Subgroups — #34058492

Order 19072.

1100 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750336	C₂	1	2	yes	maximal, complemented
2	#23750335	C₂	1	2	yes	maximal, complemented
3	#23750335	C₂	1	4	yes	maximal, complemented
4	#23750331	C₂	1	4	yes	maximal, complemented
5	Q₈×D₅₉₆	C₂	1	2	yes	maximal, complemented
6	#23750290	C₂	1	2	yes	maximal, complemented
7	#23750253	C₂	1	2	yes	maximal, complemented
8	#23750250	C₂	1	4	yes	maximal, complemented
9	#23750182	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊Q₈×D₁₄₉	C₂	1	4	yes	maximal, complemented
11	#23750099	C₂	1	2	yes	maximal
12	C₂×C₄×D₅₉₆	C₂	1	1	yes	maximal, characteristic
13	#533×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₁₄₉	C₂²	1	4	yes	complemented
15	C₂²×D₅₉₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₄₉	C₂²	1	2	yes	—
17	#18011911	C₂²	1	1	yes	characteristic, complemented
18	#18011911	C₂²	1	1	yes	characteristic, complemented
19	#18011911	C₂²	1	4	yes	complemented
20	#18011910	C₂²	1	4	yes	complemented
21	#18011903	C₂²	1	2	yes	complemented
22	#18011903	C₂²	1	4	yes	complemented
23	#18011902	C₂²	1	4	yes	complemented
24	#18011891	C₂²	1	8	yes	complemented
25	#18011891	C₂²	1	4	yes	complemented
26	#18011890	C₂²	1	2	yes	—
27	#18011890	C₂²	1	2	yes	—
28	#18011889	C₂²	1	8	yes	complemented
29	#18011889	C₂²	1	8	yes	complemented
30	#18011889	C₂²	1	8	yes	complemented
31	#18011888	C₂²	1	8	yes	—
32	#18011887	C₂²	1	8	yes	complemented
33	#18011887	C₂²	1	4	yes	complemented
34	C₄×Dic₂₉₈	C₂²	1	2	yes	complemented
35	C₄×Dic₂₉₈	C₂²	1	2	yes	complemented
36	C₄×D₅₉₆	C₂²	1	2	yes	—
37	C₄×D₅₉₆	C₂²	1	2	yes	—
38	C₄×D₅₉₆	C₂²	1	4	yes	—
39	C₄×D₅₉₆	C₂²	1	2	yes	—
40	C₄×D₅₉₆	C₂²	1	2	yes	—
41	C₄×D₅₉₆	C₂²	1	4	yes	—
42	#18011873	C₂²	1	4	yes	—
43	#18011873	C₂²	1	4	yes	—
44	#18011870	C₂²	1	2	yes	—
45	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
46	C₄⋊C₄×D₁₄₉	C₂²	1	8	yes	complemented
47	C₄⋊C₄×D₁₄₉	C₂²	1	4	yes	complemented
48	C₂×#14558713	C₂²	1	2	yes	—
49	C₂²⋊C₄×D₁₄₉	C₂²	1	8	yes	—
50	C₄⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic, complemented
51	C₄⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic, complemented
52	#125×C₁₄₉	C₂²	1	2	yes	complemented
53	C₂²⋊Q₈×C₁₄₉	C₂²	1	2	yes	—
54	C₂²⋊Q₈×C₁₄₉	C₂²	1	2	yes	—
55	C₄×Q₈×C₁₄₉	C₂²	1	2	yes	complemented
56	C₄×Q₈×C₁₄₉	C₂²	1	2	yes	complemented
57	#117×C₁₄₉	C₂²	1	2	yes	—
58	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
59	C₂³×D₁₄₉	C₂³	1	2	yes	—
60	C₂×Dic₂₉₈	C₂³	1	2	yes	—
61	C₂×Dic₂₉₈	C₂³	1	8	yes	—
62	C₂×Dic₂₉₈	C₂³	1	2	yes	—
63	Q₈×D₁₄₉	—	4	8	no	—
64	C₂×D₅₉₆	Q₈	1	8	yes	complemented
65	C₂×D₅₉₆	C₂³	1	2	yes	—
66	C₂×D₅₉₆	C₂³	1	2	yes	—
67	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
68	C₂×C₄×D₁₄₉	—	4	4	no	—
69	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
70	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
71	C₂×C₄×D₁₄₉	C₂³	1	8	yes	—
72	C₂×C₄×D₁₄₉	C₂³	1	8	yes	—
73	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
74	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
75	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
76	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
77	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
78	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
79	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
80	#14558715	C₂³	1	8	yes	—
81	#14558715	C₂³	1	4	yes	—
82	#14558715	C₂³	1	8	yes	—
83	#14558715	C₂³	1	4	yes	—
84	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
85	#14558713	C₂³	1	8	yes	—
86	#14558713	C₂³	1	8	yes	—
87	#14558713	C₂³	1	4	yes	—
88	#14558713	C₂³	1	4	yes	—
89	C₄×Dic₁₄₉	C₂³	1	4	yes	—
90	C₂×Q₈×C₁₄₉	C₂³	1	2	yes	—
91	C₂×Q₈×C₁₄₉	C₂³	1	2	yes	—
92	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
96	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
97	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
98	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
99	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
100	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
101	C₂²⋊C₄×C₁₄₉	C₂³	1	4	yes	—
102	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
103	C₄²×C₁₄₉	C₂³	1	4	yes	—
104	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
105	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
106	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
107	C₂²×D₁₄₉	—	4	2	no	—
108	C₂²×D₁₄₉	—	4	2	no	—
109	C₂²×D₁₄₉	C₂×Q₈	1	8	yes	complemented
110	C₂²×D₁₄₉	C₂⁴	1	4	yes	—
111	Dic₂₉₈	—	2	4	no	—
112	Dic₂₉₈	—	4	8	no	—
113	Dic₂₉₈	—	4	2	no	—
114	D₅₉₆	C₂×Q₈	1	8	yes	complemented
115	D₅₉₆	C₂×Q₈	1	8	yes	complemented
116	C₂×Dic₁₄₉	—	4	2	no	—
117	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
118	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
119	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
120	C₄×D₁₄₉	—	2	8	no	—
121	C₄×D₁₄₉	—	4	4	no	—
122	C₄×D₁₄₉	—	4	8	no	—
123	C₄×D₁₄₉	—	4	8	no	—
124	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
125	Q₈×C₁₄₉	—	2	4	no	—
126	Q₈×C₁₄₉	—	4	2	no	—
127	C₂×C₄×C₁₄₉	—	2	2	no	—
128	C₂×C₄×C₁₄₉	—	4	1	no	—
129	C₂×C₄×C₁₄₉	C₂×Q₈	1	4	yes	complemented
130	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
131	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
132	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
133	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
134	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
135	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
136	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
137	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
138	C₂²×C₁₄₉	—	2	1	no	—
139	C₂²×C₁₄₉	—	2	1	no	—
140	C₂²×C₁₄₉	C₂²×Q₈	1	2	yes	—
141	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
142	C₂×D₁₄₉	—	2	4	no	—
143	C₂×D₁₄₉	—	2	4	no	—
144	C₂×D₁₄₉	—	4	8	no	—
145	C₂×D₁₄₉	C₂²×Q₈	1	8	yes	—
146	C₄×C₁₄₉	—	2	1	no	—
147	C₄×C₁₄₉	—	2	4	no	—
148	C₄×C₁₄₉	—	2	1	no	—
149	C₄×C₁₄₉	—	2	1	no	—
150	C₄×C₁₄₉	—	2	4	no	—
151	C₄×C₁₄₉	C₂²×Q₈	1	2	yes	—
152	C₄×C₁₄₉	C₂²×Q₈	1	2	yes	—
153	C₄×C₁₄₉	H(5,𝔽₂)	1	2	yes	—
154	Dic₁₄₉	—	2	4	no	—
155	Dic₁₄₉	—	2	4	no	—
156	Dic₁₄₉	—	2	4	no	—
157	C₂×C₁₄₉	—	2	2	no	—
158	C₂×C₁₄₉	C₂³×Q₈	1	1	yes	characteristic
159	C₂×C₁₄₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₁₄₉	#585	1	1	yes	characteristic
161	D₁₄₉	—	2	8	no	—
162	C₁₄₉	#3501	1	1	yes	characteristic, complemented, Sylow
163	#3501	—	149	1	no	maximal, Sylow
164	#554	—	149	2	no	—
165	#554	—	149	4	no	—
166	#554	—	149	2	no	—
167	#554	—	149	4	no	—
168	D₄×Q₈	—	149	2	no	—
169	D₄×Q₈	—	149	2	no	—
170	#544	—	149	2	no	—
171	#543	—	149	4	no	—
172	#533	—	149	1	no	—
173	#533	D₁₄₉	1	1	yes	characteristic, complemented
174	C₂×C₂²⋊Q₈	—	149	4	no	—
175	#518	—	149	2	no	—
176	C₂×C₄×D₄	—	149	1	no	—
177	C₂²×Q₈	—	149	4	no	—
178	C₂²×D₄	—	149	1	no	—
179	C₂³×C₄	—	149	2	no	—
180	C₄⋊Q₈	—	149	4	no	—
181	C₄⋊Q₈	—	149	1	no	—
182	C₄⋊Q₈	—	149	1	no	—
183	C₄⋊Q₈	—	149	4	no	—
184	C₄⋊Q₈	C₂×D₁₄₉	1	1	yes	characteristic, complemented
185	C₄⋊Q₈	C₂×D₁₄₉	1	1	yes	characteristic, complemented
186	#125	—	149	4	no	—
187	#125	—	149	2	no	—
188	#125	—	149	4	no	—
189	#125	C₂×D₁₄₉	1	2	yes	complemented
190	C₂²⋊Q₈	—	149	4	no	—
191	C₂²⋊Q₈	—	149	8	no	—
192	C₂²⋊Q₈	—	149	2	no	—
193	C₂²⋊Q₈	—	149	2	no	—
194	C₂²⋊Q₈	—	149	4	no	—
195	C₂²⋊Q₈	—	149	8	no	—
196	C₂²⋊Q₈	—	149	8	no	—
197	C₂²⋊Q₈	—	149	8	no	—
198	C₂²⋊Q₈	—	149	8	no	—
199	C₂²⋊Q₈	—	149	8	no	—
200	C₂²⋊Q₈	C₂×D₁₄₉	1	2	yes	—
201	C₂²⋊Q₈	C₂×D₁₄₉	1	2	yes	—
202	C₄×Q₈	—	149	2	no	—
203	C₄×Q₈	—	149	2	no	—
204	C₄×Q₈	C₂×D₁₄₉	1	2	yes	complemented
205	C₄×Q₈	C₂×D₁₄₉	1	2	yes	complemented
206	C₄×D₄	—	149	4	no	—
207	C₄×D₄	—	149	4	no	—
208	C₄×D₄	—	149	2	no	—
209	C₄×D₄	—	149	2	no	—
210	C₄×D₄	—	149	2	no	—
211	C₄×D₄	—	149	2	no	—
212	C₄×D₄	—	149	4	no	—
213	C₄×D₄	—	149	4	no	—
214	#117	—	149	2	no	—
215	#117	C₂×D₁₄₉	1	2	yes	—
216	C₂×C₄⋊C₄	—	149	1	no	—
217	C₂×C₄⋊C₄	—	149	8	no	—
218	C₂×C₄⋊C₄	—	149	4	no	—
219	C₂×C₂²⋊C₄	—	149	2	no	—
220	C₂×C₂²⋊C₄	—	149	8	no	—
221	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
222	C₂⁴	—	149	2	no	—
223	C₂×Q₈	—	596	8	no	—
224	C₂×Q₈	—	149	2	no	—
225	C₂×Q₈	—	149	2	no	—
226	C₂×Q₈	—	149	8	no	—
227	C₂×Q₈	C₂²×D₁₄₉	1	2	yes	—
228	C₂×Q₈	C₂²×D₁₄₉	1	2	yes	—
229	C₂×D₄	—	149	8	no	—
230	C₂×D₄	—	149	2	no	—
231	C₂×D₄	—	149	2	no	—
232	C₂²×C₄	—	596	4	no	—
233	C₂²×C₄	—	149	4	no	—
234	C₂²×C₄	—	149	2	no	—
235	C₂²×C₄	—	149	8	no	—
236	C₂²×C₄	—	149	8	no	—
237	C₂²×C₄	—	149	4	no	—
238	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
241	C₄⋊C₄	—	149	4	no	—
242	C₄⋊C₄	—	149	2	no	—
243	C₄⋊C₄	—	149	2	no	—
244	C₄⋊C₄	—	149	2	no	—
245	C₄⋊C₄	—	149	8	no	—
246	C₄⋊C₄	—	149	8	no	—
247	C₄⋊C₄	—	149	8	no	—
248	C₄⋊C₄	—	149	2	no	—
249	C₄⋊C₄	—	149	2	no	—
250	C₄⋊C₄	—	149	4	no	—
251	C₄⋊C₄	—	149	2	no	—
252	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
253	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
254	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
255	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
256	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
257	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
258	C₂²⋊C₄	—	149	4	no	—
259	C₂²⋊C₄	—	149	8	no	—
260	C₂²⋊C₄	—	149	8	no	—
261	C₂²⋊C₄	—	149	4	no	—
262	C₂²⋊C₄	—	149	4	no	—
263	C₂²⋊C₄	C₂²×D₁₄₉	1	4	yes	—
264	C₄²	—	149	4	no	—
265	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
266	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
267	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
268	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
269	C₄²	C₂²×D₁₄₉	1	4	yes	—
270	C₂³	—	596	2	no	—
271	C₂³	—	596	2	no	—
272	C₂³	—	149	4	no	—
273	C₂³	—	149	8	no	—
274	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	298	4	no	—
278	Q₈	—	596	8	no	—
279	Q₈	—	596	2	no	—
280	D₄	—	149	8	no	—
281	D₄	—	149	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	298	8	no	—
285	C₂×C₄	—	596	4	no	—
286	C₂×C₄	—	596	8	no	—
287	C₂×C₄	—	596	8	no	—
288	C₂×C₄	—	596	2	no	—
289	C₂×C₄	—	149	4	no	—
290	C₂×C₄	—	149	4	no	—
291	C₂×C₄	—	149	4	no	—
292	C₂×C₄	Q₈×D₁₄₉	1	4	yes	complemented
293	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
297	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
298	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
299	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	298	4	no	—
304	C₂²	—	298	4	no	—
305	C₂²	—	596	8	no	—
306	C₂²	—	149	8	no	—
307	C₂²	C₂×Q₈×D₁₄₉	1	2	yes	—
308	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	4	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	4	no	—
314	C₄	—	298	4	no	—
315	C₄	—	298	4	no	—
316	C₄	—	298	4	no	—
317	C₄	C₂×Q₈×D₁₄₉	1	2	yes	—
318	C₄	C₂×Q₈×D₁₄₉	1	2	yes	—
319	C₄	#18011957	1	2	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	298	8	no	—
322	C₂	C₂²×Q₈×D₁₄₉	1	1	yes	characteristic
323	C₂	C₂×#18011957	1	1	yes	characteristic
324	C₂	#23750461	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting	#533×C₁₄₉	C₂
prime radical	C₂²×D₅₉₆	C₂²
hypercenter	#533	D₁₄₉
hypocenter	C₁₄₉	#3501
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. #23750336	C₂	semidirect
2. #23750335	C₂	semidirect
3. #23750335	C₂	semidirect
4. #23750331	C₂	semidirect
5. Q₈×D₅₉₆	C₂	semidirect
6. #23750290	C₂	semidirect
7. #23750253	C₂	semidirect
8. #23750250	C₂	semidirect
9. #23750182	C₂	semidirect
10. C₂²⋊Q₈×D₁₄₉	C₂	semidirect
13. #533×C₁₄₉	C₂	semidirect
14. C₂×Q₈×D₁₄₉	C₂²	semidirect
17. #18011911	C₂²	semidirect
18. #18011911	C₂²	semidirect
19. #18011911	C₂²	semidirect
20. #18011910	C₂²	semidirect
21. #18011903	C₂²	semidirect
22. #18011903	C₂²	semidirect
23. #18011902	C₂²	semidirect
24. #18011891	C₂²	semidirect
25. #18011891	C₂²	semidirect
28. #18011889	C₂²	semidirect
29. #18011889	C₂²	semidirect
30. #18011889	C₂²	semidirect
32. #18011887	C₂²	semidirect
33. #18011887	C₂²	semidirect
34. C₄×Dic₂₉₈	C₂²	semidirect
35. C₄×Dic₂₉₈	C₂²	semidirect
46. C₄⋊C₄×D₁₄₉	C₂²	semidirect
47. C₄⋊C₄×D₁₄₉	C₂²	semidirect
50. C₄⋊Q₈×C₁₄₉	C₂²	semidirect
51. C₄⋊Q₈×C₁₄₉	C₂²	semidirect
52. #125×C₁₄₉	C₂²	semidirect
55. C₄×Q₈×C₁₄₉	C₂²	semidirect
56. C₄×Q₈×C₁₄₉	C₂²	semidirect
64. C₂×D₅₉₆	Q₈	semidirect
109. C₂²×D₁₄₉	C₂×Q₈	semidirect
114. D₅₉₆	C₂×Q₈	semidirect
115. D₅₉₆	C₂×Q₈	semidirect
129. C₂×C₄×C₁₄₉	C₂×Q₈	semidirect
162. C₁₄₉	#3501	semidirect
173. #533	D₁₄₉	semidirect
184. C₄⋊Q₈	C₂×D₁₄₉	semidirect
185. C₄⋊Q₈	C₂×D₁₄₉	semidirect
189. #125	C₂×D₁₄₉	semidirect
204. C₄×Q₈	C₂×D₁₄₉	semidirect
205. C₄×Q₈	C₂×D₁₄₉	semidirect
292. C₂×C₄	Q₈×D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#23750336	2
#23750335	2
#23750331	2
Q₈×D₅₉₆	2
#23750290	2
#23750253	2
#23750250	2
#23750182	2
C₂²⋊Q₈×D₁₄₉	2
#23750099	2
C₂×C₄×D₅₉₆	2
#533×C₁₄₉	2
#3501	149