Subgroups — #13136467

Order 1728.

412 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	C₂₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11720195	C₂	1	2	yes	maximal
3	#11720193	C₂	1	2	yes	maximal, complemented
4	#11720145	C₂	1	2	yes	maximal
5	#11720141	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×D₂₇	C₂	1	1	yes	maximal, characteristic
7	#11720139	C₂	1	1	yes	maximal, characteristic, complemented
8	#11720138	C₂	1	2	yes	maximal, complemented
9	#11720137	C₂	1	1	yes	maximal, characteristic, complemented
10	#11720121	C₂	1	1	yes	maximal, characteristic
11	#11720119	C₂	1	1	yes	maximal, characteristic, complemented
12	#10589831	—	3	1	no	maximal
13	C₂×Q₈×C₂₇	C₂²	1	2	yes	—
14	C₂×C₄×D₂₇	C₂²	1	2	yes	—
15	C₂×C₄×D₂₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₄	C₂²	1	2	yes	—
17	C₄⋊C₄×C₂₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₇	C₂²	1	2	yes	—
19	C₄⋊C₄×C₂₇	C₂²	1	1	yes	characteristic
20	C₄²×C₂₇	C₂²	1	1	yes	characteristic
21	(C₂×D₂₇)⋊C₄	C₂²	1	1	yes	characteristic
22	(C₂×D₂₇)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₂₇)⋊C₄	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₂₇	C₂²	1	2	yes	—
25	C₄⋊Dic₂₇	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₂₇	C₂²	1	1	yes	characteristic
27	Dic₂₇⋊C₄	C₂²	1	1	yes	characteristic
28	Dic₂₇⋊C₄	C₂²	1	2	yes	—
29	Dic₂₇⋊C₄	C₂²	1	2	yes	—
30	Dic₂₇⋊C₄	C₂²	1	2	yes	—
31	Dic₂₇⋊C₄	C₂²	1	2	yes	—
32	Dic₂₇⋊C₄	C₂²	1	2	yes	—
33	Dic₂₇⋊C₄	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₇	C₂²	1	2	yes	—
35	C₄×Dic₂₇	C₂²	1	1	yes	characteristic
36	C₉×C₄⋊Q₈	D₃	1	1	yes	characteristic
37	#63508	—	3	2	no	—
38	#63506	—	3	2	no	—
39	#63458	—	3	2	no	—
40	#63454	—	3	1	no	—
41	C₄⋊C₄×D₉	—	3	1	no	—
42	#63452	—	3	1	no	—
43	#63451	—	3	2	no	—
44	#63450	—	3	1	no	—
45	#63434	—	3	1	no	—
46	#63432	—	3	1	no	—
47	C₂²×D₂₇	C₂³	1	1	yes	characteristic
48	Q₈×C₂₇	—	4	2	no	—
49	C₂×C₄×C₂₇	C₂³	1	2	yes	—
50	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₂₇	C₂³	1	2	yes	—
53	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
54	C₂×Dic₂₇	C₂³	1	2	yes	—
55	C₂×Dic₂₇	C₂³	1	1	yes	characteristic
56	C₂×Dic₂₇	C₂³	1	1	yes	characteristic
57	C₂×Dic₂₇	C₂³	1	2	yes	—
58	C₂×Dic₂₇	C₂³	1	1	yes	characteristic
59	C₄×D₂₇	—	4	2	no	—
60	C₄×D₂₇	Q₈	1	4	yes	complemented
61	Dic₅₄	—	4	2	no	—
62	#5729	—	9	1	no	—
63	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
64	C₂×C₄×D₉	—	3	1	no	—
65	C₂×C₄×D₉	—	3	2	no	—
66	C₂×Dic₁₈	—	3	2	no	—
67	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
68	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
69	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
70	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
71	(C₂×D₉)⋊C₄	—	3	1	no	—
72	(C₂×D₉)⋊C₄	—	3	2	no	—
73	(C₂×D₉)⋊C₄	—	3	1	no	—
74	C₄⋊Dic₉	—	3	2	no	—
75	C₄⋊Dic₉	—	3	1	no	—
76	C₄⋊Dic₉	—	3	1	no	—
77	Dic₉⋊C₄	—	3	1	no	—
78	Dic₉⋊C₄	—	3	2	no	—
79	Dic₉⋊C₄	—	3	2	no	—
80	Dic₉⋊C₄	—	3	2	no	—
81	Dic₉⋊C₄	—	3	2	no	—
82	Dic₉⋊C₄	—	3	2	no	—
83	Dic₉⋊C₄	—	3	1	no	—
84	C₄×Dic₉	—	3	2	no	—
85	C₄×Dic₉	—	3	1	no	—
86	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
87	C₂×D₂₇	—	2	1	no	—
88	C₂×D₂₇	—	2	1	no	—
89	C₂×D₂₇	C₂×Q₈	1	2	yes	—
90	C₄×C₂₇	—	2	2	no	—
91	C₄×C₂₇	—	2	2	no	—
92	C₄×C₂₇	—	2	1	no	—
93	C₄×C₂₇	—	2	1	no	—
94	C₄×C₂₇	C₂×Q₈	1	2	yes	—
95	Dic₂₇	—	2	1	no	—
96	Dic₂₇	—	2	2	no	—
97	Dic₂₇	—	2	2	no	—
98	Dic₂₇	—	2	1	no	—
99	Dic₂₇	C₂×Q₈	1	2	yes	—
100	C₃×C₄⋊Q₈	D₉	1	1	yes	characteristic
101	#946	—	9	2	no	—
102	#944	—	9	2	no	—
103	#896	—	9	2	no	—
104	#892	—	9	1	no	—
105	D₃×C₄⋊C₄	—	9	1	no	—
106	#890	—	9	1	no	—
107	#889	—	9	2	no	—
108	#888	—	9	1	no	—
109	#872	—	9	1	no	—
110	#870	—	9	1	no	—
111	C₂²×D₉	—	3	1	no	—
112	Q₈×C₉	—	4	2	no	—
113	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
114	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
117	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
118	C₂×Dic₉	—	3	1	no	—
119	C₂×Dic₉	—	3	2	no	—
120	C₂×Dic₉	—	3	1	no	—
121	C₂×Dic₉	—	3	1	no	—
122	C₂×Dic₉	—	3	2	no	—
123	C₄×D₉	—	3	4	no	—
124	C₄×D₉	—	12	2	no	—
125	Dic₁₈	—	12	2	no	—
126	#544	—	27	1	no	Sylow
127	C₂×C₂₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₂₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₂₇	2₋¹⁺⁴	1	1	yes	characteristic
130	D₂₇	—	2	2	no	—
131	C₂×C₃×Q₈	C₂×D₉	1	2	yes	—
132	C₂×C₄×D₃	—	9	1	no	—
133	C₂×C₄×D₃	—	9	2	no	—
134	C₂×Dic₆	—	9	2	no	—
135	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
136	C₃×C₄⋊C₄	C₂×D₉	1	2	yes	—
137	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
138	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
139	(C₂×D₃)⋊C₄	—	9	1	no	—
140	(C₂×D₃)⋊C₄	—	9	1	no	—
141	(C₂×D₃)⋊C₄	—	9	2	no	—
142	C₄⋊Dic₃	—	9	1	no	—
143	C₄⋊Dic₃	—	9	1	no	—
144	C₄⋊Dic₃	—	9	2	no	—
145	Dic₃⋊C₄	—	9	2	no	—
146	Dic₃⋊C₄	—	9	2	no	—
147	Dic₃⋊C₄	—	9	1	no	—
148	Dic₃⋊C₄	—	9	1	no	—
149	Dic₃⋊C₄	—	9	2	no	—
150	Dic₃⋊C₄	—	9	2	no	—
151	Dic₃⋊C₄	—	9	2	no	—
152	C₄×Dic₃	—	9	1	no	—
153	C₄×Dic₃	—	9	2	no	—
154	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
155	C₂×D₉	—	3	2	no	—
156	C₂×D₉	—	6	1	no	—
157	C₂×D₉	—	6	1	no	—
158	C₄×C₉	—	2	2	no	—
159	C₄×C₉	—	2	2	no	—
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	D₃×Q₈	1	2	yes	—
163	Dic₉	—	3	2	no	—
164	Dic₉	—	6	1	no	—
165	Dic₉	—	6	2	no	—
166	Dic₉	—	6	2	no	—
167	Dic₉	—	6	1	no	—
168	C₄⋊Q₈	—	27	2	no	—
169	C₄⋊Q₈	—	27	1	no	—
170	C₄⋊Q₈	D₂₇	1	1	yes	characteristic, complemented
171	#125	—	27	2	no	—
172	#125	—	27	1	no	—
173	#125	—	27	1	no	—
174	C₂²⋊Q₈	—	27	2	no	—
175	C₂²⋊Q₈	—	27	2	no	—
176	#117	—	27	1	no	—
177	#117	—	27	1	no	—
178	C₂×C₄⋊C₄	—	27	1	no	—
179	C₂₇	#544	1	1	yes	characteristic, complemented, Sylow
180	C₂²×D₃	—	9	1	no	—
181	C₃×Q₈	—	4	2	no	—
182	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
183	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
187	C₂×Dic₃	—	9	1	no	—
188	C₂×Dic₃	—	9	1	no	—
189	C₂×Dic₃	—	9	2	no	—
190	C₂×Dic₃	—	9	1	no	—
191	C₂×Dic₃	—	9	2	no	—
192	C₄×D₃	—	9	4	no	—
193	C₄×D₃	—	36	2	no	—
194	Dic₆	—	36	2	no	—
195	C₂×C₉	#1004	1	1	yes	characteristic
196	C₂×C₉	C₂×D₃×Q₈	1	1	yes	characteristic
197	C₂×C₉	#1007	1	1	yes	characteristic
198	D₉	—	6	2	no	—
199	C₂×Q₈	—	27	2	no	—
200	C₂×Q₈	C₂×D₂₇	1	2	yes	—
201	C₂²×C₄	—	27	1	no	—
202	C₂²×C₄	—	27	2	no	—
203	C₄⋊C₄	—	27	2	no	—
204	C₄⋊C₄	—	27	1	no	—
205	C₄⋊C₄	—	27	1	no	—
206	C₄⋊C₄	—	27	1	no	—
207	C₄⋊C₄	—	27	1	no	—
208	C₄⋊C₄	—	27	2	no	—
209	C₄⋊C₄	—	27	2	no	—
210	C₄⋊C₄	—	27	2	no	—
211	C₄⋊C₄	—	27	2	no	—
212	C₄⋊C₄	—	27	2	no	—
213	C₄⋊C₄	C₂×D₂₇	1	1	yes	characteristic
214	C₄⋊C₄	C₂×D₂₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂×D₂₇	1	2	yes	—
216	C₂²⋊C₄	—	27	2	no	—
217	C₂²⋊C₄	—	27	1	no	—
218	C₂²⋊C₄	—	27	1	no	—
219	C₄²	—	27	2	no	—
220	C₄²	—	27	1	no	—
221	C₄²	C₂×D₂₇	1	1	yes	characteristic
222	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
223	C₂×D₃	—	9	2	no	—
224	C₂×D₃	—	18	1	no	—
225	C₂×D₃	—	18	1	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	Q₈×D₉	1	2	yes	—
231	Dic₃	—	9	2	no	—
232	Dic₃	—	18	1	no	—
233	Dic₃	—	18	2	no	—
234	Dic₃	—	18	1	no	—
235	Dic₃	—	18	2	no	—
236	C₉	#5729	1	1	yes	characteristic
237	C₂³	—	27	1	no	—
238	Q₈	—	4	2	no	—
239	Q₈	—	108	2	no	—
240	C₂×C₄	—	27	1	no	—
241	C₂×C₄	—	27	4	no	—
242	C₂×C₄	—	27	1	no	—
243	C₂×C₄	—	27	2	no	—
244	C₂×C₄	—	27	1	no	—
245	C₂×C₄	—	27	2	no	—
246	C₂×C₄	—	108	2	no	—
247	C₂×C₄	C₂²×D₂₇	1	2	yes	—
248	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
250	C₂×C₄	C₂²×D₂₇	1	2	yes	—
251	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
252	C₂×C₃	#63710	1	1	yes	characteristic
253	C₂×C₃	C₂×Q₈×D₉	1	1	yes	characteristic
254	C₂×C₃	#63713	1	1	yes	characteristic
255	D₃	—	18	2	no	—
256	C₂²	—	27	2	no	—
257	C₂²	—	54	1	no	—
258	C₂²	—	54	1	no	—
259	C₂²	C₂³×D₂₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	—	27	2	no	—
265	C₄	—	54	2	no	—
266	C₄	—	54	1	no	—
267	C₄	—	54	1	no	—
268	C₄	—	54	2	no	—
269	C₄	Q₈×D₂₇	1	2	yes	—
270	C₃	#10589831	1	1	yes	characteristic
271	C₂	—	54	2	no	—
272	C₂	#11720670	1	1	yes	characteristic
273	C₂	C₂×Q₈×D₂₇	1	1	yes	characteristic
274	C₂	#11720673	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₇
commutator, cosocle	C₂²×C₂₇	C₂⁴
Frattini	C₂²×C₉	C₂³×D₃
Fitting	C₂₇×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₂³×D₉
prime radical	C₂²×D₂₇	C₂³
hypercenter	C₄⋊Q₈	D₂₇
hypocenter	C₂₇	#544
LCS	G ▷ C₂²×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂₇×C₄⋊Q₈	C₂	semidirect
3. #11720193	C₂	semidirect
7. #11720139	C₂	semidirect
8. #11720138	C₂	semidirect
9. #11720137	C₂	semidirect
11. #11720119	C₂	semidirect
60. C₄×D₂₇	Q₈	semidirect
170. C₄⋊Q₈	D₂₇	semidirect
179. C₂₇	#544	semidirect

Maximal subgroups

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

subgroup	index
C₂₇×C₄⋊Q₈	2
#11720195	2
#11720193	2
#11720145	2
#11720141	2
C₄⋊C₄×D₂₇	2
#11720139	2
#11720138	2
#11720137	2
#11720121	2
#11720119	2
#10589831	3