes	complemented
322	C₂	—	2	4	no	—
323	C₂	—	596	4	no	—
324	C₂	C₂×#18011871	1	1	yes	characteristic
325	C₂	C₂×C₄×D₅₉₆	1	1	yes	characteristic
326	C₂	#23750105	1	1	yes	characteristic
327	C₂	D₄×D₅₉₆	1	1	yes	characteristic
328	C₂	#23750279	1	1	yes	characteristic
329	C₂	#23750284	1	1	yes	characteristic
330	C₂	#23750321	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₄₉
commutator	C₂²×C₁₄₉	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₁₄₉	C₂
socle, cosocle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂×#18011885	C₂
hypercenter	C₄×C₂²⋊C₄	D₁₄₉
hypocenter	C₁₄₉	#2361
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₅₉₆	C₂	semidirect
3. C₂×#18011871	C₂	semidirect
4. C₂²×C₄⋊Dic₁₄₉	C₂	semidirect
5. #23749527	C₂	semidirect
6. #23749523	C₂	semidirect
7. #23749517	C₂	semidirect
8. #23749502	C₂	semidirect
9. #23749499	C₂	semidirect
10. C₄×C₄⋊Dic₁₄₉	C₂	semidirect
11. C₄×C₂²⋊C₄×C₁₄₉	C₂	semidirect
16. #18011885	C₄	semidirect
42. C₂×D₅₉₆	C₂×C₄	semidirect
45. C₂×C₂²⋊D₁₄₉	C₂×C₄	semidirect
57. C₄⋊Dic₁₄₉	C₂×C₄	semidirect
58. C₄⋊Dic₁₄₉	D₄	semidirect
63. #14558713	C₂×C₄	semidirect
68. C₂²×C₄×C₁₄₉	C₂×C₄	semidirect
76. C₂²⋊C₄×C₁₄₉	D₄	semidirect
132. C₂²×C₁₄₉	C₄×D₄	semidirect
152. C₄×C₁₄₉	C₄×D₄	semidirect
165. C₁₄₉	#2361	semidirect
177. C₄×C₂²⋊C₄	D₁₄₉	semidirect
224. C₂²×C₄	C₄×D₁₄₉	semidirect
241. C₂²⋊C₄	D₅₉₆	semidirect
305. C₂²	C₄×D₅₉₆	semidirect
321. C₄	#18011871	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18011885	2
C₂×C₄×D₅₉₆	2
C₂×#18011871	2
C₂²×C₄⋊Dic₁₄₉	2
#23749527	2
#23749523	2
#23749517	2
#23749502	2
#23749499	2
C₄×C₄⋊Dic₁₄₉	2
C₄×C₂²⋊C₄×C₁₄₉	2
#2361	149