Subgroups — #34061262

Order 19072.

556 conjugacy classes of subgroups.

The subgroup lattice has 216 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
2	#23750590	C₂	1	2	yes	maximal, complemented
3	#23750589	C₂	1	4	yes	maximal, complemented
4	C₄×#14558746	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#14558745	C₂	1	2	yes	maximal, complemented
6	#23750584	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#18011966	C₂	1	2	yes	maximal, complemented
8	C₄×#14558735	C₂	1	1	yes	maximal, characteristic
9	#18011998	C₄	1	16	yes	complemented
10	C₂×#14558746	C₂²	1	2	yes	—
11	C₂×#14558745	C₂²	1	4	yes	—
12	C₂×D₁₄₉⋊C₈	C₂²	1	2	yes	—
13	C₂²×#11924753	C₂²	1	4	yes	—
14	#18011966	C₂²	1	1	yes	characteristic
15	#18011966	C₂²	1	2	yes	—
16	#18011966	C₂²	1	4	yes	—
17	#18011966	C₂²	1	1	yes	characteristic
18	#18011966	C₂²	1	2	yes	—
19	C₄×#11924753	C₂²	1	4	yes	—
20	C₄×#11924753	C₂²	1	2	yes	—
21	C₂×#14558735	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₉₈	C₄	1	1	yes	characteristic
23	#18011876	C₄	1	1	yes	characteristic
24	D₄×Dic₁₄₉	C₄	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₄₉	C₄	1	2	yes	—
26	#18011871	C₄	1	2	yes	—
27	#18011869	C₂²	1	1	yes	characteristic
28	#18011867	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
30	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₄₉	C₄	1	1	yes	characteristic
32	#14558746	C₂×C₄	1	8	yes	complemented
33	#14558745	C₂×C₄	1	16	yes	complemented
34	D₁₄₉⋊C₈	C₂×C₄	1	8	yes	complemented
35	C₂×#11924753	C₂×C₄	1	16	yes	complemented
36	C₂×#11924753	C₂³	1	4	yes	—
37	C₂×#11924753	C₂³	1	2	yes	—
38	C₂×#11924753	C₂³	1	2	yes	—
39	#14558735	C₂×C₄	1	8	yes	—
40	C₂×Dic₂₉₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
42	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
43	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
44	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
48	#14558715	C₂×C₄	1	2	yes	—
49	#14558715	C₂×C₄	1	2	yes	—
50	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
51	#14558713	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₄₉	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₄₉	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
56	C₄×Dic₁₄₉	C₂³	1	2	yes	—
57	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₄₉	C₂×C₄	1	2	yes	—
60	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
62	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
63	#11924753	C₂²×C₄	1	4	yes	—
64	#11924753	C₂²×C₄	1	8	yes	—
65	#11924753	C₂²×C₄	1	4	yes	—
66	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
67	Dic₂₉₈	C₄²	1	4	yes	—
68	C₂²⋊D₁₄₉	C₄²	1	8	yes	—
69	C₂×Dic₁₄₉	—	4	2	no	—
70	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₄₉	C₂²×C₄	1	8	yes	—
75	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
76	C₄×D₁₄₉	—	4	1	no	—
77	C₄×D₁₄₉	—	4	1	no	—
78	C₄×D₁₄₉	C₂²×C₄	1	4	yes	—
79	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
80	D₄×C₁₄₉	C₄²	1	4	yes	—
81	C₂×C₄×C₁₄₉	—	4	2	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	2	yes	—
85	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₁₄₉	—	2	2	no	—
87	C₂²×C₁₄₉	—	2	2	no	—
88	C₂²×C₁₄₉	C₂×C₄²	1	4	yes	—
89	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₁₄₉	—	2	1	no	—
91	C₂×D₁₄₉	—	2	1	no	—
92	C₂×D₁₄₉	C₂×C₄²	1	2	yes	—
93	C₄×C₁₄₉	—	2	1	no	—
94	C₄×C₁₄₉	—	2	2	no	—
95	C₄×C₁₄₉	—	2	1	no	—
96	C₄×C₁₄₉	C₂×C₄²	1	2	yes	—
97	Dic₁₄₉	—	2	1	no	—
98	Dic₁₄₉	—	2	1	no	—
99	Dic₁₄₉	—	2	2	no	—
100	Dic₁₄₉	C₂×C₄²	1	4	yes	—
101	Dic₁₄₉	C₂³×C₄	1	2	yes	—
102	C₂×C₁₄₉	—	2	4	no	—
103	C₂×C₁₄₉	C₂²×C₄²	1	1	yes	characteristic
104	C₂×C₁₄₉	#568	1	1	yes	characteristic
105	C₂×C₁₄₉	#568	1	1	yes	characteristic
106	D₁₄₉	—	2	2	no	—
107	C₁₄₉	#2875	1	1	yes	characteristic, complemented, Sylow
108	#2875	—	149	1	no	maximal, Sylow
109	C₂×#131	—	149	2	no	—
110	C₄×#41	—	149	1	no	—
111	#405	—	149	4	no	—
112	#405	—	149	2	no	—
113	C₄×M₁₆	—	149	2	no	—
114	C₄×M₁₆	—	149	1	no	—
115	C₂×#97	—	149	2	no	—
116	C₂×#97	—	149	1	no	—
117	C₂×#41	—	149	1	no	—
118	#131	—	149	16	no	—
119	C₂×M₁₆	—	149	4	no	—
120	C₂×M₁₆	—	149	2	no	—
121	C₂²×C₈	—	149	4	no	—
122	C₂²×C₈	—	149	2	no	—
123	C₄×Q₈	—	149	1	no	—
124	C₄×Q₈	—	149	1	no	—
125	C₄×D₄	—	149	1	no	—
126	C₄×D₄	—	149	2	no	—
127	C₄×D₄	—	149	2	no	—
128	C₄×D₄	#10597010	1	1	yes	characteristic
129	#117	—	149	1	no	—
130	#117	—	149	2	no	—
131	C₂×C₄²	—	149	2	no	—
132	C₂×C₄²	—	149	1	no	—
133	#97	—	149	1	no	—
134	#97	—	149	2	no	—
135	#97	—	149	1	no	—
136	#97	—	149	4	no	—
137	#97	—	149	2	no	—
138	C₄×C₈	—	149	4	no	—
139	C₄×C₈	—	149	2	no	—
140	#41	—	149	8	no	—
141	C₂×Q₈	—	149	1	no	—
142	C₂×D₄	—	149	2	no	—
143	C₂×D₄	C₂×#10597010	1	1	yes	characteristic
144	C₂²×C₄	—	149	2	no	—
145	C₂²×C₄	—	149	1	no	—
146	C₂²×C₄	—	149	1	no	—
147	C₂²×C₄	—	149	2	no	—
148	C₂²×C₄	—	149	1	no	—
149	C₂²×C₄	C₂×#10597010	1	2	yes	—
150	M₁₆	—	149	16	no	—
151	M₁₆	—	149	8	no	—
152	C₂×C₈	—	149	16	no	—
153	C₂×C₈	—	149	4	no	—
154	C₂×C₈	—	149	2	no	—
155	C₂×C₈	—	149	2	no	—
156	C₂×C₈	—	149	8	no	—
157	C₄⋊C₄	—	149	2	no	—
158	C₄⋊C₄	—	149	2	no	—
159	C₄⋊C₄	—	149	1	no	—
160	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
161	C₂²⋊C₄	—	149	2	no	—
162	C₂²⋊C₄	—	149	2	no	—
163	C₂²⋊C₄	C₂×#10597010	1	2	yes	—
164	C₄²	—	149	2	no	—
165	C₄²	—	149	1	no	—
166	C₄²	—	149	1	no	—
167	C₄²	—	149	1	no	—
168	C₄²	—	149	2	no	—
169	C₄²	—	149	2	no	—
170	C₄²	C₂×#10597010	1	1	yes	characteristic
171	C₂³	—	149	1	no	—
172	C₂³	C₂²×#10597010	1	2	yes	—
173	Q₈	—	149	4	no	—
174	D₄	—	149	8	no	—
175	D₄	C₄×#10597010	1	4	yes	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	596	1	no	—
178	C₂×C₄	—	596	1	no	—
179	C₂×C₄	—	596	2	no	—
180	C₂×C₄	—	149	1	no	—
181	C₂×C₄	—	149	1	no	—
182	C₂×C₄	—	149	2	no	—
183	C₂×C₄	—	149	8	no	—
184	C₂×C₄	—	149	2	no	—
185	C₂×C₄	—	149	1	no	—
186	C₂×C₄	—	149	4	no	—
187	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
188	C₂×C₄	C₂²×#10597010	1	2	yes	—
189	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
190	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
191	C₈	—	149	4	no	—
192	C₈	—	149	8	no	—
193	C₈	—	149	4	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	298	1	no	—
197	C₂²	—	298	1	no	—
198	C₂²	—	149	2	no	—
199	C₂²	C₂×C₄×#10597010	1	4	yes	—
200	C₂²	C₂³×#10597010	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	298	1	no	—
205	C₄	—	298	1	no	—
206	C₄	—	298	2	no	—
207	C₄	—	149	2	no	—
208	C₄	—	149	4	no	—
209	C₄	C₂×C₄×#10597010	1	2	yes	—
210	C₂	—	2	4	no	—
211	C₂	—	298	2	no	—
212	C₂	C₂²×C₄×#10597010	1	1	yes	characteristic
213	C₂	#23750703	1	1	yes	characteristic
214	C₂	#23750704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597010
commutator	C₂×C₁₄₉	C₂²×C₄²
Fitting	C₄×D₄×C₁₄₉	C₄
socle	C₂²×C₁₄₉	C₂³×C₄
prime radical	C₂×#14558735	C₂²
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	C₄×D₄	#10597010
hypocenter	C₁₄₉	#2875
LCS	G ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #23750590	C₂	semidirect
3. #23750589	C₂	semidirect
4. C₄×#14558746	C₂	semidirect
5. C₄×#14558745	C₂	semidirect
6. #23750584	C₂	semidirect
7. C₂×#18011966	C₂	semidirect
9. #18011998	C₄	semidirect
32. #14558746	C₂×C₄	semidirect
33. #14558745	C₂×C₄	semidirect
34. D₁₄₉⋊C₈	C₂×C₄	semidirect
35. C₂×#11924753	C₂×C₄	semidirect
107. C₁₄₉	#2875	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
#23750590	2
#23750589	2
C₄×#14558746	2
C₄×#14558745	2
#23750584	2
C₂×#18011966	2
C₄×#14558735	2
#2875	149