Subgroups — #30864498

Order 17152.

822 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863526	C₂	1	2	yes	maximal
2	C₄×#14524043	C₂	1	1	yes	maximal, characteristic
3	#22561145	C₂	1	2	yes	maximal
4	#22560927	C₂	1	4	yes	maximal
5	#22560361	C₂	1	1	yes	maximal, characteristic
6	#22560076	C₂	1	4	yes	maximal
7	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524043	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
10	#16863526	—	2	8	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×Dic₆₇⋊C₈	C₂²	1	4	yes	—
14	C₂×#14523953	C₂²	1	4	yes	—
15	#16863302	C₄	1	2	yes	—
16	#16863294	C₄	1	4	yes	—
17	#16863266	C₄	1	1	yes	characteristic
18	C₄×C₄⋊Dic₆₇	C₂²	1	4	yes	—
19	#16863150	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
24	C₄³×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
26	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
27	#14524043	—	2	4	no	—
28	C₄×Dic₁₃₄	—	2	4	no	—
29	C₄×Dic₁₃₄	—	2	4	no	—
30	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
33	C₂×Dic₆₇⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
35	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
37	Dic₆₇⋊C₈	—	2	8	no	—
38	#14523953	—	2	8	no	—
39	C₆₇⋊#95	C₂×C₄	1	8	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
48	C₄×C₈×C₆₇	—	2	2	no	—
49	C₄×C₈×C₆₇	—	2	4	no	—
50	C₄×C₈×C₆₇	D₄	1	4	yes	—
51	C₄×C₈×C₆₇	Q₈	1	8	yes	complemented
52	C₂×Dic₁₃₄	—	2	4	no	—
53	C₂×Dic₁₃₄	—	2	4	no	—
54	C₂×Dic₁₃₄	—	2	4	no	—
55	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
56	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	8	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	2	8	no	—
61	C₄⋊Dic₆₇	—	2	8	no	—
62	Dic₆₇⋊C₄	—	2	8	no	—
63	C₄×Dic₆₇	—	2	4	no	—
64	C₄×Dic₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
76	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
77	C₄²×C₆₇	—	2	2	no	—
78	C₄²×C₆₇	—	2	4	no	—
79	C₄²×C₆₇	M₁₆	1	4	yes	—
80	C₄²×C₆₇	C₂×D₄	1	2	yes	—
81	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
82	C₄²×C₆₇	#41	1	2	yes	—
83	C₄²×C₆₇	#41	1	4	yes	—
84	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
91	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
92	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
94	C₂×C₄×C₆₇	#131	1	2	yes	—
95	C₂×C₄×C₆₇	#131	1	4	yes	—
96	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
100	C₂×Dic₆₇	—	2	4	no	—
101	C₂×Dic₆₇	—	2	4	no	—
102	C₂×Dic₆₇	—	2	4	no	—
103	C₂×Dic₆₇	—	2	8	no	—
104	C₂×Dic₆₇	—	2	4	no	—
105	Dic₁₃₄	—	4	8	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	—	2	4	no	—
108	C₈×C₆₇	C₄⋊Q₈	1	4	yes	complemented
109	C₆₇⋊C₈	—	4	4	no	—
110	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂²×C₆₇	C₂×C₄×Q₈	1	2	yes	—
112	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×#131	1	2	yes	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	2	4	no	—
117	C₄×C₆₇	#436	1	4	yes	—
118	C₄×C₆₇	#446	1	8	yes	complemented
119	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₄×C₆₇	#532	1	1	yes	characteristic
121	C₄×C₆₇	#533	1	2	yes	—
122	Dic₆₇	—	4	4	no	—
123	Dic₆₇	—	4	4	no	—
124	#21712	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
127	C₂×C₆₇	#2931	1	1	yes	characteristic
128	C₂×C₆₇	C₂×#446	1	2	yes	—
129	C₂×C₆₇	#2960	1	2	yes	—
130	C₂×#446	—	67	2	no	—
131	C₄×C₄⋊Q₈	—	67	1	no	—
132	#1984	—	67	2	no	—
133	#1939	—	67	4	no	—
134	#1842	—	67	1	no	—
135	#1770	—	67	4	no	—
136	C₄²×C₈	D₆₇	1	1	yes	characteristic
137	C₆₇	#21712	1	1	yes	characteristic, complemented, Sylow
138	C₂×C₄⋊Q₈	—	67	1	no	—
139	C₂×C₄×Q₈	—	67	2	no	—
140	#446	—	134	8	no	—
141	C₂×C₄⋊C₈	—	67	4	no	—
142	C₂×C₄⋊C₈	—	67	2	no	—
143	C₂×C₄⋊C₈	—	67	4	no	—
144	C₂×#97	—	67	4	no	—
145	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
149	#391	—	67	2	no	—
150	#389	—	67	4	no	—
151	#384	—	67	1	no	—
152	C₄×C₄⋊C₄	—	67	4	no	—
153	C₄³	C₂×D₆₇	1	1	yes	characteristic
154	#336	—	67	8	no	—
155	C₂²×Q₈	—	67	2	no	—
156	C₂²×C₈	—	67	4	no	—
157	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
159	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
160	C₄⋊Q₈	—	134	4	no	—
161	C₄×Q₈	—	134	4	no	—
162	C₄×Q₈	—	134	4	no	—
163	C₂×C₄⋊C₄	—	67	4	no	—
164	C₂×C₄⋊C₄	—	67	2	no	—
165	C₂×C₄⋊C₄	—	67	4	no	—
166	C₂×C₄⋊C₄	—	67	4	no	—
167	C₂×C₄²	—	67	4	no	—
168	C₂×C₄²	C₄×D₆₇	1	2	yes	—
169	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
173	C₄⋊C₈	—	134	8	no	—
174	C₄⋊C₈	—	134	4	no	—
175	C₄⋊C₈	—	134	8	no	—
176	#97	—	134	8	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	4	no	—
179	C₄×C₈	Dic₁₃₄	1	8	yes	complemented
180	C₄×C₈	D₂₆₈	1	4	yes	—
181	#95	—	67	8	no	—
182	C₂×Q₈	—	134	4	no	—
183	C₂×Q₈	—	134	4	no	—
184	C₂×Q₈	—	134	4	no	—
185	C₂²×C₄	—	67	4	no	—
186	C₂²×C₄	—	67	4	no	—
187	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
189	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
191	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	134	4	no	—
196	C₂×C₈	—	134	8	no	—
197	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
198	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
199	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
200	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
201	C₄⋊C₄	—	134	8	no	—
202	C₄⋊C₄	—	134	4	no	—
203	C₄⋊C₄	—	134	8	no	—
204	C₄⋊C₄	—	134	8	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	134	4	no	—
208	C₄²	—	134	4	no	—
209	C₄²	#11761611	1	4	yes	—
210	C₄²	C₂×D₂₆₈	1	2	yes	—
211	C₄²	C₂×Dic₁₃₄	1	4	yes	—
212	C₄²	#11761628	1	2	yes	—
213	C₄²	#11761628	1	4	yes	—
214	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
215	Q₈	—	268	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	134	4	no	—
220	C₂×C₄	—	134	4	no	—
221	C₂×C₄	—	134	4	no	—
222	C₂×C₄	—	134	8	no	—
223	C₂×C₄	—	134	4	no	—
224	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
225	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
226	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
227	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
228	C₂×C₄	C₂×#11761611	1	4	yes	—
229	C₂×C₄	C₂×#11761611	1	2	yes	—
230	C₂×C₄	#14524049	1	4	yes	—
231	C₂×C₄	#14524049	1	2	yes	—
232	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
233	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
234	C₂×C₄	C₂×#11761628	1	2	yes	—
235	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	4	no	—
238	C₈	—	268	4	no	—
239	C₈	#14524043	1	4	yes	complemented
240	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
241	C₂²	C₂×C₄×Dic₁₃₄	1	2	yes	—
242	C₂²	C₂²×#11761611	1	1	yes	characteristic
243	C₂²	C₂×#14524049	1	2	yes	—
244	C₂²	C₂×#14524049	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	268	4	no	—
248	C₄	—	268	4	no	—
249	C₄	#16863475	1	4	yes	—
250	C₄	#16863526	1	8	yes	complemented
251	C₄	C₂×#14524043	1	1	yes	characteristic
252	C₄	#16863946	1	1	yes	characteristic
253	C₄	#16863953	1	2	yes	—
254	C₂	C₄×#14524043	1	1	yes	characteristic
255	C₂	C₂×#16863475	1	1	yes	characteristic
256	C₂	#22568922	1	1	yes	characteristic
257	C₂	C₂×#16863526	1	2	yes	—
258	C₂	#22569143	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄²×C₈×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#21712
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
51. C₄×C₈×C₆₇	Q₈	semidirect
108. C₈×C₆₇	C₄⋊Q₈	semidirect
118. C₄×C₆₇	#446	semidirect
137. C₆₇	#21712	semidirect
179. C₄×C₈	Dic₁₃₄	semidirect
239. C₈	#14524043	semidirect
250. C₄	#16863526	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863526	2
C₄×#14524043	2
#22561145	2
#22560927	2
#22560361	2
#22560076	2
C₄²×C₈×C₆₇	2
#21712	67