Subgroups — #30826540

Order 17152.

726 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570788	C₂	1	1	yes	maximal, characteristic
2	#22568922	C₂	1	2	yes	maximal, complemented
3	#22559040	C₂	1	8	yes	maximal, complemented
4	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558661	C₂	1	2	yes	maximal, complemented
6	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14524049	—	2	2	no	—
8	#16863946	C₂²	1	2	yes	—
9	#16863801	C₄	1	1	yes	characteristic
10	#16863773	C₄	1	1	yes	characteristic
11	#16863743	C₄	1	2	yes	—
12	#16863632	C₄	1	4	yes	—
13	C₄×D₅₃₆	C₂²	1	4	yes	—
14	#16863475	—	2	2	no	—
15	#16863475	C₂²	1	4	yes	—
16	C₈×D₂₆₈	—	2	2	no	—
17	C₈×D₂₆₈	C₂²	1	4	yes	—
18	#16863420	C₂²	1	2	yes	—
19	#16863362	—	2	2	no	—
20	#16863149	C₂²	1	4	yes	—
21	#16863125	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#322×C₆₇	C₂²	1	4	yes	—
26	C₂×#11761628	—	2	2	no	—
27	C₂×D₅₃₆	C₂×C₄	1	4	yes	—
28	#14524049	—	4	8	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761611	—	2	4	no	—
31	C₂×C₈×D₆₇	—	2	4	no	—
32	#14524043	C₂×C₄	1	2	yes	—
33	#14524040	C₂×C₄	1	2	yes	—
34	#14524031	—	2	2	no	—
35	#14524017	—	2	2	no	—
36	#14524017	C₂×C₄	1	4	yes	—
37	C₄×D₂₆₈	—	2	2	no	—
38	C₄×D₂₆₈	C₂³	1	4	yes	—
39	#14523977	C₂×C₄	1	4	yes	—
40	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
42	#14523965	C₂×C₄	1	8	yes	—
43	(C₂×D₆₇)⋊C₈	—	2	4	no	—
44	(C₂×D₆₇)⋊C₈	—	2	4	no	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₆₇	—	2	1	no	—
51	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	D₄	1	2	yes	—
54	C₄×C₈×C₆₇	C₂³	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄×C₈×C₆₇	C₂³	1	2	yes	—
57	#11761628	—	4	4	no	—
58	#11761628	—	4	4	no	—
59	C₂×Dic₁₃₄	—	2	2	no	—
60	C₂×D₂₆₈	—	2	2	no	—
61	C₂×D₂₆₈	—	2	2	no	—
62	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
63	C₂×C₂²⋊D₆₇	—	2	4	no	—
64	C₂×C₄×D₆₇	—	2	4	no	—
65	D₅₃₆	—	4	4	no	—
66	#11761612	—	4	4	no	—
67	#11761611	—	4	8	no	—
68	C₂×C₆₇⋊C₈	—	2	4	no	—
69	C₈×D₆₇	—	4	8	no	—
70	C₄⋊Dic₆₇	—	2	2	no	—
71	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
72	(C₂×D₆₇)⋊C₄	—	2	4	no	—
73	(C₂×D₆₇)⋊C₄	—	2	4	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	4	2	no	—
80	C₂×C₈×C₆₇	—	4	2	no	—
81	C₂×C₈×C₆₇	D₈	1	4	yes	complemented
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₆₇	#41	1	1	yes	characteristic
89	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
90	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
91	C₂²×D₆₇	—	2	4	no	—
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×D₈	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	C₂²⋊D₆₇	—	4	8	no	—
103	C₂×Dic₆₇	—	2	4	no	—
104	D₂₆₈	—	2	8	no	—
105	D₂₆₈	—	4	4	no	—
106	D₂₆₈	—	4	4	no	—
107	C₄×D₆₇	—	4	4	no	—
108	C₄×D₆₇	—	4	4	no	—
109	Dic₁₃₄	—	4	4	no	—
110	C₈×C₆₇	—	2	2	no	—
111	C₈×C₆₇	—	2	2	no	—
112	C₈×C₆₇	—	2	2	no	—
113	C₈×C₆₇	—	4	2	no	—
114	C₈×C₆₇	C₂×D₈	1	4	yes	—
115	C₆₇⋊C₈	—	4	4	no	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	C₄×D₈	1	2	yes	—
119	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂×D₆₇	—	4	4	no	—
121	C₂×D₆₇	—	4	4	no	—
122	C₂×D₆₇	—	4	4	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	C₄×D₈	1	2	yes	—
126	C₄×C₆₇	#444	1	2	yes	—
127	C₄×C₆₇	#568	1	2	yes	—
128	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
129	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
130	Dic₆₇	—	4	4	no	—
131	#17975	—	67	1	no	maximal, Sylow
132	C₂×C₆₇	—	2	2	no	—
133	C₂×C₆₇	#2932	1	1	yes	characteristic
134	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
136	D₆₇	—	8	4	no	—
137	#3169	—	67	1	no	—
138	#2931	—	67	2	no	—
139	#1581	—	67	8	no	—
140	#1567	—	67	1	no	—
141	#1492	—	67	2	no	—
142	C₂×#322	D₆₇	1	1	yes	characteristic, complemented
143	C₆₇	#17975	1	1	yes	characteristic, complemented, Sylow
144	C₂×#131	—	134	2	no	—
145	#532	—	67	2	no	—
146	#500	—	67	1	no	—
147	#493	—	67	1	no	—
148	#486	—	67	2	no	—
149	#466	—	67	4	no	—
150	C₄×D₈	—	67	4	no	—
151	#436	—	67	4	no	—
152	#436	—	134	2	no	—
153	C₈×D₄	—	67	4	no	—
154	C₈×D₄	—	134	2	no	—
155	#423	—	67	2	no	—
156	#408	—	134	2	no	—
157	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	#335	—	67	4	no	—
161	D₄⋊C₈	—	67	8	no	—
162	#322	C₂×D₆₇	1	4	yes	—
163	C₂×#41	—	134	2	no	—
164	C₂×D₈	—	67	4	no	—
165	#131	—	268	8	no	—
166	C₂×M₁₆	—	134	2	no	—
167	C₂×M₁₆	—	134	4	no	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	—	134	4	no	—
170	C₂²×C₈	C₄×D₆₇	1	2	yes	—
171	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
172	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
173	C₄⋊Q₈	—	67	2	no	—
174	#127	—	67	2	no	—
175	#124	—	134	2	no	—
176	#121	—	67	4	no	—
177	#121	—	134	2	no	—
178	C₄×D₄	—	67	4	no	—
179	C₄×D₄	—	134	2	no	—
180	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
181	#107	—	67	4	no	—
182	C₄⋊C₈	—	67	4	no	—
183	C₄⋊C₈	—	134	2	no	—
184	#102	—	67	8	no	—
185	C₂²⋊C₈	—	134	4	no	—
186	C₂²⋊C₈	—	134	4	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
189	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
190	C₄×C₈	D₂₆₈	1	2	yes	—
191	C₄×C₈	C₂²×D₆₇	1	2	yes	—
192	C₄×C₈	C₂²×D₆₇	1	2	yes	—
193	C₄×C₈	C₂²×D₆₇	1	2	yes	—
194	#41	—	268	4	no	—
195	#41	—	268	4	no	—
196	C₂×Q₈	—	134	2	no	—
197	C₂×D₄	—	67	4	no	—
198	C₂×D₄	—	134	4	no	—
199	C₂×D₄	—	134	2	no	—
200	C₂×D₄	—	134	2	no	—
201	C₂²×C₄	—	134	4	no	—
202	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
204	C₂²×C₄	#11761628	1	1	yes	characteristic
205	D₈	—	268	4	no	—
206	M₁₆	—	268	4	no	—
207	M₁₆	—	268	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	268	8	no	—
213	C₂×C₈	—	134	4	no	—
214	C₂×C₈	D₅₃₆	1	4	yes	complemented
215	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
216	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
217	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
218	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
219	C₄⋊C₄	—	67	4	no	—
220	C₄⋊C₄	—	134	2	no	—
221	C₂²⋊C₄	—	134	4	no	—
222	C₂²⋊C₄	—	134	4	no	—
223	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
224	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
225	C₄²	#11761628	1	1	yes	characteristic
226	C₄²	C₂³×D₆₇	1	1	yes	characteristic
227	C₂³	—	134	4	no	—
228	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
229	Q₈	—	268	4	no	—
230	D₄	—	268	8	no	—
231	D₄	—	268	4	no	—
232	D₄	—	268	4	no	—
233	D₄	—	134	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	268	4	no	—
237	C₂×C₄	—	268	4	no	—
238	C₂×C₄	—	134	4	no	—
239	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
240	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
241	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
242	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
243	C₂×C₄	#14524065	1	2	yes	—
244	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
245	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
246	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	—	268	4	no	—
252	C₈	C₂×D₅₃₆	1	4	yes	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	268	4	no	—
256	C₂²	—	268	4	no	—
257	C₂²	—	268	4	no	—
258	C₂²	C₄×D₅₃₆	1	2	yes	—
259	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	268	4	no	—
263	C₄	C₄×D₅₃₆	1	2	yes	—
264	C₄	#16863519	1	2	yes	—
265	C₄	#16864162	1	2	yes	—
266	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
267	C₄	C₂×#14524065	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	536	4	no	—
270	C₂	#22568928	1	1	yes	characteristic
271	C₂	C₂×C₄×D₅₃₆	1	1	yes	characteristic
272	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570788	C₂
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#17975
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #22568922	C₂	semidirect
3. #22559040	C₂	semidirect
4. #22558968	C₂	semidirect
5. #22558661	C₂	semidirect
6. C₂×#322×C₆₇	C₂	semidirect
81. C₂×C₈×C₆₇	D₈	semidirect
142. C₂×#322	D₆₇	semidirect
143. C₆₇	#17975	semidirect
214. C₂×C₈	D₅₃₆	semidirect

Maximal subgroups

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

subgroup	index
#22570788	2
#22568922	2
#22559040	2
#22558968	2
#22558661	2
C₂×#322×C₆₇	2
#17975	67