Subgroups — C₆₇×#21940 #30701191

Order 17152.

1266 conjugacy classes of subgroups.

The subgroup lattice has 272 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	1	yes	maximal, characteristic
2	C₂×#380×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#1852	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1808	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
8	C₂²×C₂²⋊C₄×C₆₇	—	2	2	no	—
9	C₂×#107×C₆₇	C₂²	1	8	yes	complemented
10	C₂×C₂²⋊C₈×C₆₇	C₄	1	8	yes	complemented
11	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
12	#380×C₆₇	—	2	2	no	—
13	#380×C₆₇	—	2	2	no	—
14	#380×C₆₇	C₂²	1	4	yes	—
15	#380×C₆₇	C₂²	1	4	yes	—
16	C₂×#95×C₆₇	—	2	2	no	—
17	#340×C₆₇	C₂²	1	8	yes	complemented
18	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₆₇	—	2	4	no	—
20	C₂³×C₄×C₆₇	—	4	4	no	—
21	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₆₇	D₄	1	2	yes	—
23	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	—	4	4	no	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
29	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
30	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
32	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
33	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
36	#107×C₆₇	—	4	8	no	—
37	C₂²⋊C₈×C₆₇	C₂×C₄	1	16	yes	complemented
38	#95×C₆₇	—	2	4	no	—
39	#95×C₆₇	—	2	4	no	—
40	C₂⁴×C₆₇	—	4	2	no	—
41	C₂⁴×C₆₇	—	4	2	no	—
42	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
43	C₂⁴×C₆₇	D₈	1	4	yes	—
44	C₂⁴×C₆₇	Q₁₆	1	4	yes	—
45	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₆₇	—	2	8	no	—
48	C₂²×C₄×C₆₇	—	2	4	no	—
49	C₂²×C₄×C₆₇	—	2	8	no	—
50	C₂²×C₄×C₆₇	—	4	8	no	—
51	C₂²×C₄×C₆₇	—	4	8	no	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	—	4	4	no	—
54	C₂²×C₄×C₆₇	—	4	8	no	—
55	C₂²×C₄×C₆₇	—	4	8	no	—
56	C₂²×C₄×C₆₇	—	4	8	no	—
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
58	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₆₇	—	2	8	no	—
67	C₂×C₈×C₆₇	—	4	8	no	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
69	C₄⋊C₄×C₆₇	—	2	16	no	—
70	C₄⋊C₄×C₆₇	—	4	8	no	—
71	C₂²⋊C₄×C₆₇	—	4	8	no	—
72	C₂²⋊C₄×C₆₇	—	4	8	no	—
73	C₂³×C₆₇	—	2	4	no	—
74	C₂³×C₆₇	—	2	2	no	—
75	C₂³×C₆₇	—	2	4	no	—
76	C₂³×C₆₇	—	2	2	no	—
77	C₂³×C₆₇	—	4	4	no	—
78	C₂³×C₆₇	—	4	4	no	—
79	C₂³×C₆₇	—	4	8	no	—
80	C₂³×C₆₇	—	4	8	no	—
81	C₂³×C₆₇	#107	1	16	yes	complemented
82	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
84	C₂³×C₆₇	C₂×D₈	1	2	yes	—
85	C₂³×C₆₇	C₂×D₈	1	4	yes	—
86	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
87	C₂³×C₆₇	C₂×Q₁₆	1	4	yes	—
88	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	—	2	8	no	—
93	C₂×C₄×C₆₇	—	2	8	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	—	4	8	no	—
97	C₂×C₄×C₆₇	—	4	8	no	—
98	C₂×C₄×C₆₇	—	4	8	no	—
99	C₂×C₄×C₆₇	—	4	8	no	—
100	C₂×C₄×C₆₇	—	8	16	no	—
101	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
106	C₈×C₆₇	—	2	8	no	—
107	C₂²×C₆₇	—	2	2	no	—
108	C₂²×C₆₇	—	2	4	no	—
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	2	8	no	—
112	C₂²×C₆₇	—	2	4	no	—
113	C₂²×C₆₇	—	2	8	no	—
114	C₂²×C₆₇	—	2	4	no	—
115	C₂²×C₆₇	—	4	16	no	—
116	C₂²×C₆₇	C₂×#107	1	4	yes	—
117	C₂²×C₆₇	C₂×#107	1	8	yes	—
118	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₆₇	#534	1	2	yes	—
120	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
121	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
122	C₂²×C₆₇	#575	1	2	yes	—
123	C₄×C₆₇	—	2	4	no	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	8	8	no	—
126	C₄×C₆₇	#518	1	2	yes	—
127	C₄×C₆₇	#543	1	2	yes	—
128	#21940	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₆₇	—	2	8	no	—
130	C₂×C₆₇	—	2	4	no	—
131	C₂×C₆₇	#2315	1	1	yes	characteristic
132	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
133	C₂×C₆₇	#2911	1	1	yes	characteristic
134	C₂×C₆₇	#3186	1	2	yes	—
135	C₂×C₆₇	#3189	1	2	yes	—
136	C₂²×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
137	C₂×#380	C₂×C₆₇	1	2	yes	—
138	#1852	C₂×C₆₇	1	8	yes	complemented
139	#1808	C₂×C₆₇	1	4	yes	complemented
140	C₆₇	#21940	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
142	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
143	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
144	C₂²×C₂²⋊C₄	—	2	2	no	—
145	C₂×#107	C₂²×C₆₇	1	8	yes	complemented
146	C₂×C₂²⋊C₈	C₄×C₆₇	1	8	yes	complemented
147	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
148	#380	—	2	2	no	—
149	#380	—	2	2	no	—
150	#380	C₂²×C₆₇	1	4	yes	—
151	#380	C₂²×C₆₇	1	4	yes	—
152	C₂×#95	—	2	2	no	—
153	#340	C₂²×C₆₇	1	8	yes	complemented
154	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
155	C₂³×C₄	—	2	4	no	—
156	C₂³×C₄	—	4	4	no	—
157	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
158	C₂³×C₄	D₄×C₆₇	1	2	yes	—
159	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
160	C₂³×C₄	Q₈×C₆₇	1	4	yes	—
161	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
162	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
163	C₂²×C₈	—	4	4	no	—
164	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
165	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	—	4	8	no	—
168	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
169	C₂×C₂²⋊C₄	—	2	4	no	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₂²⋊C₄	—	4	8	no	—
172	#107	—	4	8	no	—
173	C₂²⋊C₈	C₂×C₄×C₆₇	1	16	yes	complemented
174	#95	—	2	4	no	—
175	#95	—	2	4	no	—
176	C₂⁴	—	4	2	no	—
177	C₂⁴	—	4	2	no	—
178	C₂⁴	C₄⋊C₄×C₆₇	1	4	yes	—
179	C₂⁴	D₈×C₆₇	1	4	yes	—
180	C₂⁴	Q₁₆×C₆₇	1	4	yes	—
181	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
182	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	4	8	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	4	8	no	—
191	C₂²×C₄	—	4	8	no	—
192	C₂²×C₄	—	4	8	no	—
193	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
194	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
199	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
200	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
201	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	4	8	no	—
204	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
205	C₄⋊C₄	—	2	16	no	—
206	C₄⋊C₄	—	4	8	no	—
207	C₂²⋊C₄	—	4	8	no	—
208	C₂²⋊C₄	—	4	8	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	4	8	no	—
216	C₂³	—	4	8	no	—
217	C₂³	#107×C₆₇	1	16	yes	complemented
218	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
219	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
220	C₂³	C₂×D₈×C₆₇	1	2	yes	—
221	C₂³	C₂×D₈×C₆₇	1	4	yes	—
222	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
223	C₂³	C₂×Q₁₆×C₆₇	1	4	yes	—
224	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	8	16	no	—
237	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
239	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
240	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
241	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
242	C₈	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	8	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	4	16	no	—
252	C₂²	C₂×#107×C₆₇	1	4	yes	—
253	C₂²	C₂×#107×C₆₇	1	8	yes	—
254	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
255	C₂²	#534×C₆₇	1	2	yes	—
256	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
257	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
258	C₂²	#575×C₆₇	1	2	yes	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	8	8	no	—
262	C₄	#518×C₆₇	1	2	yes	—
263	C₄	#543×C₆₇	1	2	yes	—
264	C₂	—	2	8	no	—
265	C₂	—	2	4	no	—
266	C₂	C₆₇×#2315	1	1	yes	characteristic
267	C₂	C₂²×#107×C₆₇	1	1	yes	characteristic
268	C₂	C₆₇×#2911	1	1	yes	characteristic
269	C₂	C₆₇×#3186	1	2	yes	—
270	C₂	C₆₇×#3189	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁵×C₆₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1852	C₂	semidirect
4. C₆₇×#1808	C₂	semidirect
9. C₂×#107×C₆₇	C₂²	semidirect
10. C₂×C₂²⋊C₈×C₆₇	C₄	semidirect
17. #340×C₆₇	C₂²	semidirect
28. C₂²×C₈×C₆₇	C₂×C₄	semidirect
37. C₂²⋊C₈×C₆₇	C₂×C₄	semidirect
81. C₂³×C₆₇	#107	semidirect
128. #21940	C₆₇	direct (N × H)
138. #1852	C₂×C₆₇	semidirect
139. #1808	C₂×C₆₇	semidirect
140. C₆₇	#21940	direct (N × H)
145. C₂×#107	C₂²×C₆₇	semidirect
146. C₂×C₂²⋊C₈	C₄×C₆₇	semidirect
153. #340	C₂²×C₆₇	semidirect
164. C₂²×C₈	C₂×C₄×C₆₇	semidirect
173. C₂²⋊C₈	C₂×C₄×C₆₇	semidirect
217. C₂³	#107×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
C₂×#380×C₆₇	2
C₆₇×#1852	2
C₆₇×#1808	2
#21940	67

Subgroups — C67×#21940 #30701191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21940 #30701191