Subgroups — #30826364

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570795	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	2	yes	maximal
3	#22559107	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₂₆₈	C₂	1	4	yes	maximal, complemented
5	#22558968	C₂	1	1	yes	maximal, characteristic
6	#22558665	C₂	1	2	yes	maximal
7	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₂²	1	2	yes	—
9	#16863801	C₄	1	1	yes	characteristic
10	#16863776	C₄	1	1	yes	characteristic
11	#16863747	C₄	1	2	yes	—
12	#16863637	C₄	1	4	yes	—
13	#16863526	—	2	2	no	—
14	#16863526	C₂²	1	4	yes	—
15	C₈×Dic₁₃₄	—	2	2	no	—
16	C₈×Dic₁₃₄	C₂²	1	4	yes	—
17	C₄×Dic₂₆₈	C₂²	1	4	yes	—
18	#16863422	—	2	2	no	—
19	#16863420	C₂²	1	2	yes	—
20	#16863351	—	2	2	no	—
21	#16863149	C₂²	1	4	yes	—
22	#16863128	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₈²×C₆₇	C₂²	1	2	yes	—
27	C₈²×C₆₇	C₂²	1	2	yes	—
28	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
29	C₂×#11761612	—	2	2	no	—
30	#14524043	C₂×C₄	1	2	yes	—
31	#14524043	C₂×C₄	1	2	yes	—
32	#14524035	—	2	2	no	—
33	#14524022	—	2	2	no	—
34	#14524022	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₃₄	—	2	2	no	—
36	C₄×Dic₁₃₄	C₂³	1	4	yes	—
37	#14524002	—	2	2	no	—
38	#14523977	C₂×C₄	1	4	yes	—
39	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
41	Dic₆₇⋊C₈	—	2	4	no	—
42	Dic₆₇⋊C₈	—	2	4	no	—
43	#14523968	C₂×C₄	1	8	yes	—
44	#14523953	—	2	4	no	—
45	C₈×Dic₆₇	—	2	4	no	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₆₇	—	2	1	no	—
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂³	1	2	yes	—
57	C₄×C₈×C₆₇	C₂³	1	2	yes	—
58	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₃₄	—	2	2	no	—
61	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
62	Dic₂₆₈	—	2	8	no	—
63	#11761612	—	4	4	no	—
64	C₂×C₆₇⋊C₈	—	2	4	no	—
65	C₄⋊Dic₆₇	—	2	2	no	—
66	C₄⋊Dic₆₇	—	2	2	no	—
67	C₄⋊Dic₆₇	—	2	2	no	—
68	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
69	Dic₆₇⋊C₄	—	2	4	no	—
70	Dic₆₇⋊C₄	—	2	4	no	—
71	C₂²⋊Dic₆₇	—	2	4	no	—
72	C₄×Dic₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
76	C₂×C₈×C₆₇	—	2	4	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
81	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂×D₄	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	—	2	2	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×Q₁₆	1	2	yes	—
98	C₂×C₄×C₆₇	#135	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	Dic₁₃₄	—	2	8	no	—
105	Dic₁₃₄	—	4	4	no	—
106	C₈×C₆₇	—	2	4	no	—
107	C₈×C₆₇	—	2	2	no	—
108	C₈×C₆₇	—	2	1	no	—
109	C₈×C₆₇	—	2	1	no	—
110	C₈×C₆₇	#131	1	4	yes	—
111	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
112	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
113	C₆₇⋊C₈	—	4	4	no	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	2	1	no	—
116	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
117	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
121	C₄×C₆₇	#443	1	2	yes	—
122	C₄×C₆₇	C₂×#131	1	2	yes	—
123	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
124	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
125	Dic₆₇	—	4	4	no	—
126	Dic₆₇	—	4	4	no	—
127	#17958	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	—	2	2	no	—
129	C₂×C₆₇	#2931	1	1	yes	characteristic
130	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
131	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
132	#3170	—	67	1	no	—
133	#2960	—	67	2	no	—
134	#1591	—	67	4	no	—
135	C₈×Q₁₆	—	67	4	no	—
136	#1567	—	67	1	no	—
137	#1493	—	67	2	no	—
138	C₂×C₈²	D₆₇	1	1	yes	characteristic
139	C₆₇	#17958	1	1	yes	characteristic, complemented, Sylow
140	#533	—	67	2	no	—
141	#500	—	67	1	no	—
142	#494	—	67	1	no	—
143	#487	—	67	2	no	—
144	#467	—	67	4	no	—
145	#446	—	67	4	no	—
146	#446	—	134	2	no	—
147	C₈×Q₈	—	67	4	no	—
148	C₈×Q₈	—	134	2	no	—
149	C₄×Q₁₆	—	67	4	no	—
150	#424	—	134	2	no	—
151	#423	—	67	2	no	—
152	#405	—	134	2	no	—
153	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
156	#335	—	67	4	no	—
157	Q₈⋊C₈	—	67	8	no	—
158	C₈²	C₂×D₆₇	1	2	yes	—
159	C₈²	C₂×D₆₇	1	2	yes	—
160	C₂×Q₁₆	—	67	4	no	—
161	C₂×M₁₆	—	134	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₄×D₆₇	1	2	yes	—
164	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
165	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
166	C₄⋊Q₈	—	67	2	no	—
167	C₄⋊Q₈	—	67	2	no	—
168	#125	—	134	2	no	—
169	C₂²⋊Q₈	—	67	4	no	—
170	C₂²⋊Q₈	—	134	2	no	—
171	C₄×Q₈	—	67	4	no	—
172	C₄×Q₈	—	134	2	no	—
173	#117	—	134	2	no	—
174	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
175	#107	—	67	4	no	—
176	C₄⋊C₈	—	67	4	no	—
177	C₄⋊C₈	—	134	4	no	—
178	C₄⋊C₈	—	134	4	no	—
179	C₄⋊C₈	—	134	2	no	—
180	#103	—	67	8	no	—
181	#97	—	134	4	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	134	4	no	—
184	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
185	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
186	C₄×C₈	D₂₆₈	1	1	yes	characteristic
187	C₄×C₈	D₂₆₈	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₆₇	1	2	yes	—
189	C₄×C₈	C₂²×D₆₇	1	2	yes	—
190	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
192	C₂×Q₈	—	67	4	no	—
193	C₂×Q₈	—	134	2	no	—
194	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
196	C₂²×C₄	#11761628	1	1	yes	characteristic
197	Q₁₆	—	134	8	no	—
198	M₁₆	—	268	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	134	4	no	—
204	C₂×C₈	Dic₂₆₈	1	4	yes	complemented
205	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
207	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
208	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
209	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
210	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
211	C₄⋊C₄	—	67	4	no	—
212	C₄⋊C₄	—	134	2	no	—
213	C₄⋊C₄	—	134	2	no	—
214	C₄⋊C₄	—	134	4	no	—
215	C₄⋊C₄	—	134	2	no	—
216	C₄⋊C₄	—	134	4	no	—
217	C₂²⋊C₄	—	134	4	no	—
218	C₄²	—	134	4	no	—
219	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
220	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
221	C₄²	#11761628	1	1	yes	characteristic
222	C₄²	C₂³×D₆₇	1	1	yes	characteristic
223	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
224	Q₈	—	268	4	no	—
225	Q₈	—	134	8	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	134	4	no	—
229	C₂×C₄	—	134	4	no	—
230	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
231	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
232	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
234	C₂×C₄	#14524065	1	2	yes	—
235	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
237	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	268	4	no	—
243	C₈	#14524049	1	4	yes	—
244	C₈	C₂×Dic₂₆₈	1	2	yes	—
245	C₈	C₂×Dic₂₆₈	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×Dic₂₆₈	1	2	yes	—
249	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	268	4	no	—
253	C₄	—	268	4	no	—
254	C₄	C₄×Dic₂₆₈	1	2	yes	—
255	C₄	#16863514	1	2	yes	—
256	C₄	C₂×#14524049	1	2	yes	—
257	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
258	C₄	C₂×#14524065	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#22568922	1	1	yes	characteristic
261	C₂	C₂×C₄×Dic₂₆₈	1	1	yes	characteristic
262	C₂	C₂×#16863514	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#17958
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #22559107	C₂	semidirect
4. C₈×Dic₂₆₈	C₂	semidirect
80. C₂×C₈×C₆₇	Q₁₆	semidirect
139. C₆₇	#17958	semidirect
204. C₂×C₈	Dic₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
#22570795	2
#22569143	2
#22559107	2
C₈×Dic₂₆₈	2
#22558968	2
#22558665	2
C₂×C₈²×C₆₇	2
#17958	67