Subgroups — #30825675

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569292	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22559117	C₂	1	4	yes	maximal, complemented
4	#22559105	C₂	1	4	yes	maximal, complemented
5	#22558603	C₂	1	2	yes	maximal
6	#22558492	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
10	#16863526	—	2	4	no	—
11	#16863523	C₂²	1	8	yes	—
12	C₄×Dic₂₆₈	C₄	1	2	yes	—
13	C₄×Dic₂₆₈	C₄	1	2	yes	—
14	C₄×Dic₂₆₈	C₂²	1	2	yes	—
15	C₄×Dic₂₆₈	C₂²	1	2	yes	—
16	#16863450	—	2	4	no	—
17	#16863449	C₂²	1	1	yes	characteristic
18	#16863449	C₂²	1	1	yes	characteristic
19	C₂×#14523977	C₄	1	1	yes	characteristic
20	C₂×#14523968	C₄	1	2	yes	—
21	#16863126	C₂²	1	8	yes	—
22	#16863117	C₂²	1	2	yes	—
23	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
24	#431×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#335×C₆₇	C₂²	1	2	yes	—
27	#334×C₆₇	C₂²	1	2	yes	—
28	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₆₈	—	2	4	no	—
30	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
31	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₃₄	—	2	2	no	—
33	C₄×Dic₁₃₄	—	2	2	no	—
34	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₃₄	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
37	C₂×Dic₆₇⋊C₄	—	2	2	no	—
38	C₂×C₄×Dic₆₇	—	2	2	no	—
39	#14523977	C₂×C₄	1	2	yes	—
40	#14523977	C₂×C₄	1	2	yes	—
41	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
42	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
43	Dic₆₇⋊C₈	—	2	8	no	—
44	#14523968	C₂×C₄	1	4	yes	—
45	#14523968	C₂×C₄	1	4	yes	—
46	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
47	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
48	#14523953	—	2	8	no	—
49	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
50	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
55	C₂²⋊C₈×C₆₇	—	2	2	no	—
56	C₄×C₈×C₆₇	—	2	2	no	—
57	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₃₄	—	2	4	no	—
62	C₂×Dic₁₃₄	—	2	4	no	—
63	C₂×Dic₁₃₄	—	2	2	no	—
64	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
65	C₂²×Dic₆₇	—	2	2	no	—
66	C₂²×Dic₆₇	—	2	2	no	—
67	Dic₂₆₈	—	2	4	no	—
68	Dic₂₆₈	—	2	4	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
71	C₄⋊Dic₆₇	—	2	2	no	—
72	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
73	Dic₆₇⋊C₄	—	2	4	no	—
74	Dic₆₇⋊C₄	—	2	4	no	—
75	C₄×Dic₆₇	—	2	4	no	—
76	C₄×Dic₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	#41	1	1	yes	characteristic
90	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	4	1	no	—
94	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₆₇	#131	1	2	yes	—
99	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₆₇	—	2	4	no	—
103	C₂×Dic₆₇	—	2	4	no	—
104	C₂×Dic₆₇	—	4	4	no	—
105	C₂×Dic₆₇	—	8	2	no	—
106	Dic₁₃₄	—	2	4	no	—
107	Dic₁₃₄	—	2	4	no	—
108	Dic₁₃₄	—	4	4	no	—
109	C₈×C₆₇	—	2	1	no	—
110	C₈×C₆₇	—	2	1	no	—
111	C₈×C₆₇	—	4	4	no	—
112	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
113	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
114	C₆₇⋊C₈	—	2	4	no	—
115	C₆₇⋊C₈	—	4	2	no	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	#436	1	2	yes	—
119	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	#436	1	2	yes	—
124	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
125	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
126	C₄×C₆₇	#578	1	2	yes	—
127	Dic₆₇	—	4	4	no	—
128	Dic₆₇	—	4	4	no	—
129	#17897	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	—	2	2	no	—
131	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
132	C₂×C₆₇	#2950	1	1	yes	characteristic
133	C₂×C₆₇	#2957	1	1	yes	characteristic
134	#2983	—	67	2	no	—
135	C₂×C₄×Q₁₆	—	67	1	no	—
136	#1593	—	67	4	no	—
137	#1591	—	67	4	no	—
138	#1564	D₆₇	1	1	yes	characteristic
139	#1479	—	67	2	no	—
140	#1450	—	67	1	no	—
141	C₆₇	#17897	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₁₆	—	67	1	no	—
143	C₂×C₄×Q₈	—	67	2	no	—
144	#446	—	67	8	no	—
145	#446	—	134	4	no	—
146	C₄×Q₁₆	—	67	2	no	—
147	C₄×Q₁₆	—	67	2	no	—
148	C₄×Q₁₆	—	67	2	no	—
149	C₄×Q₁₆	—	67	2	no	—
150	#432	—	134	4	no	—
151	#431	—	67	1	no	—
152	#431	—	67	1	no	—
153	#431	C₂×D₆₇	1	2	yes	—
154	C₂×#107	—	67	1	no	—
155	C₂×#103	—	67	2	no	—
156	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
157	#335	C₂×D₆₇	1	2	yes	—
158	#334	C₂×D₆₇	1	2	yes	—
159	Q₈⋊C₈	—	67	8	no	—
160	#322	—	67	2	no	—
161	C₈²	—	67	2	no	—
162	C₂²×Q₈	—	67	2	no	—
163	C₂×Q₁₆	—	67	2	no	—
164	C₂×Q₁₆	—	67	2	no	—
165	C₂×Q₁₆	—	134	4	no	—
166	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
168	C₄×Q₈	—	67	4	no	—
169	C₄×Q₈	—	67	4	no	—
170	C₄×Q₈	—	134	2	no	—
171	C₄×Q₈	—	134	2	no	—
172	C₂×C₄⋊C₄	—	67	2	no	—
173	C₂×C₄⋊C₄	—	134	2	no	—
174	C₂×C₄²	—	134	2	no	—
175	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
176	#107	—	67	2	no	—
177	#107	—	67	2	no	—
178	C₄⋊C₈	—	67	2	no	—
179	C₄⋊C₈	—	134	8	no	—
180	C₄⋊C₈	—	134	1	no	—
181	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
182	#103	—	67	4	no	—
183	#103	—	67	4	no	—
184	C₂²⋊C₈	—	2	2	no	—
185	C₂²⋊C₈	—	67	2	no	—
186	C₂²⋊C₈	—	134	1	no	—
187	#97	—	134	8	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	67	2	no	—
190	C₄×C₈	—	67	2	no	—
191	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
192	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
195	C₂×Q₈	—	67	4	no	—
196	C₂×Q₈	—	134	4	no	—
197	C₂×Q₈	—	134	4	no	—
198	C₂×Q₈	—	134	2	no	—
199	C₂²×C₄	—	134	2	no	—
200	C₂²×C₄	—	134	2	no	—
201	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
203	C₂²×C₄	#11761629	1	1	yes	characteristic
204	Q₁₆	—	134	4	no	—
205	Q₁₆	—	134	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	67	4	no	—
209	C₂×C₈	—	134	2	no	—
210	C₂×C₈	#11761611	1	4	yes	—
211	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₄⋊C₄	—	67	4	no	—
216	C₄⋊C₄	—	134	2	no	—
217	C₄⋊C₄	—	134	4	no	—
218	C₄⋊C₄	—	134	4	no	—
219	C₄²	—	134	4	no	—
220	C₄²	—	134	4	no	—
221	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
222	C₄²	D₄×D₆₇	1	1	yes	characteristic
223	C₄²	#11761629	1	1	yes	characteristic
224	C₄²	C₂³×D₆₇	1	1	yes	characteristic
225	C₂³	#14524005	1	1	yes	characteristic
226	Q₈	—	268	4	no	—
227	Q₈	—	134	4	no	—
228	Q₈	—	134	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	268	4	no	—
232	C₂×C₄	—	536	2	no	—
233	C₂×C₄	—	134	4	no	—
234	C₂×C₄	—	134	4	no	—
235	C₂×C₄	#14524005	1	1	yes	characteristic
236	C₂×C₄	#14524005	1	1	yes	characteristic
237	C₂×C₄	#14524005	1	1	yes	characteristic
238	C₂×C₄	C₂×#11761611	1	2	yes	—
239	C₂×C₄	#14524051	1	2	yes	—
240	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
242	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	4	no	—
246	C₈	—	268	2	no	—
247	C₈	—	134	4	no	—
248	C₈	C₂×#11761611	1	2	yes	—
249	C₈	C₂×#11761611	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#16863473	1	2	yes	—
253	C₂²	C₂×#14524005	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	268	4	no	—
258	C₄	—	268	4	no	—
259	C₄	#16863473	1	2	yes	—
260	C₄	C₂²×#11761611	1	1	yes	characteristic
261	C₄	C₂×#14524051	1	1	yes	characteristic
262	C₄	#16864210	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×#16863473	1	1	yes	characteristic
265	C₂	#22569076	1	1	yes	characteristic
266	C₂	#22569125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#17897
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #22559117	C₂	semidirect
4. #22559105	C₂	semidirect
141. C₆₇	#17897	semidirect

Maximal subgroups

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

subgroup	index
#22569292	2
C₂×C₄×Dic₂₆₈	2
#22559117	2
#22559105	2
#22558603	2
#22558492	2
C₆₇×#1564	2
#17897	67