Subgroups — C₆₇×#33249 #30712500

Order 17152.

838 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#491×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#2642	C₂	1	2	yes	maximal
5	C₆₇×#1971	C₂	1	4	yes	maximal
6	C₆₇×#1969	C₂	1	4	yes	maximal
7	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
11	C₂×#125×C₆₇	C₂²	1	2	yes	—
12	#499×C₆₇	—	2	4	no	—
13	#494×C₆₇	—	2	4	no	—
14	#491×C₆₇	—	2	8	no	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#106×C₆₇	C₂²	1	2	yes	—
17	C₂×#103×C₆₇	C₂²	1	8	yes	—
18	C₂×#97×C₆₇	C₂²	1	2	yes	—
19	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#400×C₆₇	—	2	8	no	—
23	#391×C₆₇	C₂²	1	8	yes	—
24	#376×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
26	C₂×Q₁₆×C₆₇	—	2	4	no	—
27	C₂×Q₁₆×C₆₇	—	2	4	no	—
28	C₂×Q₁₆×C₆₇	—	2	8	no	—
29	C₂×Q₁₆×C₆₇	—	2	8	no	—
30	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₆₇	—	2	4	no	—
34	C₄⋊Q₈×C₆₇	—	2	4	no	—
35	#125×C₆₇	—	2	4	no	—
36	#125×C₆₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#107×C₆₇	—	2	4	no	—
43	#106×C₆₇	—	2	4	no	—
44	#103×C₆₇	—	2	16	no	—
45	#97×C₆₇	—	2	4	no	—
46	#97×C₆₇	D₄	1	4	yes	—
47	C₄×C₈×C₆₇	—	2	4	no	—
48	C₄×C₈×C₆₇	D₄	1	4	yes	—
49	#95×C₆₇	—	2	16	no	—
50	#95×C₆₇	D₄	1	4	yes	—
51	C₂×Q₈×C₆₇	—	2	4	no	—
52	C₂×Q₈×C₆₇	—	2	4	no	—
53	C₂×Q₈×C₆₇	—	2	8	no	—
54	C₂×Q₈×C₆₇	—	2	8	no	—
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	2	4	no	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
60	Q₁₆×C₆₇	—	4	8	no	—
61	Q₁₆×C₆₇	—	4	8	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₆₇	#41	1	4	yes	—
68	C₂×C₈×C₆₇	#41	1	4	yes	—
69	C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₄⋊C₄×C₆₇	—	4	8	no	—
72	C₄⋊C₄×C₆₇	—	4	8	no	—
73	C₄²×C₆₇	—	2	2	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	C₂×D₄	1	2	yes	—
76	C₄²×C₆₇	C₂×D₄	1	2	yes	—
77	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₆₇	—	4	4	no	—
79	Q₈×C₆₇	—	4	4	no	—
80	Q₈×C₆₇	—	4	8	no	—
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	2	8	no	—
84	C₂×C₄×C₆₇	—	4	4	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	4	8	no	—
87	C₂×C₄×C₆₇	—	4	8	no	—
88	C₂×C₄×C₆₇	#127	1	4	yes	—
89	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
93	C₈×C₆₇	—	2	4	no	—
94	C₈×C₆₇	#124	1	4	yes	—
95	C₈×C₆₇	#124	1	4	yes	—
96	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
97	C₂²×C₆₇	#532	1	2	yes	—
98	C₂²×C₆₇	#578	1	4	yes	—
99	C₄×C₆₇	—	2	4	no	—
100	C₄×C₆₇	—	4	4	no	—
101	C₄×C₆₇	—	8	4	no	—
102	C₄×C₆₇	—	8	4	no	—
103	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
104	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
105	C₄×C₆₇	#532	1	2	yes	—
106	#33249	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₆₇	#2858	1	1	yes	characteristic
108	C₂×C₆₇	#3156	1	2	yes	—
109	C₂×C₆₇	#3177	1	4	yes	—
110	C₂×#499	C₂×C₆₇	1	1	yes	characteristic
111	C₂×#494	C₂×C₆₇	1	1	yes	characteristic
112	C₂×#491	C₂×C₆₇	1	2	yes	—
113	#2642	C₂×C₆₇	1	2	yes	—
114	#1971	C₂×C₆₇	1	4	yes	—
115	#1969	C₂×C₆₇	1	4	yes	—
116	#1726	C₂×C₆₇	1	1	yes	characteristic
117	C₆₇	#33249	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
119	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
120	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
121	C₂×#125	C₂²×C₆₇	1	2	yes	—
122	#499	—	2	4	no	—
123	#494	—	2	4	no	—
124	#491	—	2	8	no	—
125	C₂×#107	C₂²×C₆₇	1	2	yes	—
126	C₂×#106	C₂²×C₆₇	1	2	yes	—
127	C₂×#103	C₂²×C₆₇	1	8	yes	—
128	C₂×#97	C₂²×C₆₇	1	2	yes	—
129	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
132	#400	—	2	8	no	—
133	#391	C₂²×C₆₇	1	8	yes	—
134	#376	C₂²×C₆₇	1	1	yes	characteristic
135	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
136	C₂×Q₁₆	—	2	4	no	—
137	C₂×Q₁₆	—	2	4	no	—
138	C₂×Q₁₆	—	2	8	no	—
139	C₂×Q₁₆	—	2	8	no	—
140	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
141	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	—	2	4	no	—
145	#125	—	2	4	no	—
146	#125	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
150	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
151	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
152	#107	—	2	4	no	—
153	#106	—	2	4	no	—
154	#103	—	2	16	no	—
155	#97	—	2	4	no	—
156	#97	D₄×C₆₇	1	4	yes	—
157	C₄×C₈	—	2	4	no	—
158	C₄×C₈	D₄×C₆₇	1	4	yes	—
159	#95	—	2	16	no	—
160	#95	D₄×C₆₇	1	4	yes	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	—	2	8	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
168	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
169	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
170	Q₁₆	—	4	8	no	—
171	Q₁₆	—	4	8	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
175	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
176	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
177	C₂×C₈	#41×C₆₇	1	4	yes	—
178	C₂×C₈	#41×C₆₇	1	4	yes	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	8	no	—
182	C₄⋊C₄	—	4	8	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂×D₄×C₆₇	1	2	yes	—
186	C₄²	C₂×D₄×C₆₇	1	2	yes	—
187	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
188	Q₈	—	4	4	no	—
189	Q₈	—	4	4	no	—
190	Q₈	—	4	8	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	8	no	—
197	C₂×C₄	—	4	8	no	—
198	C₂×C₄	#127×C₆₇	1	4	yes	—
199	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
200	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
201	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
202	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
203	C₈	—	2	4	no	—
204	C₈	#124×C₆₇	1	4	yes	—
205	C₈	#124×C₆₇	1	4	yes	—
206	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
207	C₂²	#532×C₆₇	1	2	yes	—
208	C₂²	#578×C₆₇	1	4	yes	—
209	C₄	—	2	4	no	—
210	C₄	—	4	4	no	—
211	C₄	—	8	4	no	—
212	C₄	—	8	4	no	—
213	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
214	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
215	C₄	#532×C₆₇	1	2	yes	—
216	C₂	C₆₇×#2858	1	1	yes	characteristic
217	C₂	C₆₇×#3156	1	2	yes	—
218	C₂	C₆₇×#3177	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
106. #33249	C₆₇	direct (N × H)
117. C₆₇	#33249	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×#499×C₆₇	2
C₂×#494×C₆₇	2
C₂×#491×C₆₇	2
C₆₇×#2642	2
C₆₇×#1971	2
C₆₇×#1969	2
C₆₇×#1726	2
#33249	67

Subgroups — C67×#33249 #30712500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33249 #30712500