Subgroups — C₂×#23750288 #34058162

Order 19072.

1568 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₉₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Q₈×D₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750288	C₂	1	16	yes	maximal, direct product factor
4	C₂×#18011910	C₂	1	2	yes	maximal, complemented
5	C₂×#18011909	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#18011891	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#18011889	C₂	1	2	yes	maximal, complemented
8	C₂×#18011888	C₂	1	2	yes	maximal, complemented
9	C₂×#18011876	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₂²⋊D₁₄₉	C₂	1	1	yes	maximal, characteristic
11	C₂×#18011871	C₂	1	2	yes	maximal
12	C₂×C₂²⋊Q₈×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₉₈	—	2	8	no	—
14	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
16	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
17	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₉₈	C₂²	1	8	yes	complemented
19	C₂×Q₈×D₁₄₉	—	2	8	no	—
20	C₂×Q₈×D₁₄₉	C₂²	1	8	yes	complemented
21	C₂²×C₂²⋊D₁₄₉	C₂²	1	1	yes	characteristic
22	C₂³×Dic₁₄₉	C₂²	1	2	yes	—
23	C₂²×C₄×D₁₄₉	C₂²	1	2	yes	—
24	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
25	#18011910	C₂²	1	16	yes	complemented
26	#18011909	C₂²	1	8	yes	complemented
27	#18011891	C₂²	1	8	yes	complemented
28	#18011889	C₂²	1	16	yes	complemented
29	#18011888	C₂²	1	16	yes	complemented
30	#18011876	C₂²	1	8	yes	complemented
31	C₄×C₂²⋊D₁₄₉	C₂²	1	8	yes	—
32	#18011871	C₂²	1	16	yes	—
33	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
34	C₂×#14558715	C₂²	1	2	yes	—
35	C₂×#14558715	C₂²	1	2	yes	—
36	C₂×#14558715	C₂²	1	1	yes	characteristic
37	C₂×#14558715	C₂²	1	2	yes	—
38	C₂×C₂²⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
39	C₂×#14558713	C₂²	1	2	yes	—
40	C₂×#14558713	C₂²	1	1	yes	characteristic
41	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
42	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
43	C₂²×Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
44	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
45	C₂²⋊Q₈×C₁₄₉	C₂²	1	8	yes	complemented
46	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	—
49	C₂³×D₁₄₉	C₂³	1	1	yes	characteristic
50	C₂×Dic₂₉₈	—	2	32	no	—
51	C₂×Dic₂₉₈	—	2	8	no	—
52	C₂×Dic₂₉₈	—	2	8	no	—
53	C₂×Dic₂₉₈	—	2	4	no	—
54	C₂×Dic₂₉₈	D₄	1	8	yes	complemented
55	C₂×Dic₂₉₈	C₂³	1	4	yes	—
56	C₂×Dic₂₉₈	C₂³	1	8	yes	—
57	C₂×Dic₂₉₈	C₂³	1	8	yes	—
58	C₂×Dic₂₉₈	C₂³	1	4	yes	—
59	Q₈×D₁₄₉	—	2	32	no	—
60	C₂×C₂²⋊D₁₄₉	Q₈	1	8	yes	complemented
61	C₂×C₂²⋊D₁₄₉	C₂³	1	4	yes	—
62	C₂²×Dic₁₄₉	—	2	8	no	—
63	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
65	C₂²×Dic₁₄₉	C₂³	1	8	yes	—
66	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
67	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
68	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₄₉	—	2	8	no	—
70	C₂×C₄×D₁₄₉	—	2	4	no	—
71	C₂×C₄×D₁₄₉	C₂³	1	8	yes	—
72	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
73	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
74	#14558715	C₂³	1	8	yes	—
75	#14558715	C₂³	1	8	yes	—
76	#14558715	C₂³	1	4	yes	—
77	#14558715	C₂³	1	8	yes	—
78	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
79	#14558713	C₂³	1	8	yes	—
80	#14558713	C₂³	1	4	yes	—
81	C₄×Dic₁₄₉	C₂³	1	4	yes	—
82	C₄×Dic₁₄₉	C₂³	1	8	yes	—
83	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
84	C₂×Q₈×C₁₄₉	—	2	4	no	—
85	C₂×Q₈×C₁₄₉	C₂³	1	4	yes	—
86	C₂²×C₄×C₁₄₉	—	2	4	no	—
87	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
88	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
90	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
91	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₁₄₉	C₂³	1	8	yes	—
93	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
94	C₂²⋊C₄×C₁₄₉	C₂³	1	8	yes	—
95	C₂²×D₁₄₉	—	2	2	no	—
96	C₂²×D₁₄₉	—	2	2	no	—
97	C₂²×D₁₄₉	C₂×Q₈	1	4	yes	complemented
98	C₂²×D₁₄₉	C₂⁴	1	2	yes	—
99	Dic₂₉₈	—	2	8	no	—
100	Dic₂₉₈	—	2	16	no	—
101	Dic₂₉₈	—	2	16	no	—
102	Dic₂₉₈	C₂×D₄	1	16	yes	complemented
103	C₂²⋊D₁₄₉	C₂×Q₈	1	16	yes	complemented
104	C₂×Dic₁₄₉	—	2	2	no	—
105	C₂×Dic₁₄₉	—	2	4	no	—
106	C₂×Dic₁₄₉	—	2	2	no	—
107	C₂×Dic₁₄₉	—	2	16	no	—
108	C₂×Dic₁₄₉	C₂×D₄	1	8	yes	—
109	C₂×Dic₁₄₉	C₂×Q₈	1	4	yes	complemented
110	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
111	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
112	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
113	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
114	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
115	C₄×D₁₄₉	—	2	8	no	—
116	C₄×D₁₄₉	—	2	16	no	—
117	C₂³×C₁₄₉	—	2	2	no	—
118	C₂³×C₁₄₉	—	2	2	no	—
119	C₂³×C₁₄₉	C₂×Q₈	1	4	yes	complemented
120	C₂³×C₁₄₉	C₂⁴	1	2	yes	—
121	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
122	Q₈×C₁₄₉	—	2	8	no	—
123	C₂×C₄×C₁₄₉	—	2	4	no	—
124	C₂×C₄×C₁₄₉	—	2	2	no	—
125	C₂×C₄×C₁₄₉	—	2	4	no	—
126	C₂×C₄×C₁₄₉	—	2	8	no	—
127	C₂×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
128	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
129	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
130	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
131	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
132	C₂²×C₁₄₉	—	2	8	no	—
133	C₂²×C₁₄₉	—	2	2	no	—
134	C₂²×C₁₄₉	—	2	2	no	—
135	C₂²×C₁₄₉	C₂²×D₄	1	2	yes	—
136	C₂²×C₁₄₉	C₂²×Q₈	1	2	yes	—
137	C₂²×C₁₄₉	C₂²×Q₈	1	4	yes	complemented
138	C₂²×C₁₄₉	2₋¹⁺⁴	1	2	yes	—
139	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
140	C₂×D₁₄₉	—	2	8	no	—
141	C₂×D₁₄₉	—	2	2	no	—
142	C₂×D₁₄₉	—	2	2	no	—
143	C₂×D₁₄₉	C₂²×Q₈	1	4	yes	complemented
144	C₄×C₁₄₉	—	2	2	no	—
145	C₄×C₁₄₉	—	2	4	no	—
146	C₄×C₁₄₉	—	2	4	no	—
147	C₄×C₁₄₉	C₂²×D₄	1	4	yes	—
148	Dic₁₄₉	—	2	2	no	—
149	Dic₁₄₉	—	2	4	no	—
150	Dic₁₄₉	—	2	2	no	—
151	Dic₁₄₉	C₂²×D₄	1	8	yes	—
152	Dic₁₄₉	C₂²×Q₈	1	4	yes	complemented
153	C₂×C₁₄₉	—	2	4	no	—
154	C₂×C₁₄₉	D₄×Q₈	1	4	yes	complemented
155	C₂×C₁₄₉	C₂³×D₄	1	1	yes	characteristic
156	C₂×C₁₄₉	C₂³×Q₈	1	1	yes	characteristic
157	C₂×C₁₄₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
158	D₁₄₉	—	2	4	no	—
159	C₁₄₉	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
160	C₂×D₄×Q₈	—	149	1	no	maximal, Sylow
161	C₂³×Q₈	—	149	1	no	—
162	C₂³×Q₈	—	149	1	no	—
163	D₄×Q₈	—	149	16	no	—
164	C₂×C₄⋊Q₈	—	149	1	no	—
165	C₂×C₄⋊Q₈	—	149	2	no	—
166	C₂×C₂²⋊Q₈	—	149	2	no	—
167	C₂×C₂²⋊Q₈	—	149	1	no	—
168	C₂×C₂²⋊Q₈	—	149	2	no	—
169	C₂×C₂²⋊Q₈	D₁₄₉	1	1	yes	characteristic, complemented
170	C₂×C₄×Q₈	—	149	1	no	—
171	C₂×C₄×D₄	—	149	1	no	—
172	C₂×C₄×D₄	—	149	2	no	—
173	C₂²×Q₈	—	298	8	no	—
174	C₂²×Q₈	—	298	8	no	—
175	C₂²×Q₈	—	149	8	no	—
176	C₂²×Q₈	—	149	1	no	—
177	C₂²×Q₈	—	149	2	no	—
178	C₂²×Q₈	—	149	2	no	—
179	C₂²×Q₈	—	149	8	no	—
180	C₂²×Q₈	—	149	1	no	—
181	C₂²×Q₈	C₂×D₁₄₉	1	1	yes	characteristic
182	C₂²×D₄	—	149	1	no	—
183	C₂³×C₄	—	149	2	no	—
184	C₂³×C₄	—	149	2	no	—
185	C₂³×C₄	—	149	1	no	—
186	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
187	C₄⋊Q₈	—	149	8	no	—
188	C₄⋊Q₈	—	149	16	no	—
189	C₂²⋊Q₈	—	149	16	no	—
190	C₂²⋊Q₈	—	149	8	no	—
191	C₂²⋊Q₈	—	149	16	no	—
192	C₂²⋊Q₈	C₂×D₁₄₉	1	8	yes	complemented
193	C₄×Q₈	—	149	8	no	—
194	C₄×D₄	—	149	8	no	—
195	C₄×D₄	—	149	16	no	—
196	C₂×C₄⋊C₄	—	149	1	no	—
197	C₂×C₄⋊C₄	—	149	2	no	—
198	C₂×C₄⋊C₄	—	149	2	no	—
199	C₂×C₄⋊C₄	—	149	2	no	—
200	C₂×C₄⋊C₄	—	149	2	no	—
201	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
203	C₂×C₂²⋊C₄	—	149	1	no	—
204	C₂×C₂²⋊C₄	—	149	1	no	—
205	C₂×C₂²⋊C₄	—	149	2	no	—
206	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	2	yes	—
207	C₂×C₄²	—	149	1	no	—
208	C₂×C₄²	—	149	2	no	—
209	C₂⁴	—	149	1	no	—
210	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	298	32	no	—
213	C₂×Q₈	—	298	8	no	—
214	C₂×Q₈	—	298	8	no	—
215	C₂×Q₈	—	298	32	no	—
216	C₂×Q₈	—	298	4	no	—
217	C₂×Q₈	—	149	8	no	—
218	C₂×Q₈	—	149	4	no	—
219	C₂×Q₈	—	149	8	no	—
220	C₂×Q₈	—	149	8	no	—
221	C₂×Q₈	—	149	4	no	—
222	C₂×Q₈	C₂²×D₁₄₉	1	4	yes	—
223	C₂×D₄	—	149	8	no	—
224	C₂×D₄	—	149	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	298	8	no	—
227	C₂²×C₄	—	298	8	no	—
228	C₂²×C₄	—	298	4	no	—
229	C₂²×C₄	—	149	8	no	—
230	C₂²×C₄	—	149	1	no	—
231	C₂²×C₄	—	149	1	no	—
232	C₂²×C₄	—	149	2	no	—
233	C₂²×C₄	—	149	8	no	—
234	C₂²×C₄	—	149	4	no	—
235	C₂²×C₄	—	149	1	no	—
236	C₂²×C₄	—	149	2	no	—
237	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
238	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
239	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
241	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
242	C₄⋊C₄	—	149	4	no	—
243	C₄⋊C₄	—	149	8	no	—
244	C₄⋊C₄	—	149	8	no	—
245	C₄⋊C₄	—	149	8	no	—
246	C₄⋊C₄	—	149	8	no	—
247	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
248	C₄⋊C₄	C₂²×D₁₄₉	1	8	yes	—
249	C₂²⋊C₄	—	149	4	no	—
250	C₂²⋊C₄	—	149	4	no	—
251	C₂²⋊C₄	—	149	8	no	—
252	C₂²⋊C₄	C₂²×D₁₄₉	1	8	yes	—
253	C₄²	—	149	4	no	—
254	C₄²	—	149	8	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	298	2	no	—
258	C₂³	—	298	2	no	—
259	C₂³	—	149	4	no	—
260	C₂³	—	149	2	no	—
261	C₂³	Q₈×D₁₄₉	1	4	yes	complemented
262	C₂³	C₂³×D₁₄₉	1	2	yes	—
263	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
264	Q₈	—	2	8	no	—
265	Q₈	—	298	16	no	—
266	Q₈	—	298	16	no	—
267	Q₈	—	298	8	no	—
268	Q₈	—	149	16	no	—
269	D₄	—	149	16	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	298	16	no	—
275	C₂×C₄	—	298	2	no	—
276	C₂×C₄	—	298	4	no	—
277	C₂×C₄	—	298	2	no	—
278	C₂×C₄	—	298	16	no	—
279	C₂×C₄	—	298	8	no	—
280	C₂×C₄	—	149	2	no	—
281	C₂×C₄	—	149	2	no	—
282	C₂×C₄	—	149	4	no	—
283	C₂×C₄	—	149	8	no	—
284	C₂×C₄	—	149	4	no	—
285	C₂×C₄	—	149	2	no	—
286	C₂×C₄	—	149	4	no	—
287	C₂×C₄	D₄×D₁₄₉	1	4	yes	—
288	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
289	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
290	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
291	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	8	no	—
295	C₂²	—	298	2	no	—
296	C₂²	—	298	2	no	—
297	C₂²	—	298	8	no	—
298	C₂²	—	149	4	no	—
299	C₂²	C₂×D₄×D₁₄₉	1	2	yes	—
300	C₂²	C₂×Q₈×D₁₄₉	1	2	yes	—
301	C₂²	C₂×Q₈×D₁₄₉	1	4	yes	complemented
302	C₂²	#18011958	1	2	yes	—
303	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	4	no	—
307	C₄	—	298	4	no	—
308	C₄	—	298	2	no	—
309	C₄	—	298	2	no	—
310	C₄	—	149	8	no	—
311	C₄	—	149	4	no	—
312	C₄	C₂×D₄×D₁₄₉	1	4	yes	—
313	C₂	—	2	4	no	—
314	C₂	—	298	4	no	—
315	C₂	#23750288	1	4	yes	direct product factor
316	C₂	C₂²×D₄×D₁₄₉	1	1	yes	characteristic
317	C₂	C₂²×Q₈×D₁₄₉	1	1	yes	characteristic
318	C₂	C₂×#18011958	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₄₉
commutator, cosocle	C₂²×C₁₄₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₄₉
Fitting	C₂×C₂²⋊Q₈×C₁₄₉	C₂
socle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂²×C₂²⋊D₁₄₉	C₂²
hypercenter	C₂×C₂²⋊Q₈	D₁₄₉
hypocenter	C₁₄₉	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₉₈	C₂	semidirect
2. C₂²×Q₈×D₁₄₉	C₂	semidirect
3. #23750288	C₂	direct (N × H)
4. C₂×#18011910	C₂	semidirect
5. C₂×#18011909	C₂	semidirect
6. C₂×#18011891	C₂	semidirect
7. C₂×#18011889	C₂	semidirect
8. C₂×#18011888	C₂	semidirect
9. C₂×#18011876	C₂	semidirect
12. C₂×C₂²⋊Q₈×C₁₄₉	C₂	semidirect
18. C₂²×Dic₂₉₈	C₂²	semidirect
20. C₂×Q₈×D₁₄₉	C₂²	semidirect
25. #18011910	C₂²	semidirect
26. #18011909	C₂²	semidirect
27. #18011891	C₂²	semidirect
28. #18011889	C₂²	semidirect
29. #18011888	C₂²	semidirect
30. #18011876	C₂²	semidirect
45. C₂²⋊Q₈×C₁₄₉	C₂²	semidirect
54. C₂×Dic₂₉₈	D₄	semidirect
60. C₂×C₂²⋊D₁₄₉	Q₈	semidirect
97. C₂²×D₁₄₉	C₂×Q₈	semidirect
102. Dic₂₉₈	C₂×D₄	semidirect
103. C₂²⋊D₁₄₉	C₂×Q₈	semidirect
109. C₂×Dic₁₄₉	C₂×Q₈	semidirect
119. C₂³×C₁₄₉	C₂×Q₈	semidirect
137. C₂²×C₁₄₉	C₂²×Q₈	semidirect
143. C₂×D₁₄₉	C₂²×Q₈	semidirect
152. Dic₁₄₉	C₂²×Q₈	semidirect
154. C₂×C₁₄₉	D₄×Q₈	semidirect
159. C₁₄₉	C₂×D₄×Q₈	semidirect
169. C₂×C₂²⋊Q₈	D₁₄₉	semidirect
192. C₂²⋊Q₈	C₂×D₁₄₉	semidirect
261. C₂³	Q₈×D₁₄₉	semidirect
301. C₂²	C₂×Q₈×D₁₄₉	semidirect
315. C₂	#23750288	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₉₈	2
C₂²×Q₈×D₁₄₉	2
#23750288	2
C₂×#18011910	2
C₂×#18011909	2
C₂×#18011891	2
C₂×#18011889	2
C₂×#18011888	2
C₂×#18011876	2
C₂×C₄×C₂²⋊D₁₄₉	2
C₂×#18011871	2
C₂×C₂²⋊Q₈×C₁₄₉	2
C₂×D₄×Q₈	149

Subgroups — C2×#23750288 #34058162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23750288 #34058162