Subgroups — #34061433

Order 19072.

492 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18011998	C₂	1	2	yes	maximal, complemented
2	#23750660	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750659	C₂	1	1	yes	maximal, characteristic, complemented
4	#23750657	C₂	1	2	yes	maximal, complemented
5	#23750654	C₂	1	2	yes	maximal, complemented
6	#23750653	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×#11924753	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊#11924753	C₂	1	1	yes	maximal, characteristic, complemented
9	#23750598	C₂	1	2	yes	maximal, complemented
10	C₄×#14558745	C₂	1	1	yes	maximal, characteristic, complemented
11	#23750170	C₂	1	1	yes	maximal, characteristic
12	#18011998	—	2	8	no	—
13	C₂×#14558746	C₂²	1	2	yes	—
14	C₂×#14558745	C₂²	1	2	yes	—
15	C₂×#14558745	C₂²	1	1	yes	characteristic
16	C₂×#14558745	C₂²	1	1	yes	characteristic
17	C₂×D₁₄₉⋊C₈	C₂²	1	2	yes	—
18	C₂²×#11924753	C₂²	1	2	yes	—
19	C₂²×#11924753	C₂²	1	2	yes	—
20	C₄⋊#11924753	C₂²	1	2	yes	—
21	C₄⋊#11924753	C₂²	1	1	yes	characteristic
22	C₄⋊#11924753	C₂²	1	1	yes	characteristic
23	#18011968	C₂²	1	2	yes	—
24	#18011968	C₂²	1	2	yes	—
25	C₂²⋊#11924753	C₂²	1	2	yes	—
26	C₂²⋊#11924753	C₂²	1	2	yes	—
27	#18011966	C₂²	1	1	yes	characteristic
28	#18011966	C₂²	1	1	yes	characteristic
29	C₄×#11924753	C₂²	1	1	yes	characteristic
30	C₄×#11924753	C₂²	1	1	yes	characteristic
31	C₂×#14558735	C₂²	1	2	yes	—
32	#18011910	C₄	1	1	yes	characteristic
33	#18011907	C₄	1	1	yes	characteristic
34	#18011897	C₄	1	2	yes	—
35	#18011885	C₄	1	1	yes	characteristic
36	#18011883	C₄	1	2	yes	—
37	D₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
38	#18011873	C₂²	1	1	yes	characteristic
39	#18011871	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
41	#121×C₁₄₉	C₄	1	1	yes	characteristic
42	#14558746	—	2	4	no	—
43	#14558745	—	2	4	no	—
44	#14558745	—	2	2	no	—
45	#14558745	D₄	1	4	yes	complemented
46	D₁₄₉⋊C₈	—	2	4	no	—
47	C₂×#11924753	—	2	4	no	—
48	C₂×#11924753	—	2	4	no	—
49	C₂×#11924753	C₂³	1	1	yes	characteristic
50	C₂×#11924753	C₂³	1	1	yes	characteristic
51	C₂×#11924753	C₂³	1	2	yes	—
52	C₂×#11924753	C₂³	1	2	yes	—
53	C₂×#11924753	C₂³	1	1	yes	characteristic
54	C₂×#11924753	C₂³	1	1	yes	characteristic
55	#14558735	—	2	4	no	—
56	#14558735	—	2	4	no	—
57	C₂×Dic₂₉₈	C₂×C₄	1	2	yes	—
58	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
62	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
64	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
66	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
67	#14558715	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
69	#14558713	C₂×C₄	1	2	yes	—
70	#14558713	C₂×C₄	1	2	yes	—
71	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
74	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
76	C₂×D₄×C₁₄₉	C₂×C₄	1	2	yes	—
77	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
80	#11924753	—	2	2	no	—
81	#11924753	—	2	2	no	—
82	#11924753	—	2	1	no	—
83	#11924753	—	2	1	no	—
84	#11924753	C₂×D₄	1	2	yes	—
85	#11924753	C₂×D₄	1	2	yes	—
86	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
87	Dic₂₉₈	—	2	4	no	—
88	D₅₉₆	—	2	2	no	—
89	C₂²⋊D₁₄₉	—	2	4	no	—
90	C₂²⋊D₁₄₉	—	2	4	no	—
91	C₂²⋊D₁₄₉	—	4	2	no	—
92	C₂×Dic₁₄₉	—	2	2	no	—
93	C₂×Dic₁₄₉	—	2	2	no	—
94	C₂×Dic₁₄₉	—	4	1	no	—
95	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
98	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₄₉	C₂×D₄	1	2	yes	—
100	C₂×Dic₁₄₉	#41	1	2	yes	—
101	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
102	C₄×D₁₄₉	—	2	2	no	—
103	C₄×D₁₄₉	—	2	2	no	—
104	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
105	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
106	D₄×C₁₄₉	—	2	4	no	—
107	D₄×C₁₄₉	—	2	2	no	—
108	C₂×C₄×C₁₄₉	—	2	2	no	—
109	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₁₄₉	—	2	1	no	—
113	C₂²×C₁₄₉	—	2	1	no	—
114	C₂²×C₁₄₉	—	2	2	no	—
115	C₂²×C₁₄₉	—	2	2	no	—
116	C₂²×C₁₄₉	—	2	2	no	—
117	C₂²×C₁₄₉	C₄×D₄	1	2	yes	—
118	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₁₄₉	—	2	2	no	—
120	C₂×D₁₄₉	—	2	2	no	—
121	C₂×D₁₄₉	—	2	2	no	—
122	C₄×C₁₄₉	—	2	2	no	—
123	C₄×C₁₄₉	—	2	1	no	—
124	C₄×C₁₄₉	#131	1	2	yes	complemented
125	Dic₁₄₉	—	2	2	no	—
126	Dic₁₄₉	—	2	1	no	—
127	Dic₁₄₉	—	2	1	no	—
128	Dic₁₄₉	C₄×D₄	1	2	yes	—
129	Dic₁₄₉	C₂²×D₄	1	1	yes	characteristic
130	Dic₁₄₉	C₂×#41	1	1	yes	characteristic
131	C₂×C₁₄₉	—	2	2	no	—
132	C₂×C₁₄₉	—	4	2	no	—
133	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
135	C₂×C₁₄₉	#568	1	1	yes	characteristic
136	D₁₄₉	—	4	2	no	—
137	C₁₄₉	#2935	1	1	yes	characteristic, complemented, Sylow
138	#2935	—	149	1	no	maximal, Sylow
139	C₂×#131	—	149	2	no	—
140	#532	—	149	1	no	—
141	#436	—	149	1	no	—
142	#436	—	149	1	no	—
143	#435	—	149	2	no	—
144	#435	—	149	2	no	—
145	C₈×D₄	—	149	1	no	—
146	C₈×D₄	—	149	1	no	—
147	C₂×C₄⋊C₈	—	149	1	no	—
148	#408	—	149	2	no	—
149	C₄×M₁₆	—	149	1	no	—
150	C₂×#41	—	149	2	no	—
151	#131	—	298	8	no	—
152	C₂×M₁₆	—	149	2	no	—
153	C₂×M₁₆	—	149	2	no	—
154	C₂×M₁₆	—	149	1	no	—
155	C₂×M₁₆	—	149	1	no	—
156	C₂²×C₈	—	149	2	no	—
157	C₂²×C₈	—	149	2	no	—
158	C₂²×C₈	—	149	2	no	—
159	C₄⋊Q₈	—	149	1	no	—
160	#127	—	149	1	no	—
161	#124	—	149	2	no	—
162	#121	—	149	2	no	—
163	#121	—	149	1	no	—
164	#121	#10597010	1	1	yes	characteristic
165	C₄×D₄	—	149	1	no	—
166	C₄×D₄	—	149	1	no	—
167	C₄×D₄	—	149	2	no	—
168	C₂×C₄²	—	149	1	no	—
169	C₄⋊C₈	—	149	1	no	—
170	C₄⋊C₈	—	149	2	no	—
171	C₄⋊C₈	—	149	1	no	—
172	C₂²⋊C₈	—	149	2	no	—
173	C₂²⋊C₈	—	149	2	no	—
174	C₂²⋊C₈	—	149	2	no	—
175	C₂²⋊C₈	—	149	2	no	—
176	#97	—	149	1	no	—
177	#97	—	149	1	no	—
178	C₄×C₈	—	149	1	no	—
179	C₄×C₈	—	149	1	no	—
180	#41	—	298	4	no	—
181	#41	—	298	4	no	—
182	C₂×Q₈	—	149	2	no	—
183	C₂×D₄	—	149	2	no	—
184	C₂×D₄	—	149	2	no	—
185	C₂×D₄	—	149	2	no	—
186	C₂×D₄	—	149	1	no	—
187	C₂×D₄	C₂×#10597010	1	1	yes	characteristic
188	C₂×D₄	C₂×#10597010	1	2	yes	—
189	C₂²×C₄	—	149	2	no	—
190	C₂²×C₄	—	149	2	no	—
191	C₂²×C₄	—	149	1	no	—
192	C₂²×C₄	—	149	1	no	—
193	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
194	M₁₆	—	298	2	no	—
195	M₁₆	—	298	4	no	—
196	M₁₆	—	298	4	no	—
197	M₁₆	—	149	4	no	—
198	C₂×C₈	—	298	4	no	—
199	C₂×C₈	—	298	4	no	—
200	C₂×C₈	—	298	4	no	—
201	C₂×C₈	—	149	1	no	—
202	C₂×C₈	—	149	1	no	—
203	C₂×C₈	—	149	2	no	—
204	C₂×C₈	—	149	2	no	—
205	C₂×C₈	—	149	1	no	—
206	C₂×C₈	—	149	1	no	—
207	C₄⋊C₄	—	149	2	no	—
208	C₄⋊C₄	—	149	1	no	—
209	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
210	C₂²⋊C₄	—	149	2	no	—
211	C₂²⋊C₄	—	149	2	no	—
212	C₂²⋊C₄	—	149	2	no	—
213	C₂²⋊C₄	C₂×#10597010	1	2	yes	—
214	C₄²	—	149	1	no	—
215	C₄²	—	149	1	no	—
216	C₄²	—	149	1	no	—
217	C₄²	—	149	1	no	—
218	C₂³	—	149	2	no	—
219	C₂³	C₂²×#10597010	1	2	yes	—
220	C₂³	C₂²×#10597010	1	1	yes	characteristic
221	Q₈	—	298	4	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	D₄	—	298	4	no	—
225	D₄	—	298	4	no	—
226	D₄	—	298	2	no	—
227	D₄	—	596	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	298	2	no	—
230	C₂×C₄	—	298	2	no	—
231	C₂×C₄	—	298	2	no	—
232	C₂×C₄	—	298	2	no	—
233	C₂×C₄	—	596	1	no	—
234	C₂×C₄	—	149	1	no	—
235	C₂×C₄	—	149	1	no	—
236	C₂×C₄	—	149	2	no	—
237	C₂×C₄	—	149	2	no	—
238	C₂×C₄	—	149	2	no	—
239	C₂×C₄	—	149	1	no	—
240	C₂×C₄	—	149	1	no	—
241	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
242	C₂×C₄	C₂²×#10597010	1	2	yes	—
243	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
244	C₈	—	298	2	no	—
245	C₈	—	298	2	no	—
246	C₈	—	298	1	no	—
247	C₈	—	298	1	no	—
248	C₈	—	149	2	no	—
249	C₈	—	149	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	298	2	no	—
256	C₂²	—	298	2	no	—
257	C₂²	—	298	2	no	—
258	C₂²	D₄×#10597010	1	2	yes	—
259	C₂²	C₂³×#10597010	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	298	2	no	—
263	C₄	—	298	1	no	—
264	C₄	—	298	1	no	—
265	C₄	—	149	1	no	—
266	C₄	—	149	1	no	—
267	C₄	—	149	2	no	—
268	C₄	#18011998	1	2	yes	complemented
269	C₂	—	2	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	596	2	no	—
272	C₂	C₂×D₄×#10597010	1	1	yes	characteristic
273	C₂	C₂×#18011998	1	1	yes	characteristic
274	C₂	#23750704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597010
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#121×C₁₄₉	C₄
prime radical	#23750170	C₂
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	#121	#10597010
hypocenter	C₁₄₉	#2935
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂×#18011998	C₂	semidirect
2. #23750660	C₂	semidirect
3. #23750659	C₂	semidirect
4. #23750657	C₂	semidirect
5. #23750654	C₂	semidirect
6. #23750653	C₂	semidirect
7. D₄×#11924753	C₂	semidirect
8. C₂×C₄⋊#11924753	C₂	semidirect
9. #23750598	C₂	semidirect
10. C₄×#14558745	C₂	semidirect
45. #14558745	D₄	semidirect
124. C₄×C₁₄₉	#131	semidirect
137. C₁₄₉	#2935	semidirect
268. C₄	#18011998	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18011998	2
#23750660	2
#23750659	2
#23750657	2
#23750654	2
#23750653	2
D₄×#11924753	2
C₂×C₄⋊#11924753	2
#23750598	2
C₄×#14558745	2
#23750170	2
#2935	149