Subgroups — C₆₇×#21931 #30701182

Order 17152.

1154 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2280	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1850	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1798	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
7	C₂×#117×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	complemented
9	C₂×C₂²⋊C₈×C₆₇	—	2	8	no	—
10	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
12	#381×C₆₇	C₄	1	2	yes	—
13	#380×C₆₇	C₄	1	4	yes	—
14	#379×C₆₇	C₄	1	2	yes	—
15	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
16	#336×C₆₇	C₂²	1	8	yes	complemented
17	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₆₇	—	4	4	no	—
19	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₇	D₄	1	4	yes	—
22	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
23	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
25	C₂²×C₈×C₆₇	—	2	8	no	—
26	C₂²×C₈×C₆₇	—	4	8	no	—
27	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
28	#117×C₆₇	—	4	4	no	—
29	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
31	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
33	C₄⋊C₈×C₆₇	—	4	8	no	—
34	C₂²⋊C₈×C₆₇	—	2	8	no	—
35	C₂²⋊C₈×C₆₇	—	2	8	no	—
36	#95×C₆₇	C₂×C₄	1	8	yes	—
37	C₂⁴×C₆₇	—	2	8	no	—
38	C₂⁴×C₆₇	—	4	2	no	—
39	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
40	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	—	2	4	no	—
43	C₂²×C₄×C₆₇	—	2	8	no	—
44	C₂²×C₄×C₆₇	—	4	8	no	—
45	C₂²×C₄×C₆₇	—	4	8	no	—
46	C₂²×C₄×C₆₇	—	4	4	no	—
47	C₂²×C₄×C₆₇	—	4	2	no	—
48	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₆₇	—	2	8	no	—
59	C₂×C₈×C₆₇	—	2	16	no	—
60	C₂×C₈×C₆₇	—	4	16	no	—
61	C₄⋊C₄×C₆₇	—	4	4	no	—
62	C₂²⋊C₄×C₆₇	—	2	16	no	—
63	C₂²⋊C₄×C₆₇	—	4	4	no	—
64	C₄²×C₆₇	—	2	8	no	—
65	C₂³×C₆₇	—	2	4	no	—
66	C₂³×C₆₇	—	2	4	no	—
67	C₂³×C₆₇	—	2	2	no	—
68	C₂³×C₆₇	—	2	4	no	—
69	C₂³×C₆₇	—	2	16	no	—
70	C₂³×C₆₇	—	4	4	no	—
71	C₂³×C₆₇	—	4	16	no	—
72	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
73	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂³×C₆₇	#131	1	8	yes	—
75	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	2	8	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	8	no	—
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	8	no	—
83	C₂×C₄×C₆₇	—	2	16	no	—
84	C₂×C₄×C₆₇	—	4	8	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
91	C₈×C₆₇	—	4	8	no	—
92	C₂²×C₆₇	—	2	4	no	—
93	C₂²×C₆₇	—	2	4	no	—
94	C₂²×C₆₇	—	2	16	no	—
95	C₂²×C₆₇	—	2	2	no	—
96	C₂²×C₆₇	—	2	4	no	—
97	C₂²×C₆₇	—	4	16	no	—
98	C₂²×C₆₇	—	4	4	no	—
99	C₂²×C₆₇	#424	1	8	yes	—
100	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₆₇	#518	1	2	yes	—
102	C₂²×C₆₇	C₂×#131	1	4	yes	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	2	4	no	—
105	C₄×C₆₇	—	4	8	no	—
106	C₄×C₆₇	#534	1	2	yes	—
107	C₄×C₆₇	#543	1	2	yes	—
108	#21931	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₆₇	—	2	8	no	—
110	C₂×C₆₇	—	4	2	no	—
111	C₂×C₆₇	#2315	1	1	yes	characteristic
112	C₂×C₆₇	C₂×#424	1	2	yes	—
113	C₂×C₆₇	#2977	1	4	yes	—
114	C₂²×C₂²⋊C₈	C₂×C₆₇	1	2	yes	—
115	#2280	C₂×C₆₇	1	1	yes	characteristic
116	#1850	C₂×C₆₇	1	8	yes	complemented
117	#1798	C₂×C₆₇	1	4	yes	complemented
118	C₆₇	#21931	1	1	yes	characteristic, direct product factor, Sylow
119	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
121	C₂×#117	C₂²×C₆₇	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	complemented
123	C₂×C₂²⋊C₈	—	2	8	no	—
124	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
125	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
126	#381	C₄×C₆₇	1	2	yes	—
127	#380	C₄×C₆₇	1	4	yes	—
128	#379	C₄×C₆₇	1	2	yes	—
129	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
130	#336	C₂²×C₆₇	1	8	yes	complemented
131	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
132	C₂³×C₄	—	4	4	no	—
133	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
134	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
135	C₂³×C₄	D₄×C₆₇	1	4	yes	—
136	C₂³×C₄	Q₈×C₆₇	1	4	yes	—
137	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
138	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
139	C₂²×C₈	—	2	8	no	—
140	C₂²×C₈	—	4	8	no	—
141	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
142	#117	—	4	4	no	—
143	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
144	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
145	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
146	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
147	C₄⋊C₈	—	4	8	no	—
148	C₂²⋊C₈	—	2	8	no	—
149	C₂²⋊C₈	—	2	8	no	—
150	#95	C₂×C₄×C₆₇	1	8	yes	—
151	C₂⁴	—	2	8	no	—
152	C₂⁴	—	4	2	no	—
153	C₂⁴	C₄⋊C₄×C₆₇	1	4	yes	—
154	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
155	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	8	no	—
158	C₂²×C₄	—	4	8	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	—	4	4	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
163	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
164	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
165	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
168	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
169	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
170	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
171	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
172	C₂×C₈	—	2	8	no	—
173	C₂×C₈	—	2	16	no	—
174	C₂×C₈	—	4	16	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₂²⋊C₄	—	2	16	no	—
177	C₂²⋊C₄	—	4	4	no	—
178	C₄²	—	2	8	no	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	16	no	—
184	C₂³	—	4	4	no	—
185	C₂³	—	4	16	no	—
186	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
187	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
188	C₂³	#131×C₆₇	1	8	yes	—
189	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	16	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
201	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
202	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
203	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
204	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
205	C₈	—	4	8	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	16	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	16	no	—
212	C₂²	—	4	4	no	—
213	C₂²	#424×C₆₇	1	8	yes	—
214	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
215	C₂²	#518×C₆₇	1	2	yes	—
216	C₂²	C₂×#131×C₆₇	1	4	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	—	4	8	no	—
220	C₄	#534×C₆₇	1	2	yes	—
221	C₄	#543×C₆₇	1	2	yes	—
222	C₂	—	2	8	no	—
223	C₂	—	4	2	no	—
224	C₂	C₆₇×#2315	1	1	yes	characteristic
225	C₂	C₂×#424×C₆₇	1	2	yes	—
226	C₂	C₆₇×#2977	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1850	C₂	semidirect
4. C₆₇×#1798	C₂	semidirect
8. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
16. #336×C₆₇	C₂²	semidirect
108. #21931	C₆₇	direct (N × H)
116. #1850	C₂×C₆₇	semidirect
117. #1798	C₂×C₆₇	semidirect
118. C₆₇	#21931	direct (N × H)
122. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
130. #336	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊C₈×C₆₇	2
C₆₇×#2280	2
C₆₇×#1850	2
C₆₇×#1798	2
#21931	67

Subgroups — C67×#21931 #30701182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21931 #30701182