Subgroups — #34061523

Order 19072.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750676	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11924753	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750660	C₂	1	1	yes	maximal, characteristic, complemented
4	#23750658	C₂	1	2	yes	maximal, complemented
5	#23750654	C₂	1	2	yes	maximal, complemented
6	D₄×#11924753	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750649	C₂	1	2	yes	maximal, complemented
8	#23750644	C₂	1	1	yes	maximal, characteristic, complemented
9	#23750589	C₂	1	2	yes	maximal, complemented
10	C₄×#14558746	C₂	1	1	yes	maximal, characteristic, complemented
11	#23750154	C₂	1	1	yes	maximal, characteristic
12	C₂×#14558746	C₂²	1	1	yes	characteristic
13	C₂×#14558746	C₂²	1	1	yes	characteristic
14	C₂×#14558745	C₂²	1	2	yes	—
15	C₂²×#11924753	C₂²	1	2	yes	—
16	#18011978	C₂²	1	2	yes	—
17	C₄⋊#11924753	C₂²	1	1	yes	characteristic
18	C₄⋊#11924753	C₂²	1	1	yes	characteristic
19	C₄⋊#11924753	C₂²	1	2	yes	—
20	#18011968	C₂²	1	2	yes	—
21	C₂²⋊#11924753	C₂²	1	2	yes	—
22	C₂²⋊#11924753	C₂²	1	2	yes	—
23	#18011966	C₂²	1	2	yes	—
24	#18011966	C₂²	1	1	yes	characteristic
25	#18011966	C₂²	1	1	yes	characteristic
26	C₄×#11924753	C₂²	1	2	yes	—
27	C₄×#11924753	C₂²	1	1	yes	characteristic
28	C₄×#11924753	C₂²	1	1	yes	characteristic
29	C₄×#11924753	C₂²	1	2	yes	—
30	#18011904	C₄	1	2	yes	—
31	#18011903	C₄	1	1	yes	characteristic
32	#18011893	C₄	1	2	yes	—
33	#18011887	C₄	1	1	yes	characteristic
34	#18011882	C₄	1	1	yes	characteristic
35	Q₈×Dic₁₄₉	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
37	#18011871	C₂²	1	2	yes	—
38	#18011867	C₂²	1	2	yes	—
39	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
40	#124×C₁₄₉	C₄	1	1	yes	characteristic
41	#14558746	—	2	2	no	—
42	#14558746	—	2	2	no	—
43	#14558745	—	2	4	no	—
44	C₂×#11924753	—	2	4	no	—
45	C₂×#11924753	C₂³	1	2	yes	—
46	C₂×#11924753	C₂³	1	1	yes	characteristic
47	C₂×#11924753	C₂³	1	1	yes	characteristic
48	C₂×#11924753	C₂³	1	2	yes	—
49	C₂×#11924753	C₂³	1	1	yes	characteristic
50	C₂×#11924753	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
52	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
53	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
56	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
59	#14558715	C₂×C₄	1	2	yes	—
60	#14558715	C₂×C₄	1	2	yes	—
61	#14558715	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
63	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
64	#14558713	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
68	C₄×Dic₁₄₉	C₂³	1	2	yes	—
69	C₂×Q₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
70	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
71	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
72	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
73	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
74	#11924753	—	2	2	no	—
75	#11924753	—	2	2	no	—
76	#11924753	—	2	1	no	—
77	#11924753	—	2	1	no	—
78	#11924753	#41	1	2	yes	—
79	#11924753	#41	1	2	yes	—
80	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊D₁₄₉	—	4	2	no	—
82	C₂×Dic₁₄₉	—	2	2	no	—
83	C₂×Dic₁₄₉	—	2	2	no	—
84	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
89	C₄×D₁₄₉	—	2	2	no	—
90	C₄×D₁₄₉	—	2	1	no	—
91	C₄×D₁₄₉	—	2	1	no	—
92	C₄×D₁₄₉	—	4	1	no	—
93	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
94	Q₈×C₁₄₉	—	2	2	no	—
95	D₄×C₁₄₉	—	2	2	no	—
96	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₁₄₉	—	2	2	no	—
101	C₂²×C₁₄₉	—	2	2	no	—
102	C₂²×C₁₄₉	—	2	2	no	—
103	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₁₄₉	—	2	1	no	—
105	C₂×D₁₄₉	—	2	1	no	—
106	C₂×D₁₄₉	—	2	1	no	—
107	C₄×C₁₄₉	—	2	1	no	—
108	C₄×C₁₄₉	—	2	2	no	—
109	C₄×C₁₄₉	—	2	1	no	—
110	C₄×C₁₄₉	#131	1	2	yes	—
111	Dic₁₄₉	—	2	2	no	—
112	Dic₁₄₉	—	2	1	no	—
113	Dic₁₄₉	—	2	1	no	—
114	Dic₁₄₉	—	2	2	no	—
115	Dic₁₄₉	C₂×#41	1	1	yes	characteristic
116	Dic₁₄₉	C₂×#41	1	1	yes	characteristic
117	C₂×C₁₄₉	—	4	2	no	—
118	C₂×C₁₄₉	C₄×#41	1	1	yes	characteristic
119	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
120	C₂×C₁₄₉	#568	1	1	yes	characteristic
121	D₁₄₉	—	4	1	no	—
122	C₁₄₉	#2969	1	1	yes	characteristic, complemented, Sylow
123	#2969	—	149	1	no	maximal, Sylow
124	#529	—	149	1	no	—
125	#446	—	149	1	no	—
126	C₈×Q₈	—	149	1	no	—
127	#436	—	149	2	no	—
128	#436	—	149	1	no	—
129	#435	—	149	2	no	—
130	C₈×D₄	—	149	1	no	—
131	#433	—	149	2	no	—
132	#431	—	149	1	no	—
133	#405	—	149	2	no	—
134	C₄×M₁₆	—	149	1	no	—
135	C₂×M₁₆	—	149	2	no	—
136	C₂×M₁₆	—	149	1	no	—
137	C₂×M₁₆	—	149	1	no	—
138	C₂²×C₈	—	149	2	no	—
139	#126	—	149	2	no	—
140	#125	—	149	1	no	—
141	#124	#10597010	1	1	yes	characteristic
142	#123	—	149	2	no	—
143	C₂²⋊Q₈	—	149	1	no	—
144	#121	—	149	1	no	—
145	C₄×Q₈	—	149	1	no	—
146	C₄×D₄	—	149	1	no	—
147	C₄×D₄	—	149	2	no	—
148	#117	—	149	2	no	—
149	C₂×C₄²	—	149	1	no	—
150	C₄⋊C₈	—	149	1	no	—
151	C₄⋊C₈	—	149	1	no	—
152	C₄⋊C₈	—	149	2	no	—
153	C₄⋊C₈	—	149	2	no	—
154	C₂²⋊C₈	—	149	2	no	—
155	C₂²⋊C₈	—	149	2	no	—
156	C₂²⋊C₈	—	149	2	no	—
157	#97	—	149	1	no	—
158	#97	—	149	1	no	—
159	#97	—	149	2	no	—
160	C₄×C₈	—	149	1	no	—
161	C₄×C₈	—	149	1	no	—
162	C₄×C₈	—	149	2	no	—
163	C₄×C₈	—	149	2	no	—
164	C₂×Q₈	C₂×#10597010	1	1	yes	characteristic
165	C₂×D₄	—	149	2	no	—
166	C₂×D₄	C₂×#10597010	1	1	yes	characteristic
167	C₂²×C₄	—	149	2	no	—
168	C₂²×C₄	—	149	1	no	—
169	C₂²×C₄	—	149	1	no	—
170	C₂²×C₄	—	149	1	no	—
171	M₁₆	—	298	4	no	—
172	M₁₆	—	298	2	no	—
173	M₁₆	—	298	2	no	—
174	C₂×C₈	—	298	4	no	—
175	C₂×C₈	—	149	2	no	—
176	C₂×C₈	—	149	1	no	—
177	C₂×C₈	—	149	1	no	—
178	C₂×C₈	—	149	2	no	—
179	C₂×C₈	—	149	1	no	—
180	C₂×C₈	—	149	1	no	—
181	C₄⋊C₄	—	149	2	no	—
182	C₄⋊C₄	—	149	2	no	—
183	C₄⋊C₄	—	149	1	no	—
184	C₄⋊C₄	—	149	2	no	—
185	C₄⋊C₄	—	149	1	no	—
186	C₄⋊C₄	—	149	2	no	—
187	C₂²⋊C₄	—	149	2	no	—
188	C₂²⋊C₄	—	149	2	no	—
189	C₂²⋊C₄	—	149	2	no	—
190	C₂²⋊C₄	C₂×#10597010	1	2	yes	—
191	C₂²⋊C₄	C₂×#10597010	1	2	yes	—
192	C₄²	—	149	1	no	—
193	C₄²	—	149	1	no	—
194	C₄²	—	149	2	no	—
195	C₄²	—	149	1	no	—
196	C₄²	C₂×#10597010	1	1	yes	characteristic
197	C₂³	—	149	1	no	—
198	C₂³	C₂²×#10597010	1	2	yes	—
199	Q₈	—	2	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	596	2	no	—
202	C₂×C₄	—	298	1	no	—
203	C₂×C₄	—	298	1	no	—
204	C₂×C₄	—	298	2	no	—
205	C₂×C₄	—	298	2	no	—
206	C₂×C₄	—	298	2	no	—
207	C₂×C₄	—	596	1	no	—
208	C₂×C₄	—	149	1	no	—
209	C₂×C₄	—	149	1	no	—
210	C₂×C₄	—	149	2	no	—
211	C₂×C₄	—	149	2	no	—
212	C₂×C₄	—	149	1	no	—
213	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
214	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
215	C₂×C₄	C₂²×#10597010	1	2	yes	—
216	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
217	C₈	—	298	2	no	—
218	C₈	—	298	2	no	—
219	C₈	—	298	1	no	—
220	C₈	—	298	1	no	—
221	C₈	—	149	2	no	—
222	C₈	—	149	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	298	1	no	—
227	C₂²	—	298	1	no	—
228	C₂²	—	298	1	no	—
229	C₂²	C₂³×#10597010	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	298	2	no	—
234	C₄	—	298	1	no	—
235	C₄	—	298	1	no	—
236	C₄	—	298	2	no	—
237	C₄	—	149	1	no	—
238	C₄	—	149	1	no	—
239	C₄	#18011998	1	2	yes	—
240	C₂	—	4	2	no	—
241	C₂	—	596	1	no	—
242	C₂	#41×#10597010	1	1	yes	characteristic
243	C₂	C₂×#18011998	1	1	yes	characteristic
244	C₂	#23750704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597010
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#124×C₁₄₉	C₄
prime radical	#18011882	C₄
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	#124	#10597010
hypocenter	C₁₄₉	#2969
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #23750676	C₂	semidirect
2. Q₈×#11924753	C₂	semidirect
3. #23750660	C₂	semidirect
4. #23750658	C₂	semidirect
5. #23750654	C₂	semidirect
6. D₄×#11924753	C₂	semidirect
7. #23750649	C₂	semidirect
8. #23750644	C₂	semidirect
9. #23750589	C₂	semidirect
10. C₄×#14558746	C₂	semidirect
122. C₁₄₉	#2969	semidirect

Maximal subgroups

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

subgroup	index
#23750676	2
Q₈×#11924753	2
#23750660	2
#23750658	2
#23750654	2
D₄×#11924753	2
#23750649	2
#23750644	2
#23750589	2
C₄×#14558746	2
#23750154	2
#2969	149