Subgroups — C₆₇×#33901 #30713152

Order 17152.

548 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3334	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2232	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2224	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2215	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#366×C₆₇	C₂	1	2	yes	maximal, complemented
9	#547×C₆₇	—	2	2	no	—
10	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
12	#500×C₆₇	C₂²	1	2	yes	—
13	#480×C₆₇	—	2	4	no	—
14	#474×C₆₇	—	2	2	no	—
15	#466×C₆₇	C₂²	1	4	yes	—
16	C₄×D₈×C₆₇	C₂²	1	2	yes	—
17	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#107×C₆₇	C₂²	1	4	yes	—
19	#366×C₆₇	C₂²	1	2	yes	—
20	#366×C₆₇	C₂²	1	2	yes	—
21	#366×C₆₇	C₂²	1	1	yes	characteristic
22	#366×C₆₇	C₂²	1	4	yes	—
23	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	#357×C₆₇	C₂²	1	2	yes	—
25	#357×C₆₇	C₂²	1	4	yes	—
26	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
27	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#41×C₆₇	—	4	2	no	—
29	C₂×D₈×C₆₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	#123×C₆₇	—	4	4	no	—
33	C₂²⋊Q₈×C₆₇	—	4	2	no	—
34	#121×C₆₇	—	2	4	no	—
35	C₄×D₄×C₆₇	—	2	2	no	—
36	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
38	D₁₆×C₆₇	—	2	2	no	—
39	C₂×C₁₆×C₆₇	—	2	4	no	—
40	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
41	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
43	#107×C₆₇	—	2	4	no	—
44	#107×C₆₇	—	2	2	no	—
45	#107×C₆₇	C₂³	1	4	yes	—
46	#107×C₆₇	C₂³	1	2	yes	—
47	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
48	#102×C₆₇	—	2	4	no	—
49	#102×C₆₇	—	2	2	no	—
50	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
51	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	#41×C₆₇	—	8	8	no	—
53	C₂×Q₈×C₆₇	—	4	2	no	—
54	C₂×D₄×C₆₇	—	2	2	no	—
55	C₂×D₄×C₆₇	—	4	4	no	—
56	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	—	4	4	no	—
58	C₂²×C₄×C₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
60	D₈×C₆₇	—	2	2	no	—
61	D₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄⋊C₄×C₆₇	—	4	4	no	—
70	C₄⋊C₄×C₆₇	—	4	2	no	—
71	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
72	C₂²⋊C₄×C₆₇	—	4	2	no	—
73	C₂²⋊C₄×C₆₇	—	4	4	no	—
74	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
75	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₆₇	—	2	2	no	—
77	C₁₆×C₆₇	—	2	1	no	—
78	C₁₆×C₆₇	#41	1	2	yes	complemented
79	C₂³×C₆₇	—	4	2	no	—
80	C₂³×C₆₇	C₂×D₈	1	2	yes	—
81	Q₈×C₆₇	—	8	4	no	—
82	D₄×C₆₇	—	4	2	no	—
83	D₄×C₆₇	—	4	2	no	—
84	D₄×C₆₇	—	8	8	no	—
85	D₄×C₆₇	D₁₆	1	4	yes	complemented
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	8	8	no	—
91	C₂×C₄×C₆₇	—	8	4	no	—
92	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₆₇	—	2	2	no	—
97	C₈×C₆₇	—	2	1	no	—
98	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
99	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	—	8	2	no	—
103	C₂²×C₆₇	—	8	2	no	—
104	C₂²×C₆₇	—	8	2	no	—
105	C₂²×C₆₇	C₂×D₁₆	1	4	yes	—
106	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	8	2	no	—
110	C₄×C₆₇	—	8	4	no	—
111	C₄×C₆₇	C₂×D₁₆	1	2	yes	—
112	C₄×C₆₇	#547	1	1	yes	characteristic
113	C₄×C₆₇	#578	1	1	yes	characteristic
114	#33901	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₆₇	—	2	4	no	—
116	C₂×C₆₇	—	16	2	no	—
117	C₂×C₆₇	#3334	1	1	yes	characteristic
118	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
119	C₂×C₆₇	#3417	1	1	yes	characteristic
120	#3334	C₂×C₆₇	1	2	yes	complemented
121	#2252	C₂×C₆₇	1	1	yes	characteristic, complemented
122	#2232	C₂×C₆₇	1	4	yes	complemented
123	#2224	C₂×C₆₇	1	2	yes	complemented
124	#2215	C₂×C₆₇	1	2	yes	complemented
125	C₄×D₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
126	D₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
127	C₂×#366	C₂×C₆₇	1	2	yes	complemented
128	C₆₇	#33901	1	1	yes	characteristic, direct product factor, Sylow
129	#547	—	2	2	no	—
130	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
131	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
132	#500	C₂²×C₆₇	1	2	yes	—
133	#480	—	2	4	no	—
134	#474	—	2	2	no	—
135	#466	C₂²×C₆₇	1	4	yes	—
136	C₄×D₈	C₂²×C₆₇	1	2	yes	—
137	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
138	C₂×#107	C₂²×C₆₇	1	4	yes	—
139	#366	C₂²×C₆₇	1	1	yes	characteristic
140	#366	C₂²×C₆₇	1	2	yes	—
141	#366	C₂²×C₆₇	1	4	yes	—
142	#366	C₂²×C₆₇	1	2	yes	—
143	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
144	#357	C₂²×C₆₇	1	2	yes	—
145	#357	C₂²×C₆₇	1	4	yes	—
146	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
147	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
148	C₂×#41	—	4	2	no	—
149	C₂×D₈	C₂³×C₆₇	1	2	yes	—
150	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
151	C₄⋊Q₈	—	2	2	no	—
152	#123	—	4	4	no	—
153	C₂²⋊Q₈	—	4	2	no	—
154	#121	—	2	4	no	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	2	4	no	—
158	D₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	4	no	—
160	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
161	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
162	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
163	#107	—	2	4	no	—
164	#107	—	2	2	no	—
165	#107	C₂³×C₆₇	1	2	yes	—
166	#107	C₂³×C₆₇	1	4	yes	—
167	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
168	#102	—	2	2	no	—
169	#102	—	2	4	no	—
170	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
171	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
172	#41	—	8	8	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
180	D₈	—	2	2	no	—
181	D₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
186	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	4	4	no	—
194	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
195	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
196	C₁₆	—	2	2	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#41×C₆₇	1	2	yes	complemented
199	C₂³	—	4	2	no	—
200	C₂³	C₂×D₈×C₆₇	1	2	yes	—
201	Q₈	—	8	4	no	—
202	D₄	—	4	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	8	8	no	—
205	D₄	D₁₆×C₆₇	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	8	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
213	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
214	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
219	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	8	2	no	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	2	no	—
225	C₂²	C₂×D₁₆×C₆₇	1	4	yes	—
226	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	8	2	no	—
230	C₄	—	8	4	no	—
231	C₄	C₂×D₁₆×C₆₇	1	2	yes	—
232	C₄	#547×C₆₇	1	1	yes	characteristic
233	C₄	#578×C₆₇	1	1	yes	characteristic
234	C₂	—	2	4	no	—
235	C₂	—	16	2	no	—
236	C₂	C₆₇×#3334	1	1	yes	characteristic
237	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic
238	C₂	C₆₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×D₄×C₆₇ ◁ C₈×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3334	C₂	semidirect
2. C₆₇×#2252	C₂	semidirect
3. C₆₇×#2232	C₂	semidirect
4. C₆₇×#2224	C₂	semidirect
5. C₆₇×#2215	C₂	semidirect
6. C₄×D₁₆×C₆₇	C₂	semidirect
7. D₄×C₁₆×C₆₇	C₂	semidirect
8. C₂×#366×C₆₇	C₂	semidirect
78. C₁₆×C₆₇	#41	semidirect
85. D₄×C₆₇	D₁₆	semidirect
114. #33901	C₆₇	direct (N × H)
120. #3334	C₂×C₆₇	semidirect
121. #2252	C₂×C₆₇	semidirect
122. #2232	C₂×C₆₇	semidirect
123. #2224	C₂×C₆₇	semidirect
124. #2215	C₂×C₆₇	semidirect
125. C₄×D₁₆	C₂×C₆₇	semidirect
126. D₄×C₁₆	C₂×C₆₇	semidirect
127. C₂×#366	C₂×C₆₇	semidirect
128. C₆₇	#33901	direct (N × H)
198. C₁₆	#41×C₆₇	semidirect
205. D₄	D₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3334	2
C₆₇×#2252	2
C₆₇×#2232	2
C₆₇×#2224	2
C₆₇×#2215	2
C₄×D₁₆×C₆₇	2
D₄×C₁₆×C₆₇	2
C₂×#366×C₆₇	2
#33901	67

Subgroups — C67×#33901 #30713152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33901 #30713152