Subgroups — #30765639

Order 17152.

362 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	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22561965	C₂	1	1	yes	maximal, characteristic
3	#22558189	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
6	C₂×#14523983	—	2	1	no	—
7	C₈×Dic₁₃₄	—	2	1	no	—
8	C₈×Dic₁₃₄	—	2	1	no	—
9	C₈×Dic₁₃₄	C₄	1	2	yes	—
10	C₈×Dic₁₃₄	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₈	—	2	1	no	—
13	C₂×C₈×Dic₆₇	—	2	1	no	—
14	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₂×M₃₂×C₆₇	—	2	1	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#346×C₆₇	—	2	1	no	—
19	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂²×Dic₁₃₄	C₈	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	—	2	1	no	—
22	C₄×Dic₁₃₄	—	2	1	no	—
23	C₄×Dic₁₃₄	—	2	1	no	—
24	C₄×Dic₁₃₄	C₈	1	2	yes	—
25	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
27	C₂×Dic₆₇⋊C₄	—	2	1	no	—
28	C₂×C₄×Dic₆₇	—	2	1	no	—
29	#14523983	—	4	2	no	—
30	C₂×C₆₇⋊C₁₆	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
33	Dic₆₇⋊C₈	—	2	2	no	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	C₈×Dic₆₇	—	2	2	no	—
36	C₈×Dic₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₆₇	—	2	4	no	—
41	C₂×C₁₆×C₆₇	—	2	2	no	—
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	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₃₄	—	2	1	no	—
47	C₂×Dic₁₃₄	—	2	4	no	—
48	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
49	C₂²×Dic₆₇	—	2	1	no	—
50	C₂²×Dic₆₇	—	2	1	no	—
51	C₂×C₆₇⋊C₈	—	2	2	no	—
52	C₂×C₆₇⋊C₈	—	4	2	no	—
53	C₄⋊Dic₆₇	—	2	1	no	—
54	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
55	Dic₆₇⋊C₄	—	2	2	no	—
56	Dic₆₇⋊C₄	—	2	2	no	—
57	C₄×Dic₆₇	—	2	2	no	—
58	C₄×Dic₆₇	—	2	2	no	—
59	C₆₇⋊C₁₆	—	4	2	no	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
67	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
68	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
73	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₁₆×C₆₇	—	2	4	no	—
75	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	4	1	no	—
78	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#103	1	2	yes	—
82	C₂×C₄×C₆₇	#104	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₆₇	—	2	4	no	—
87	C₂×Dic₆₇	—	2	2	no	—
88	C₂×Dic₆₇	—	2	2	no	—
89	C₂×Dic₆₇	—	4	2	no	—
90	Dic₁₃₄	—	2	4	no	—
91	Dic₁₃₄	—	2	2	no	—
92	Dic₁₃₄	M₃₂	1	4	yes	complemented
93	C₈×C₆₇	—	2	1	no	—
94	C₈×C₆₇	—	2	1	no	—
95	C₈×C₆₇	#103	1	2	yes	—
96	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₆₇⋊C₈	—	4	2	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	—	2	1	no	—
101	C₂²×C₆₇	Q₈⋊C₈	1	2	yes	—
102	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	—	2	1	no	—
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	Q₈⋊C₈	1	2	yes	—
107	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
108	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
109	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
110	Dic₆₇	—	2	4	no	—
111	Dic₆₇	—	4	2	no	—
112	#11607	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	—	2	2	no	—
114	C₂×C₆₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
115	C₂×C₆₇	#2112	1	1	yes	characteristic
116	C₂×C₆₇	#2117	1	1	yes	characteristic
117	C₂×C₈×Q₈	—	67	1	no	—
118	#2150	—	67	1	no	—
119	C₄×M₃₂	D₆₇	1	1	yes	characteristic
120	Q₈⋊C₁₆	—	67	4	no	—
121	C₆₇	#11607	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄×Q₈	—	67	1	no	—
123	C₂×M₃₂	—	2	1	no	—
124	C₂×M₃₂	—	134	1	no	—
125	C₈×Q₈	—	67	2	no	—
126	C₈×Q₈	—	67	2	no	—
127	C₈×Q₈	—	134	1	no	—
128	C₈×Q₈	—	134	1	no	—
129	C₂×C₄⋊C₈	—	67	1	no	—
130	C₂×C₄⋊C₈	—	134	1	no	—
131	C₂×C₄×C₈	—	134	1	no	—
132	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
133	C₄⋊C₁₆	—	67	2	no	—
134	C₄⋊C₁₆	—	134	1	no	—
135	#346	—	2	1	no	—
136	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
137	C₂²×Q₈	—	67	1	no	—
138	C₂²×C₈	—	134	1	no	—
139	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
141	C₄×Q₈	—	67	2	no	—
142	C₄×Q₈	—	67	2	no	—
143	C₄×Q₈	—	134	1	no	—
144	C₄×Q₈	—	134	1	no	—
145	C₂×C₄⋊C₄	—	67	1	no	—
146	C₂×C₄⋊C₄	—	134	1	no	—
147	C₂×C₄²	—	134	1	no	—
148	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
149	M₃₂	—	2	4	no	—
150	M₃₂	—	268	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	134	2	no	—
153	C₄⋊C₈	—	67	2	no	—
154	C₄⋊C₈	—	134	2	no	—
155	C₄⋊C₈	—	134	2	no	—
156	C₄⋊C₈	—	134	1	no	—
157	C₄×C₈	—	134	2	no	—
158	C₄×C₈	—	134	2	no	—
159	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
160	C₄×C₈	D₂₆₈	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
163	C₂×Q₈	—	67	2	no	—
164	C₂×Q₈	—	134	4	no	—
165	C₂×Q₈	—	134	1	no	—
166	C₂²×C₄	—	134	1	no	—
167	C₂²×C₄	—	134	1	no	—
168	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
170	C₂²×C₄	#11761611	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	268	2	no	—
173	C₂×C₈	—	134	2	no	—
174	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
175	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
176	C₂×C₈	#11761617	1	2	yes	—
177	C₂×C₈	Dic₂₆₈	1	2	yes	—
178	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
180	C₄⋊C₄	—	67	2	no	—
181	C₄⋊C₄	—	134	1	no	—
182	C₄⋊C₄	—	134	2	no	—
183	C₄⋊C₄	—	134	2	no	—
184	C₄²	—	134	2	no	—
185	C₄²	—	134	2	no	—
186	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
187	C₄²	C₈×D₆₇	1	1	yes	characteristic
188	C₄²	#11761611	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
190	C₁₆	—	2	4	no	—
191	C₁₆	—	268	2	no	—
192	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
193	Q₈	—	67	4	no	—
194	Q₈	—	134	4	no	—
195	Q₈	—	134	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	268	2	no	—
199	C₂×C₄	—	134	2	no	—
200	C₂×C₄	—	134	4	no	—
201	C₂×C₄	—	134	2	no	—
202	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
203	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
204	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
205	C₂×C₄	#14523968	1	2	yes	—
206	C₂×C₄	#14523971	1	2	yes	—
207	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
208	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
209	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	268	2	no	—
213	C₈	#14523968	1	2	yes	—
214	C₈	C₂×#11761617	1	1	yes	characteristic
215	C₈	C₂×Dic₂₆₈	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	#16863128	1	2	yes	—
219	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	268	2	no	—
224	C₄	—	134	4	no	—
225	C₄	#16863128	1	2	yes	—
226	C₄	C₂×#14523968	1	1	yes	characteristic
227	C₄	C₂×#14523971	1	1	yes	characteristic
228	C₄	M₃₂×D₆₇	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×#16863128	1	1	yes	characteristic
231	C₂	#22561819	1	1	yes	characteristic
232	C₂	#22561836	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₇	C₂²⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#11607
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22558189	C₂	semidirect
92. Dic₁₃₄	M₃₂	semidirect
121. C₆₇	#11607	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22561965	2
#22558189	2
C₄×M₃₂×C₆₇	2
#11607	67