Subgroups — #30761531

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559942	C₂	1	2	yes	maximal
2	#22559926	C₂	1	2	yes	maximal
3	C₂×#16863153	C₂	1	2	yes	maximal
4	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	C₂²	1	4	yes	—
6	#16863163	—	2	4	no	—
7	#16863159	—	2	4	no	—
8	#16863153	C₄	1	8	yes	complemented
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
12	#405×C₆₇	—	2	4	no	—
13	C₂×#97×C₆₇	C₄	1	2	yes	complemented
14	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
15	#14524002	—	2	4	no	—
16	#14524002	—	2	4	no	—
17	#14524002	—	2	4	no	—
18	#14524002	C₂×C₄	1	8	yes	—
19	C₂×C₄⋊Dic₆₇	—	2	4	no	—
20	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
21	C₂×C₄×Dic₆₇	—	2	4	no	—
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₆₇	—	2	2	no	—
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₆₇	—	2	2	no	—
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	4	yes	—
32	#117×C₆₇	—	2	2	no	—
33	#117×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
35	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
37	#97×C₆₇	—	2	4	no	—
38	C₄×C₈×C₆₇	—	2	4	no	—
39	C₂²×Dic₆₇	—	4	4	no	—
40	C₄⋊Dic₆₇	—	2	8	no	—
41	C₄⋊Dic₆₇	—	4	4	no	—
42	C₂²⋊Dic₆₇	—	2	8	no	—
43	C₂²⋊Dic₆₇	—	4	4	no	—
44	C₄×Dic₆₇	—	2	8	no	—
45	C₄×Dic₆₇	—	4	4	no	—
46	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	#41	1	2	yes	—
53	C₂²×C₄×C₆₇	#41	1	2	yes	—
54	C₂²×C₄×C₆₇	#41	1	2	yes	—
55	C₂²×C₄×C₆₇	#41	1	2	yes	—
56	M₁₆×C₆₇	—	2	8	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	2	1	no	—
59	C₂×C₈×C₆₇	—	2	1	no	—
60	C₂×C₈×C₆₇	—	2	1	no	—
61	C₂×C₈×C₆₇	—	2	1	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	2	8	no	—
64	C₂×C₈×C₆₇	C₄²	1	4	yes	—
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₂²⋊C₄×C₆₇	—	2	4	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₂³×C₆₇	—	4	2	no	—
72	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₆₇	C₄×D₄	1	2	yes	—
74	C₂³×C₆₇	#121	1	2	yes	—
75	C₂³×C₆₇	#124	1	2	yes	—
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#117	1	4	yes	—
81	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
87	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
88	C₂×C₄×C₆₇	#123	1	4	yes	—
89	C₂×C₄×C₆₇	#125	1	2	yes	—
90	C₂×C₄×C₆₇	#126	1	4	yes	—
91	C₂×Dic₆₇	—	4	8	no	—
92	C₂×Dic₆₇	—	8	8	no	—
93	C₈×C₆₇	—	2	4	no	—
94	C₈×C₆₇	—	2	2	no	—
95	C₈×C₆₇	—	2	2	no	—
96	C₂²×C₆₇	—	2	4	no	—
97	C₂²×C₆₇	—	2	2	no	—
98	C₂²×C₆₇	—	4	2	no	—
99	C₂²×C₆₇	—	4	2	no	—
100	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₆₇	#380	1	1	yes	characteristic
102	C₂²×C₆₇	#382	1	1	yes	characteristic
103	C₂²×C₆₇	#388	1	4	yes	—
104	C₄×C₆₇	—	4	4	no	—
105	C₄×C₆₇	#376	1	1	yes	characteristic
106	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₆₇	#381	1	1	yes	characteristic
108	C₄×C₆₇	#383	1	1	yes	characteristic
109	C₄×C₆₇	#387	1	4	yes	—
110	Dic₆₇	—	8	8	no	—
111	#10814	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	—	2	4	no	—
113	C₂×C₆₇	—	2	1	no	—
114	C₂×C₆₇	—	2	1	no	—
115	C₂×C₆₇	#1434	1	1	yes	characteristic
116	C₂×C₆₇	#1812	1	2	yes	—
117	C₂×#405	D₆₇	1	1	yes	characteristic
118	#1740	—	67	2	no	—
119	#1735	—	67	2	no	—
120	C₂×H(3,ℤ/4)	—	67	2	no	—
121	C₆₇	#10814	1	1	yes	characteristic, complemented, Sylow
122	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
123	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
124	C₂×#117	—	67	4	no	—
125	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
126	#405	—	2	4	no	—
127	C₂×#97	Dic₆₇	1	2	yes	complemented
128	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
129	#342	—	134	4	no	—
130	#340	—	134	4	no	—
131	H(3,ℤ/4)	—	67	8	no	—
132	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
139	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
140	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
141	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
142	#117	—	2	2	no	—
143	#117	—	2	2	no	—
144	#117	—	67	8	no	—
145	#117	—	134	4	no	—
146	#117	—	134	4	no	—
147	#117	—	134	4	no	—
148	C₂×C₄⋊C₄	—	134	4	no	—
149	C₂×C₄⋊C₄	Dic₁₃₄	1	2	yes	—
150	C₂×C₂²⋊C₄	—	134	4	no	—
151	C₂×C₂²⋊C₄	D₂₆₈	1	2	yes	—
152	C₂×C₄²	—	134	4	no	—
153	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
154	#97	—	2	4	no	—
155	C₄×C₈	—	2	4	no	—
156	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
157	C₂²×C₄	—	268	4	no	—
158	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
159	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
160	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
162	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
163	C₂²×C₄	#11761628	1	2	yes	—
164	C₂²×C₄	#11761628	1	2	yes	—
165	C₂²×C₄	#11761628	1	2	yes	—
166	C₂²×C₄	#11761628	1	2	yes	—
167	M₁₆	—	2	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
176	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
177	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	268	4	no	—
180	C₄⋊C₄	—	134	8	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	268	4	no	—
183	C₂²⋊C₄	—	134	8	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	268	4	no	—
187	C₄²	—	134	8	no	—
188	C₂³	—	4	2	no	—
189	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
190	C₂³	C₄×D₂₆₈	1	2	yes	—
191	C₂³	#14524017	1	2	yes	—
192	C₂³	#14524031	1	2	yes	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	268	8	no	—
196	C₂×C₄	—	536	8	no	—
197	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
199	C₂×C₄	#14524000	1	4	yes	—
200	C₂×C₄	#14524002	1	1	yes	characteristic
201	C₂×C₄	#14524002	1	1	yes	characteristic
202	C₂×C₄	#14524002	1	1	yes	characteristic
203	C₂×C₄	#14524002	1	1	yes	characteristic
204	C₂×C₄	C₄×C₂²⋊D₆₇	1	4	yes	—
205	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
206	C₂×C₄	#14524022	1	2	yes	—
207	C₂×C₄	#14524026	1	4	yes	—
208	C₂×C₄	#14524035	1	2	yes	—
209	C₂×C₄	#14524037	1	4	yes	—
210	C₈	—	2	4	no	—
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
218	C₂²	#16863256	1	1	yes	characteristic
219	C₂²	#16863262	1	1	yes	characteristic
220	C₂²	#16863287	1	4	yes	—
221	C₄	—	4	4	no	—
222	C₄	—	536	8	no	—
223	C₄	#16863242	1	1	yes	characteristic
224	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
225	C₄	#16863259	1	1	yes	characteristic
226	C₄	#16863264	1	1	yes	characteristic
227	C₄	#16863280	1	4	yes	—
228	C₂	—	2	4	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	#22558436	1	1	yes	characteristic
232	C₂	#22560255	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#10814
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. #16863153	C₄	semidirect
13. C₂×#97×C₆₇	C₄	semidirect
14. C₂×C₄×C₈×C₆₇	C₄	semidirect
121. C₆₇	#10814	semidirect
127. C₂×#97	Dic₆₇	semidirect
128. C₂×C₄×C₈	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22559942	2
#22559926	2
C₂×#16863153	2
C₂×#405×C₆₇	2
#10814	67