Subgroups — #34061502

Order 19072.

412 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750677	C₂	1	2	yes	maximal, complemented
2	#23750658	C₂	1	2	yes	maximal, complemented
3	#23750657	C₂	1	2	yes	maximal, complemented
4	#23750655	C₂	1	2	yes	maximal, complemented
5	#23750644	C₂	1	1	yes	maximal, characteristic, complemented
6	#23750643	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750642	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14558746	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14558745	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₄₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#23750093	C₂	1	1	yes	maximal, characteristic
12	C₂×#14558746	C₂²	1	2	yes	—
13	C₂×#14558745	C₂²	1	2	yes	—
14	C₂×D₁₄₉⋊C₈	C₂²	1	2	yes	—
15	#18011978	C₂²	1	2	yes	—
16	#18011978	C₂²	1	2	yes	—
17	C₄⋊#11924753	C₂²	1	2	yes	—
18	#18011968	C₂²	1	2	yes	—
19	#18011968	C₂²	1	2	yes	—
20	C₂²⋊#11924753	C₂²	1	2	yes	—
21	#18011966	C₂²	1	2	yes	—
22	#18011966	C₂²	1	2	yes	—
23	C₄×#11924753	C₂²	1	2	yes	—
24	C₄×#11924753	C₂²	1	1	yes	characteristic
25	C₄×#11924753	C₂²	1	1	yes	characteristic
26	C₄×#11924753	C₂²	1	1	yes	characteristic
27	C₄×#11924753	C₂²	1	1	yes	characteristic
28	C₂×#14558736	C₄	1	1	yes	characteristic
29	#18011876	C₄	1	1	yes	characteristic
30	#18011876	C₂²	1	1	yes	characteristic
31	#18011873	C₄	1	1	yes	characteristic
32	#18011873	C₂²	1	1	yes	characteristic
33	#18011871	C₄	1	1	yes	characteristic
34	#18011871	C₄	1	1	yes	characteristic
35	#18011871	C₂²	1	1	yes	characteristic
36	#18011871	C₂²	1	1	yes	characteristic
37	#18011868	C₄	1	1	yes	characteristic
38	#18011868	C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
40	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
41	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
42	#117×C₁₄₉	C₄	1	1	yes	characteristic
43	#14558746	—	2	4	no	—
44	#14558745	—	2	4	no	—
45	D₁₄₉⋊C₈	—	2	4	no	—
46	C₂×#11924753	C₂³	1	2	yes	—
47	C₂×#11924753	C₂³	1	2	yes	—
48	C₂×#11924753	C₂³	1	2	yes	—
49	C₂×#11924753	C₂³	1	2	yes	—
50	#14558736	—	2	4	no	—
51	C₂×Dic₂₉₈	C₂×C₄	1	1	yes	characteristic
52	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
63	#14558715	C₂×C₄	1	1	yes	characteristic
64	#14558715	C₂×C₄	1	1	yes	characteristic
65	#14558715	C₂×C₄	1	1	yes	characteristic
66	#14558715	C₂×C₄	1	1	yes	characteristic
67	#14558713	C₂×C₄	1	1	yes	characteristic
68	#14558713	C₂×C₄	1	1	yes	characteristic
69	#14558713	C₂×C₄	1	1	yes	characteristic
70	#14558713	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
76	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
78	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
85	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
86	#11924753	—	2	2	no	—
87	#11924753	—	2	2	no	—
88	#11924753	—	2	2	no	—
89	#11924753	#41	1	4	yes	complemented
90	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
92	Dic₂₉₈	—	2	2	no	—
93	D₅₉₆	—	2	2	no	—
94	C₂²⋊D₁₄₉	—	2	2	no	—
95	C₂²⋊D₁₄₉	—	2	2	no	—
96	C₂×Dic₁₄₉	—	2	2	no	—
97	C₂×Dic₁₄₉	—	2	2	no	—
98	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
104	C₄×D₁₄₉	—	2	2	no	—
105	C₄×D₁₄₉	—	2	2	no	—
106	C₄×D₁₄₉	—	2	2	no	—
107	C₄×D₁₄₉	—	2	2	no	—
108	C₄×D₁₄₉	—	2	2	no	—
109	C₄×D₁₄₉	M₁₆	1	4	yes	complemented
110	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₄₉	—	2	2	no	—
112	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₁₄₉	—	2	1	no	—
119	C₂²×C₁₄₉	—	2	1	no	—
120	C₂²×C₁₄₉	—	2	1	no	—
121	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₁₄₉	—	2	1	no	—
123	C₂×D₁₄₉	—	2	1	no	—
124	C₂×D₁₄₉	—	2	1	no	—
125	C₂×D₁₄₉	—	2	1	no	—
126	C₂×D₁₄₉	—	2	1	no	—
127	C₂×D₁₄₉	C₂×M₁₆	1	2	yes	—
128	C₄×C₁₄₉	—	2	1	no	—
129	C₄×C₁₄₉	—	2	1	no	—
130	C₄×C₁₄₉	—	2	1	no	—
131	C₄×C₁₄₉	—	2	1	no	—
132	C₄×C₁₄₉	—	2	1	no	—
133	C₄×C₁₄₉	C₂×M₁₆	1	2	yes	—
134	Dic₁₄₉	—	2	1	no	—
135	Dic₁₄₉	—	2	1	no	—
136	Dic₁₄₉	—	2	1	no	—
137	Dic₁₄₉	C₂×M₁₆	1	2	yes	—
138	Dic₁₄₉	#131	1	2	yes	—
139	Dic₁₄₉	C₂×#41	1	2	yes	—
140	C₂×C₁₄₉	—	4	1	no	—
141	C₂×C₁₄₉	C₄×#41	1	1	yes	characteristic
142	C₂×C₁₄₉	C₂²×M₁₆	1	1	yes	characteristic
143	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
144	D₁₄₉	—	2	2	no	—
145	D₁₄₉	—	4	1	no	—
146	C₁₄₉	#2965	1	1	yes	characteristic, complemented, Sylow
147	#2965	—	149	1	no	maximal, Sylow
148	C₄×#41	—	149	1	no	—
149	#446	—	149	2	no	—
150	#436	—	149	2	no	—
151	#435	—	149	2	no	—
152	#435	—	149	2	no	—
153	#431	—	149	1	no	—
154	#431	—	149	1	no	—
155	#431	—	149	1	no	—
156	C₄×M₁₆	—	149	1	no	—
157	C₄×M₁₆	—	149	1	no	—
158	C₂×C₄×C₈	—	149	1	no	—
159	C₂×#41	—	149	1	no	—
160	C₂×M₁₆	—	149	2	no	—
161	C₂×M₁₆	—	149	2	no	—
162	C₂²×C₈	—	149	2	no	—
163	C₄×Q₈	—	149	1	no	—
164	C₄×Q₈	—	149	1	no	—
165	C₄×D₄	—	149	1	no	—
166	C₄×D₄	—	149	1	no	—
167	C₄×D₄	—	149	1	no	—
168	C₄×D₄	—	149	1	no	—
169	C₄×D₄	—	149	1	no	—
170	C₄×D₄	—	149	1	no	—
171	#117	—	149	1	no	—
172	#117	—	149	1	no	—
173	#117	#10597010	1	1	yes	characteristic
174	C₂×C₄²	—	149	1	no	—
175	C₂×C₄²	—	149	1	no	—
176	C₂×C₄²	—	149	1	no	—
177	C₄⋊C₈	—	149	2	no	—
178	C₄⋊C₈	—	149	2	no	—
179	C₄⋊C₈	—	149	2	no	—
180	C₂²⋊C₈	—	149	2	no	—
181	C₂²⋊C₈	—	149	2	no	—
182	C₂²⋊C₈	—	149	2	no	—
183	#97	—	149	2	no	—
184	#97	—	149	2	no	—
185	C₄×C₈	—	149	1	no	—
186	C₄×C₈	—	149	1	no	—
187	C₄×C₈	—	149	2	no	—
188	C₄×C₈	—	149	1	no	—
189	C₄×C₈	—	149	1	no	—
190	#41	—	298	4	no	—
191	C₂×Q₈	—	149	1	no	—
192	C₂×D₄	—	149	1	no	—
193	C₂×D₄	—	149	1	no	—
194	C₂×D₄	—	149	1	no	—
195	C₂²×C₄	—	149	1	no	—
196	C₂²×C₄	—	149	1	no	—
197	C₂²×C₄	—	149	1	no	—
198	C₂²×C₄	—	149	1	no	—
199	C₂²×C₄	—	149	1	no	—
200	C₂²×C₄	—	149	1	no	—
201	C₂²×C₄	—	149	1	no	—
202	C₂²×C₄	—	149	1	no	—
203	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
204	M₁₆	—	298	4	no	—
205	M₁₆	—	298	4	no	—
206	C₂×C₈	—	298	4	no	—
207	C₂×C₈	—	149	2	no	—
208	C₂×C₈	—	149	2	no	—
209	C₂×C₈	—	149	2	no	—
210	C₂×C₈	—	149	2	no	—
211	C₄⋊C₄	—	149	1	no	—
212	C₄⋊C₄	—	149	1	no	—
213	C₄⋊C₄	—	149	1	no	—
214	C₄⋊C₄	—	149	1	no	—
215	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
216	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
217	C₂²⋊C₄	—	149	1	no	—
218	C₂²⋊C₄	—	149	1	no	—
219	C₂²⋊C₄	—	149	1	no	—
220	C₂²⋊C₄	—	149	1	no	—
221	C₂²⋊C₄	C₂×#10597010	1	1	yes	characteristic
222	C₂²⋊C₄	C₂×#10597010	1	1	yes	characteristic
223	C₄²	—	149	1	no	—
224	C₄²	—	149	1	no	—
225	C₄²	—	149	1	no	—
226	C₄²	—	149	1	no	—
227	C₄²	—	149	1	no	—
228	C₄²	—	149	1	no	—
229	C₄²	—	149	1	no	—
230	C₄²	—	149	1	no	—
231	C₄²	C₂×#10597010	1	1	yes	characteristic
232	C₄²	C₂×#10597010	1	1	yes	characteristic
233	C₂³	—	149	1	no	—
234	C₂³	—	149	1	no	—
235	C₂³	C₂²×#10597010	1	1	yes	characteristic
236	Q₈	—	298	2	no	—
237	D₄	—	298	2	no	—
238	D₄	—	298	2	no	—
239	D₄	—	298	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	298	2	no	—
242	C₂×C₄	—	298	2	no	—
243	C₂×C₄	—	298	2	no	—
244	C₂×C₄	—	298	2	no	—
245	C₂×C₄	—	298	2	no	—
246	C₂×C₄	—	298	2	no	—
247	C₂×C₄	—	298	2	no	—
248	C₂×C₄	—	149	1	no	—
249	C₂×C₄	—	149	1	no	—
250	C₂×C₄	—	149	1	no	—
251	C₂×C₄	—	149	1	no	—
252	C₂×C₄	—	149	1	no	—
253	C₂×C₄	—	149	1	no	—
254	C₂×C₄	—	149	4	no	—
255	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
256	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
257	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
258	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
259	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
260	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
261	C₈	—	298	2	no	—
262	C₈	—	298	2	no	—
263	C₈	—	298	2	no	—
264	C₈	—	149	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	298	1	no	—
269	C₂²	—	298	1	no	—
270	C₂²	—	298	1	no	—
271	C₂²	—	298	1	no	—
272	C₂²	—	298	1	no	—
273	C₂²	—	149	2	no	—
274	C₂²	C₂³×#10597010	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	298	1	no	—
281	C₄	—	298	1	no	—
282	C₄	—	298	1	no	—
283	C₄	—	149	2	no	—
284	C₄	—	149	2	no	—
285	C₄	—	149	2	no	—
286	C₄	D₁₄₉⋊M₁₆	1	2	yes	—
287	C₂	—	4	1	no	—
288	C₂	—	298	2	no	—
289	C₂	—	596	1	no	—
290	C₂	#41×#10597010	1	1	yes	characteristic
291	C₂	C₂×D₁₄₉⋊M₁₆	1	1	yes	characteristic
292	C₂	D₁₄₉⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#10597010
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Frattini	C₂²	C₂³×#10597010
Fitting	#117×C₁₄₉	C₄
prime radical	C₂×#14558736	C₄
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	#117	#10597010
hypocenter	C₁₄₉	#2965
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. #23750677	C₂	semidirect
2. #23750658	C₂	semidirect
3. #23750657	C₂	semidirect
4. #23750655	C₂	semidirect
5. #23750644	C₂	semidirect
6. #23750643	C₂	semidirect
7. #23750642	C₂	semidirect
8. C₄×#14558746	C₂	semidirect
9. C₄×#14558745	C₂	semidirect
10. C₄×D₁₄₉⋊C₈	C₂	semidirect
89. #11924753	#41	semidirect
109. C₄×D₁₄₉	M₁₆	semidirect
146. C₁₄₉	#2965	semidirect

Maximal subgroups

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

subgroup	index
#23750677	2
#23750658	2
#23750657	2
#23750655	2
#23750644	2
#23750643	2
#23750642	2
C₄×#14558746	2
C₄×#14558745	2
C₄×D₁₄₉⋊C₈	2
#23750093	2
#2965	149