Subgroups — #30766773

Order 17152.

378 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562008	C₂	1	1	yes	maximal, characteristic, complemented
2	#22560337	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558218	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic
5	#16863266	—	2	1	no	—
6	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
7	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
8	C₄×C₆₇⋊C₁₆	—	2	1	no	—
9	#16863159	—	2	1	no	—
10	#16863159	C₂²	1	2	yes	—
11	#532×C₆₇	—	2	1	no	—
12	#500×C₆₇	C₄	1	1	yes	characteristic, complemented
13	#493×C₆₇	C₄	1	1	yes	characteristic, complemented
14	#486×C₆₇	—	2	1	no	—
15	#466×C₆₇	—	2	1	no	—
16	#466×C₆₇	C₂²	1	2	yes	—
17	C₄×D₈×C₆₇	—	2	1	no	—
18	C₄×D₈×C₆₇	C₄	1	2	yes	complemented
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	4	1	no	—
22	C₂×C₄⋊Dic₆₇	—	4	1	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×#41×C₆₇	—	4	1	no	—
25	C₂×D₈×C₆₇	—	2	1	no	—
26	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₆₇	—	2	1	no	—
30	#127×C₆₇	—	2	1	no	—
31	#124×C₆₇	—	4	1	no	—
32	#121×C₆₇	—	2	2	no	—
33	#121×C₆₇	—	4	1	no	—
34	C₄×D₄×C₆₇	—	2	2	no	—
35	C₄×D₄×C₆₇	—	4	1	no	—
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	#107×C₆₇	—	2	1	no	—
38	#107×C₆₇	C₂×C₄	1	2	yes	—
39	#102×C₆₇	—	2	2	no	—
40	#102×C₆₇	—	2	2	no	—
41	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×Dic₆₇	—	4	2	no	—
46	C₄⋊Dic₆₇	—	4	2	no	—
47	C₄⋊Dic₆₇	—	4	2	no	—
48	C₄⋊Dic₆₇	—	8	2	no	—
49	C₄⋊Dic₆₇	—	8	2	no	—
50	C₆₇⋊C₁₆	—	4	2	no	—
51	#41×C₆₇	—	8	2	no	—
52	#41×C₆₇	—	8	2	no	—
53	C₂×Q₈×C₆₇	—	4	1	no	—
54	C₂×D₄×C₆₇	—	2	2	no	—
55	C₂×D₄×C₆₇	—	4	2	no	—
56	C₂×D₄×C₆₇	—	4	1	no	—
57	C₂×D₄×C₆₇	—	4	1	no	—
58	C₂²×C₄×C₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	D₈×C₆₇	—	4	2	no	—
63	D₈×C₆₇	—	4	2	no	—
64	C₂×C₈×C₆₇	—	4	1	no	—
65	C₂×C₈×C₆₇	—	4	1	no	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	—	4	1	no	—
72	C₂²⋊C₄×C₆₇	—	4	2	no	—
73	C₂²⋊C₄×C₆₇	—	4	2	no	—
74	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
77	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
78	C₂³×C₆₇	—	4	2	no	—
79	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
80	Q₈×C₆₇	—	8	2	no	—
81	D₄×C₆₇	—	4	2	no	—
82	D₄×C₆₇	—	4	2	no	—
83	D₄×C₆₇	—	8	2	no	—
84	D₄×C₆₇	—	8	2	no	—
85	D₄×C₆₇	—	8	2	no	—
86	D₄×C₆₇	—	8	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	8	4	no	—
92	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
96	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
97	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×Dic₆₇	—	8	2	no	—
101	C₂×Dic₆₇	—	8	2	no	—
102	C₂×Dic₆₇	—	8	2	no	—
103	C₈×C₆₇	—	2	1	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	—	2	1	no	—
109	C₂²×C₆₇	—	8	2	no	—
110	C₂²×C₆₇	—	8	2	no	—
111	C₂²×C₆₇	—	8	2	no	—
112	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	8	2	no	—
118	C₄×C₆₇	#357	1	2	yes	—
119	C₄×C₆₇	#357	1	2	yes	—
120	C₄×C₆₇	#360	1	2	yes	—
121	C₄×C₆₇	#410	1	1	yes	characteristic
122	C₄×C₆₇	#421	1	1	yes	characteristic
123	Dic₆₇	—	16	2	no	—
124	#11856	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	—	2	1	no	—
126	C₂×C₆₇	—	2	1	no	—
127	C₂×C₆₇	—	16	2	no	—
128	C₂×C₆₇	#1517	1	1	yes	characteristic
129	C₂×C₆₇	C₂×#357	1	1	yes	characteristic
130	C₂×C₆₇	C₂×#360	1	1	yes	characteristic
131	#3169	D₆₇	1	1	yes	characteristic
132	#2162	—	67	1	no	—
133	#1834	—	67	1	no	—
134	#1347	—	67	4	no	—
135	C₆₇	#11856	1	1	yes	characteristic, complemented, Sylow
136	#532	—	2	1	no	—
137	#500	Dic₆₇	1	1	yes	characteristic, complemented
138	#493	Dic₆₇	1	1	yes	characteristic, complemented
139	#486	—	2	1	no	—
140	#466	—	2	1	no	—
141	#466	C₂×D₆₇	1	2	yes	—
142	C₄×D₈	—	2	1	no	—
143	C₄×D₈	Dic₆₇	1	2	yes	complemented
144	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
145	#384	—	134	1	no	—
146	C₄⋊C₁₆	—	134	1	no	—
147	C₂²⋊C₁₆	—	67	2	no	—
148	C₄×C₁₆	—	134	1	no	—
149	#340	—	67	2	no	—
150	#340	—	134	1	no	—
151	C₂×#41	—	4	1	no	—
152	C₂×D₈	—	2	1	no	—
153	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
154	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	1	no	—
157	#127	—	2	1	no	—
158	#124	—	4	1	no	—
159	#121	—	2	2	no	—
160	#121	—	4	1	no	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	—	4	1	no	—
163	C₂×C₄⋊C₄	—	268	1	no	—
164	C₂×C₄⋊C₄	—	268	1	no	—
165	C₂×C₄⋊C₄	—	134	2	no	—
166	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂×C₁₆	—	134	2	no	—
168	#107	—	2	1	no	—
169	#107	C₂×Dic₆₇	1	2	yes	—
170	#102	—	2	2	no	—
171	#102	—	2	2	no	—
172	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
173	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
176	#41	—	8	2	no	—
177	#41	—	8	2	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	268	2	no	—
185	C₂²×C₄	#11761613	1	1	yes	characteristic
186	C₂²×C₄	#11761613	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
188	D₈	—	4	2	no	—
189	D₈	—	4	2	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
193	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
194	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
195	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	268	2	no	—
199	C₄⋊C₄	—	268	2	no	—
200	C₄⋊C₄	—	536	2	no	—
201	C₄⋊C₄	—	536	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
205	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
206	C₄²	#11761616	1	1	yes	characteristic
207	C₄²	#11761616	1	1	yes	characteristic
208	C₁₆	—	268	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×#11761613	1	1	yes	characteristic
211	Q₈	—	8	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	536	2	no	—
224	C₂×C₄	—	536	2	no	—
225	C₂×C₄	—	536	2	no	—
226	C₂×C₄	#14523964	1	1	yes	characteristic
227	C₂×C₄	#14523964	1	1	yes	characteristic
228	C₂×C₄	#14523964	1	1	yes	characteristic
229	C₂×C₄	#14523964	1	1	yes	characteristic
230	C₂×C₄	#14523984	1	2	yes	—
231	C₂×C₄	#14523987	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
233	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₂×#14523964	1	1	yes	characteristic
244	C₂²	C₂×#14523984	1	1	yes	characteristic
245	C₂²	C₂×#14523987	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	2	no	—
249	C₄	—	1072	2	no	—
250	C₄	#16863197	1	2	yes	—
251	C₄	#16863197	1	2	yes	—
252	C₄	#16863206	1	2	yes	—
253	C₄	#16863370	1	1	yes	characteristic
254	C₄	#16863413	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	#22558769	1	1	yes	characteristic
259	C₂	C₂×#16863197	1	1	yes	characteristic
260	C₂	C₂×#16863206	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#3169	C₂
socle	C₂²×C₆₇	C₂×#102
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3169	D₆₇
hypocenter	C₆₇	#11856
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #22562008	C₂	semidirect
2. #22560337	C₂	semidirect
3. #22558218	C₂	semidirect
12. #500×C₆₇	C₄	semidirect
13. #493×C₆₇	C₄	semidirect
18. C₄×D₈×C₆₇	C₄	semidirect
135. C₆₇	#11856	semidirect
137. #500	Dic₆₇	semidirect
138. #493	Dic₆₇	semidirect
143. 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
#22562008	2
#22560337	2
#22558218	2
C₆₇×#3169	2
#11856	67