Subgroups — #97×Dic₂₆ #15861586

Order 3328.

954 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₂₆	C₂	1	2	yes	maximal
2	C₄²×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#13114704	C₂	1	4	yes	maximal
4	#13114703	C₂	1	2	yes	maximal
5	#13114690	C₂	1	2	yes	maximal
6	#13114689	C₂	1	1	yes	maximal, characteristic
7	#97×Dic₁₃	C₂	1	2	yes	maximal
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	C₄	1	2	yes	—
10	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
15	C₄³×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88055	C₂²	1	4	yes	—
17	C₂×#88054	C₂²	1	4	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
21	C₄×#6101	C₄	1	4	yes	—
22	C₄×#6101	C₂²	1	2	yes	—
23	C₄²×Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄⋊#1055	C₂²	1	2	yes	—
25	C₂×#88043	C₂²	1	2	yes	—
26	C₈×Dic₂₆	C₄	1	16	yes	complemented
27	#11718142	C₂²	1	4	yes	—
28	#11718116	C₂²	1	2	yes	—
29	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×#6101	C₂×C₄	1	4	yes	—
40	C₂×#6101	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×#1055	C₂³	1	4	yes	—
46	C₄×Dic₂₆	—	2	8	no	—
47	C₄×Dic₂₆	C₂×C₄	1	8	yes	—
48	C₁₃×#97	—	2	4	no	—
49	C₁₃×#97	—	2	2	no	—
50	C₁₃×#97	Q₈	1	4	yes	complemented
51	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
52	#88071	C₂×C₄	1	2	yes	—
53	#88071	C₂×C₄	1	4	yes	—
54	#88055	—	2	8	no	—
55	#88054	C₂×C₄	1	16	yes	—
56	C₈×Dic₁₃	C₂×C₄	1	16	yes	—
57	C₄⋊#1055	C₂×C₄	1	8	yes	—
58	#88043	—	2	4	no	—
59	Q₈×#97	—	13	1	no	maximal, Sylow
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
68	C₂×Dic₂₆	C₄²	1	4	yes	—
69	C₂×Dic₂₆	M₁₆	1	8	yes	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₁₃	#41	1	4	yes	—
76	C₄²×C₁₃	—	2	4	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	4	2	no	—
79	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
80	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
81	C₄²×C₁₃	#41	1	2	yes	—
82	C₄⋊Dic₁₃	—	2	4	no	—
83	C₄⋊Dic₁₃	C₄²	1	4	yes	—
84	#6101	—	2	8	no	—
85	#6101	C₄²	1	8	yes	—
86	C₄×Dic₁₃	—	2	4	no	—
87	C₄×Dic₁₃	—	2	8	no	—
88	C₄×Dic₁₃	—	4	4	no	—
89	C₄×Dic₁₃	C₂²×C₄	1	8	yes	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	C₂²×C₄	1	8	yes	—
93	C₂×C₈×Q₈	—	13	2	no	—
94	C₄²×Q₈	—	13	1	no	—
95	#1770	—	13	4	no	—
96	#1770	—	13	2	no	—
97	#1767	—	13	2	no	—
98	#1767	—	13	1	no	—
99	C₄×#97	—	13	2	no	—
100	C₄×#97	D₁₃	1	1	yes	characteristic
101	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂×Dic₁₃	—	2	4	no	—
117	C₂×Dic₁₃	—	2	4	no	—
118	C₂×Dic₁₃	—	2	4	no	—
119	C₂×Dic₁₃	—	2	4	no	—
120	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
121	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
122	C₂×Dic₁₃	C₂×M₁₆	1	8	yes	—
123	Dic₂₆	#97	1	16	yes	direct product factor
124	C₈×C₁₃	—	2	4	no	—
125	C₈×C₁₃	C₄×Q₈	1	8	yes	complemented
126	#1055	—	2	8	no	—
127	C₂×C₄×Q₈	—	13	2	no	—
128	C₂×C₄×Q₈	—	13	1	no	—
129	C₈×Q₈	—	13	16	no	—
130	C₂×C₄⋊C₈	—	13	4	no	—
131	C₂×C₄⋊C₈	—	13	2	no	—
132	C₂×#97	—	13	4	no	—
133	C₂×#97	—	13	2	no	—
134	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
135	C₂×#97	C₂×D₁₃	1	2	yes	—
136	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
137	C₂×C₄×C₈	—	13	4	no	—
138	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
139	C₄×C₄⋊C₄	—	13	4	no	—
140	C₄×C₄⋊C₄	—	13	2	no	—
141	C₄×C₄⋊C₄	—	13	2	no	—
142	C₄×C₄⋊C₄	—	13	1	no	—
143	C₄³	—	13	2	no	—
144	C₄³	C₂×D₁₃	1	1	yes	characteristic
145	#336	—	13	4	no	—
146	#336	—	13	2	no	—
147	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
150	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
151	C₂²×C₁₃	#568	1	2	yes	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	—	4	2	no	—
155	C₄×C₁₃	C₂×#97	1	4	yes	—
156	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
157	C₄×C₁₃	C₄×#41	1	2	yes	—
158	Dic₁₃	—	2	4	no	—
159	Dic₁₃	—	4	4	no	—
160	Dic₁₃	C₂×#97	1	8	yes	—
161	C₂²×Q₈	—	13	1	no	—
162	C₂²×C₈	—	13	4	no	—
163	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
164	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
165	C₄×Q₈	—	13	8	no	—
166	C₄×Q₈	—	26	8	no	—
167	C₂×C₄⋊C₄	—	13	2	no	—
168	C₂×C₄⋊C₄	—	13	1	no	—
169	C₂×C₄⋊C₄	—	13	4	no	—
170	C₂×C₄⋊C₄	—	13	2	no	—
171	C₂×C₄²	—	13	2	no	—
172	C₂×C₄²	—	13	4	no	—
173	C₂×C₄²	—	13	2	no	—
174	C₂×C₄²	—	13	4	no	—
175	C₂×C₄²	C₄×D₁₃	1	2	yes	—
176	C₂×C₄²	C₄×D₁₃	1	2	yes	—
177	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
180	C₄⋊C₈	—	13	16	no	—
181	C₄⋊C₈	—	13	8	no	—
182	#97	—	2	4	no	—
183	#97	—	2	2	no	—
184	#97	—	26	8	no	—
185	#97	—	26	4	no	—
186	#97	Dic₂₆	1	4	yes	direct product factor
187	C₄×C₈	—	13	16	no	—
188	C₄×C₈	C₄×D₁₃	1	8	yes	—
189	#95	—	13	4	no	—
190	#95	—	13	2	no	—
191	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
192	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
193	C₂×C₁₃	#2875	1	1	yes	characteristic
194	C₂×C₁₃	Q₈×M₁₆	1	2	yes	—
195	C₂×C₁₃	#2971	1	2	yes	—
196	C₂×Q₈	—	13	4	no	—
197	C₂×Q₈	—	13	8	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	13	2	no	—
200	C₂²×C₄	—	13	4	no	—
201	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	13	8	no	—
209	C₂×C₈	—	26	4	no	—
210	C₂×C₈	—	26	4	no	—
211	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
212	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
213	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
214	C₂×C₈	#6127	1	4	yes	—
215	C₄⋊C₄	—	13	8	no	—
216	C₄⋊C₄	—	13	4	no	—
217	C₄⋊C₄	—	26	8	no	—
218	C₄⋊C₄	—	26	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	—	13	8	no	—
223	C₄²	—	26	8	no	—
224	C₄²	—	26	4	no	—
225	C₄²	—	52	4	no	—
226	C₄²	C₂×Dic₂₆	1	2	yes	—
227	C₄²	C₂×C₄×D₁₃	1	4	yes	—
228	C₄²	#6127	1	2	yes	—
229	C₁₃	Q₈×#97	1	1	yes	characteristic, complemented, Sylow
230	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
231	Q₈	—	13	16	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	13	8	no	—
240	C₂×C₄	—	13	4	no	—
241	C₂×C₄	—	26	4	no	—
242	C₂×C₄	—	26	4	no	—
243	C₂×C₄	—	26	4	no	—
244	C₂×C₄	—	26	4	no	—
245	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
246	C₂×C₄	C₄²×D₁₃	1	2	yes	—
247	C₂×C₄	C₄²×D₁₃	1	2	yes	—
248	C₂×C₄	M₁₆×D₁₃	1	4	yes	—
249	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
251	C₂×C₄	C₂×#6127	1	2	yes	—
252	C₂×C₄	C₂×#6127	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₈	—	26	8	no	—
255	C₈	C₄×Dic₂₆	1	8	yes	complemented
256	C₂²	C₂×C₄×Dic₂₆	1	1	yes	characteristic
257	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
258	C₂²	C₄×#6127	1	1	yes	characteristic
259	C₂²	C₂×M₁₆×D₁₃	1	2	yes	—
260	C₂²	#11719450	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	4	2	no	—
264	C₄	—	13	8	no	—
265	C₄	—	26	4	no	—
266	C₄	—	52	4	no	—
267	C₄	D₁₃×#97	1	4	yes	—
268	C₄	C₂×C₄×Dic₂₆	1	2	yes	—
269	C₄	C₄×#6127	1	2	yes	—
270	C₂	C₄²×Dic₂₆	1	1	yes	characteristic
271	C₂	C₂×D₁₃×#97	1	1	yes	characteristic
272	C₂	#13123261	1	1	yes	characteristic
273	C₂	M₁₆×Dic₂₆	1	2	yes	—
274	C₂	#13123854	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₄×C₁₃×#97	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	Q₈×#97
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
26. C₈×Dic₂₆	C₄	semidirect
50. C₁₃×#97	Q₈	semidirect
123. Dic₂₆	#97	direct (N × H)
125. C₈×C₁₃	C₄×Q₈	semidirect
186. #97	Dic₂₆	direct (N × H)
229. C₁₃	Q₈×#97	semidirect
255. C₈	C₄×Dic₂₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₂₆	2
C₄²×Dic₂₆	2
#13114704	2
#13114703	2
#13114690	2
#13114689	2
#97×Dic₁₃	2
C₄×C₁₃×#97	2
Q₈×#97	13

Subgroups — #97×Dic26 #15861586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #97×Dic₂₆ #15861586