Subgroups — #30843528

Order 17152.

922 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570784	C₂	1	4	yes	maximal, complemented
2	#22570637	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559825	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559823	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559816	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559814	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559799	C₂	1	4	yes	maximal, complemented
8	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524065	—	2	2	no	—
11	#16863946	C₂²	1	2	yes	—
12	#16863799	C₂²	1	4	yes	—
13	#16863780	—	2	2	no	—
14	#16863776	C₂²	1	1	yes	characteristic
15	#16863773	C₂²	1	1	yes	characteristic
16	#16863772	C₂²	1	4	yes	—
17	#16863769	C₂²	1	2	yes	—
18	#16863744	C₂²	1	4	yes	—
19	#16863742	C₂²	1	4	yes	—
20	#16863630	—	2	8	no	—
21	C₄×#11761613	—	2	8	no	—
22	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
23	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
24	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×#135×C₆₇	—	2	2	no	—
27	#532×C₆₇	C₂²	1	2	yes	—
28	#495×C₆₇	—	2	2	no	—
29	#494×C₆₇	C₂²	1	1	yes	characteristic
30	#493×C₆₇	C₂²	1	1	yes	characteristic
31	#492×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#11761628	—	2	2	no	—
34	#14524065	—	4	8	no	—
35	C₂×Dic₂₆₈	—	2	2	no	—
36	C₂×#11761617	—	2	4	no	—
37	C₂×D₅₃₆	—	2	2	no	—
38	C₂×#11761613	—	2	8	no	—
39	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
40	#14524043	C₂³	1	2	yes	—
41	#14524040	C₂³	1	2	yes	—
42	#14524031	—	2	2	no	—
43	#14524017	—	2	4	no	—
44	C₄×D₂₆₈	—	2	4	no	—
45	#14523976	—	2	8	no	—
46	#14523964	—	2	8	no	—
47	#14523963	—	2	8	no	—
48	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
49	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
50	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
51	C₂×#41×C₆₇	—	2	2	no	—
52	#135×C₆₇	—	4	8	no	—
53	C₂×Q₁₆×C₆₇	—	2	2	no	—
54	C₂×SD₁₆×C₆₇	—	2	4	no	—
55	C₂×D₈×C₆₇	—	2	2	no	—
56	C₂²×C₈×C₆₇	D₄	1	2	yes	—
57	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
58	#127×C₆₇	C₂³	1	2	yes	—
59	#124×C₆₇	—	2	2	no	—
60	#121×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	2	4	no	—
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₆₇	D₄	1	2	yes	—
64	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	#11761628	—	4	8	no	—
67	C₂×Dic₁₃₄	—	2	2	no	—
68	C₂×D₂₆₈	—	2	4	no	—
69	C₂×D₂₆₈	—	2	2	no	—
70	C₂×C₂²⋊D₆₇	—	4	2	no	—
71	C₂×C₄×D₆₇	—	4	2	no	—
72	Dic₂₆₈	—	4	4	no	—
73	#11761617	—	4	8	no	—
74	D₅₃₆	—	4	4	no	—
75	#11761613	—	4	16	no	—
76	C₂×C₆₇⋊C₈	—	2	8	no	—
77	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
78	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
79	C₄⋊Dic₆₇	—	2	4	no	—
80	(C₂×D₆₇)⋊C₄	—	4	4	no	—
81	#41×C₆₇	—	4	8	no	—
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×D₄×C₆₇	—	2	4	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	4	2	no	—
86	C₂²×C₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₆₇	—	4	4	no	—
90	SD₁₆×C₆₇	—	4	8	no	—
91	D₈×C₆₇	—	4	4	no	—
92	C₂×C₈×C₆₇	D₈	1	8	yes	complemented
93	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
95	C₄⋊C₄×C₆₇	—	2	4	no	—
96	C₂²⋊C₄×C₆₇	—	4	4	no	—
97	C₄²×C₆₇	C₂×D₄	1	2	yes	—
98	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂³×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	#127	1	1	yes	characteristic
102	C₂²×D₆₇	—	4	2	no	—
103	Q₈×C₆₇	—	4	4	no	—
104	D₄×C₆₇	—	4	4	no	—
105	D₄×C₆₇	—	4	8	no	—
106	D₄×C₆₇	—	8	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	8	4	no	—
110	C₂×C₄×C₆₇	#127	1	2	yes	—
111	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
113	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂²⋊D₆₇	—	8	4	no	—
116	C₂×Dic₆₇	—	4	2	no	—
117	D₂₆₈	—	4	4	no	—
118	D₂₆₈	—	4	8	no	—
119	C₄×D₆₇	—	8	4	no	—
120	Dic₁₃₄	—	4	4	no	—
121	C₈×C₆₇	C₂×D₈	1	4	yes	—
122	C₈×C₆₇	C₂×D₈	1	4	yes	—
123	C₆₇⋊C₈	—	2	4	no	—
124	C₆₇⋊C₈	#135	1	8	yes	complemented
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	—	8	2	no	—
127	C₂²×C₆₇	—	8	4	no	—
128	C₂²×C₆₇	#493	1	2	yes	—
129	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
130	C₂×D₆₇	—	8	2	no	—
131	C₂×D₆₇	—	8	4	no	—
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	8	2	no	—
134	C₄×C₆₇	#493	1	2	yes	—
135	C₄×C₆₇	C₂²×D₈	1	2	yes	—
136	C₄×C₆₇	C₂×#135	1	4	yes	—
137	Dic₆₇	—	8	2	no	—
138	#19603	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	—	2	2	no	—
140	C₂×C₆₇	—	16	2	no	—
141	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
142	C₂×C₆₇	#3147	1	2	yes	—
143	D₆₇	—	16	2	no	—
144	#3169	—	67	4	no	—
145	#3147	—	67	1	no	—
146	#3147	D₆₇	1	1	yes	characteristic, complemented
147	#1713	—	67	1	no	—
148	#1712	—	67	1	no	—
149	#1709	—	67	1	no	—
150	#1709	—	67	1	no	—
151	#1706	—	67	4	no	—
152	C₂×C₈²	—	67	1	no	—
153	C₆₇	#19603	1	1	yes	characteristic, complemented, Sylow
154	C₂×#135	—	2	2	no	—
155	C₂×#135	—	134	2	no	—
156	#532	—	67	2	no	—
157	#532	C₂×D₆₇	1	2	yes	—
158	#500	—	67	4	no	—
159	#495	—	2	2	no	—
160	#495	—	134	2	no	—
161	#494	—	67	1	no	—
162	#494	C₂×D₆₇	1	1	yes	characteristic
163	#493	—	67	4	no	—
164	#493	—	67	1	no	—
165	#493	C₂×D₆₇	1	1	yes	characteristic
166	#492	—	67	2	no	—
167	#492	C₂×D₆₇	1	2	yes	—
168	#486	—	67	4	no	—
169	#486	—	67	4	no	—
170	#466	—	134	8	no	—
171	C₄×D₈	—	134	8	no	—
172	C₂×C₄×C₈	—	67	2	no	—
173	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
174	C₈²	—	67	2	no	—
175	C₈²	—	67	1	no	—
176	C₈²	—	67	1	no	—
177	C₂×#41	—	2	2	no	—
178	C₂×#41	—	134	2	no	—
179	#135	—	4	8	no	—
180	#135	—	268	8	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	134	2	no	—
183	C₂×SD₁₆	—	2	4	no	—
184	C₂×SD₁₆	—	134	4	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×D₈	—	134	8	no	—
187	C₂×D₈	—	134	2	no	—
188	C₂²×C₈	—	67	4	no	—
189	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
190	C₄⋊Q₈	—	67	2	no	—
191	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
192	#127	—	67	2	no	—
193	#127	C₂²×D₆₇	1	2	yes	—
194	#124	—	2	2	no	—
195	#124	—	134	2	no	—
196	#121	—	2	4	no	—
197	#121	—	134	4	no	—
198	C₄×D₄	—	2	4	no	—
199	C₄×D₄	—	134	4	no	—
200	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
201	#107	—	134	8	no	—
202	#102	—	134	8	no	—
203	#102	—	134	8	no	—
204	C₄×C₈	—	67	4	no	—
205	C₄×C₈	—	67	2	no	—
206	C₄×C₈	—	67	2	no	—
207	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
208	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
210	#41	—	4	8	no	—
211	#41	—	268	8	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	134	2	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	268	2	no	—
218	C₂×D₄	—	134	4	no	—
219	C₂×D₄	—	134	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	268	2	no	—
222	C₂²×C₄	D₄×D₆₇	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
224	Q₁₆	—	4	4	no	—
225	Q₁₆	—	268	4	no	—
226	SD₁₆	—	4	8	no	—
227	SD₁₆	—	268	8	no	—
228	D₈	—	4	4	no	—
229	D₈	—	268	16	no	—
230	D₈	—	268	4	no	—
231	C₂×C₈	—	67	4	no	—
232	C₂×C₈	—	67	4	no	—
233	C₂×C₈	—	134	8	no	—
234	C₂×C₈	#11761613	1	8	yes	complemented
235	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	134	4	no	—
239	C₂²⋊C₄	—	4	4	no	—
240	C₂²⋊C₄	—	268	4	no	—
241	C₄²	D₄×D₆₇	1	2	yes	—
242	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
243	C₄²	C₂³×D₆₇	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	—	268	2	no	—
246	C₂³	#14524039	1	1	yes	characteristic
247	Q₈	—	4	4	no	—
248	Q₈	—	268	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	8	no	—
251	D₄	—	8	4	no	—
252	D₄	—	268	4	no	—
253	D₄	—	268	8	no	—
254	D₄	—	536	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	268	2	no	—
259	C₂×C₄	—	536	4	no	—
260	C₂×C₄	#14524039	1	2	yes	—
261	C₂×C₄	#14524039	1	1	yes	characteristic
262	C₂×C₄	C₂×#11761613	1	4	yes	—
263	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
264	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
265	C₈	—	67	8	no	—
266	C₈	—	134	4	no	—
267	C₈	C₂×#11761613	1	4	yes	—
268	C₈	C₂×#11761613	1	4	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	536	2	no	—
273	C₂²	—	536	4	no	—
274	C₂²	#16863772	1	2	yes	—
275	C₂²	C₂×#14524039	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	536	2	no	—
279	C₄	#16863772	1	2	yes	—
280	C₄	C₂²×#11761613	1	2	yes	—
281	C₄	#135×D₆₇	1	4	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	16	2	no	—
284	C₂	—	1072	2	no	—
285	C₂	C₂×#16863772	1	1	yes	characteristic
286	C₂	#22570633	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524039
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3147	C₂
socle	C₂²×C₆₇	C₂×#127
hypercenter	#3147	D₆₇
hypocenter	C₆₇	#19603
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #22570784	C₂	semidirect
2. #22570637	C₂	semidirect
3. #22559825	C₂	semidirect
4. #22559823	C₂	semidirect
5. #22559816	C₂	semidirect
6. #22559814	C₂	semidirect
7. #22559799	C₂	semidirect
8. C₂×C₈×C₆₇⋊C₈	C₂	semidirect
9. C₆₇×#3147	C₂	semidirect
92. C₂×C₈×C₆₇	D₈	semidirect
124. C₆₇⋊C₈	#135	semidirect
146. #3147	D₆₇	semidirect
153. C₆₇	#19603	semidirect
234. C₂×C₈	#11761613	semidirect

Maximal subgroups

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

subgroup	index
#22570784	2
#22570637	2
#22559825	2
#22559823	2
#22559816	2
#22559814	2
#22559799	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#3147	2
#19603	67