Subgroups — C₆₇×#33963 #30713214

Order 17152.

532 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	C₆₇×#3396	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2249	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2246	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2210	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2207	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
9	#559×C₆₇	—	2	2	no	—
10	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×SD₃₂×C₆₇	C₂²	1	4	yes	—
12	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
14	#495×C₆₇	C₂²	1	4	yes	—
15	#493×C₆₇	C₂²	1	2	yes	—
16	#463×C₆₇	—	2	4	no	—
17	#460×C₆₇	—	2	2	no	—
18	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
20	C₄×D₈×C₆₇	C₂²	1	4	yes	—
21	#366×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
24	#358×C₆₇	C₂²	1	2	yes	—
25	#357×C₆₇	C₂²	1	4	yes	—
26	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
28	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
29	C₂×SD₁₆×C₆₇	—	2	4	no	—
30	C₂×D₈×C₆₇	—	2	2	no	—
31	C₂×D₈×C₆₇	C₂³	1	4	yes	—
32	#127×C₆₇	—	2	2	no	—
33	#124×C₆₇	—	2	4	no	—
34	#124×C₆₇	—	4	2	no	—
35	#123×C₆₇	—	4	4	no	—
36	C₄×Q₈×C₆₇	—	2	2	no	—
37	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
38	C₄×D₄×C₆₇	—	2	4	no	—
39	C₄×D₄×C₆₇	—	4	2	no	—
40	Q₃₂×C₆₇	—	2	2	no	—
41	SD₃₂×C₆₇	—	4	4	no	—
42	D₁₆×C₆₇	—	2	4	no	—
43	D₁₆×C₆₇	—	4	2	no	—
44	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
46	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	#107×C₆₇	C₂³	1	2	yes	—
48	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
50	#103×C₆₇	—	2	2	no	—
51	#102×C₆₇	—	2	4	no	—
52	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂×Q₈×C₆₇	—	2	2	no	—
55	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
56	C₂×D₄×C₆₇	—	2	4	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂×D₄×C₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	—	4	4	no	—
60	Q₁₆×C₆₇	—	2	2	no	—
61	Q₁₆×C₆₇	—	2	2	no	—
62	SD₁₆×C₆₇	—	8	4	no	—
63	D₈×C₆₇	—	2	4	no	—
64	D₈×C₆₇	—	2	4	no	—
65	D₈×C₆₇	—	8	2	no	—
66	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₄⋊C₄×C₆₇	—	4	2	no	—
71	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
72	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
73	C₂²⋊C₄×C₆₇	—	4	4	no	—
74	C₂²⋊C₄×C₆₇	—	4	4	no	—
75	C₂²⋊C₄×C₆₇	—	4	4	no	—
76	C₄²×C₆₇	—	4	2	no	—
77	C₄²×C₆₇	C₂×D₄	1	2	yes	—
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₁₆×C₆₇	—	2	1	no	—
80	C₁₆×C₆₇	—	2	2	no	—
81	C₁₆×C₆₇	#41	1	2	yes	—
82	C₂³×C₆₇	—	4	4	no	—
83	Q₈×C₆₇	—	2	2	no	—
84	Q₈×C₆₇	—	4	2	no	—
85	Q₈×C₆₇	—	4	2	no	—
86	D₄×C₆₇	—	4	4	no	—
87	D₄×C₆₇	—	4	4	no	—
88	D₄×C₆₇	—	8	4	no	—
89	D₄×C₆₇	—	16	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	16	4	no	—
93	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	2	2	no	—
100	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
101	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₆₇	—	8	4	no	—
103	C₂²×C₆₇	—	8	4	no	—
104	C₂²×C₆₇	—	8	4	no	—
105	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₆₇	—	2	2	no	—
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	8	2	no	—
110	C₄×C₆₇	—	8	2	no	—
111	C₄×C₆₇	#508	1	2	yes	—
112	C₄×C₆₇	#550	1	1	yes	characteristic
113	C₄×C₆₇	#576	1	1	yes	characteristic
114	#33963	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₆₇	—	16	4	no	—
116	C₂×C₆₇	#3391	1	1	yes	characteristic
117	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
118	C₂×C₆₇	#3415	1	1	yes	characteristic
119	#3396	C₂×C₆₇	1	2	yes	complemented
120	#2249	C₂×C₆₇	1	2	yes	complemented
121	#2246	C₂×C₆₇	1	1	yes	characteristic
122	#2210	C₂×C₆₇	1	4	yes	complemented
123	#2207	C₂×C₆₇	1	2	yes	complemented
124	Q₈×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
125	C₄×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
126	C₄×D₁₆	C₂×C₆₇	1	2	yes	complemented
127	C₆₇	#33963	1	1	yes	characteristic, direct product factor, Sylow
128	#559	—	2	2	no	—
129	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
130	C₂×SD₃₂	C₂²×C₆₇	1	4	yes	—
131	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
132	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
133	#495	C₂²×C₆₇	1	4	yes	—
134	#493	C₂²×C₆₇	1	2	yes	—
135	#463	—	2	4	no	—
136	#460	—	2	2	no	—
137	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
138	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
139	C₄×D₈	C₂²×C₆₇	1	4	yes	—
140	#366	C₂²×C₆₇	1	1	yes	characteristic
141	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
142	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
143	#358	C₂²×C₆₇	1	2	yes	—
144	#357	C₂²×C₆₇	1	4	yes	—
145	C₄×C₁₆	C₂²×C₆₇	1	2	yes	—
146	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
147	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
148	C₂×SD₁₆	—	2	4	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	C₂³×C₆₇	1	4	yes	—
151	#127	—	2	2	no	—
152	#124	—	2	4	no	—
153	#124	—	4	2	no	—
154	#123	—	4	4	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
157	C₄×D₄	—	2	4	no	—
158	C₄×D₄	—	4	2	no	—
159	Q₃₂	—	2	2	no	—
160	SD₃₂	—	4	4	no	—
161	D₁₆	—	2	4	no	—
162	D₁₆	—	4	2	no	—
163	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
164	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
165	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
166	#107	C₂³×C₆₇	1	2	yes	—
167	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
168	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
169	#103	—	2	2	no	—
170	#102	—	2	4	no	—
171	C₄×C₈	C₂³×C₆₇	1	2	yes	—
172	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂²×C₄	—	4	4	no	—
179	Q₁₆	—	2	2	no	—
180	Q₁₆	—	2	2	no	—
181	SD₁₆	—	8	4	no	—
182	D₈	—	2	4	no	—
183	D₈	—	2	4	no	—
184	D₈	—	8	2	no	—
185	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
186	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
187	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
191	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	—	4	4	no	—
194	C₂²⋊C₄	—	4	4	no	—
195	C₄²	—	4	2	no	—
196	C₄²	C₂×D₄×C₆₇	1	2	yes	—
197	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
198	C₁₆	—	2	1	no	—
199	C₁₆	—	2	2	no	—
200	C₁₆	#41×C₆₇	1	2	yes	—
201	C₂³	—	4	4	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	D₄	—	4	4	no	—
206	D₄	—	4	4	no	—
207	D₄	—	8	4	no	—
208	D₄	—	16	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	16	4	no	—
212	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
213	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
214	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
215	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
220	C₈	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
221	C₂²	—	8	4	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	8	4	no	—
224	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	1	no	—
228	C₄	—	8	2	no	—
229	C₄	—	8	2	no	—
230	C₄	#508×C₆₇	1	2	yes	—
231	C₄	#550×C₆₇	1	1	yes	characteristic
232	C₄	#576×C₆₇	1	1	yes	characteristic
233	C₂	—	16	4	no	—
234	C₂	C₆₇×#3391	1	1	yes	characteristic
235	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
236	C₂	C₆₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#2246	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3396	C₂	semidirect
2. C₆₇×#2249	C₂	semidirect
4. C₆₇×#2210	C₂	semidirect
5. C₆₇×#2207	C₂	semidirect
6. Q₈×C₁₆×C₆₇	C₂	semidirect
7. C₄×Q₃₂×C₆₇	C₂	semidirect
8. C₄×D₁₆×C₆₇	C₂	semidirect
114. #33963	C₆₇	direct (N × H)
119. #3396	C₂×C₆₇	semidirect
120. #2249	C₂×C₆₇	semidirect
122. #2210	C₂×C₆₇	semidirect
123. #2207	C₂×C₆₇	semidirect
124. Q₈×C₁₆	C₂×C₆₇	semidirect
125. C₄×Q₃₂	C₂×C₆₇	semidirect
126. C₄×D₁₆	C₂×C₆₇	semidirect
127. C₆₇	#33963	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3396	2
C₆₇×#2249	2
C₆₇×#2246	2
C₆₇×#2210	2
C₆₇×#2207	2
Q₈×C₁₆×C₆₇	2
C₄×Q₃₂×C₆₇	2
C₄×D₁₆×C₆₇	2
#33963	67

Subgroups — C67×#33963 #30713214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33963 #30713214