Subgroups — C₆₇×#33168 #30712419

Order 17152.

1122 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#463×C₆₇	C₂	1	3	yes	maximal
2	C₂×#461×C₆₇	C₂	1	3	yes	maximal, complemented
3	C₆₇×#2658	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2275	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1983	C₂	1	3	yes	maximal, complemented
6	C₆₇×#1981	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₆₇	C₂²	1	3	yes	—
9	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
11	C₂×#124×C₆₇	C₂²	1	3	yes	—
12	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
13	#463×C₆₇	—	2	12	no	—
14	#461×C₆₇	—	2	12	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
17	C₂×#103×C₆₇	C₂²	1	6	yes	—
18	C₂×#103×C₆₇	C₂²	1	3	yes	—
19	C₂×#102×C₆₇	C₂²	1	3	yes	—
20	#399×C₆₇	—	2	1	no	—
21	#399×C₆₇	—	2	3	no	—
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
23	C₄×C₂²⋊C₄×C₆₇	—	2	3	no	—
24	#376×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#95×C₆₇	—	2	1	no	—
26	C₂²×Q₈×C₆₇	C₂³	1	3	yes	—
27	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	—	2	3	no	—
30	C₂×SD₁₆×C₆₇	—	2	12	no	—
31	C₂×SD₁₆×C₆₇	—	2	6	no	—
32	C₂×D₈×C₆₇	—	2	6	no	—
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
35	C₄⋊Q₈×C₆₇	—	2	6	no	—
36	C₄⋊Q₈×C₆₇	—	2	6	no	—
37	#124×C₆₇	—	2	6	no	—
38	#124×C₆₇	—	2	6	no	—
39	C₄×D₄×C₆₇	—	2	6	no	—
40	C₄×D₄×C₆₇	D₄	1	12	yes	—
41	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
44	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
45	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
46	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
47	C₂×C₄²×C₆₇	—	2	3	no	—
48	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
49	C₄⋊C₈×C₆₇	—	2	6	no	—
50	C₄⋊C₈×C₆₇	—	2	6	no	—
51	#103×C₆₇	—	2	12	no	—
52	#103×C₆₇	—	2	6	no	—
53	#102×C₆₇	—	2	6	no	—
54	#95×C₆₇	—	2	3	no	—
55	#95×C₆₇	D₄	1	1	yes	characteristic
56	#95×C₆₇	D₄	1	3	yes	—
57	C₂⁴×C₆₇	—	2	1	no	—
58	C₂⁴×C₆₇	—	2	1	no	—
59	C₂×Q₈×C₆₇	—	2	3	no	—
60	C₂×Q₈×C₆₇	—	2	6	no	—
61	C₂×Q₈×C₆₇	—	2	3	no	—
62	C₂×Q₈×C₆₇	—	2	6	no	—
63	C₂×D₄×C₆₇	—	2	3	no	—
64	C₂×D₄×C₆₇	—	2	3	no	—
65	C₂×D₄×C₆₇	—	2	3	no	—
66	C₂×D₄×C₆₇	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₆₇	—	2	3	no	—
68	C₂²×C₄×C₆₇	—	2	12	no	—
69	C₂²×C₄×C₆₇	—	2	3	no	—
70	C₂²×C₄×C₆₇	—	8	6	no	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
74	SD₁₆×C₆₇	—	4	12	no	—
75	D₈×C₆₇	—	4	4	no	—
76	C₂×C₈×C₆₇	—	2	3	no	—
77	C₂×C₈×C₆₇	—	2	3	no	—
78	C₂×C₈×C₆₇	—	2	6	no	—
79	C₄⋊C₄×C₆₇	—	2	3	no	—
80	C₄⋊C₄×C₆₇	—	4	6	no	—
81	C₄⋊C₄×C₆₇	—	4	6	no	—
82	C₄⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
83	C₂²⋊C₄×C₆₇	—	2	12	no	—
84	C₂²⋊C₄×C₆₇	—	4	6	no	—
85	C₂²⋊C₄×C₆₇	—	4	6	no	—
86	C₄²×C₆₇	—	8	3	no	—
87	C₄²×C₆₇	C₂×D₄	1	6	yes	—
88	C₄²×C₆₇	#41	1	6	yes	—
89	C₂³×C₆₇	—	2	6	no	—
90	C₂³×C₆₇	—	4	3	no	—
91	C₂³×C₆₇	—	4	1	no	—
92	C₂³×C₆₇	—	4	3	no	—
93	C₂³×C₆₇	—	4	3	no	—
94	C₂³×C₆₇	—	4	1	no	—
95	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₆₇	—	4	3	no	—
97	Q₈×C₆₇	—	4	3	no	—
98	Q₈×C₆₇	—	4	6	no	—
99	D₄×C₆₇	—	2	6	no	—
100	D₄×C₆₇	—	4	1	no	—
101	D₄×C₆₇	—	4	3	no	—
102	D₄×C₆₇	#127	1	4	yes	—
103	C₂×C₄×C₆₇	—	2	3	no	—
104	C₂×C₄×C₆₇	—	2	6	no	—
105	C₂×C₄×C₆₇	—	2	3	no	—
106	C₂×C₄×C₆₇	—	2	6	no	—
107	C₂×C₄×C₆₇	—	4	6	no	—
108	C₂×C₄×C₆₇	—	4	3	no	—
109	C₂×C₄×C₆₇	—	4	6	no	—
110	C₂×C₄×C₆₇	—	8	12	no	—
111	C₂×C₄×C₆₇	#121	1	6	yes	—
112	C₂×C₄×C₆₇	#121	1	6	yes	—
113	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
115	C₈×C₆₇	—	4	1	no	—
116	C₈×C₆₇	—	4	3	no	—
117	C₂²×C₆₇	—	2	4	no	—
118	C₂²×C₆₇	—	4	6	no	—
119	C₂²×C₆₇	—	4	6	no	—
120	C₂²×C₆₇	—	8	1	no	—
121	C₂²×C₆₇	—	8	3	no	—
122	C₂²×C₆₇	—	8	3	no	—
123	C₂²×C₆₇	C₂×#121	1	3	yes	—
124	C₂²×C₆₇	#577	1	3	yes	—
125	C₂²×C₆₇	#578	1	1	yes	characteristic
126	C₄×C₆₇	—	2	6	no	—
127	C₄×C₆₇	—	2	6	no	—
128	C₄×C₆₇	—	4	6	no	—
129	C₄×C₆₇	—	8	3	no	—
130	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
131	C₄×C₆₇	#532	1	3	yes	—
132	#33168	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₆₇	—	4	4	no	—
134	C₂×C₆₇	—	16	1	no	—
135	C₂×C₆₇	#2856	1	1	yes	characteristic
136	C₂×C₆₇	#3042	1	3	yes	—
137	C₂×C₆₇	#3046	1	3	yes	—
138	C₂×#463	C₂×C₆₇	1	3	yes	—
139	C₂×#461	C₂×C₆₇	1	3	yes	complemented
140	#2658	C₂×C₆₇	1	1	yes	characteristic, complemented
141	#2275	C₂×C₆₇	1	1	yes	characteristic, complemented
142	#1983	C₂×C₆₇	1	3	yes	complemented
143	#1981	C₂×C₆₇	1	3	yes	complemented
144	#1843	C₂×C₆₇	1	1	yes	characteristic, complemented
145	C₆₇	#33168	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×SD₁₆	C₂²×C₆₇	1	3	yes	—
147	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
149	C₂×#124	C₂²×C₆₇	1	3	yes	—
150	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
151	#463	—	2	12	no	—
152	#461	—	2	12	no	—
153	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
154	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
155	C₂×#103	C₂²×C₆₇	1	3	yes	—
156	C₂×#103	C₂²×C₆₇	1	6	yes	—
157	C₂×#102	C₂²×C₆₇	1	3	yes	—
158	#399	—	2	1	no	—
159	#399	—	2	3	no	—
160	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
161	C₄×C₂²⋊C₄	—	2	3	no	—
162	#376	C₂²×C₆₇	1	1	yes	characteristic
163	C₂×#95	—	2	1	no	—
164	C₂²×Q₈	C₂³×C₆₇	1	3	yes	—
165	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
166	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
167	C₂³×C₄	—	2	3	no	—
168	C₂×SD₁₆	—	2	12	no	—
169	C₂×SD₁₆	—	2	6	no	—
170	C₂×D₈	—	2	6	no	—
171	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
173	C₄⋊Q₈	—	2	6	no	—
174	C₄⋊Q₈	—	2	6	no	—
175	#124	—	2	6	no	—
176	#124	—	2	6	no	—
177	C₄×D₄	—	2	6	no	—
178	C₄×D₄	D₄×C₆₇	1	12	yes	—
179	C₂×C₄⋊C₄	—	2	3	no	—
180	C₂×C₄⋊C₄	—	2	3	no	—
181	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
182	C₂×C₂²⋊C₄	—	2	3	no	—
183	C₂×C₂²⋊C₄	—	2	3	no	—
184	C₂×C₂²⋊C₄	—	2	3	no	—
185	C₂×C₄²	—	2	3	no	—
186	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
187	C₄⋊C₈	—	2	6	no	—
188	C₄⋊C₈	—	2	6	no	—
189	#103	—	2	12	no	—
190	#103	—	2	6	no	—
191	#102	—	2	6	no	—
192	#95	—	2	3	no	—
193	#95	D₄×C₆₇	1	1	yes	characteristic
194	#95	D₄×C₆₇	1	3	yes	—
195	C₂⁴	—	2	1	no	—
196	C₂⁴	—	2	1	no	—
197	C₂×Q₈	—	2	3	no	—
198	C₂×Q₈	—	2	6	no	—
199	C₂×Q₈	—	2	3	no	—
200	C₂×Q₈	—	2	6	no	—
201	C₂×D₄	—	2	3	no	—
202	C₂×D₄	—	2	3	no	—
203	C₂×D₄	—	2	3	no	—
204	C₂×D₄	C₂×D₄×C₆₇	1	6	yes	—
205	C₂²×C₄	—	2	3	no	—
206	C₂²×C₄	—	2	12	no	—
207	C₂²×C₄	—	2	3	no	—
208	C₂²×C₄	—	8	6	no	—
209	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
210	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
211	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
212	SD₁₆	—	4	12	no	—
213	D₈	—	4	4	no	—
214	C₂×C₈	—	2	3	no	—
215	C₂×C₈	—	2	3	no	—
216	C₂×C₈	—	2	6	no	—
217	C₄⋊C₄	—	2	3	no	—
218	C₄⋊C₄	—	4	6	no	—
219	C₄⋊C₄	—	4	6	no	—
220	C₄⋊C₄	C₂×D₄×C₆₇	1	6	yes	—
221	C₂²⋊C₄	—	2	12	no	—
222	C₂²⋊C₄	—	4	6	no	—
223	C₂²⋊C₄	—	4	6	no	—
224	C₄²	—	8	3	no	—
225	C₄²	C₂×D₄×C₆₇	1	6	yes	—
226	C₄²	#41×C₆₇	1	6	yes	—
227	C₂³	—	2	6	no	—
228	C₂³	—	4	3	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	3	no	—
231	C₂³	—	4	3	no	—
232	C₂³	—	4	1	no	—
233	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
234	Q₈	—	4	3	no	—
235	Q₈	—	4	3	no	—
236	Q₈	—	4	6	no	—
237	D₄	—	2	6	no	—
238	D₄	—	4	1	no	—
239	D₄	—	4	3	no	—
240	D₄	#127×C₆₇	1	4	yes	—
241	C₂×C₄	—	2	3	no	—
242	C₂×C₄	—	2	6	no	—
243	C₂×C₄	—	2	3	no	—
244	C₂×C₄	—	2	6	no	—
245	C₂×C₄	—	4	6	no	—
246	C₂×C₄	—	4	3	no	—
247	C₂×C₄	—	4	6	no	—
248	C₂×C₄	—	8	12	no	—
249	C₂×C₄	#121×C₆₇	1	6	yes	—
250	C₂×C₄	#121×C₆₇	1	6	yes	—
251	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
252	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
253	C₈	—	4	1	no	—
254	C₈	—	4	3	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	6	no	—
257	C₂²	—	4	6	no	—
258	C₂²	—	8	3	no	—
259	C₂²	—	8	3	no	—
260	C₂²	—	8	1	no	—
261	C₂²	C₂×#121×C₆₇	1	3	yes	—
262	C₂²	#577×C₆₇	1	3	yes	—
263	C₂²	#578×C₆₇	1	1	yes	characteristic
264	C₄	—	2	6	no	—
265	C₄	—	2	6	no	—
266	C₄	—	4	6	no	—
267	C₄	—	8	3	no	—
268	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
269	C₄	#532×C₆₇	1	3	yes	—
270	C₂	—	4	4	no	—
271	C₂	—	16	1	no	—
272	C₂	C₆₇×#2856	1	1	yes	characteristic
273	C₂	C₆₇×#3042	1	3	yes	—
274	C₂	C₆₇×#3046	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
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
2. C₂×#461×C₆₇	C₂	semidirect
3. C₆₇×#2658	C₂	semidirect
4. C₆₇×#2275	C₂	semidirect
5. C₆₇×#1983	C₂	semidirect
6. C₆₇×#1981	C₂	semidirect
7. C₆₇×#1843	C₂	semidirect
132. #33168	C₆₇	direct (N × H)
139. C₂×#461	C₂×C₆₇	semidirect
140. #2658	C₂×C₆₇	semidirect
141. #2275	C₂×C₆₇	semidirect
142. #1983	C₂×C₆₇	semidirect
143. #1981	C₂×C₆₇	semidirect
144. #1843	C₂×C₆₇	semidirect
145. C₆₇	#33168	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×#463×C₆₇	2
C₂×#461×C₆₇	2
C₆₇×#2658	2
C₆₇×#2275	2
C₆₇×#1983	2
C₆₇×#1981	2
C₆₇×#1843	2
#33168	67

Subgroups — C67×#33168 #30712419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33168 #30712419