Subgroups — #30755210

Order 17152.

502 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560373	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558769	C₂	1	4	yes	maximal, complemented
3	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
5	#16863449	C₂²	1	2	yes	—
6	C₂×#14523977	—	2	1	no	—
7	C₂×#14523975	—	2	1	no	—
8	#16863262	C₂²	1	2	yes	—
9	#16863130	—	2	8	no	—
10	#16863117	—	2	1	no	—
11	C₈×C₆₇⋊C₈	—	2	1	no	—
12	C₂×#135×C₆₇	—	2	1	no	—
13	C₂×#135×C₆₇	—	2	1	no	—
14	#532×C₆₇	C₂²	1	2	yes	—
15	#495×C₆₇	—	2	1	no	—
16	#495×C₆₇	—	2	1	no	—
17	#494×C₆₇	C₄	1	1	yes	characteristic, complemented
18	#493×C₆₇	C₄	1	1	yes	characteristic, complemented
19	#492×C₆₇	C₄	1	2	yes	complemented
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	#14523977	—	4	2	no	—
23	#14523975	—	4	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
26	C₄×C₆₇⋊C₈	—	2	2	no	—
27	C₆₇⋊#95	—	2	4	no	—
28	C₂×#41×C₆₇	—	2	2	no	—
29	#135×C₆₇	—	4	2	no	—
30	#135×C₆₇	—	4	2	no	—
31	#135×C₆₇	—	4	2	no	—
32	#135×C₆₇	—	4	2	no	—
33	C₂×Q₁₆×C₆₇	—	2	1	no	—
34	C₂×Q₁₆×C₆₇	—	2	1	no	—
35	C₂×SD₁₆×C₆₇	—	2	2	no	—
36	C₂×SD₁₆×C₆₇	—	2	2	no	—
37	C₂×D₈×C₆₇	—	2	1	no	—
38	C₂×D₈×C₆₇	—	2	1	no	—
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
42	#127×C₆₇	C₂×C₄	1	2	yes	—
43	#124×C₆₇	—	2	2	no	—
44	#121×C₆₇	—	2	4	no	—
45	C₄×D₄×C₆₇	—	2	4	no	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×Dic₆₇	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₄⋊Dic₆₇	—	4	2	no	—
55	#41×C₆₇	—	4	2	no	—
56	#41×C₆₇	—	4	2	no	—
57	#41×C₆₇	—	4	2	no	—
58	#41×C₆₇	—	4	2	no	—
59	C₂×Q₈×C₆₇	—	2	2	no	—
60	C₂×D₄×C₆₇	—	2	2	no	—
61	C₂×D₄×C₆₇	—	2	4	no	—
62	C₂×D₄×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₆₇	—	4	2	no	—
67	Q₁₆×C₆₇	—	4	2	no	—
68	SD₁₆×C₆₇	—	4	4	no	—
69	SD₁₆×C₆₇	—	4	4	no	—
70	D₈×C₆₇	—	4	2	no	—
71	D₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	D₈	1	2	yes	—
77	C₂×C₈×C₆₇	D₈	1	2	yes	—
78	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
79	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
80	C₄⋊C₄×C₆₇	—	2	4	no	—
81	C₂²⋊C₄×C₆₇	—	4	4	no	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	2	yes	—
85	C₂³×C₆₇	—	4	2	no	—
86	C₂³×C₆₇	#127	1	1	yes	characteristic
87	Q₈×C₆₇	—	4	2	no	—
88	Q₈×C₆₇	—	4	2	no	—
89	D₄×C₆₇	—	4	2	no	—
90	D₄×C₆₇	—	4	2	no	—
91	D₄×C₆₇	—	8	4	no	—
92	D₄×C₆₇	—	8	2	no	—
93	D₄×C₆₇	—	8	2	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	8	2	no	—
97	C₂×C₄×C₆₇	—	8	2	no	—
98	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₆₇	#121	1	2	yes	—
101	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×Dic₆₇	—	8	4	no	—
107	C₂×Dic₆₇	—	8	2	no	—
108	C₈×C₆₇	#102	1	2	yes	—
109	C₈×C₆₇	#102	1	2	yes	—
110	C₈×C₆₇	#102	1	2	yes	—
111	C₈×C₆₇	#102	1	2	yes	—
112	C₆₇⋊C₈	—	4	4	no	—
113	C₂²×C₆₇	—	2	2	no	—
114	C₂²×C₆₇	—	8	2	no	—
115	C₂²×C₆₇	—	8	4	no	—
116	C₂²×C₆₇	#390	1	1	yes	characteristic
117	C₂²×C₆₇	#492	1	1	yes	characteristic
118	C₂²×C₆₇	#493	1	1	yes	characteristic
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	—	2	2	no	—
121	C₄×C₆₇	—	8	2	no	—
122	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
123	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
124	C₄×C₆₇	#486	1	1	yes	characteristic
125	C₄×C₆₇	#486	1	1	yes	characteristic
126	Dic₆₇	—	16	2	no	—
127	#9756	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	—	2	1	no	—
129	C₂×C₆₇	—	2	1	no	—
130	C₂×C₆₇	—	16	2	no	—
131	C₂×C₆₇	#1957	1	1	yes	characteristic
132	C₂×C₆₇	#1976	1	2	yes	—
133	#3147	D₆₇	1	1	yes	characteristic
134	#1846	—	67	1	no	—
135	#1517	—	67	4	no	—
136	#1450	—	67	1	no	—
137	C₆₇	#9756	1	1	yes	characteristic, complemented, Sylow
138	C₂×#135	—	2	1	no	—
139	C₂×#135	—	2	1	no	—
140	#532	C₂×D₆₇	1	2	yes	—
141	#495	—	2	1	no	—
142	#495	—	2	1	no	—
143	#494	Dic₆₇	1	1	yes	characteristic, complemented
144	#493	Dic₆₇	1	1	yes	characteristic, complemented
145	#492	Dic₆₇	1	2	yes	complemented
146	#431	—	67	2	no	—
147	C₂×#107	—	134	1	no	—
148	C₂×#106	—	134	1	no	—
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	#382	—	67	2	no	—
151	#327	—	134	8	no	—
152	#322	—	134	1	no	—
153	C₈²	—	134	1	no	—
154	C₂×#41	—	2	2	no	—
155	#135	—	4	2	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	#135	—	4	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
168	#127	C₂×Dic₆₇	1	2	yes	—
169	#124	—	2	2	no	—
170	#121	—	2	4	no	—
171	C₄×D₄	—	2	4	no	—
172	C₂×C₄⋊C₄	—	134	2	no	—
173	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
174	#107	—	268	2	no	—
175	#106	—	268	2	no	—
176	C₄⋊C₈	—	134	2	no	—
177	C₂²⋊C₈	—	67	4	no	—
178	C₄×C₈	—	134	2	no	—
179	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
180	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
183	#95	—	134	4	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	268	2	no	—
194	C₂²×C₄	D₄×D₆₇	1	2	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
196	Q₁₆	—	4	2	no	—
197	Q₁₆	—	4	2	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	4	2	no	—
201	D₈	—	4	2	no	—
202	C₂×C₈	—	134	4	no	—
203	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
207	C₂×C₈	#11761613	1	2	yes	—
208	C₂×C₈	#11761613	1	2	yes	—
209	C₂×C₈	#11761616	1	2	yes	—
210	C₂×C₈	#11761616	1	2	yes	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	268	2	no	—
213	C₄⋊C₄	—	268	2	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
217	C₄²	#11761627	1	2	yes	—
218	C₂³	—	4	2	no	—
219	C₂³	#14524039	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	4	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	536	4	no	—
232	C₂×C₄	—	536	2	no	—
233	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
235	C₂×C₄	#14524014	1	2	yes	—
236	C₂×C₄	#14524030	1	1	yes	characteristic
237	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
238	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
239	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
240	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
241	C₈	—	268	4	no	—
242	C₈	#14523964	1	2	yes	—
243	C₈	#14523964	1	2	yes	—
244	C₈	#14523964	1	2	yes	—
245	C₈	#14523964	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	#16863299	1	1	yes	characteristic
250	C₂²	#16863768	1	1	yes	characteristic
251	C₂²	#16863772	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	1072	2	no	—
256	C₄	C₂×#14523964	1	1	yes	characteristic
257	C₄	C₂×#14523964	1	1	yes	characteristic
258	C₄	#16863742	1	1	yes	characteristic
259	C₄	#16863742	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	#22561016	1	1	yes	characteristic
264	C₂	#22561108	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863299
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting, prime radical	C₆₇×#3147	C₂
socle	C₂²×C₆₇	#390
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#3147	D₆₇
hypocenter	C₆₇	#9756
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #22560373	C₂	semidirect
2. #22558769	C₂	semidirect
3. #22558492	C₂	semidirect
17. #494×C₆₇	C₄	semidirect
18. #493×C₆₇	C₄	semidirect
19. #492×C₆₇	C₄	semidirect
137. C₆₇	#9756	semidirect
143. #494	Dic₆₇	semidirect
144. #493	Dic₆₇	semidirect
145. #492	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22560373	2
#22558769	2
#22558492	2
C₆₇×#3147	2
#9756	67