Subgroups — #30749574

Order 17152.

422 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560326	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863148	C₂	1	2	yes	maximal
4	C₂×#16863146	C₂	1	2	yes	maximal
5	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863151	—	2	2	no	—
14	#16863148	—	2	4	no	—
15	#16863146	—	2	4	no	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₄³×C₆₇	C₂²	1	1	yes	characteristic
18	#336×C₆₇	—	2	2	no	—
19	C₂²×C₆₇⋊C₈	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄×C₆₇⋊C₈	—	2	2	no	—
27	C₄×C₆₇⋊C₈	—	2	2	no	—
28	C₄×C₆₇⋊C₈	—	2	2	no	—
29	C₄×C₆₇⋊C₈	C₈	1	4	yes	complemented
30	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
31	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
32	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
33	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
34	C₂²×C₈×C₆₇	—	2	2	no	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	D₄	1	2	yes	—
38	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	1	no	—
43	C₂×C₆₇⋊C₈	—	2	1	no	—
44	C₂×C₆₇⋊C₈	—	2	1	no	—
45	C₂×C₆₇⋊C₈	—	2	1	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	4	no	—
48	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
51	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
52	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
53	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
54	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
55	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₆₇	—	4	2	no	—
62	C₂×C₈×C₆₇	—	4	4	no	—
63	C₄²×C₆₇	—	2	4	no	—
64	C₄²×C₆₇	—	2	1	no	—
65	C₄²×C₆₇	—	2	1	no	—
66	C₄²×C₆₇	—	2	4	no	—
67	C₄²×C₆₇	C₂×C₈	1	2	yes	—
68	C₄²×C₆₇	M₁₆	1	2	yes	—
69	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₆₇	#108	1	4	yes	—
81	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#135	1	4	yes	—
92	C₈×C₆₇	—	8	2	no	—
93	C₆₇⋊C₈	—	2	2	no	—
94	C₆₇⋊C₈	—	2	2	no	—
95	C₆₇⋊C₈	—	8	2	no	—
96	C₆₇⋊C₈	C₄⋊C₈	1	4	yes	—
97	C₆₇⋊C₈	C₄⋊C₈	1	4	yes	—
98	C₂²×C₆₇	#384	1	1	yes	characteristic
99	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₆₇	#423	1	1	yes	characteristic
101	C₂²×C₆₇	#427	1	2	yes	—
102	C₂²×C₆₇	C₂×#108	1	2	yes	—
103	C₄×C₆₇	—	2	4	no	—
104	C₄×C₆₇	—	2	4	no	—
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₄×C₆₇	#423	1	1	yes	characteristic
110	C₄×C₆₇	#423	1	1	yes	characteristic
111	C₄×C₆₇	#495	1	2	yes	—
112	C₄×C₆₇	#499	1	2	yes	—
113	#8807	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	#1564	1	2	yes	—
115	C₂×C₆₇	#1568	1	2	yes	—
116	C₂×C₆₇	#1842	1	1	yes	characteristic
117	C₂×C₆₇	#1846	1	1	yes	characteristic
118	C₂×C₆₇	#1848	1	1	yes	characteristic
119	#1831	—	67	1	no	—
120	#1831	D₆₇	1	1	yes	characteristic
121	C₄²×C₈	—	67	1	no	—
122	C₂×#335	—	67	2	no	—
123	C₂×#334	—	67	2	no	—
124	C₆₇	#8807	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄⋊C₈	—	67	2	no	—
126	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
127	C₂×C₄×C₈	—	67	1	no	—
128	C₂×C₄×C₈	—	67	1	no	—
129	C₂×C₄×C₈	—	67	1	no	—
130	C₂×C₄×C₈	—	67	1	no	—
131	C₂×C₄×C₈	—	67	1	no	—
132	C₂×C₄×C₈	—	67	1	no	—
133	C₄³	C₂×D₆₇	1	1	yes	characteristic
134	#336	—	2	2	no	—
135	#336	—	134	2	no	—
136	#335	—	134	4	no	—
137	#334	—	134	4	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	67	1	no	—
140	C₂²×C₈	—	67	1	no	—
141	C₂²×C₈	—	67	1	no	—
142	C₂²×C₈	—	67	1	no	—
143	C₂²×C₈	—	134	2	no	—
144	C₂×C₄²	Dic₁₃₄	1	2	yes	—
145	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
148	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	134	2	no	—
152	C₄⋊C₈	—	134	2	no	—
153	C₄×C₈	—	67	4	no	—
154	C₄×C₈	—	67	2	no	—
155	C₄×C₈	—	67	2	no	—
156	C₄×C₈	—	67	2	no	—
157	C₄×C₈	—	67	2	no	—
158	C₄×C₈	—	134	2	no	—
159	C₄×C₈	—	134	2	no	—
160	C₄×C₈	—	134	2	no	—
161	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
162	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
163	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	268	2	no	—
169	C₂×C₈	—	268	4	no	—
170	C₂×C₈	—	67	2	no	—
171	C₂×C₈	—	67	2	no	—
172	C₂×C₈	—	67	2	no	—
173	C₂×C₈	—	67	2	no	—
174	C₂×C₈	—	67	2	no	—
175	C₂×C₈	—	67	2	no	—
176	C₂×C₈	—	67	2	no	—
177	C₂×C₈	—	67	2	no	—
178	C₂×C₈	—	134	1	no	—
179	C₂×C₈	—	134	1	no	—
180	C₂×C₈	—	134	1	no	—
181	C₂×C₈	—	134	1	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	4	no	—
186	C₄²	C₈×D₆₇	1	2	yes	—
187	C₄²	#11761611	1	2	yes	—
188	C₄²	D₄×D₆₇	1	1	yes	characteristic
189	C₄²	D₄×D₆₇	1	1	yes	characteristic
190	C₄²	Q₈×D₆₇	1	1	yes	characteristic
191	C₄²	Q₈×D₆₇	1	1	yes	characteristic
192	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	Dic₆₇⋊C₈	1	4	yes	—
199	C₂×C₄	#14523978	1	4	yes	—
200	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
201	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
202	C₂×C₄	#14524039	1	1	yes	characteristic
203	C₂×C₄	#14524041	1	1	yes	characteristic
204	C₂×C₄	#14524042	1	1	yes	characteristic
205	C₂×C₄	#14524042	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
208	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
209	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
210	C₂×C₄	#14524063	1	4	yes	—
211	C₈	—	8	2	no	—
212	C₈	—	536	2	no	—
213	C₈	—	67	4	no	—
214	C₈	—	67	4	no	—
215	C₈	—	134	2	no	—
216	C₈	—	134	2	no	—
217	C₂²	#16863265	1	1	yes	characteristic
218	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
219	C₂²	#16863419	1	1	yes	characteristic
220	C₂²	#16863432	1	2	yes	—
221	C₂²	C₂×#14523978	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
227	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
228	C₄	#16863418	1	1	yes	characteristic
229	C₄	#16863418	1	1	yes	characteristic
230	C₄	#16863778	1	2	yes	—
231	C₄	#16863795	1	2	yes	—
232	C₂	#22558955	1	2	yes	—
233	C₂	#22558970	1	2	yes	—
234	C₂	#22560360	1	1	yes	characteristic
235	C₂	#22560371	1	1	yes	characteristic
236	C₂	#22560380	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1831	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1831	D₆₇
hypocenter	C₆₇	#8807
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
29. C₄×C₆₇⋊C₈	C₈	semidirect
124. C₆₇	#8807	semidirect

Maximal subgroups

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

subgroup	index
#22560326	2
C₄²×C₆₇⋊C₈	2
C₂×#16863148	2
C₂×#16863146	2
C₆₇×#1831	2
#8807	67