Subgroups — #30772028

Order 17152.

358 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863226	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561993	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560263	C₂	1	2	yes	maximal
4	#22558324	C₂	1	2	yes	maximal, complemented
5	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	C₂²	1	2	yes	—
7	#16863434	C₂²	1	2	yes	—
8	#16863413	—	2	2	no	—
9	#16863226	—	2	1	no	—
10	#16863226	—	2	1	no	—
11	#16863224	—	2	1	no	—
12	#16863222	—	2	1	no	—
13	#16863168	—	2	2	no	—
14	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
15	#504×C₆₇	—	2	1	no	—
16	#504×C₆₇	—	2	1	no	—
17	#504×C₆₇	C₄	1	2	yes	complemented
18	#504×C₆₇	C₄	1	2	yes	complemented
19	C₂×M₃₂×C₆₇	—	2	1	no	—
20	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₇	—	2	1	no	—
22	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	—	2	2	no	—
24	#14524002	—	2	2	no	—
25	#14523979	—	2	2	no	—
26	#14523979	—	2	2	no	—
27	#14523977	—	2	2	no	—
28	#14523975	—	2	2	no	—
29	#14523970	—	4	4	no	—
30	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
31	#131×C₆₇	—	2	1	no	—
32	#131×C₆₇	—	2	1	no	—
33	#131×C₆₇	C₂×C₄	1	2	yes	—
34	#131×C₆₇	Q₈	1	2	yes	—
35	C₂×M₁₆×C₆₇	—	2	1	no	—
36	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	—	2	1	no	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	M₃₂×C₆₇	—	2	1	no	—
40	M₃₂×C₆₇	—	2	2	no	—
41	M₃₂×C₆₇	—	2	2	no	—
42	M₃₂×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₆₇	—	2	1	no	—
44	C₂×C₁₆×C₆₇	—	2	2	no	—
45	C₂×C₁₆×C₆₇	—	2	2	no	—
46	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
52	#11761612	—	4	2	no	—
53	#11761612	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₄⋊Dic₆₇	—	4	2	no	—
56	C₂²⋊Dic₆₇	—	4	2	no	—
57	C₄×Dic₆₇	—	4	2	no	—
58	#41×C₆₇	—	2	1	no	—
59	#41×C₆₇	—	2	1	no	—
60	#41×C₆₇	C₄⋊C₄	1	2	yes	—
61	#41×C₆₇	D₈	1	2	yes	—
62	C₂×Q₈×C₆₇	Q₁₆	1	1	yes	characteristic
63	C₂×D₄×C₆₇	—	2	1	no	—
64	C₂×D₄×C₆₇	Q₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	M₁₆×C₆₇	—	2	1	no	—
68	M₁₆×C₆₇	—	2	2	no	—
69	M₁₆×C₆₇	—	2	2	no	—
70	M₁₆×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
79	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
80	C₁₆×C₆₇	—	2	2	no	—
81	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂³×C₆₇	—	2	1	no	—
86	C₂³×C₆₇	#135	1	1	yes	characteristic
87	Q₈×C₆₇	—	2	1	no	—
88	Q₈×C₆₇	#107	1	2	yes	—
89	D₄×C₆₇	—	2	2	no	—
90	D₄×C₆₇	—	2	2	no	—
91	D₄×C₆₇	—	2	1	no	—
92	D₄×C₆₇	#107	1	2	yes	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
102	C₂×Dic₆₇	—	8	2	no	—
103	C₈×C₆₇	—	2	2	no	—
104	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₆₇	#117	1	1	yes	characteristic
106	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₆₇⋊C₈	—	8	2	no	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	4	1	no	—
111	C₂²×C₆₇	—	4	2	no	—
112	C₂²×C₆₇	#427	1	1	yes	characteristic
113	C₂²×C₆₇	#466	1	1	yes	characteristic
114	C₂²×C₆₇	#467	1	1	yes	characteristic
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	#380	1	1	yes	characteristic
117	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
118	C₄×C₆₇	#466	1	1	yes	characteristic
119	C₄×C₆₇	#467	1	1	yes	characteristic
120	Dic₆₇	—	16	2	no	—
121	#12841	—	67	1	no	maximal, Sylow
122	C₂×C₆₇	—	2	1	no	—
123	C₂×C₆₇	—	2	1	no	—
124	C₂×C₆₇	—	2	1	no	—
125	C₂×C₆₇	—	4	2	no	—
126	C₂×C₆₇	#1808	1	1	yes	characteristic
127	C₂×#504	D₆₇	1	1	yes	characteristic
128	C₂×#368	—	67	1	no	—
129	#2158	—	67	1	no	—
130	#1814	—	67	2	no	—
131	#1389	—	67	2	no	—
132	C₆₇	#12841	1	1	yes	characteristic, complemented, Sylow
133	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
134	#504	—	2	1	no	—
135	#504	—	2	1	no	—
136	#504	Dic₆₇	1	2	yes	complemented
137	#504	Dic₆₇	1	2	yes	complemented
138	C₂×M₃₂	—	2	1	no	—
139	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
140	C₂²×C₁₆	—	2	1	no	—
141	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
142	C₂×#108	—	67	2	no	—
143	#427	—	67	2	no	—
144	#421	—	134	2	no	—
145	#368	—	134	1	no	—
146	#368	—	134	1	no	—
147	#367	—	134	1	no	—
148	#366	—	134	1	no	—
149	#344	—	134	2	no	—
150	C₂×#41	D₂₆₈	1	1	yes	characteristic
151	#131	—	2	1	no	—
152	#131	—	2	1	no	—
153	#131	Dic₁₃₄	1	2	yes	—
154	#131	C₂×Dic₆₇	1	2	yes	—
155	C₂×M₁₆	—	2	1	no	—
156	C₂×M₁₆	—	134	2	no	—
157	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
160	#117	—	134	2	no	—
161	M₃₂	—	2	1	no	—
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	C₂×Dic₆₇	1	2	yes	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
169	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
170	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
171	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
172	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
174	#108	—	134	2	no	—
175	#108	—	134	2	no	—
176	#107	—	134	2	no	—
177	#106	—	134	2	no	—
178	#104	—	268	4	no	—
179	#41	—	2	1	no	—
180	#41	—	2	1	no	—
181	#41	C₄⋊Dic₆₇	1	2	yes	—
182	#41	D₅₃₆	1	2	yes	—
183	C₂×Q₈	Dic₂₆₈	1	1	yes	characteristic
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	Dic₂₆₈	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
188	M₁₆	—	2	1	no	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	268	2	no	—
192	M₁₆	—	268	2	no	—
193	M₁₆	C₄⋊Dic₆₇	1	2	yes	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	268	2	no	—
198	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
202	C₂×C₈	#11761628	1	1	yes	characteristic
203	C₂×C₈	#11761628	1	1	yes	characteristic
204	C₄⋊C₄	—	268	2	no	—
205	C₂²⋊C₄	—	268	2	no	—
206	C₄²	—	268	2	no	—
207	C₁₆	—	2	2	no	—
208	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
209	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
210	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
211	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	#14524065	1	1	yes	characteristic
214	Q₈	—	2	1	no	—
215	Q₈	#14523977	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	#14523977	1	2	yes	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	536	2	no	—
224	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
225	C₂×C₄	#14524017	1	1	yes	characteristic
226	C₂×C₄	#14524017	1	1	yes	characteristic
227	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
228	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
229	C₂×C₄	#14524065	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	536	2	no	—
232	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
233	C₈	#14524002	1	1	yes	characteristic
234	C₈	#14524022	1	1	yes	characteristic
235	C₈	#14524022	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	#16863434	1	1	yes	characteristic
240	C₂²	#16863632	1	1	yes	characteristic
241	C₂²	#16863637	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	1072	2	no	—
244	C₄	#16863256	1	1	yes	characteristic
245	C₄	C₂×#14523977	1	1	yes	characteristic
246	C₄	#16863632	1	1	yes	characteristic
247	C₄	#16863637	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	4	2	no	—
252	C₂	#22560238	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863632
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂×C₆₇	#1808
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#12841
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16863226	C₂	semidirect
2. #22561993	C₂	semidirect
4. #22558324	C₂	semidirect
17. #504×C₆₇	C₄	semidirect
18. #504×C₆₇	C₄	semidirect
132. C₆₇	#12841	semidirect
136. #504	Dic₆₇	semidirect
137. #504	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863226	2
#22561993	2
#22560263	2
#22558324	2
C₂×#504×C₆₇	2
#12841	67