Subgroups — #34053676

Order 19072.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750159	C₂	1	1	yes	maximal, characteristic
2	#23749753	C₂	1	2	yes	maximal, complemented
3	#23749701	C₂	1	2	yes	maximal, complemented
4	#23749698	C₂	1	2	yes	maximal, complemented
5	#23749694	C₂	1	2	yes	maximal, complemented
6	#23749684	C₂	1	2	yes	maximal, complemented
7	#23749682	C₂	1	1	yes	maximal, characteristic, complemented
8	#23749579	C₂	1	2	yes	maximal, complemented
9	C₄×M₁₆×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14558719	C₂²	1	2	yes	—
11	C₂×C₈×D₁₄₉	C₂²	1	2	yes	—
12	#18011905	C₄	1	2	yes	—
13	#18011903	C₄	1	1	yes	characteristic
14	#18011899	C₄	1	1	yes	characteristic
15	#18011894	C₄	1	2	yes	—
16	#18011890	C₄	1	1	yes	characteristic
17	#18011885	C₄	1	1	yes	characteristic
18	C₄×Dic₂₉₈	C₂²	1	1	yes	characteristic
19	C₄×D₅₉₆	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₄₉	C₂²	1	2	yes	—
21	#18011868	C₂²	1	2	yes	—
22	C₄⋊#11924746	C₂²	1	2	yes	—
23	#18011840	C₂²	1	2	yes	—
24	#18011840	C₂²	1	2	yes	—
25	C₂²⋊#11924746	C₂²	1	1	yes	characteristic
26	C₂²⋊#11924746	C₂²	1	1	yes	characteristic
27	#18011823	C₂²	1	2	yes	—
28	#18011823	C₂²	1	2	yes	—
29	#18011822	C₂²	1	2	yes	—
30	#18011821	C₂²	1	2	yes	—
31	C₈×Dic₁₄₉	C₂²	1	2	yes	—
32	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
34	#97×C₁₄₉	C₂²	1	2	yes	—
35	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
37	C₂×Dic₂₉₈	C₂×C₄	1	1	yes	characteristic
38	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
41	#14558719	—	2	4	no	—
42	C₂×#11924746	C₂³	1	2	yes	—
43	C₂×#11924746	C₂³	1	2	yes	—
44	C₈×D₁₄₉	—	2	4	no	—
45	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
47	#14558715	C₂×C₄	1	2	yes	—
48	#14558715	C₂×C₄	1	2	yes	—
49	#14558715	C₂×C₄	1	2	yes	—
50	#14558715	C₂×C₄	1	2	yes	—
51	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
52	#14558713	C₂×C₄	1	2	yes	—
53	#14558713	C₂×C₄	1	2	yes	—
54	#14558713	C₂×C₄	1	2	yes	—
55	#14558713	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₄₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
60	M₁₆×C₁₄₉	—	2	4	no	—
61	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
62	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
63	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
66	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
67	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
68	Dic₂₉₈	—	4	1	no	—
69	D₅₉₆	—	4	1	no	—
70	C₂²⋊D₁₄₉	—	4	2	no	—
71	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
74	C₄×D₁₄₉	—	2	2	no	—
75	C₄×D₁₄₉	—	2	2	no	—
76	#11924746	—	2	2	no	—
77	#11924746	—	2	2	no	—
78	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₄₉	—	2	2	no	—
80	C₂×C₄×C₁₄₉	—	4	1	no	—
81	C₂×C₄×C₁₄₉	—	4	1	no	—
82	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
88	C₈×C₁₄₉	—	2	2	no	—
89	C₈×C₁₄₉	#41	1	4	yes	—
90	C₂²×C₁₄₉	—	2	1	no	—
91	C₂²×C₁₄₉	—	2	1	no	—
92	C₂²×C₁₄₉	—	2	1	no	—
93	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₁₄₉	—	2	2	no	—
95	C₂×D₁₄₉	—	2	2	no	—
96	C₂×D₁₄₉	—	2	2	no	—
97	C₄×C₁₄₉	—	2	1	no	—
98	C₄×C₁₄₉	—	2	1	no	—
99	C₄×C₁₄₉	—	2	1	no	—
100	C₄×C₁₄₉	—	2	1	no	—
101	C₄×C₁₄₉	—	2	1	no	—
102	C₄×C₁₄₉	C₂×#41	1	2	yes	—
103	Dic₁₄₉	—	2	2	no	—
104	Dic₁₄₉	—	2	2	no	—
105	Dic₁₄₉	—	2	2	no	—
106	C₂×C₁₄₉	—	4	1	no	—
107	C₂×C₁₄₉	C₄×#41	1	1	yes	characteristic
108	C₂×C₁₄₉	#568	1	1	yes	characteristic
109	C₂×C₁₄₉	#568	1	1	yes	characteristic
110	D₁₄₉	—	4	2	no	—
111	C₁₄₉	#2967	1	1	yes	characteristic, complemented, Sylow
112	#2967	—	149	1	no	maximal, Sylow
113	#529	—	149	1	no	—
114	#446	—	149	2	no	—
115	#436	—	149	2	no	—
116	#435	—	149	2	no	—
117	#435	—	149	2	no	—
118	#433	—	149	2	no	—
119	#432	—	149	1	no	—
120	#405	—	149	2	no	—
121	C₄×M₁₆	D₁₄₉	1	1	yes	characteristic, complemented
122	C₂×M₁₆	—	149	2	no	—
123	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
124	C₂²×C₈	—	149	2	no	—
125	#126	—	149	2	no	—
126	#125	—	149	1	no	—
127	#124	—	149	1	no	—
128	#123	—	149	2	no	—
129	C₂²⋊Q₈	—	149	1	no	—
130	#121	—	149	1	no	—
131	C₄×Q₈	—	149	1	no	—
132	C₄×D₄	—	149	2	no	—
133	C₄×D₄	—	149	1	no	—
134	#117	—	149	2	no	—
135	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
136	C₄⋊C₈	—	149	2	no	—
137	C₄⋊C₈	—	149	2	no	—
138	C₄⋊C₈	—	149	2	no	—
139	C₂²⋊C₈	—	149	1	no	—
140	C₂²⋊C₈	—	149	1	no	—
141	C₂²⋊C₈	—	149	2	no	—
142	C₂²⋊C₈	—	149	2	no	—
143	#97	—	149	2	no	—
144	#97	—	149	2	no	—
145	#97	C₂×D₁₄₉	1	2	yes	—
146	C₄×C₈	—	149	2	no	—
147	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
148	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
149	C₂×Q₈	—	149	1	no	—
150	C₂×D₄	—	149	2	no	—
151	C₂×D₄	—	149	1	no	—
152	C₂²×C₄	—	149	2	no	—
153	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
156	M₁₆	—	2	4	no	—
157	M₁₆	—	298	4	no	—
158	C₂×C₈	—	298	4	no	—
159	C₂×C₈	—	149	2	no	—
160	C₂×C₈	—	149	2	no	—
161	C₂×C₈	C₂²×D₁₄₉	1	2	yes	—
162	C₂×C₈	C₂²×D₁₄₉	1	2	yes	—
163	C₄⋊C₄	—	149	2	no	—
164	C₄⋊C₄	—	149	2	no	—
165	C₄⋊C₄	—	149	1	no	—
166	C₄⋊C₄	—	149	2	no	—
167	C₄⋊C₄	—	149	1	no	—
168	C₄⋊C₄	—	149	2	no	—
169	C₂²⋊C₄	—	149	2	no	—
170	C₂²⋊C₄	—	149	2	no	—
171	C₂²⋊C₄	—	149	2	no	—
172	C₂²⋊C₄	—	149	2	no	—
173	C₂²⋊C₄	—	149	2	no	—
174	C₄²	—	149	2	no	—
175	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
176	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
177	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
178	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
179	C₂³	—	149	2	no	—
180	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
181	Q₈	—	596	1	no	—
182	D₄	—	596	1	no	—
183	D₄	—	596	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	298	2	no	—
188	C₂×C₄	—	298	2	no	—
189	C₂×C₄	—	149	2	no	—
190	C₂×C₄	—	149	2	no	—
191	C₂×C₄	—	149	2	no	—
192	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	298	2	no	—
200	C₈	—	298	2	no	—
201	C₈	#14558736	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	298	2	no	—
206	C₂²	—	298	2	no	—
207	C₂²	—	298	2	no	—
208	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	298	2	no	—
215	C₄	—	298	2	no	—
216	C₄	—	298	2	no	—
217	C₄	C₂×#14558736	1	2	yes	—
218	C₂	—	4	1	no	—
219	C₂	—	596	2	no	—
220	C₂	C₄×#14558736	1	1	yes	characteristic
221	C₂	#23750391	1	1	yes	characteristic
222	C₂	#23750391	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₄₉
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	C₄×M₁₆×C₁₄₉	C₂
prime radical	#18011885	C₄
cosocle	C₂×C₄×C₁₄₉	C₂⁴
hypercenter	C₄×M₁₆	D₁₄₉
hypocenter	C₁₄₉	#2967
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #23749753	C₂	semidirect
3. #23749701	C₂	semidirect
4. #23749698	C₂	semidirect
5. #23749694	C₂	semidirect
6. #23749684	C₂	semidirect
7. #23749682	C₂	semidirect
8. #23749579	C₂	semidirect
9. C₄×M₁₆×C₁₄₉	C₂	semidirect
111. C₁₄₉	#2967	semidirect
121. C₄×M₁₆	D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#23750159	2
#23749753	2
#23749701	2
#23749698	2
#23749694	2
#23749684	2
#23749682	2
#23749579	2
C₄×M₁₆×C₁₄₉	2
#2967	149