Subgroups — C₂×C₆₇×#1333 #30690846

Order 17152.

462 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	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1333	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₆₇	—	2	1	no	—
7	C₂²×C₁₆×C₆₇	—	2	1	no	—
8	C₄×D₈×C₆₇	—	2	1	no	—
9	C₄×D₈×C₆₇	—	2	1	no	—
10	C₄×D₈×C₆₇	C₄	1	2	yes	—
11	C₄×D₈×C₆₇	C₂²	1	2	yes	—
12	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
14	C₂×#102×C₆₇	—	2	1	no	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#346×C₆₇	—	2	1	no	—
17	#346×C₆₇	C₂²	1	2	yes	complemented
18	#334×C₆₇	—	2	1	no	—
19	#334×C₆₇	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₆₇	—	2	1	no	—
21	C₂³×C₄×C₆₇	—	4	1	no	—
22	C₂×D₈×C₆₇	—	2	1	no	—
23	C₂×D₈×C₆₇	—	2	2	no	—
24	C₂×D₈×C₆₇	C₈	1	4	yes	complemented
25	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₇	—	4	1	no	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₄×D₄×C₆₇	—	2	2	no	—
30	C₄×D₄×C₆₇	—	2	2	no	—
31	C₄×D₄×C₆₇	—	4	1	no	—
32	C₄×D₄×C₆₇	—	4	1	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₆₇	—	2	4	no	—
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	#107×C₆₇	—	2	1	no	—
39	#107×C₆₇	C₂×C₄	1	2	yes	—
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	4	1	no	—
42	#102×C₆₇	—	2	2	no	—
43	#102×C₆₇	—	2	2	no	—
44	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂⁴×C₆₇	—	4	1	no	—
49	C₂×D₄×C₆₇	—	2	4	no	—
50	C₂×D₄×C₆₇	—	2	2	no	—
51	C₂×D₄×C₆₇	—	4	1	no	—
52	C₂×D₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	—	4	1	no	—
54	C₂²×C₄×C₆₇	—	4	2	no	—
55	C₂²×C₄×C₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	—	8	4	no	—
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
60	D₈×C₆₇	—	2	4	no	—
61	D₈×C₆₇	—	2	2	no	—
62	D₈×C₆₇	C₂×C₈	1	4	yes	complemented
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	—	4	2	no	—
65	C₂×C₈×C₆₇	—	8	2	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	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₄⋊C₄×C₆₇	—	4	1	no	—
74	C₂²⋊C₄×C₆₇	—	4	2	no	—
75	C₂²⋊C₄×C₆₇	—	4	2	no	—
76	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₆₇	D₈	1	1	yes	characteristic
78	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
79	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₁₆×C₆₇	—	2	4	no	—
81	C₂³×C₆₇	—	4	4	no	—
82	C₂³×C₆₇	—	4	2	no	—
83	C₂³×C₆₇	—	8	2	no	—
84	C₂³×C₆₇	—	8	2	no	—
85	C₂³×C₆₇	#102	1	1	yes	characteristic
86	D₄×C₆₇	—	2	4	no	—
87	D₄×C₆₇	—	4	4	no	—
88	D₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	8	4	no	—
93	C₂×C₄×C₆₇	—	8	4	no	—
94	C₂×C₄×C₆₇	—	8	2	no	—
95	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#104	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
103	C₈×C₆₇	—	2	1	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	8	2	no	—
106	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
107	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
108	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
109	C₂²×C₆₇	—	4	4	no	—
110	C₂²×C₆₇	—	8	8	no	—
111	C₂²×C₆₇	—	8	2	no	—
112	C₂²×C₆₇	—	8	2	no	—
113	C₂²×C₆₇	D₄⋊C₈	1	2	yes	—
114	C₂²×C₆₇	#360	1	2	yes	—
115	C₂²×C₆₇	#361	1	2	yes	—
116	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	8	2	no	—
122	C₄×C₆₇	D₄⋊C₈	1	2	yes	—
123	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
125	C₂×#1333	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₆₇	—	8	4	no	—
127	C₂×C₆₇	#1333	1	4	yes	direct product factor
128	C₂×C₆₇	C₂×D₄⋊C₈	1	1	yes	characteristic
129	C₂×C₆₇	C₂×#360	1	1	yes	characteristic
130	C₂×C₆₇	C₂×#361	1	1	yes	characteristic
131	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
132	C₂×#346	C₂×C₆₇	1	1	yes	characteristic, complemented
133	C₂×#334	C₂×C₆₇	1	1	yes	characteristic, complemented
134	#1333	C₂×C₆₇	1	4	yes	direct product factor
135	C₆₇	C₂×#1333	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic
137	C₂×C₄×D₄	—	2	1	no	—
138	C₂²×C₁₆	—	2	1	no	—
139	C₄×D₈	—	2	1	no	—
140	C₄×D₈	—	2	1	no	—
141	C₄×D₈	C₄×C₆₇	1	2	yes	—
142	C₄×D₈	C₂²×C₆₇	1	2	yes	—
143	C₂×#107	C₄×C₆₇	1	1	yes	characteristic
144	C₂×C₄⋊C₈	—	2	1	no	—
145	C₂×#102	—	2	1	no	—
146	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
147	#346	—	2	1	no	—
148	#346	C₂²×C₆₇	1	2	yes	complemented
149	#334	—	2	1	no	—
150	#334	C₂²×C₆₇	1	2	yes	complemented
151	C₂²×D₄	—	2	1	no	—
152	C₂³×C₄	—	4	1	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂×D₈	C₈×C₆₇	1	4	yes	complemented
156	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
157	C₂²×C₈	—	4	1	no	—
158	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
160	C₄×D₄	—	2	2	no	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	—	4	1	no	—
163	C₄×D₄	—	4	1	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₂²⋊C₄	—	4	1	no	—
166	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
167	C₂×C₁₆	—	2	4	no	—
168	C₂×C₁₆	—	2	2	no	—
169	#107	—	2	1	no	—
170	#107	C₂×C₄×C₆₇	1	2	yes	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	4	1	no	—
173	#102	—	2	2	no	—
174	#102	—	2	2	no	—
175	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
176	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
177	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
178	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
179	C₂⁴	—	4	1	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	8	4	no	—
188	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
189	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
190	C₂²×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
191	D₈	—	2	2	no	—
192	D₈	—	2	4	no	—
193	D₈	C₂×C₈×C₆₇	1	4	yes	complemented
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	8	2	no	—
197	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂×C₈×C₆₇	1	2	yes	—
200	C₂×C₈	M₁₆×C₆₇	1	2	yes	—
201	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
202	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
208	C₄²	D₈×C₆₇	1	1	yes	characteristic
209	C₄²	SD₁₆×C₆₇	1	1	yes	characteristic
210	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
211	C₁₆	—	2	4	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	8	2	no	—
215	C₂³	—	8	2	no	—
216	C₂³	#102×C₆₇	1	1	yes	characteristic
217	D₄	—	2	4	no	—
218	D₄	—	4	4	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₂²⋊C₈×C₆₇	1	2	yes	—
227	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
228	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
229	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
230	C₂×C₄	#104×C₆₇	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
233	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	8	2	no	—
237	C₈	C₂²⋊C₈×C₆₇	1	2	yes	—
238	C₈	C₂²×C₈×C₆₇	1	1	yes	characteristic
239	C₈	C₂×M₁₆×C₆₇	1	1	yes	characteristic
240	C₂²	—	4	4	no	—
241	C₂²	—	8	8	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	8	2	no	—
244	C₂²	D₄⋊C₈×C₆₇	1	2	yes	—
245	C₂²	#360×C₆₇	1	2	yes	—
246	C₂²	#361×C₆₇	1	2	yes	—
247	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	8	2	no	—
253	C₄	D₄⋊C₈×C₆₇	1	2	yes	—
254	C₄	C₂×C₂²⋊C₈×C₆₇	1	1	yes	characteristic
255	C₄	C₂×#104×C₆₇	1	1	yes	characteristic
256	C₂	—	8	4	no	—
257	C₂	C₆₇×#1333	1	4	yes	direct product factor
258	C₂	C₂×D₄⋊C₈×C₆₇	1	1	yes	characteristic
259	C₂	C₂×#360×C₆₇	1	1	yes	characteristic
260	C₂	C₂×#361×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#102
commutator	C₈	C₂²×C₈×C₆₇
Frattini, cosocle	C₄×C₈	C₂³×C₆₇
prime radical	C₂²×D₈×C₆₇	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#346×C₆₇	C₂	semidirect
3. C₂×#334×C₆₇	C₂	semidirect
4. C₆₇×#1333	C₂	direct (N × H)
17. #346×C₆₇	C₂²	semidirect
19. #334×C₆₇	C₂²	semidirect
24. C₂×D₈×C₆₇	C₈	semidirect
62. D₈×C₆₇	C₂×C₈	semidirect
125. C₂×#1333	C₆₇	direct (N × H)
127. C₂×C₆₇	#1333	direct (N × H)
132. C₂×#346	C₂×C₆₇	semidirect
133. C₂×#334	C₂×C₆₇	semidirect
134. #1333	C₂×C₆₇	direct (N × H)
135. C₆₇	C₂×#1333	direct (N × H)
148. #346	C₂²×C₆₇	semidirect
150. #334	C₂²×C₆₇	semidirect
155. C₂×D₈	C₈×C₆₇	semidirect
193. D₈	C₂×C₈×C₆₇	semidirect
257. C₂	C₆₇×#1333	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₆₇	2
C₂×#346×C₆₇	2
C₂×#334×C₆₇	2
C₆₇×#1333	2
C₂×#1333	67

Subgroups — C2×C67×#1333 #30690846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1333 #30690846