Subgroups — #30770693

Order 17152.

586 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	C₂²×#14523983	C₂	1	2	yes	maximal
2	#22558293	C₂	1	4	yes	maximal, complemented
3	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	—	2	4	no	—
5	C₂×#14523983	—	2	2	no	—
6	C₂×#14523983	C₄	1	8	yes	complemented
7	C₂×#14523983	C₂²	1	4	yes	—
8	C₂²×C₆₇⋊C₁₆	—	2	2	no	—
9	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
12	#405×C₆₇	—	2	2	no	—
13	#405×C₆₇	—	2	2	no	—
14	C₂×#97×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
16	#14523983	—	2	8	no	—
17	#14523983	—	2	4	no	—
18	#14523983	C₂×C₄	1	8	yes	—
19	C₂×C₆₇⋊C₁₆	—	2	8	no	—
20	C₂×C₆₇⋊C₁₆	—	2	4	no	—
21	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	—	2	2	no	—
24	C₂×M₁₆×C₆₇	C₈	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	D₄	1	2	yes	—
29	C₂²×C₈×C₆₇	D₄	1	2	yes	—
30	C₂²×C₈×C₆₇	D₄	1	2	yes	—
31	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	#117×C₆₇	—	2	2	no	—
34	#117×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₆₇	C₈	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	#97×C₆₇	—	2	4	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	C₄×C₈×C₆₇	—	2	2	no	—
41	C₆₇⋊C₁₆	—	2	8	no	—
42	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
50	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₆₇	—	2	8	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	C₄²	1	2	yes	—
56	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
57	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
58	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
61	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
62	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₂²⋊C₄×C₆₇	—	2	4	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₂³×C₆₇	—	4	2	no	—
73	C₂³×C₆₇	C₄×C₈	1	2	yes	—
74	C₂³×C₆₇	C₂²⋊C₈	1	2	yes	—
75	C₂³×C₆₇	C₄⋊C₈	1	2	yes	—
76	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	#95	1	4	yes	—
80	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
81	C₂×C₄×C₆₇	#97	1	4	yes	—
82	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
84	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
92	C₈×C₆₇	—	2	4	no	—
93	C₈×C₆₇	#95	1	2	yes	—
94	C₈×C₆₇	#95	1	2	yes	—
95	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
96	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂²×C₆₇	—	2	2	no	—
98	C₂²×C₆₇	—	2	4	no	—
99	C₂²×C₆₇	—	4	2	no	—
100	C₂²×C₆₇	—	4	2	no	—
101	C₂²×C₆₇	#336	1	4	yes	—
102	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₄×C₆₇	—	4	4	no	—
106	C₄×C₆₇	#336	1	4	yes	—
107	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
108	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
109	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
111	#12551	—	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	C₂×#336	1	1	yes	characteristic
116	C₂×C₆₇	#2115	1	2	yes	—
117	C₂²×M₃₂	—	67	2	no	—
118	C₂×#405	D₆₇	1	1	yes	characteristic
119	#1377	—	67	4	no	—
120	C₆₇	#12551	1	1	yes	characteristic, complemented, Sylow
121	C₂²×M₁₆	Dic₆₇	1	1	yes	characteristic
122	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
123	C₂×#117	Dic₆₇	1	1	yes	characteristic
124	C₂×M₃₂	—	67	8	no	—
125	C₂×M₃₂	—	67	4	no	—
126	C₂×M₃₂	—	134	4	no	—
127	C₂×M₃₂	—	134	2	no	—
128	C₂²×C₁₆	—	134	2	no	—
129	#405	—	2	2	no	—
130	#405	—	2	2	no	—
131	C₂×#97	C₂×D₆₇	1	2	yes	—
132	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
133	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	C₆₇⋊C₈	1	4	yes	—
137	C₂²×C₈	Dic₁₃₄	1	2	yes	—
138	C₂²×C₈	C₄×D₆₇	1	4	yes	—
139	C₂²×C₈	D₂₆₈	1	2	yes	—
140	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
141	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
143	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
144	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
145	#117	—	2	2	no	—
146	#117	—	2	2	no	—
147	C₂×C₄⋊C₄	C₆₇⋊C₈	1	2	yes	—
148	C₂×C₂²⋊C₄	C₆₇⋊C₈	1	2	yes	—
149	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
150	M₃₂	—	67	8	no	—
151	M₃₂	—	134	8	no	—
152	M₃₂	—	134	4	no	—
153	C₂×C₁₆	—	134	8	no	—
154	C₂×C₁₆	—	134	4	no	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₂⁴	C₂×C₆₇⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
160	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
161	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
162	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
163	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
165	C₂²×C₄	#11761612	1	2	yes	—
166	C₂²×C₄	#11761612	1	2	yes	—
167	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
168	M₁₆	—	2	8	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
172	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
173	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
174	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
176	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
177	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
178	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
179	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
180	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
181	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
182	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
183	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
184	C₄⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₁₆	—	134	8	no	—
189	C₂³	—	4	2	no	—
190	C₂³	C₄×C₆₇⋊C₈	1	2	yes	—
191	C₂³	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
192	C₂³	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
193	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₆₇⋊#95	1	4	yes	—
197	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
198	C₂×C₄	#14523954	1	4	yes	—
199	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
200	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
201	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
202	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
203	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
208	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
209	C₈	—	2	4	no	—
210	C₈	C₆₇⋊#95	1	2	yes	—
211	C₈	C₆₇⋊#95	1	2	yes	—
212	C₈	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
213	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	#16863151	1	4	yes	—
219	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
220	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
221	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
222	C₄	—	4	4	no	—
223	C₄	#16863151	1	4	yes	—
224	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
225	C₄	C₂×#14523954	1	1	yes	characteristic
226	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
227	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	4	no	—
230	C₂	—	2	1	no	—
231	C₂	C₂×#16863151	1	1	yes	characteristic
232	C₂	#22561830	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#12551
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22558293	C₂	semidirect
6. C₂×#14523983	C₄	semidirect
120. C₆₇	#12551	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
#22558293	2
C₂×#405×C₆₇	2
#12551	67