Subgroups — #15761692

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114988	C₂	1	1	yes	maximal, characteristic
2	C₄²×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#1055	C₂	1	4	yes	maximal
4	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₁₃×#391	C₄	1	2	yes	—
8	C₁₃×#389	—	2	2	no	—
9	C₁₃×#384	C₄	1	1	yes	characteristic
10	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
11	C₄³×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	—	2	2	no	—
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×C₄×#1055	—	2	2	no	—
15	C₂×C₄×#1055	C₂²	1	2	yes	—
16	Q₈⋊#1055	—	2	8	no	—
17	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₁₃×C₄⋊Q₈	—	2	2	no	—
19	C₁₃×C₄⋊Q₈	C₈	1	4	yes	complemented
20	C₄×Q₈×C₁₃	—	2	4	no	—
21	C₄×Q₈×C₁₃	—	2	4	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	2	yes	—
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×#1055	—	2	2	no	—
33	C₂²×#1055	—	2	2	no	—
34	C₁₃×#95	—	2	4	no	—
35	C₄⋊#1055	—	2	4	no	—
36	C₄⋊#1055	—	4	4	no	—
37	C₄×#1055	—	2	8	no	—
38	C₄×#1055	—	2	2	no	—
39	C₄×#1055	D₄	1	4	yes	—
40	#9637	—	13	1	no	maximal, Sylow
41	C₂×Q₈×C₁₃	—	2	2	no	—
42	C₂×Q₈×C₁₃	—	2	4	no	—
43	C₂×Q₈×C₁₃	—	2	4	no	—
44	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	—	2	8	no	—
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₄⋊C₄	—	4	4	no	—
55	C₁₃×C₄⋊C₄	—	4	4	no	—
56	C₄²×C₁₃	—	2	4	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	4	2	no	—
59	C₄²×C₁₃	—	4	2	no	—
60	C₄²×C₁₃	C₂×C₈	1	2	yes	—
61	C₄²×C₁₃	M₁₆	1	2	yes	—
62	C₄²×C₁₃	SD₁₆	1	4	yes	—
63	C₄²×C₁₃	Q₁₆	1	4	yes	—
64	C₄²×C₁₃	C₂×D₄	1	2	yes	—
65	C₄²×C₁₃	#41	1	2	yes	—
66	C₂×#1055	—	2	8	no	—
67	C₂×#1055	—	2	4	no	—
68	C₂×#1055	—	4	2	no	—
69	C₂×#1055	—	4	4	no	—
70	C₄×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
71	#1842	—	13	1	no	—
72	C₄²×C₈	—	13	1	no	—
73	C₂×Q₈⋊C₈	—	13	4	no	—
74	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	Q₈×C₁₃	—	2	8	no	—
76	Q₈×C₁₃	—	4	4	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	4	no	—
81	C₂×C₄×C₁₃	—	4	2	no	—
82	C₂×C₄×C₁₃	—	4	2	no	—
83	C₂×C₄×C₁₃	—	4	2	no	—
84	C₂×C₄×C₁₃	—	4	4	no	—
85	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
86	C₂×C₄×C₁₃	#103	1	4	yes	—
87	C₂×C₄×C₁₃	#103	1	4	yes	—
88	C₂×C₄×C₁₃	#104	1	4	yes	—
89	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₁₃	#121	1	2	yes	—
91	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#131	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
98	#1055	—	2	8	no	—
99	#1055	—	8	2	no	—
100	C₂×C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
101	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
102	C₂×C₄⋊C₈	—	13	2	no	—
103	C₂×C₄⋊C₈	—	26	2	no	—
104	C₂×C₄×C₈	—	13	2	no	—
105	C₂×C₄×C₈	—	26	2	no	—
106	#391	Dic₁₃	1	2	yes	—
107	#389	—	2	2	no	—
108	#384	Dic₁₃	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	2	2	no	—
110	C₄³	C₂×D₁₃	1	1	yes	characteristic
111	Q₈⋊C₈	—	26	8	no	—
112	C₂²×C₁₃	#390	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₁₃	#408	1	1	yes	characteristic
115	C₂²×C₁₃	C₂×#103	1	2	yes	—
116	C₂²×C₁₃	C₂×#104	1	2	yes	—
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	2	4	no	—
119	C₄×C₁₃	—	4	4	no	—
120	C₄×C₁₃	—	8	2	no	—
121	C₄×C₁₃	Q₈⋊C₈	1	8	yes	complemented
122	C₄×C₁₃	C₈×D₄	1	2	yes	—
123	C₄×C₁₃	#436	1	2	yes	—
124	C₄×C₁₃	#487	1	2	yes	—
125	C₄×C₁₃	#492	1	1	yes	characteristic
126	C₄×C₁₃	#494	1	1	yes	characteristic
127	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
128	C₂²×C₈	—	26	2	no	—
129	C₂²×C₈	—	26	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	#1055	1	4	yes	complemented
132	C₄×Q₈	—	2	4	no	—
133	C₄×Q₈	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
140	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
141	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
142	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
143	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
144	C₄⋊C₈	—	26	4	no	—
145	C₄⋊C₈	—	52	4	no	—
146	C₄×C₈	—	13	4	no	—
147	C₄×C₈	—	26	8	no	—
148	C₄×C₈	—	26	2	no	—
149	#95	—	2	4	no	—
150	C₂×C₁₃	C₂×Q₈⋊C₈	1	2	yes	—
151	C₂×C₁₃	#1493	1	2	yes	—
152	C₂×C₁₃	#1954	1	1	yes	characteristic
153	C₂×C₁₃	#1956	1	1	yes	characteristic
154	C₂×C₁₃	#1962	1	1	yes	characteristic
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	C₂×#1055	1	4	yes	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
162	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
166	C₂×C₈	—	26	8	no	—
167	C₂×C₈	—	26	4	no	—
168	C₂×C₈	—	52	2	no	—
169	C₂×C₈	—	52	4	no	—
170	C₄⋊C₄	—	2	8	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	4	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂×#1055	1	2	yes	—
179	C₄²	#6099	1	2	yes	—
180	C₄²	#6106	1	4	yes	—
181	C₄²	#6107	1	4	yes	—
182	C₄²	D₄×D₁₃	1	2	yes	—
183	C₄²	#6129	1	2	yes	—
184	C₁₃	#9637	1	1	yes	characteristic, complemented, Sylow
185	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
186	Q₈	—	2	8	no	—
187	Q₈	—	4	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂²⋊#1055	1	4	yes	—
197	C₂×C₄	#88075	1	4	yes	—
198	C₂×C₄	#88075	1	4	yes	—
199	C₂×C₄	#88077	1	4	yes	—
200	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
201	C₂×C₄	C₂×#6099	1	1	yes	characteristic
202	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
203	C₂×C₄	#88190	1	1	yes	characteristic
204	C₂×C₄	#88192	1	2	yes	—
205	C₂×C₄	#88194	1	1	yes	characteristic
206	C₂×C₄	C₂×#6106	1	2	yes	—
207	C₂×C₄	C₂×#6107	1	2	yes	—
208	C₂×C₄	#88202	1	2	yes	—
209	C₈	—	26	8	no	—
210	C₈	—	104	2	no	—
211	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
212	C₂²	#11718882	1	1	yes	characteristic
213	C₂²	C₂×#88075	1	2	yes	—
214	C₂²	#11718895	1	1	yes	characteristic
215	C₂²	C₂×#88077	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	2	no	—
220	C₄	Q₈⋊#1055	1	8	yes	complemented
221	C₄	D₄×#1055	1	2	yes	—
222	C₄	#11718673	1	2	yes	—
223	C₄	#11718742	1	2	yes	—
224	C₄	#11718746	1	1	yes	characteristic
225	C₄	#11718753	1	1	yes	characteristic
226	C₂	C₂×Q₈⋊#1055	1	2	yes	—
227	C₂	#13113291	1	2	yes	—
228	C₂	#13115631	1	1	yes	characteristic
229	C₂	#13115639	1	1	yes	characteristic
230	C₂	#13115665	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#9637
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
19. C₁₃×C₄⋊Q₈	C₈	semidirect
121. C₄×C₁₃	Q₈⋊C₈	semidirect
131. C₄⋊Q₈	#1055	semidirect
184. C₁₃	#9637	semidirect
220. C₄	Q₈⋊#1055	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
C₄²×#1055	2
C₂×Q₈⋊#1055	2
C₄×C₁₃×C₄⋊Q₈	2
#9637	13