Subgroups — #14577453

Order 2432.

1060 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	Q₈×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927270	C₂	1	2	yes	maximal, complemented
3	#11927255	C₂	1	3	yes	maximal, complemented
4	#11927254	C₂	1	3	yes	maximal, complemented
5	#11927209	C₂	1	6	yes	maximal, complemented
6	#11927171	C₂	1	6	yes	maximal, complemented
7	#11927098	C₂	1	3	yes	maximal, complemented
8	C₄⋊Q₈×D₁₉	C₂	1	3	yes	maximal, complemented
9	#11927026	C₂	1	2	yes	maximal
10	C₄×#64545	C₂	1	1	yes	maximal, characteristic
11	Q₈²×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#64545	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₉	C₂²	1	6	yes	—
14	C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
15	C₁₉×C₄⋊Q₈	C₂²	1	6	yes	—
16	C₄×Q₈×C₁₉	C₂²	1	3	yes	—
17	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
19	#10597392	C₂²	1	3	yes	—
20	#10597392	C₂²	1	12	yes	—
21	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₉	C₂²	1	2	yes	—
23	#10597390	C₂²	1	6	yes	—
24	#10597390	C₂²	1	1	yes	characteristic
25	#10597343	C₂²	1	3	yes	—
26	#10597342	C₂²	1	6	yes	—
27	#10597342	C₂²	1	12	yes	—
28	#10597339	C₂²	1	3	yes	—
29	#10597339	C₂²	1	6	yes	—
30	#10597338	C₂²	1	3	yes	—
31	C₄⋊C₄×D₁₉	C₂²	1	6	yes	—
32	C₄⋊C₄×D₁₉	C₂²	1	6	yes	—
33	#10597336	C₂²	1	6	yes	—
34	#10597335	C₂²	1	6	yes	—
35	#10597334	C₂²	1	3	yes	—
36	#10597334	C₂²	1	6	yes	—
37	#10597334	C₂²	1	6	yes	—
38	C₄×D₇₆	C₂²	1	6	yes	—
39	C₄×D₇₆	C₂²	1	3	yes	—
40	#10597318	C₂²	1	6	yes	—
41	C₄²×D₁₉	C₂²	1	3	yes	—
42	#10597316	C₂²	1	6	yes	—
43	#10597315	C₂²	1	3	yes	—
44	C₄×Dic₃₈	C₂²	1	6	yes	—
45	C₄×Dic₃₈	C₂²	1	3	yes	—
46	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₁₉	C₂³	1	6	yes	—
48	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
49	#64545	Q₈	1	8	yes	complemented
50	Q₈×D₁₉	—	4	12	no	—
51	C₂×D₇₆	C₂³	1	3	yes	—
52	C₂×C₄×D₁₉	C₂³	1	3	yes	—
53	C₂×C₄×D₁₉	C₂³	1	3	yes	—
54	C₂×C₄×D₁₉	C₂³	1	12	yes	—
55	C₂×C₄×D₁₉	C₂³	1	3	yes	—
56	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
57	C₂×Dic₃₈	C₂³	1	6	yes	—
58	C₂×Dic₃₈	C₂³	1	6	yes	—
59	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
60	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
61	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
62	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
63	C₄²×C₁₉	C₂³	1	3	yes	—
64	C₄²×C₁₉	C₂³	1	6	yes	—
65	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
66	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
67	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
68	C₄⋊Dic₁₉	C₂³	1	6	yes	—
69	C₄⋊Dic₁₉	C₂³	1	3	yes	—
70	C₄⋊Dic₁₉	C₂³	1	6	yes	—
71	C₄⋊Dic₁₉	C₂³	1	3	yes	—
72	Dic₁₉⋊C₄	C₂³	1	6	yes	—
73	Dic₁₉⋊C₄	C₂³	1	12	yes	—
74	Dic₁₉⋊C₄	C₂³	1	2	yes	—
75	Dic₁₉⋊C₄	C₂³	1	2	yes	—
76	Dic₁₉⋊C₄	C₂³	1	6	yes	—
77	Dic₁₉⋊C₄	C₂³	1	6	yes	—
78	C₄×Dic₁₉	C₂³	1	3	yes	—
79	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
80	C₄×Dic₁₉	C₂³	1	3	yes	—
81	C₄×Dic₁₉	C₂³	1	6	yes	—
82	C₂²×D₁₉	C₂⁴	1	3	yes	—
83	Q₈×C₁₉	—	2	2	no	—
84	Q₈×C₁₉	—	4	6	no	—
85	Q₈×C₁₉	C₂×Q₈	1	4	yes	complemented
86	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
87	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
88	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
89	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
90	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₁₉	C₂⁴	1	2	yes	—
92	C₂×Dic₁₉	C₂⁴	1	3	yes	—
93	C₂×Dic₁₉	C₂⁴	1	6	yes	—
94	C₂×Dic₁₉	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₁₉	C₂⁴	1	1	yes	characteristic
96	D₇₆	C₂×Q₈	1	12	yes	complemented
97	C₄×D₁₉	—	2	6	no	—
98	C₄×D₁₉	—	2	6	no	—
99	C₄×D₁₉	—	4	12	no	—
100	C₄×D₁₉	C₂×Q₈	1	12	yes	complemented
101	Dic₃₈	—	2	2	no	—
102	Dic₃₈	—	2	12	no	—
103	Dic₃₈	—	4	6	no	—
104	#3503	—	19	1	no	maximal, Sylow
105	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
106	C₂×D₁₉	—	2	3	no	—
107	C₂×D₁₉	—	2	3	no	—
108	C₂×D₁₉	C₂²×Q₈	1	6	yes	—
109	C₄×C₁₉	—	2	6	no	—
110	C₄×C₁₉	—	2	3	no	—
111	C₄×C₁₉	—	2	2	no	—
112	C₄×C₁₉	C₂²×Q₈	1	6	yes	—
113	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
114	Dic₁₉	—	2	6	no	—
115	Dic₁₉	—	2	2	no	—
116	Dic₁₉	—	2	3	no	—
117	Dic₁₉	C₂²×Q₈	1	2	yes	—
118	Dic₁₉	H(5,𝔽₂)	1	2	yes	—
119	Q₈²	—	19	1	no	—
120	Q₈²	D₁₉	1	1	yes	characteristic, complemented
121	#557	—	19	2	no	—
122	#554	—	19	3	no	—
123	#554	—	19	3	no	—
124	D₄×Q₈	—	19	6	no	—
125	#544	—	19	6	no	—
126	#533	—	19	3	no	—
127	C₂×C₄⋊Q₈	—	19	3	no	—
128	#520	—	19	2	no	—
129	C₄×#41	—	19	1	no	—
130	C₂×C₁₉	C₂³×Q₈	1	1	yes	characteristic
131	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
133	D₁₉	—	2	6	no	—
134	C₂×#41	—	19	1	no	—
135	C₂²×Q₈	—	19	6	no	—
136	C₄⋊Q₈	—	19	6	no	—
137	C₄⋊Q₈	—	19	1	no	—
138	C₄⋊Q₈	—	19	3	no	—
139	C₄⋊Q₈	—	19	6	no	—
140	C₄⋊Q₈	—	19	6	no	—
141	C₄⋊Q₈	—	19	3	no	—
142	C₄⋊Q₈	C₂×D₁₉	1	3	yes	—
143	C₄⋊Q₈	C₂×D₁₉	1	6	yes	—
144	#125	—	19	6	no	—
145	#125	—	19	6	no	—
146	#125	—	19	6	no	—
147	C₂²⋊Q₈	—	19	3	no	—
148	C₂²⋊Q₈	—	19	12	no	—
149	C₂²⋊Q₈	—	19	6	no	—
150	C₂²⋊Q₈	—	19	3	no	—
151	C₂²⋊Q₈	—	19	12	no	—
152	C₄×Q₈	—	19	1	no	—
153	C₄×Q₈	—	19	2	no	—
154	C₄×Q₈	—	19	6	no	—
155	C₄×Q₈	—	19	3	no	—
156	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
157	C₄×Q₈	C₂×D₁₉	1	2	yes	—
158	C₄×Q₈	C₂×D₁₉	1	3	yes	—
159	C₄×D₄	—	19	3	no	—
160	C₄×D₄	—	19	6	no	—
161	C₄×D₄	—	19	3	no	—
162	C₄×D₄	—	19	6	no	—
163	#117	—	19	3	no	—
164	#117	—	19	6	no	—
165	C₂×C₄⋊C₄	—	19	6	no	—
166	C₂×C₄⋊C₄	—	19	6	no	—
167	C₂×C₄²	—	19	3	no	—
168	C₁₉	#3503	1	1	yes	characteristic, complemented, Sylow
169	#41	—	19	8	no	—
170	C₂×Q₈	—	19	1	no	—
171	C₂×Q₈	—	19	6	no	—
172	C₂×Q₈	—	19	6	no	—
173	C₂×Q₈	—	76	12	no	—
174	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
175	C₂×Q₈	C₂²×D₁₉	1	6	yes	—
176	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
177	C₂×D₄	—	19	3	no	—
178	C₂²×C₄	—	19	3	no	—
179	C₂²×C₄	—	19	3	no	—
180	C₂²×C₄	—	19	3	no	—
181	C₂²×C₄	—	19	12	no	—
182	C₄⋊C₄	—	19	6	no	—
183	C₄⋊C₄	—	19	12	no	—
184	C₄⋊C₄	—	19	2	no	—
185	C₄⋊C₄	—	19	2	no	—
186	C₄⋊C₄	—	19	6	no	—
187	C₄⋊C₄	—	19	6	no	—
188	C₄⋊C₄	—	19	3	no	—
189	C₄⋊C₄	—	19	6	no	—
190	C₄⋊C₄	—	19	3	no	—
191	C₄⋊C₄	—	19	6	no	—
192	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
193	C₄⋊C₄	C₂²×D₁₉	1	6	yes	—
194	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
195	C₄⋊C₄	C₂²×D₁₉	1	6	yes	—
196	C₂²⋊C₄	—	19	6	no	—
197	C₂²⋊C₄	—	19	6	no	—
198	C₂²⋊C₄	—	19	6	no	—
199	C₄²	—	19	1	no	—
200	C₄²	—	19	3	no	—
201	C₄²	—	19	3	no	—
202	C₄²	—	19	6	no	—
203	C₄²	C₂²×D₁₉	1	3	yes	—
204	C₄²	C₂²×D₁₉	1	6	yes	—
205	C₂³	—	19	3	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	4	6	no	—
208	Q₈	—	38	12	no	—
209	Q₈	—	38	2	no	—
210	Q₈	—	76	6	no	—
211	Q₈	Q₈×D₁₉	1	4	yes	complemented
212	D₄	—	19	12	no	—
213	C₂×C₄	—	19	1	no	—
214	C₂×C₄	—	19	1	no	—
215	C₂×C₄	—	19	12	no	—
216	C₂×C₄	—	19	2	no	—
217	C₂×C₄	—	19	3	no	—
218	C₂×C₄	—	19	6	no	—
219	C₂×C₄	—	38	6	no	—
220	C₂×C₄	—	38	6	no	—
221	C₂×C₄	—	76	12	no	—
222	C₂×C₄	C₂³×D₁₉	1	3	yes	—
223	C₂×C₄	C₂³×D₁₉	1	2	yes	—
224	C₂×C₄	C₂³×D₁₉	1	6	yes	—
225	C₂×C₄	C₂³×D₁₉	1	3	yes	—
226	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
227	C₂²	—	19	6	no	—
228	C₂²	—	38	3	no	—
229	C₂²	—	38	3	no	—
230	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
231	C₄	—	2	3	no	—
232	C₄	—	2	6	no	—
233	C₄	—	2	2	no	—
234	C₄	—	19	2	no	—
235	C₄	—	19	2	no	—
236	C₄	—	38	6	no	—
237	C₄	—	38	3	no	—
238	C₄	—	38	2	no	—
239	C₄	C₂×Q₈×D₁₉	1	6	yes	—
240	C₄	#10597433	1	2	yes	—
241	C₂	—	38	6	no	—
242	C₂	C₂²×Q₈×D₁₉	1	1	yes	characteristic
243	C₂	H(5,𝔽₂)×D₁₉	1	1	yes	characteristic
244	C₂	C₂×#10597433	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	Q₈²×C₁₉	C₂
prime radical	C₂×#64545	C₂²
hypercenter	Q₈²	D₁₉
hypocenter	C₁₉	#3503
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ Q₈²

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₈	C₂	semidirect
2. #11927270	C₂	semidirect
3. #11927255	C₂	semidirect
4. #11927254	C₂	semidirect
5. #11927209	C₂	semidirect
6. #11927171	C₂	semidirect
7. #11927098	C₂	semidirect
8. C₄⋊Q₈×D₁₉	C₂	semidirect
11. Q₈²×C₁₉	C₂	semidirect
49. #64545	Q₈	semidirect
85. Q₈×C₁₉	C₂×Q₈	semidirect
96. D₇₆	C₂×Q₈	semidirect
100. C₄×D₁₉	C₂×Q₈	semidirect
120. Q₈²	D₁₉	semidirect
168. C₁₉	#3503	semidirect
211. Q₈	Q₈×D₁₉	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
#11927270	2
#11927255	2
#11927254	2
#11927209	2
#11927171	2
#11927098	2
C₄⋊Q₈×D₁₉	2
#11927026	2
C₄×#64545	2
Q₈²×C₁₉	2
#3503	19