Subgroups — #14577261

Order 2432.

1202 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927259	C₂	1	3	yes	maximal, complemented
3	#11927257	C₂	1	6	yes	maximal, complemented
4	#11927216	C₂	1	1	yes	maximal, characteristic, complemented
5	#11927215	C₂	1	3	yes	maximal, complemented
6	#11927213	C₂	1	2	yes	maximal, complemented
7	#11927094	C₂	1	3	yes	maximal, complemented
8	#11927091	C₂	1	6	yes	maximal, complemented
9	C₄×#64545	C₂	1	2	yes	maximal, complemented
10	C₄×#64541	C₂	1	3	yes	maximal, complemented
11	C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×#64545	C₂²	1	2	yes	—
14	C₂×#64541	C₂²	1	3	yes	—
15	C₂×#64541	C₂²	1	6	yes	—
16	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₉	C₂²	1	3	yes	—
19	C₄×D₄×C₁₉	C₂²	1	3	yes	—
20	C₁₉×#117	C₂²	1	3	yes	—
21	C₂×C₄²×C₁₉	C₂²	1	3	yes	—
22	Q₈×Dic₁₉	C₂²	1	2	yes	—
23	#10597386	C₂²	1	6	yes	—
24	#10597384	C₂²	1	6	yes	—
25	#10597376	C₂²	1	6	yes	—
26	#10597376	C₂²	1	6	yes	—
27	C₄×C₂²⋊D₁₉	C₂²	1	6	yes	—
28	C₄×C₂²⋊D₁₉	C₂²	1	6	yes	—
29	#10597372	C₂²	1	3	yes	—
30	#10597341	C₂²	1	6	yes	—
31	#10597339	C₂²	1	6	yes	—
32	#10597338	C₂²	1	6	yes	—
33	#10597334	C₂²	1	6	yes	—
34	#10597331	C₂²	1	12	yes	—
35	#10597330	C₂²	1	12	yes	—
36	#10597321	C₂²	1	6	yes	—
37	#10597320	C₂²	1	3	yes	—
38	C₄×D₇₆	C₂²	1	6	yes	—
39	C₄×D₇₆	C₂²	1	3	yes	—
40	C₄×D₇₆	C₂²	1	3	yes	—
41	#10597318	C₂²	1	6	yes	—
42	C₄²×D₁₉	C₂²	1	6	yes	—
43	#10597315	C₂²	1	3	yes	—
44	C₄×Dic₃₈	C₂²	1	3	yes	—
45	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
46	#41×C₁₉	—	2	4	no	—
47	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₉	C₂³	1	3	yes	—
49	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
50	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
51	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₁₉	C₂³	1	6	yes	—
53	C₂×C₂²⋊D₁₉	C₂³	1	12	yes	—
54	#64545	—	2	8	no	—
55	#64541	—	2	12	no	—
56	#64541	—	4	12	no	—
57	C₂×D₇₆	C₂³	1	3	yes	—
58	C₂×D₇₆	C₂³	1	6	yes	—
59	C₂×D₇₆	C₂³	1	6	yes	—
60	C₂×C₄×D₁₉	C₂³	1	6	yes	—
61	C₂×C₄×D₁₉	C₂³	1	6	yes	—
62	C₂×C₄×D₁₉	C₂³	1	6	yes	—
63	C₂×Dic₃₈	C₂³	1	6	yes	—
64	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
66	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
67	C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
68	C₄²×C₁₉	C₂³	1	1	yes	characteristic
69	C₄²×C₁₉	C₂³	1	3	yes	—
70	C₄²×C₁₉	C₂³	1	3	yes	—
71	C₄²×C₁₉	C₂³	1	3	yes	—
72	C₂²⋊Dic₁₉	C₂³	1	6	yes	—
73	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
74	(C₂×D₁₉)⋊C₄	C₂³	1	12	yes	—
75	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
76	C₄⋊Dic₁₉	C₂³	1	3	yes	—
77	C₄⋊Dic₁₉	C₂³	1	6	yes	—
78	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
79	Dic₁₉⋊C₄	C₂³	1	2	yes	—
80	Dic₁₉⋊C₄	C₂³	1	6	yes	—
81	C₄×Dic₁₉	C₂³	1	6	yes	—
82	C₄×Dic₁₉	C₂³	1	2	yes	—
83	C₄×Dic₁₉	C₂³	1	6	yes	—
84	C₂³×C₁₉	C₂⁴	1	3	yes	—
85	C₂²×D₁₉	C₂⁴	1	6	yes	—
86	Q₈×C₁₉	#41	1	4	yes	complemented
87	D₄×C₁₉	—	2	6	no	—
88	C₂×C₄×C₁₉	—	2	6	no	—
89	C₂×C₄×C₁₉	—	2	6	no	—
90	C₂×C₄×C₁₉	—	2	6	no	—
91	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
92	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
94	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
95	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₉	—	2	12	no	—
98	C₂²⋊D₁₉	—	4	12	no	—
99	C₂×Dic₁₉	C₂⁴	1	2	yes	—
100	C₂×Dic₁₉	C₂⁴	1	6	yes	—
101	C₂×Dic₁₉	C₂⁴	1	2	yes	—
102	D₇₆	—	2	6	no	—
103	D₇₆	—	2	12	no	—
104	D₇₆	—	4	6	no	—
105	C₄×D₁₉	—	2	12	no	—
106	C₄×D₁₉	—	2	12	no	—
107	C₄×D₁₉	—	2	12	no	—
108	Dic₃₈	—	4	6	no	—
109	Dic₃₈	#41	1	4	yes	complemented
110	#3483	—	19	1	no	maximal, Sylow
111	C₂²×C₁₉	—	2	3	no	—
112	C₂²×C₁₉	—	2	3	no	—
113	C₂²×C₁₉	—	2	3	no	—
114	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
115	C₂×D₁₉	—	2	6	no	—
116	C₂×D₁₉	—	2	6	no	—
117	C₂×D₁₉	—	2	6	no	—
118	C₄×C₁₉	—	2	3	no	—
119	C₄×C₁₉	—	2	1	no	—
120	C₄×C₁₉	—	2	3	no	—
121	C₄×C₁₉	C₂×#41	1	6	yes	—
122	C₄×C₁₉	C₂×#41	1	2	yes	—
123	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
124	Dic₁₉	—	2	2	no	—
125	Dic₁₉	—	2	6	no	—
126	Dic₁₉	C₂×#41	1	4	yes	—
127	Q₈²	—	19	1	no	—
128	#555	—	19	3	no	—
129	#555	—	19	6	no	—
130	#550	—	19	2	no	—
131	#550	—	19	1	no	—
132	#550	—	19	3	no	—
133	#532	—	19	6	no	—
134	#532	—	19	3	no	—
135	C₄×#41	—	19	2	no	—
136	C₄×#41	—	19	3	no	—
137	C₄×#41	D₁₉	1	1	yes	characteristic, complemented
138	C₂×C₁₉	—	4	3	no	—
139	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
140	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
141	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
142	D₁₉	—	4	6	no	—
143	C₂×#41	—	19	2	no	—
144	C₂×#41	—	19	3	no	—
145	C₂×#41	—	19	6	no	—
146	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
147	C₄⋊Q₈	—	19	6	no	—
148	C₄⋊Q₈	—	19	3	no	—
149	#127	—	19	6	no	—
150	#127	—	19	3	no	—
151	#124	—	19	12	no	—
152	#124	—	19	6	no	—
153	#121	—	19	6	no	—
154	#121	—	19	6	no	—
155	#121	—	19	6	no	—
156	#121	—	19	6	no	—
157	#121	—	19	12	no	—
158	C₄×Q₈	—	19	2	no	—
159	C₄×Q₈	—	19	1	no	—
160	C₄×Q₈	—	19	3	no	—
161	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
162	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
163	C₄×D₄	—	19	6	no	—
164	C₄×D₄	—	19	6	no	—
165	C₄×D₄	—	19	6	no	—
166	C₄×D₄	—	19	6	no	—
167	C₄×D₄	—	19	3	no	—
168	C₄×D₄	—	19	3	no	—
169	C₄×D₄	C₂×D₁₉	1	3	yes	—
170	C₄×D₄	C₂×D₁₉	1	3	yes	—
171	#117	—	19	3	no	—
172	#117	—	19	6	no	—
173	#117	—	19	6	no	—
174	#117	C₂×D₁₉	1	3	yes	—
175	C₂×C₄²	—	19	6	no	—
176	C₂×C₄²	C₂×D₁₉	1	3	yes	—
177	C₁₉	#3483	1	1	yes	characteristic, complemented, Sylow
178	#41	—	2	4	no	—
179	#41	—	38	12	no	—
180	#41	—	38	8	no	—
181	#41	—	76	12	no	—
182	C₂×Q₈	—	19	1	no	—
183	C₂×Q₈	—	19	6	no	—
184	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
185	C₂×D₄	—	19	6	no	—
186	C₂×D₄	—	19	12	no	—
187	C₂×D₄	—	19	6	no	—
188	C₂×D₄	—	19	3	no	—
189	C₂×D₄	—	19	6	no	—
190	C₂×D₄	C₂²×D₁₉	1	3	yes	—
191	C₂²×C₄	—	19	6	no	—
192	C₂²×C₄	—	19	6	no	—
193	C₂²×C₄	—	19	6	no	—
194	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
195	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
196	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
197	C₄⋊C₄	—	19	2	no	—
198	C₄⋊C₄	—	19	6	no	—
199	C₄⋊C₄	—	19	1	no	—
200	C₄⋊C₄	—	19	6	no	—
201	C₄⋊C₄	—	19	3	no	—
202	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
203	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
204	C₂²⋊C₄	—	19	6	no	—
205	C₂²⋊C₄	—	19	6	no	—
206	C₂²⋊C₄	—	19	12	no	—
207	C₂²⋊C₄	—	19	6	no	—
208	C₂²⋊C₄	C₂²×D₁₉	1	6	yes	—
209	C₄²	—	19	6	no	—
210	C₄²	—	19	6	no	—
211	C₄²	—	19	2	no	—
212	C₄²	C₂²×D₁₉	1	3	yes	—
213	C₄²	C₂²×D₁₉	1	3	yes	—
214	C₄²	C₂²×D₁₉	1	3	yes	—
215	C₄²	C₂²×D₁₉	1	1	yes	characteristic
216	C₂³	—	19	6	no	—
217	C₂³	C₂³×D₁₉	1	3	yes	—
218	Q₈	—	19	4	no	—
219	Q₈	—	76	6	no	—
220	Q₈	#64541	1	4	yes	complemented
221	D₄	—	2	6	no	—
222	D₄	—	38	12	no	—
223	D₄	—	38	12	no	—
224	D₄	—	38	6	no	—
225	D₄	—	76	6	no	—
226	D₄	—	76	12	no	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	2	6	no	—
229	C₂×C₄	—	2	6	no	—
230	C₂×C₄	—	19	2	no	—
231	C₂×C₄	—	19	2	no	—
232	C₂×C₄	—	19	6	no	—
233	C₂×C₄	—	38	12	no	—
234	C₂×C₄	—	38	12	no	—
235	C₂×C₄	—	38	12	no	—
236	C₂×C₄	C₂³×D₁₉	1	3	yes	—
237	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₁₉	1	3	yes	—
239	C₂×C₄	C₂³×D₁₉	1	3	yes	—
240	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
241	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
242	C₂²	—	2	3	no	—
243	C₂²	—	2	3	no	—
244	C₂²	—	2	3	no	—
245	C₂²	—	38	6	no	—
246	C₂²	—	38	6	no	—
247	C₂²	—	38	6	no	—
248	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
249	C₄	—	2	3	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	3	no	—
252	C₄	—	19	4	no	—
253	C₄	—	38	2	no	—
254	C₄	—	38	6	no	—
255	C₄	C₂×#64541	1	6	yes	—
256	C₄	#41×D₁₉	1	2	yes	—
257	C₄	#10597435	1	2	yes	—
258	C₂	—	4	3	no	—
259	C₂	—	76	6	no	—
260	C₂	C₂×#41×D₁₉	1	1	yes	characteristic
261	C₂	C₂²×#64541	1	1	yes	characteristic
262	C₂	C₂×#10597435	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₄×#41×C₁₉	C₂
hypercenter	C₄×#41	D₁₉
hypocenter	C₁₉	#3483
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₈	C₂	semidirect
2. #11927259	C₂	semidirect
3. #11927257	C₂	semidirect
4. #11927216	C₂	semidirect
5. #11927215	C₂	semidirect
6. #11927213	C₂	semidirect
7. #11927094	C₂	semidirect
8. #11927091	C₂	semidirect
9. C₄×#64545	C₂	semidirect
10. C₄×#64541	C₂	semidirect
11. C₄×#41×C₁₉	C₂	semidirect
86. Q₈×C₁₉	#41	semidirect
109. Dic₃₈	#41	semidirect
137. C₄×#41	D₁₉	semidirect
177. C₁₉	#3483	semidirect
220. Q₈	#64541	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₈	2
#11927259	2
#11927257	2
#11927216	2
#11927215	2
#11927213	2
#11927094	2
#11927091	2
C₄×#64545	2
C₄×#64541	2
C₄×#41×C₁₉	2
#3483	19