Subgroups — #14577720

Order 2432.

1104 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927279	C₂	1	3	yes	maximal, complemented
2	Q₈×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927254	C₂	1	3	yes	maximal, complemented
4	#11927216	C₂	1	1	yes	maximal, characteristic, complemented
5	#11927205	C₂	1	2	yes	maximal, complemented
6	#11927195	C₂	1	6	yes	maximal, complemented
7	#11927166	C₂	1	6	yes	maximal, complemented
8	#11927094	C₂	1	3	yes	maximal, complemented
9	#11927078	C₂	1	3	yes	maximal, complemented
10	#11927026	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×#64545	C₂²	1	2	yes	—
14	C₂×#64541	C₂²	1	6	yes	—
15	C₄×Q₈×C₁₉	C₂²	1	3	yes	—
16	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₉	C₂²	1	3	yes	—
18	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
20	C₂×C₄²×C₁₉	C₂²	1	3	yes	—
21	#10597393	C₂²	1	6	yes	—
22	#10597392	C₂²	1	6	yes	—
23	Q₈×Dic₁₉	C₂²	1	2	yes	—
24	#10597376	C₂²	1	3	yes	—
25	#10597376	C₂²	1	6	yes	—
26	#10597375	C₂²	1	6	yes	—
27	C₄×C₂²⋊D₁₉	C₂²	1	6	yes	—
28	#10597370	C₂²	1	1	yes	characteristic
29	#10597342	C₂²	1	12	yes	—
30	#10597341	C₂²	1	6	yes	—
31	#10597340	C₂²	1	12	yes	—
32	#10597339	C₂²	1	6	yes	—
33	C₄⋊C₄×D₁₉	C₂²	1	6	yes	—
34	#10597334	C₂²	1	6	yes	—
35	#10597322	C₂²	1	6	yes	—
36	#10597321	C₂²	1	3	yes	—
37	#10597321	C₂²	1	6	yes	—
38	#10597320	C₂²	1	3	yes	—
39	C₄×D₇₆	C₂²	1	3	yes	—
40	C₄×D₇₆	C₂²	1	6	yes	—
41	C₄×D₇₆	C₂²	1	6	yes	—
42	#10597318	C₂²	1	6	yes	—
43	#10597316	C₂²	1	3	yes	—
44	#10597315	C₂²	1	3	yes	—
45	C₄×Dic₃₈	C₂²	1	3	yes	—
46	C₂×Q₈×C₁₉	—	2	4	no	—
47	C₂×Q₈×C₁₉	C₂³	1	3	yes	—
48	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
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	#64545	—	2	8	no	—
54	#64541	—	4	12	no	—
55	C₂×D₇₆	C₂³	1	6	yes	—
56	C₂×D₇₆	C₂³	1	3	yes	—
57	C₂×D₇₆	C₂³	1	6	yes	—
58	C₂×C₄×D₁₉	C₂³	1	6	yes	—
59	C₂×C₄×D₁₉	C₂³	1	12	yes	—
60	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₃₈	C₂³	1	6	yes	—
62	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
63	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
64	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
65	C₄²×C₁₉	C₂³	1	3	yes	—
66	C₄²×C₁₉	C₂³	1	3	yes	—
67	C₄²×C₁₉	C₂³	1	3	yes	—
68	C₄²×C₁₉	C₂³	1	3	yes	—
69	C₂²⋊Dic₁₉	C₂³	1	2	yes	—
70	(C₂×D₁₉)⋊C₄	C₂³	1	12	yes	—
71	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
72	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
73	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
74	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
75	C₄⋊Dic₁₉	C₂³	1	6	yes	—
76	C₄⋊Dic₁₉	C₂³	1	3	yes	—
77	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
78	Dic₁₉⋊C₄	C₂³	1	6	yes	—
79	Dic₁₉⋊C₄	C₂³	1	6	yes	—
80	Dic₁₉⋊C₄	C₂³	1	6	yes	—
81	Dic₁₉⋊C₄	C₂³	1	2	yes	—
82	C₄×Dic₁₉	C₂³	1	6	yes	—
83	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
84	C₂²×D₁₉	C₂⁴	1	6	yes	—
85	Q₈×C₁₉	—	2	6	no	—
86	Q₈×C₁₉	#41	1	4	yes	complemented
87	C₂×C₄×C₁₉	—	2	6	no	—
88	C₂×C₄×C₁₉	—	2	6	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
92	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
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	C₂²⋊D₁₉	—	2	12	no	—
97	C₂×Dic₁₉	C₂⁴	1	2	yes	—
98	C₂×Dic₁₉	C₂⁴	1	6	yes	—
99	C₂×Dic₁₉	C₂⁴	1	2	yes	—
100	D₇₆	—	2	12	no	—
101	D₇₆	—	2	6	no	—
102	D₇₆	—	4	6	no	—
103	C₄×D₁₉	—	2	12	no	—
104	C₄×D₁₉	—	4	12	no	—
105	Dic₃₈	—	2	2	no	—
106	Dic₃₈	—	4	6	no	—
107	#3522	—	19	1	no	maximal, Sylow
108	C₂²×C₁₉	—	2	1	no	—
109	C₂²×C₁₉	—	2	1	no	—
110	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
111	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
112	C₂×D₁₉	—	2	6	no	—
113	C₂×D₁₉	—	2	6	no	—
114	C₂×D₁₉	—	2	6	no	—
115	C₄×C₁₉	—	2	3	no	—
116	C₄×C₁₉	—	2	3	no	—
117	C₄×C₁₉	—	2	3	no	—
118	C₄×C₁₉	—	2	1	no	—
119	C₄×C₁₉	C₂×#41	1	6	yes	—
120	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
121	Dic₁₉	—	2	2	no	—
122	Dic₁₉	—	2	6	no	—
123	Dic₁₉	—	2	2	no	—
124	#559	—	19	3	no	—
125	Q₈²	—	19	1	no	—
126	#554	—	19	3	no	—
127	#550	—	19	1	no	—
128	#548	—	19	2	no	—
129	#547	—	19	6	no	—
130	#542	—	19	6	no	—
131	#532	—	19	3	no	—
132	#529	—	19	3	no	—
133	#520	—	19	2	no	—
134	C₂×C₄×Q₈	D₁₉	1	1	yes	characteristic, complemented
135	C₂×C₁₉	—	2	2	no	—
136	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
137	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
139	D₁₉	—	4	6	no	—
140	C₂×#41	—	19	2	no	—
141	C₂×#41	—	19	6	no	—
142	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
143	C₄⋊Q₈	—	19	6	no	—
144	C₄⋊Q₈	—	19	3	no	—
145	#127	—	19	3	no	—
146	#126	—	19	6	no	—
147	#125	—	19	3	no	—
148	#124	—	19	6	no	—
149	#124	—	19	3	no	—
150	#124	—	19	6	no	—
151	#123	—	19	6	no	—
152	#123	—	19	12	no	—
153	C₂²⋊Q₈	—	19	6	no	—
154	C₂²⋊Q₈	—	19	1	no	—
155	C₂²⋊Q₈	—	19	12	no	—
156	#121	—	19	3	no	—
157	#121	—	19	6	no	—
158	#121	—	19	6	no	—
159	C₄×Q₈	—	19	2	no	—
160	C₄×Q₈	—	19	3	no	—
161	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
162	C₄×Q₈	C₂×D₁₉	1	3	yes	—
163	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
164	C₄×Q₈	C₂×D₁₉	1	3	yes	—
165	C₄×D₄	—	19	6	no	—
166	C₄×D₄	—	19	6	no	—
167	C₄×D₄	—	19	6	no	—
168	C₄×D₄	—	19	3	no	—
169	C₄×D₄	—	19	6	no	—
170	#117	—	19	6	no	—
171	C₂×C₄⋊C₄	—	19	6	no	—
172	C₂×C₄⋊C₄	C₂×D₁₉	1	3	yes	—
173	C₂×C₄²	C₂×D₁₉	1	3	yes	—
174	C₁₉	#3522	1	1	yes	characteristic, complemented, Sylow
175	#41	—	38	8	no	—
176	#41	—	76	12	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	19	1	no	—
179	C₂×Q₈	—	19	6	no	—
180	C₂×Q₈	C₂²×D₁₉	1	3	yes	—
181	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
182	C₂×D₄	—	19	6	no	—
183	C₂×D₄	—	19	3	no	—
184	C₂×D₄	—	19	6	no	—
185	C₂×D₄	—	19	6	no	—
186	C₂²×C₄	—	19	6	no	—
187	C₂²×C₄	—	19	12	no	—
188	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
190	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
191	C₄⋊C₄	—	19	6	no	—
192	C₄⋊C₄	—	19	2	no	—
193	C₄⋊C₄	—	19	6	no	—
194	C₄⋊C₄	—	19	6	no	—
195	C₄⋊C₄	—	19	6	no	—
196	C₄⋊C₄	—	19	1	no	—
197	C₄⋊C₄	—	19	3	no	—
198	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
199	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
200	C₄⋊C₄	C₂²×D₁₉	1	6	yes	—
201	C₂²⋊C₄	—	19	2	no	—
202	C₂²⋊C₄	—	19	6	no	—
203	C₂²⋊C₄	—	19	6	no	—
204	C₂²⋊C₄	—	19	6	no	—
205	C₂²⋊C₄	—	19	12	no	—
206	C₂²⋊C₄	—	19	6	no	—
207	C₄²	—	19	6	no	—
208	C₄²	C₂²×D₁₉	1	3	yes	—
209	C₄²	C₂²×D₁₉	1	3	yes	—
210	C₄²	C₂²×D₁₉	1	3	yes	—
211	C₄²	C₂²×D₁₉	1	3	yes	—
212	C₂³	—	19	6	no	—
213	C₂³	C₂³×D₁₉	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	38	2	no	—
216	Q₈	—	76	6	no	—
217	Q₈	#64541	1	4	yes	complemented
218	D₄	—	38	12	no	—
219	D₄	—	38	12	no	—
220	D₄	—	38	6	no	—
221	D₄	—	76	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	19	2	no	—
226	C₂×C₄	—	19	6	no	—
227	C₂×C₄	—	19	2	no	—
228	C₂×C₄	—	38	12	no	—
229	C₂×C₄	—	76	12	no	—
230	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
231	C₂×C₄	C₂³×D₁₉	1	3	yes	—
232	C₂×C₄	C₂³×D₁₉	1	3	yes	—
233	C₂×C₄	C₂³×D₁₉	1	3	yes	—
234	C₂×C₄	C₂³×D₁₉	1	3	yes	—
235	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	38	6	no	—
239	C₂²	—	38	6	no	—
240	C₂²	—	38	6	no	—
241	C₂²	#10597435	1	2	yes	—
242	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
243	C₄	—	2	3	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	2	1	no	—
247	C₄	—	38	2	no	—
248	C₄	—	38	6	no	—
249	C₄	—	38	2	no	—
250	C₄	C₂×#64541	1	6	yes	—
251	C₄	#10597433	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	76	6	no	—
254	C₂	C₂²×#64541	1	1	yes	characteristic
255	C₂	C₂×#10597435	1	1	yes	characteristic
256	C₂	C₂×#10597433	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₂×C₄×Q₈×C₁₉	C₂
hypercenter	C₂×C₄×Q₈	D₁₉
hypocenter	C₁₉	#3522
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #11927279	C₂	semidirect
2. Q₈×Dic₃₈	C₂	semidirect
3. #11927254	C₂	semidirect
4. #11927216	C₂	semidirect
5. #11927205	C₂	semidirect
6. #11927195	C₂	semidirect
7. #11927166	C₂	semidirect
8. #11927094	C₂	semidirect
9. #11927078	C₂	semidirect
10. #11927026	C₂	semidirect
11. C₂×C₄×Q₈×C₁₉	C₂	semidirect
86. Q₈×C₁₉	#41	semidirect
134. C₂×C₄×Q₈	D₁₉	semidirect
174. C₁₉	#3522	semidirect
217. Q₈	#64541	semidirect

Maximal subgroups

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

subgroup	index
#11927279	2
Q₈×Dic₃₈	2
#11927254	2
#11927216	2
#11927205	2
#11927195	2
#11927166	2
#11927094	2
#11927078	2
#11927026	2
C₂×C₄×Q₈×C₁₉	2
#3522	19