Subgroups — #30866261

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	2	yes	maximal
2	#22560985	C₂	1	8	yes	maximal, complemented
3	#22560247	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863453	—	2	2	no	—
6	#16863453	—	2	2	no	—
7	#16863449	—	2	2	no	—
8	#16863449	—	2	2	no	—
9	C₂×#14523979	C₂²	1	8	yes	—
10	#16863420	C₂²	1	4	yes	—
11	#16863360	C₂²	1	4	yes	—
12	C₄×#11761612	C₂²	1	4	yes	—
13	#16863161	C₂²	1	8	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₇	C₄	1	4	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
20	C₂×#11761612	C₂³	1	8	yes	—
21	#14523979	—	4	8	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
26	#14523954	—	2	4	no	—
27	C₄×C₆₇⋊C₈	—	2	4	no	—
28	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	—	4	4	no	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	2	yes	—
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
43	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
44	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
45	#11761612	—	2	8	no	—
46	#11761612	—	2	8	no	—
47	C₂×C₆₇⋊C₈	—	2	8	no	—
48	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	—	2	2	no	—
51	C₂²×C₄×C₆₇	—	2	2	no	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	8	no	—
68	C₂×C₈×C₆₇	—	4	4	no	—
69	C₂×C₈×C₆₇	—	4	4	no	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
71	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₆₇	C₂×D₄	1	2	yes	—
78	C₄²×C₆₇	C₂×D₄	1	2	yes	—
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	2	2	no	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	4	1	no	—
97	C₂×C₄×C₆₇	—	4	1	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	#108	1	4	yes	complemented
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
109	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
110	C₈×C₆₇	—	2	8	no	—
111	C₆₇⋊C₈	—	4	8	no	—
112	C₂²×C₆₇	—	2	8	no	—
113	C₂²×C₆₇	—	2	1	no	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	2	1	no	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	—	2	1	no	—
119	C₂²×C₆₇	C₂×#108	1	2	yes	—
120	C₂²×C₆₇	#430	1	2	yes	—
121	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₆₇	#544	1	2	yes	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	C₂×#108	1	4	yes	—
129	C₄×C₆₇	#518	1	1	yes	characteristic
130	C₄×C₆₇	#519	1	1	yes	characteristic
131	C₄×C₆₇	#537	1	2	yes	—
132	#21969	—	67	1	no	maximal, Sylow
133	C₂×C₆₇	—	2	2	no	—
134	C₂×C₆₇	—	2	2	no	—
135	C₂×C₆₇	—	2	2	no	—
136	C₂×C₆₇	#2316	1	1	yes	characteristic
137	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
138	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
139	#2920	—	67	2	no	—
140	C₂²×C₄⋊C₈	D₆₇	1	1	yes	characteristic
141	#1950	—	67	8	no	—
142	#1810	—	67	4	no	—
143	C₆₇	#21969	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₈	C₂×D₆₇	1	2	yes	—
145	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
146	#432	—	134	2	no	—
147	#432	—	134	2	no	—
148	#431	—	134	2	no	—
149	#431	—	134	2	no	—
150	C₂×#108	—	67	8	no	—
151	#423	—	67	4	no	—
152	C₂×C₄⋊C₈	Dic₆₇	1	4	yes	—
153	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
154	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
155	C₂×C₄⋊C₈	C₂×D₆₇	1	4	yes	—
156	#407	—	67	4	no	—
157	C₄×M₁₆	—	67	4	no	—
158	#341	—	67	8	no	—
159	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
160	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
161	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
162	C₂×M₁₆	—	67	8	no	—
163	C₂²×C₈	—	4	4	no	—
164	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
165	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
166	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
167	C₂×C₄²	Dic₁₃₄	1	2	yes	—
168	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
169	C₂×C₄²	D₂₆₈	1	2	yes	—
170	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
172	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
175	#108	—	268	8	no	—
176	C₄⋊C₈	—	134	4	no	—
177	C₄⋊C₈	—	134	4	no	—
178	C₄⋊C₈	C₂×Dic₆₇	1	8	yes	—
179	C₄⋊C₈	C₂×Dic₆₇	1	8	yes	—
180	C₂²⋊C₈	—	134	4	no	—
181	C₂²⋊C₈	—	134	4	no	—
182	#97	—	134	4	no	—
183	C₄×C₈	—	134	4	no	—
184	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
191	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
192	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
201	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
202	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
203	M₁₆	—	134	8	no	—
204	M₁₆	—	134	8	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	134	8	no	—
209	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
210	C₄²	C₄⋊Dic₆₇	1	2	yes	—
211	C₄²	C₄⋊Dic₆₇	1	2	yes	—
212	C₄²	C₄⋊Dic₆₇	1	2	yes	—
213	C₄²	C₄⋊Dic₆₇	1	2	yes	—
214	C₄²	C₂²×Dic₆₇	1	2	yes	—
215	C₄²	C₂²×Dic₆₇	1	2	yes	—
216	C₄²	C₂×D₂₆₈	1	2	yes	—
217	C₄²	C₂×D₂₆₈	1	2	yes	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
222	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
223	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	#14523979	1	4	yes	complemented
239	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
245	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
247	C₂×C₄	#14524088	1	2	yes	—
248	C₂×C₄	#14524091	1	2	yes	—
249	C₈	—	2	8	no	—
250	C₈	—	268	8	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	C₂×#14523979	1	2	yes	—
259	C₂²	#16863445	1	2	yes	—
260	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
261	C₂²	#16864027	1	2	yes	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	C₂×#14523979	1	4	yes	—
268	C₄	#16863871	1	1	yes	characteristic
269	C₄	#16863874	1	1	yes	characteristic
270	C₄	#16863977	1	2	yes	—
271	C₂	—	2	2	no	—
272	C₂	—	2	2	no	—
273	C₂	—	2	2	no	—
274	C₂	#22562827	1	1	yes	characteristic
275	C₂	C₂²×#14523979	1	1	yes	characteristic
276	C₂	C₂×#16863445	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×C₄⋊C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#21969
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #22560985	C₂	semidirect
99. C₂×C₄×C₆₇	#108	semidirect
143. C₆₇	#21969	semidirect
238. C₂×C₄	#14523979	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
#22560985	2
#22560247	2
C₂²×C₄⋊C₈×C₆₇	2
#21969	67