Subgroups — #30860238

Order 17152.

722 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568860	C₂	1	2	yes	maximal, complemented
2	#22561098	C₂	1	2	yes	maximal
3	#22560958	C₂	1	2	yes	maximal, complemented
4	#22560922	C₂	1	4	yes	maximal, complemented
5	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560019	C₂	1	2	yes	maximal, complemented
7	#22559895	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863457	—	2	4	no	—
10	#16863453	—	2	4	no	—
11	C₂×#14523979	C₂²	1	2	yes	—
12	#16863423	C₂²	1	4	yes	—
13	C₂×#14523970	C₂²	1	4	yes	—
14	#16863363	C₂²	1	4	yes	—
15	C₄×#11761612	C₂²	1	2	yes	—
16	C₂×#14523954	C₂²	1	1	yes	characteristic
17	C₂×#14523953	C₂²	1	2	yes	—
18	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
19	#16863242	C₂²	1	1	yes	characteristic
20	#16863161	C₂²	1	2	yes	—
21	#16863155	C₂²	1	4	yes	—
22	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×#41×C₆₇	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₆₇	C₄	1	2	yes	complemented
25	C₄×SD₁₆×C₆₇	C₄	1	4	yes	complemented
26	C₄×D₈×C₆₇	C₄	1	2	yes	complemented
27	#427×C₆₇	C₂²	1	1	yes	characteristic
28	#416×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#11761612	C₂³	1	4	yes	—
31	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
32	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
33	#14523979	—	2	4	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
35	#14523970	—	2	8	no	—
36	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
37	#14523954	—	2	2	no	—
38	#14523954	D₄	1	2	yes	—
39	#14523954	D₄	1	2	yes	—
40	#14523953	—	2	4	no	—
41	C₄×C₆₇⋊C₈	—	2	2	no	—
42	C₈×Dic₆₇	—	2	4	no	—
43	C₆₇⋊#95	—	2	2	no	—
44	C₂×#41×C₆₇	C₂³	1	2	yes	—
45	#135×C₆₇	—	2	4	no	—
46	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
52	C₄×D₄×C₆₇	—	2	4	no	—
53	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
54	#117×C₆₇	—	2	2	no	—
55	#117×C₆₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₆₇	—	2	2	no	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	#107×C₆₇	C₂×C₄	1	2	yes	—
59	#106×C₆₇	C₂×C₄	1	2	yes	—
60	#103×C₆₇	C₂×C₄	1	4	yes	—
61	#102×C₆₇	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
64	C₂²×Dic₆₇	—	2	2	no	—
65	#11761612	—	2	4	no	—
66	#11761612	—	2	4	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	—	2	2	no	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
71	C₄×Dic₆₇	—	2	2	no	—
72	C₄×Dic₆₇	#41	1	4	yes	—
73	#41×C₆₇	—	2	4	no	—
74	#41×C₆₇	—	2	4	no	—
75	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂×D₄×C₆₇	—	2	2	no	—
77	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	#41	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	Q₁₆×C₆₇	—	2	2	no	—
83	SD₁₆×C₆₇	—	2	4	no	—
84	D₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	#41	1	4	yes	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
92	C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₂²⋊C₄×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	2	4	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
101	Q₈×C₆₇	—	2	2	no	—
102	Q₈×C₆₇	—	2	2	no	—
103	D₄×C₆₇	—	2	2	no	—
104	D₄×C₆₇	—	2	2	no	—
105	D₄×C₆₇	—	4	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	8	4	no	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
117	C₂×Dic₆₇	—	2	4	no	—
118	C₂×Dic₆₇	—	4	4	no	—
119	C₈×C₆₇	—	2	2	no	—
120	C₈×C₆₇	#117	1	2	yes	—
121	C₈×C₆₇	#117	1	2	yes	—
122	C₆₇⋊C₈	—	2	4	no	—
123	C₆₇⋊C₈	—	4	4	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	4	4	no	—
126	C₂²×C₆₇	—	4	2	no	—
127	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂²×C₆₇	#529	1	2	yes	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	4	4	no	—
132	C₄×C₆₇	—	4	2	no	—
133	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
134	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
135	C₄×C₆₇	#532	1	2	yes	—
136	Dic₆₇	—	4	4	no	—
137	#21294	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	—	2	2	no	—
139	C₂×C₆₇	—	8	2	no	—
140	C₂×C₆₇	#2293	1	1	yes	characteristic
141	C₂×C₆₇	#2957	1	2	yes	—
142	C₄×#135	D₆₇	1	1	yes	characteristic
143	#2923	—	67	2	no	—
144	#1974	—	67	2	no	—
145	#1945	—	67	2	no	—
146	#1938	—	67	4	no	—
147	C₄×#108	—	67	1	no	—
148	#1759	—	67	2	no	—
149	#1726	—	67	1	no	—
150	C₆₇	#21294	1	1	yes	characteristic, complemented, Sylow
151	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
152	C₄×#41	C₂×D₆₇	1	2	yes	—
153	C₄×Q₁₆	Dic₆₇	1	2	yes	complemented
154	C₄×SD₁₆	Dic₆₇	1	4	yes	complemented
155	C₄×D₈	Dic₆₇	1	2	yes	complemented
156	#433	—	134	4	no	—
157	#432	—	134	4	no	—
158	C₂×#108	—	67	2	no	—
159	#427	C₂×D₆₇	1	1	yes	characteristic
160	#424	—	67	4	no	—
161	C₂×#104	—	67	4	no	—
162	#416	C₂×D₆₇	1	2	yes	—
163	#408	—	67	4	no	—
164	C₄×M₁₆	—	67	2	no	—
165	C₂×#97	—	67	1	no	—
166	C₂×#97	—	67	2	no	—
167	C₂×C₄×C₈	—	67	2	no	—
168	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
169	#376	—	67	1	no	—
170	#341	—	67	2	no	—
171	#339	—	67	4	no	—
172	C₂×#41	C₂²×D₆₇	1	2	yes	—
173	#135	—	2	4	no	—
174	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
175	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
176	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
177	C₂×M₁₆	—	67	4	no	—
178	C₂²×C₈	—	67	2	no	—
179	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
181	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	C₂×Dic₆₇	1	4	yes	—
184	#117	—	2	2	no	—
185	#117	C₂²×D₆₇	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	67	2	no	—
188	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
189	#108	—	134	4	no	—
190	#107	C₂×Dic₆₇	1	2	yes	—
191	#106	C₂×Dic₆₇	1	2	yes	—
192	C₄⋊C₈	—	134	8	no	—
193	#104	—	134	8	no	—
194	#103	C₂×Dic₆₇	1	4	yes	—
195	#102	C₂×Dic₆₇	1	4	yes	—
196	C₂²⋊C₈	—	134	8	no	—
197	#97	—	67	2	no	—
198	#97	—	67	2	no	—
199	#97	—	134	4	no	—
200	#97	—	134	2	no	—
201	C₄×C₈	—	134	4	no	—
202	C₄×C₈	—	134	2	no	—
203	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
204	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
205	#95	—	134	2	no	—
206	#41	—	2	4	no	—
207	#41	—	2	4	no	—
208	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	134	2	no	—
214	C₂²×C₄	#11761627	1	2	yes	—
215	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	SD₁₆	—	2	4	no	—
218	D₈	—	2	2	no	—
219	M₁₆	—	134	4	no	—
220	M₁₆	—	134	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	67	4	no	—
223	C₂×C₈	—	134	4	no	—
224	C₂×C₈	—	134	2	no	—
225	C₂×C₈	—	134	2	no	—
226	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
227	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
229	C₂×C₈	#11761628	1	4	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	67	4	no	—
237	C₄²	—	134	2	no	—
238	C₄²	C₂²×Dic₆₇	1	2	yes	—
239	C₄²	D₄×D₆₇	1	1	yes	characteristic
240	C₄²	D₄×D₆₇	1	1	yes	characteristic
241	C₂³	—	2	2	no	—
242	C₂³	C₂×#11761627	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	—	268	4	no	—
254	C₂×C₄	—	134	4	no	—
255	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
256	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
257	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
259	C₂×C₄	#41×D₆₇	1	2	yes	—
260	C₂×C₄	C₂×#11761628	1	2	yes	—
261	C₈	—	2	2	no	—
262	C₈	—	268	4	no	—
263	C₈	—	134	4	no	—
264	C₈	#14524002	1	2	yes	—
265	C₈	#14524002	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	4	no	—
269	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
270	C₂²	#16863928	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	268	4	no	—
276	C₄	C₂×#14524002	1	1	yes	characteristic
277	C₄	#41×Dic₆₇	1	1	yes	characteristic
278	C₄	#16863943	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	8	2	no	—
281	C₂	#22562674	1	1	yes	characteristic
282	C₂	#22569126	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#135×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#135	D₆₇
hypocenter	C₆₇	#21294
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #22568860	C₂	semidirect
3. #22560958	C₂	semidirect
4. #22560922	C₂	semidirect
5. C₄×#14523979	C₂	semidirect
6. #22560019	C₂	semidirect
7. #22559895	C₂	semidirect
24. C₄×Q₁₆×C₆₇	C₄	semidirect
25. C₄×SD₁₆×C₆₇	C₄	semidirect
26. C₄×D₈×C₆₇	C₄	semidirect
150. C₆₇	#21294	semidirect
153. C₄×Q₁₆	Dic₆₇	semidirect
154. C₄×SD₁₆	Dic₆₇	semidirect
155. C₄×D₈	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568860	2
#22561098	2
#22560958	2
#22560922	2
C₄×#14523979	2
#22560019	2
#22559895	2
C₄×#135×C₆₇	2
#21294	67