Subgroups — #30774246

Order 17152.

410 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561993	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561945	C₂	1	2	yes	maximal, complemented
3	#22561805	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561108	C₂	1	2	yes	maximal, complemented
5	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	1	yes	characteristic
7	#16863434	C₂²	1	2	yes	—
8	#16863413	—	2	2	no	—
9	#16863370	—	2	2	no	—
10	#16863351	C₂²	1	1	yes	characteristic
11	#16863224	—	2	1	no	—
12	#16863222	—	2	1	no	—
13	#16863206	—	2	4	no	—
14	#16863171	—	2	1	no	—
15	C₁₆×Dic₆₇	—	2	1	no	—
16	C₂×#135×C₆₇	C₂²	1	2	yes	—
17	#508×C₆₇	—	2	1	no	—
18	#508×C₆₇	—	2	1	no	—
19	#508×C₆₇	—	2	1	no	—
20	#508×C₆₇	—	2	1	no	—
21	C₂×Q₃₂×C₆₇	C₄	1	1	yes	characteristic, complemented
22	C₂×SD₃₂×C₆₇	C₄	1	2	yes	complemented
23	C₂×D₁₆×C₆₇	C₄	1	1	yes	characteristic, complemented
24	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	D₄	1	1	yes	characteristic
26	#14524002	—	2	2	no	—
27	#14524002	D₄	1	1	yes	characteristic
28	#14523983	—	2	1	no	—
29	#14523983	D₄	1	2	yes	—
30	C₂×C₆₇⋊C₁₆	—	2	1	no	—
31	#14523977	—	2	2	no	—
32	#14523975	—	2	2	no	—
33	#14523970	—	4	4	no	—
34	#14523958	—	4	4	no	—
35	#14523953	—	2	1	no	—
36	C₈×Dic₆₇	—	2	1	no	—
37	C₂×#41×C₆₇	—	2	2	no	—
38	#135×C₆₇	—	2	2	no	—
39	#135×C₆₇	—	2	2	no	—
40	#135×C₆₇	—	2	2	no	—
41	#135×C₆₇	—	2	2	no	—
42	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	Q₃₂×C₆₇	—	2	1	no	—
47	Q₃₂×C₆₇	—	2	1	no	—
48	SD₃₂×C₆₇	—	2	2	no	—
49	SD₃₂×C₆₇	—	2	2	no	—
50	D₁₆×C₆₇	—	2	1	no	—
51	D₁₆×C₆₇	—	2	1	no	—
52	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
58	#11761612	—	2	1	no	—
59	#11761612	D₈	1	2	yes	—
60	C₂×C₆₇⋊C₈	—	2	1	no	—
61	C₄⋊Dic₆₇	—	2	1	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₂²⋊Dic₆₇	—	4	2	no	—
64	C₂²⋊Dic₆₇	D₈	1	2	yes	—
65	C₄×Dic₆₇	—	2	1	no	—
66	C₄×Dic₆₇	—	4	2	no	—
67	C₆₇⋊C₁₆	—	2	2	no	—
68	#41×C₆₇	—	4	2	no	—
69	#41×C₆₇	—	4	2	no	—
70	#41×C₆₇	—	4	2	no	—
71	#41×C₆₇	—	4	2	no	—
72	C₂×Q₈×C₆₇	—	2	2	no	—
73	C₂×D₄×C₆₇	—	2	2	no	—
74	C₂×D₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	—	4	2	no	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₆₇	—	2	2	no	—
78	Q₁₆×C₆₇	—	2	2	no	—
79	SD₁₆×C₆₇	—	4	2	no	—
80	SD₁₆×C₆₇	—	4	2	no	—
81	D₈×C₆₇	—	2	2	no	—
82	D₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	C₂²×C₄	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	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
88	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
89	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₆₇	—	4	2	no	—
94	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₆₇	—	4	2	no	—
96	Q₈×C₆₇	—	4	2	no	—
97	D₄×C₆₇	—	4	2	no	—
98	D₄×C₆₇	—	4	2	no	—
99	D₄×C₆₇	—	8	2	no	—
100	D₄×C₆₇	—	8	2	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	—	8	2	no	—
103	C₂×C₄×C₆₇	—	8	2	no	—
104	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	8	2	no	—
112	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
114	C₈×C₆₇	#121	1	1	yes	characteristic
115	C₈×C₆₇	#124	1	1	yes	characteristic
116	C₆₇⋊C₈	—	2	2	no	—
117	C₂²×C₆₇	—	4	1	no	—
118	C₂²×C₆₇	—	8	2	no	—
119	C₂²×C₆₇	—	16	2	no	—
120	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
121	C₂²×C₆₇	#466	1	1	yes	characteristic
122	C₂²×C₆₇	#493	1	1	yes	characteristic
123	C₄×C₆₇	—	8	2	no	—
124	C₄×C₆₇	#390	1	1	yes	characteristic
125	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
126	C₄×C₆₇	#466	1	1	yes	characteristic
127	C₄×C₆₇	#495	1	1	yes	characteristic
128	Dic₆₇	—	4	2	no	—
129	Dic₆₇	—	16	2	no	—
130	#13243	—	67	1	no	maximal, Sylow
131	C₂×C₆₇	—	2	1	no	—
132	C₂×C₆₇	—	2	1	no	—
133	C₂×C₆₇	—	2	1	no	—
134	C₂×C₆₇	—	16	2	no	—
135	C₂×C₆₇	#1970	1	1	yes	characteristic
136	C₂×#508	D₆₇	1	1	yes	characteristic
137	#2158	—	67	1	no	—
138	#2145	—	67	2	no	—
139	#2108	—	67	1	no	—
140	#1976	—	67	2	no	—
141	C₆₇	#13243	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	C₂×D₆₇	1	2	yes	—
143	#508	—	2	1	no	—
144	#508	—	2	1	no	—
145	#508	—	2	1	no	—
146	#508	—	2	1	no	—
147	C₂×Q₃₂	Dic₆₇	1	1	yes	characteristic, complemented
148	C₂×SD₃₂	Dic₆₇	1	2	yes	complemented
149	C₂×D₁₆	Dic₆₇	1	1	yes	characteristic, complemented
150	C₂×M₃₂	—	67	1	no	—
151	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
152	#427	—	67	2	no	—
153	#421	—	134	2	no	—
154	#410	—	134	2	no	—
155	#405	—	67	1	no	—
156	#367	—	134	1	no	—
157	#366	—	134	1	no	—
158	#360	—	134	4	no	—
159	#346	—	134	1	no	—
160	C₄×C₁₆	—	134	1	no	—
161	C₂×#41	—	2	2	no	—
162	#135	—	2	2	no	—
163	#135	—	2	2	no	—
164	#135	—	2	2	no	—
165	#135	—	2	2	no	—
166	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
169	C₂×M₁₆	—	67	1	no	—
170	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
171	#117	—	67	1	no	—
172	#117	—	134	2	no	—
173	Q₃₂	—	2	1	no	—
174	Q₃₂	—	2	1	no	—
175	SD₃₂	—	2	2	no	—
176	SD₃₂	—	2	2	no	—
177	D₁₆	—	2	1	no	—
178	D₁₆	—	2	1	no	—
179	M₃₂	—	67	2	no	—
180	M₃₂	—	134	1	no	—
181	C₂×C₁₆	—	134	1	no	—
182	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
183	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
184	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
186	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
188	#107	—	134	2	no	—
189	#106	—	134	2	no	—
190	#104	—	268	4	no	—
191	#99	—	268	4	no	—
192	#97	—	134	1	no	—
193	C₄×C₈	—	134	1	no	—
194	#41	—	4	2	no	—
195	#41	—	4	2	no	—
196	#41	—	4	2	no	—
197	#41	—	4	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
203	Q₁₆	—	2	2	no	—
204	Q₁₆	—	2	2	no	—
205	SD₁₆	—	4	2	no	—
206	SD₁₆	—	4	2	no	—
207	D₈	—	2	2	no	—
208	D₈	—	2	2	no	—
209	M₁₆	—	67	2	no	—
210	M₁₆	—	134	1	no	—
211	C₂×C₈	—	134	1	no	—
212	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
213	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
216	C₂×C₈	#11761627	1	1	yes	characteristic
217	C₂×C₈	#11761627	1	1	yes	characteristic
218	C₄⋊C₄	—	268	2	no	—
219	C₄⋊C₄	—	134	1	no	—
220	C₂²⋊C₄	—	268	2	no	—
221	C₂²⋊C₄	—	67	2	no	—
222	C₄²	—	268	2	no	—
223	C₄²	—	134	1	no	—
224	C₁₆	—	134	2	no	—
225	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
226	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
227	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
228	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
229	C₂³	—	4	2	no	—
230	C₂³	D₈×D₆₇	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	536	2	no	—
241	C₂×C₄	—	134	2	no	—
242	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
243	C₂×C₄	#14524014	1	1	yes	characteristic
244	C₂×C₄	#14524039	1	1	yes	characteristic
245	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
246	C₂×C₄	#14524062	1	1	yes	characteristic
247	C₂×C₄	#14524062	1	1	yes	characteristic
248	C₈	—	134	2	no	—
249	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
250	C₈	D₄×Dic₆₇	1	1	yes	characteristic
251	C₈	#14524014	1	1	yes	characteristic
252	C₈	#14524030	1	1	yes	characteristic
253	C₂²	—	4	1	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	D₈×Dic₆₇	1	1	yes	characteristic
257	C₂²	#16863629	1	1	yes	characteristic
258	C₂²	#16863771	1	1	yes	characteristic
259	C₄	—	8	2	no	—
260	C₄	—	268	2	no	—
261	C₄	—	1072	2	no	—
262	C₄	#16863299	1	1	yes	characteristic
263	C₄	D₈×Dic₆₇	1	1	yes	characteristic
264	C₄	#16863629	1	1	yes	characteristic
265	C₄	#16863779	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	16	2	no	—
270	C₂	#22561076	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863629
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₂×#508×C₆₇	C₂
socle	C₂×C₆₇	#1970
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#508	D₆₇
hypocenter	C₆₇	#13243
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #22561993	C₂	semidirect
2. #22561945	C₂	semidirect
3. #22561805	C₂	semidirect
4. #22561108	C₂	semidirect
21. C₂×Q₃₂×C₆₇	C₄	semidirect
22. C₂×SD₃₂×C₆₇	C₄	semidirect
23. C₂×D₁₆×C₆₇	C₄	semidirect
141. C₆₇	#13243	semidirect
147. C₂×Q₃₂	Dic₆₇	semidirect
148. C₂×SD₃₂	Dic₆₇	semidirect
149. C₂×D₁₆	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22561993	2
#22561945	2
#22561805	2
#22561108	2
C₂×#508×C₆₇	2
#13243	67