Subgroups — #34058543

Order 19072.

1140 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750358	C₂	1	4	yes	maximal, complemented
2	#23750345	C₂	1	2	yes	maximal, complemented
3	#23750340	C₂	1	2	yes	maximal, complemented
4	#23750325	C₂	1	4	yes	maximal, complemented
5	#23750296	C₂	1	2	yes	maximal, complemented
6	#23750280	C₂	1	2	yes	maximal, complemented
7	#23750213	C₂	1	4	yes	maximal, complemented
8	#23750192	C₂	1	2	yes	maximal, complemented
9	#23750182	C₂	1	1	yes	maximal, characteristic, complemented
10	#23750175	C₂	1	1	yes	maximal, characteristic, complemented
11	#23750136	C₂	1	4	yes	maximal, complemented
12	#23750106	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×D₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#14558736	C₂²	1	2	yes	—
15	C₂×#14558736	C₂²	1	4	yes	complemented
16	#18011911	C₂²	1	1	yes	characteristic, complemented
17	#18011911	C₂²	1	1	yes	characteristic, complemented
18	#18011911	C₂²	1	1	yes	characteristic, complemented
19	#18011908	C₂²	1	1	yes	characteristic, complemented
20	#18011907	C₂²	1	4	yes	—
21	#18011904	C₂²	1	8	yes	complemented
22	#18011903	C₂²	1	2	yes	complemented
23	#18011899	C₂²	1	2	yes	complemented
24	#18011898	C₂²	1	4	yes	—
25	#18011897	C₂²	1	8	yes	complemented
26	#18011894	C₂²	1	8	yes	complemented
27	#18011893	C₂²	1	8	yes	complemented
28	#18011890	C₂²	1	2	yes	—
29	#18011890	C₂²	1	2	yes	—
30	#18011890	C₂²	1	4	yes	complemented
31	#18011885	C₂²	1	2	yes	—
32	#18011885	C₂²	1	2	yes	—
33	#18011885	C₂²	1	4	yes	complemented
34	#18011884	C₂²	1	8	yes	complemented
35	#18011882	C₂²	1	4	yes	—
36	#18011882	C₂²	1	4	yes	—
37	#18011880	C₂²	1	8	yes	—
38	C₄×Dic₂₉₈	C₂²	1	2	yes	complemented
39	C₄×Dic₂₉₈	C₂²	1	2	yes	complemented
40	#18011876	C₂²	1	4	yes	complemented
41	C₄×D₅₉₆	C₂²	1	2	yes	complemented
42	C₄×D₅₉₆	C₂²	1	2	yes	complemented
43	#18011873	C₂²	1	4	yes	complemented
44	C₄×C₂²⋊D₁₄₉	C₂²	1	8	yes	—
45	C₄×C₂²⋊D₁₄₉	C₂²	1	8	yes	complemented
46	#18011870	C₂²	1	2	yes	—
47	#18011870	C₂²	1	4	yes	complemented
48	#18011869	C₂²	1	4	yes	complemented
49	C₄⋊C₄×D₁₄₉	C₂²	1	4	yes	complemented
50	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
51	C₂³×C₄×C₁₄₉	C₂²	1	2	yes	—
52	C₄×D₄×C₁₄₉	C₂²	1	2	yes	—
53	C₄×D₄×C₁₄₉	C₂²	1	2	yes	—
54	C₄×D₄×C₁₄₉	C₂²	1	2	yes	—
55	C₄×D₄×C₁₄₉	C₂²	1	2	yes	—
56	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	—
58	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
59	#14558736	—	2	8	no	—
60	#14558736	—	4	8	no	—
61	C₂×Dic₂₉₈	C₂³	1	2	yes	—
62	C₂×Dic₂₉₈	C₂³	1	2	yes	—
63	C₂×Dic₂₉₈	C₂³	1	2	yes	—
64	C₂×D₅₉₆	C₂³	1	2	yes	—
65	C₂×D₅₉₆	C₂³	1	2	yes	—
66	C₂×D₅₉₆	C₂³	1	2	yes	—
67	C₂×C₂²⋊D₁₄₉	C₂³	1	4	yes	—
68	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
69	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
70	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
71	C₂×C₄×D₁₄₉	C₂³	1	8	yes	—
72	C₄⋊Dic₁₄₉	C₂³	1	2	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	2	yes	—
78	#14558715	C₂³	1	4	yes	—
79	#14558715	C₂³	1	8	yes	—
80	#14558715	C₂³	1	4	yes	—
81	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
82	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
83	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
84	#14558713	C₂³	1	8	yes	—
85	#14558713	C₂³	1	4	yes	—
86	#14558713	C₂³	1	4	yes	—
87	C₄×Dic₁₄₉	C₂³	1	8	yes	—
88	C₄×Dic₁₄₉	C₂³	1	4	yes	—
89	C₂⁴×C₁₄₉	C₂³	1	2	yes	—
90	C₂×D₄×C₁₄₉	—	2	4	no	—
91	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
92	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
93	C₂²×C₄×C₁₄₉	—	4	4	no	—
94	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
95	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
98	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
99	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
101	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
102	C₂²⋊C₄×C₁₄₉	C₂³	1	4	yes	—
103	C₂²⋊C₄×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	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
108	C₂²×D₁₄₉	C₂⁴	1	4	yes	—
109	Dic₂₉₈	—	2	4	no	—
110	Dic₂₉₈	—	2	4	no	—
111	Dic₂₉₈	—	4	2	no	—
112	D₅₉₆	—	2	4	no	—
113	D₅₉₆	—	2	4	no	—
114	D₅₉₆	—	4	2	no	—
115	C₂²⋊D₁₄₉	—	2	8	no	—
116	C₂²⋊D₁₄₉	—	4	8	no	—
117	C₂²⋊D₁₄₉	—	4	8	no	—
118	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
119	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
120	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
121	C₄×D₁₄₉	—	2	8	no	—
122	C₄×D₁₄₉	—	4	8	no	—
123	C₂³×C₁₄₉	—	2	4	no	—
124	C₂³×C₁₄₉	—	4	2	no	—
125	C₂³×C₁₄₉	—	4	2	no	—
126	C₂³×C₁₄₉	C₂⁴	1	4	yes	—
127	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
128	D₄×C₁₄₉	—	2	4	no	—
129	D₄×C₁₄₉	—	2	4	no	—
130	C₂×C₄×C₁₄₉	—	2	8	no	—
131	C₂×C₄×C₁₄₉	—	2	2	no	—
132	C₂×C₄×C₁₄₉	—	4	4	no	—
133	C₂×C₄×C₁₄₉	—	4	2	no	—
134	C₂×C₄×C₁₄₉	—	4	1	no	—
135	C₂×C₄×C₁₄₉	#41	1	4	yes	complemented
136	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
137	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
138	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
139	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
140	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
141	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
142	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
143	C₂²×C₁₄₉	—	2	1	no	—
144	C₂²×C₁₄₉	—	2	1	no	—
145	C₂²×C₁₄₉	—	2	4	no	—
146	C₂²×C₁₄₉	—	2	4	no	—
147	C₂²×C₁₄₉	—	2	4	no	—
148	C₂²×C₁₄₉	—	4	8	no	—
149	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
150	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
151	C₂×D₁₄₉	—	2	4	no	—
152	C₂×D₁₄₉	—	2	4	no	—
153	C₂×D₁₄₉	—	2	4	no	—
154	C₄×C₁₄₉	—	2	1	no	—
155	C₄×C₁₄₉	—	2	4	no	—
156	C₄×C₁₄₉	—	2	1	no	—
157	C₄×C₁₄₉	—	2	1	no	—
158	C₄×C₁₄₉	C₂×#41	1	2	yes	—
159	C₄×C₁₄₉	C₂×#41	1	2	yes	—
160	C₄×C₁₄₉	H(5,𝔽₂)	1	2	yes	—
161	Dic₁₄₉	—	2	4	no	—
162	Dic₁₄₉	—	2	4	no	—
163	Dic₁₄₉	—	2	4	no	—
164	C₂×C₁₄₉	—	2	2	no	—
165	C₂×C₁₄₉	—	4	4	no	—
166	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
167	C₂×C₁₄₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₁₄₉	#585	1	1	yes	characteristic
169	D₁₄₉	—	4	4	no	—
170	C₁₄₉	#3506	1	1	yes	characteristic, complemented, Sylow
171	#3506	—	149	1	no	maximal, Sylow
172	#559	—	149	4	no	—
173	#556	—	149	2	no	—
174	#555	—	149	2	no	—
175	#553	—	149	4	no	—
176	#550	—	149	2	no	—
177	#547	—	149	2	no	—
178	#538	—	149	4	no	—
179	#535	—	149	2	no	—
180	#533	—	149	1	no	—
181	#532	—	149	1	no	—
182	#525	—	149	4	no	—
183	#520	—	149	2	no	—
184	C₂×C₄×D₄	D₁₄₉	1	1	yes	characteristic, complemented
185	C₂×#41	—	149	2	no	—
186	C₂×#41	—	149	4	no	—
187	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
188	C₂³×C₄	C₂×D₁₄₉	1	2	yes	—
189	C₄⋊Q₈	—	149	1	no	—
190	C₄⋊Q₈	—	149	1	no	—
191	C₄⋊Q₈	—	149	1	no	—
192	#127	—	149	4	no	—
193	#127	—	149	1	no	—
194	#126	—	149	8	no	—
195	#125	—	149	2	no	—
196	#124	—	149	4	no	—
197	#124	—	149	8	no	—
198	#124	—	149	2	no	—
199	#123	—	149	8	no	—
200	#123	—	149	8	no	—
201	C₂²⋊Q₈	—	149	2	no	—
202	C₂²⋊Q₈	—	149	4	no	—
203	C₂²⋊Q₈	—	149	2	no	—
204	#121	—	149	4	no	—
205	#121	—	149	4	no	—
206	#121	—	149	2	no	—
207	#121	—	149	4	no	—
208	#121	—	149	2	no	—
209	#121	—	149	8	no	—
210	#120	—	149	8	no	—
211	C₄×Q₈	—	149	4	no	—
212	C₄×Q₈	—	149	2	no	—
213	C₄×Q₈	—	149	2	no	—
214	C₄×D₄	—	149	8	no	—
215	C₄×D₄	—	149	8	no	—
216	C₄×D₄	—	149	4	no	—
217	C₄×D₄	—	149	2	no	—
218	C₄×D₄	—	149	2	no	—
219	C₄×D₄	C₂×D₁₄₉	1	2	yes	—
220	C₄×D₄	C₂×D₁₄₉	1	2	yes	—
221	C₄×D₄	C₂×D₁₄₉	1	2	yes	—
222	C₄×D₄	C₂×D₁₄₉	1	2	yes	—
223	#117	—	149	2	no	—
224	#117	—	149	4	no	—
225	#117	—	149	4	no	—
226	C₂×C₄⋊C₄	—	149	4	no	—
227	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	2	yes	—
229	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
230	C₂⁴	C₂²×D₁₄₉	1	2	yes	—
231	#41	—	298	8	no	—
232	#41	—	596	8	no	—
233	C₂×Q₈	—	149	2	no	—
234	C₂×Q₈	—	149	2	no	—
235	C₂×Q₈	—	149	2	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	149	4	no	—
238	C₂×D₄	—	149	8	no	—
239	C₂×D₄	—	149	8	no	—
240	C₂×D₄	—	149	2	no	—
241	C₂×D₄	—	149	2	no	—
242	C₂×D₄	—	149	2	no	—
243	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
244	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	149	4	no	—
247	C₂²×C₄	—	149	8	no	—
248	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
250	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
252	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
253	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
254	C₄⋊C₄	—	149	2	no	—
255	C₄⋊C₄	—	149	8	no	—
256	C₄⋊C₄	—	149	2	no	—
257	C₄⋊C₄	—	149	2	no	—
258	C₄⋊C₄	—	149	2	no	—
259	C₄⋊C₄	—	149	4	no	—
260	C₄⋊C₄	—	149	2	no	—
261	C₄⋊C₄	—	149	4	no	—
262	C₄⋊C₄	—	149	2	no	—
263	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
264	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
265	C₂²⋊C₄	—	149	4	no	—
266	C₂²⋊C₄	—	149	8	no	—
267	C₂²⋊C₄	—	149	8	no	—
268	C₂²⋊C₄	—	149	8	no	—
269	C₂²⋊C₄	—	149	4	no	—
270	C₂²⋊C₄	—	149	4	no	—
271	C₂²⋊C₄	C₂²×D₁₄₉	1	4	yes	—
272	C₂²⋊C₄	C₂²×D₁₄₉	1	4	yes	—
273	C₄²	—	149	8	no	—
274	C₄²	—	149	4	no	—
275	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
276	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
277	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
278	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
279	C₂³	—	2	4	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	149	4	no	—
283	C₂³	C₂³×D₁₄₉	1	4	yes	—
284	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
285	Q₈	—	298	4	no	—
286	Q₈	—	298	4	no	—
287	Q₈	—	596	2	no	—
288	D₄	—	2	4	no	—
289	D₄	—	2	4	no	—
290	D₄	—	298	8	no	—
291	D₄	—	298	4	no	—
292	D₄	—	298	4	no	—
293	D₄	—	596	2	no	—
294	D₄	—	596	8	no	—
295	D₄	—	596	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	298	8	no	—
302	C₂×C₄	—	596	8	no	—
303	C₂×C₄	—	149	4	no	—
304	C₂×C₄	—	149	4	no	—
305	C₂×C₄	—	149	4	no	—
306	C₂×C₄	#14558736	1	4	yes	complemented
307	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
308	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
309	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	298	4	no	—
321	C₂²	—	298	4	no	—
322	C₂²	—	298	4	no	—
323	C₂²	C₂×#14558736	1	2	yes	—
324	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	4	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	298	4	no	—
330	C₄	—	298	4	no	—
331	C₄	—	298	4	no	—
332	C₄	C₂×#14558736	1	2	yes	—
333	C₄	C₂×#14558736	1	2	yes	—
334	C₄	#18011956	1	2	yes	—
335	C₂	—	2	2	no	—
336	C₂	—	4	4	no	—
337	C₂	—	596	4	no	—
338	C₂	C₂²×#14558736	1	1	yes	characteristic
339	C₂	C₂×#18011956	1	1	yes	characteristic
340	C₂	#23750461	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting	C₂×C₄×D₄×C₁₄₉	C₂
hypercenter	C₂×C₄×D₄	D₁₄₉
hypocenter	C₁₄₉	#3506
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #23750358	C₂	semidirect
2. #23750345	C₂	semidirect
3. #23750340	C₂	semidirect
4. #23750325	C₂	semidirect
5. #23750296	C₂	semidirect
6. #23750280	C₂	semidirect
7. #23750213	C₂	semidirect
8. #23750192	C₂	semidirect
9. #23750182	C₂	semidirect
10. #23750175	C₂	semidirect
11. #23750136	C₂	semidirect
12. #23750106	C₂	semidirect
13. C₂×C₄×D₄×C₁₄₉	C₂	semidirect
15. C₂×#14558736	C₂²	semidirect
16. #18011911	C₂²	semidirect
17. #18011911	C₂²	semidirect
18. #18011911	C₂²	semidirect
19. #18011908	C₂²	semidirect
21. #18011904	C₂²	semidirect
22. #18011903	C₂²	semidirect
23. #18011899	C₂²	semidirect
25. #18011897	C₂²	semidirect
26. #18011894	C₂²	semidirect
27. #18011893	C₂²	semidirect
30. #18011890	C₂²	semidirect
33. #18011885	C₂²	semidirect
34. #18011884	C₂²	semidirect
38. C₄×Dic₂₉₈	C₂²	semidirect
39. C₄×Dic₂₉₈	C₂²	semidirect
40. #18011876	C₂²	semidirect
41. C₄×D₅₉₆	C₂²	semidirect
42. C₄×D₅₉₆	C₂²	semidirect
43. #18011873	C₂²	semidirect
45. C₄×C₂²⋊D₁₄₉	C₂²	semidirect
47. #18011870	C₂²	semidirect
48. #18011869	C₂²	semidirect
49. C₄⋊C₄×D₁₄₉	C₂²	semidirect
135. C₂×C₄×C₁₄₉	#41	semidirect
170. C₁₄₉	#3506	semidirect
184. C₂×C₄×D₄	D₁₄₉	semidirect
306. C₂×C₄	#14558736	semidirect

Maximal subgroups

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

subgroup	index
#23750358	2
#23750345	2
#23750340	2
#23750325	2
#23750296	2
#23750280	2
#23750213	2
#23750192	2
#23750182	2
#23750175	2
#23750136	2
#23750106	2
C₂×C₄×D₄×C₁₄₉	2
#3506	149