Subgroups — C₆₇×#33167 #30712418

Order 17152.

1162 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#464×C₆₇	C₂	1	3	yes	maximal, complemented
2	C₂×#460×C₆₇	C₂	1	3	yes	maximal, complemented
3	C₆₇×#2657	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2276	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1983	C₂	1	3	yes	maximal, complemented
6	C₆₇×#1982	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₆₇	C₂²	1	3	yes	—
10	C₂×#127×C₆₇	C₂²	1	3	yes	—
11	C₂×#124×C₆₇	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
13	#464×C₆₇	—	2	12	no	—
14	#460×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	3	yes	—
18	C₂×#102×C₆₇	C₂²	1	6	yes	—
19	C₂×#102×C₆₇	C₂²	1	3	yes	—
20	#392×C₆₇	—	2	1	no	—
21	#392×C₆₇	—	2	3	no	—
22	C₄×C₄⋊C₄×C₆₇	—	2	3	no	—
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
24	#376×C₆₇	—	2	1	no	—
25	#376×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×D₄×C₆₇	—	2	3	no	—
29	C₂²×D₄×C₆₇	—	2	3	no	—
30	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
31	C₂×Q₁₆×C₆₇	—	2	6	no	—
32	C₂×SD₁₆×C₆₇	—	2	12	no	—
33	C₂×SD₁₆×C₆₇	—	2	6	no	—
34	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	#127×C₆₇	—	2	6	no	—
37	#127×C₆₇	—	2	6	no	—
38	#124×C₆₇	—	2	6	no	—
39	#124×C₆₇	—	2	6	no	—
40	C₄×Q₈×C₆₇	—	2	6	no	—
41	C₄×Q₈×C₆₇	D₄	1	12	yes	complemented
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×C₄²×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	6	no	—
52	#102×C₆₇	—	2	12	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	3	no	—
58	C₂×Q₈×C₆₇	—	2	3	no	—
59	C₂×Q₈×C₆₇	—	2	3	no	—
60	C₂×Q₈×C₆₇	—	2	3	no	—
61	C₂×Q₈×C₆₇	C₂×D₄	1	6	yes	—
62	C₂×D₄×C₆₇	—	2	3	no	—
63	C₂×D₄×C₆₇	—	2	6	no	—
64	C₂×D₄×C₆₇	—	2	3	no	—
65	C₂×D₄×C₆₇	—	2	6	no	—
66	C₂×D₄×C₆₇	—	4	6	no	—
67	C₂×D₄×C₆₇	—	4	6	no	—
68	C₂×D₄×C₆₇	—	8	3	no	—
69	C₂×D₄×C₆₇	—	8	6	no	—
70	C₂×D₄×C₆₇	—	8	3	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	3	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	Q₁₆×C₆₇	—	4	4	no	—
78	SD₁₆×C₆₇	—	4	12	no	—
79	C₂×C₈×C₆₇	—	2	6	no	—
80	C₂×C₈×C₆₇	—	2	3	no	—
81	C₂×C₈×C₆₇	—	2	3	no	—
82	C₄⋊C₄×C₆₇	—	2	12	no	—
83	C₄⋊C₄×C₆₇	—	2	3	no	—
84	C₄⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
85	C₂²⋊C₄×C₆₇	—	4	6	no	—
86	C₂²⋊C₄×C₆₇	—	4	6	no	—
87	C₄²×C₆₇	—	2	12	no	—
88	C₄²×C₆₇	—	8	3	no	—
89	C₄²×C₆₇	C₂×D₄	1	6	yes	—
90	C₄²×C₆₇	#41	1	6	yes	—
91	C₂³×C₆₇	—	4	3	no	—
92	C₂³×C₆₇	—	4	6	no	—
93	C₂³×C₆₇	—	4	3	no	—
94	C₂³×C₆₇	—	4	3	no	—
95	C₂³×C₆₇	—	4	6	no	—
96	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₆₇	—	2	6	no	—
98	Q₈×C₆₇	—	4	1	no	—
99	Q₈×C₆₇	—	4	3	no	—
100	Q₈×C₆₇	#127	1	4	yes	—
101	D₄×C₆₇	—	4	3	no	—
102	D₄×C₆₇	—	4	3	no	—
103	D₄×C₆₇	—	4	6	no	—
104	D₄×C₆₇	—	8	12	no	—
105	D₄×C₆₇	—	8	12	no	—
106	C₂×C₄×C₆₇	—	2	6	no	—
107	C₂×C₄×C₆₇	—	2	3	no	—
108	C₂×C₄×C₆₇	—	2	6	no	—
109	C₂×C₄×C₆₇	—	2	3	no	—
110	C₂×C₄×C₆₇	—	2	6	no	—
111	C₂×C₄×C₆₇	—	4	3	no	—
112	C₂×C₄×C₆₇	—	4	3	no	—
113	C₂×C₄×C₆₇	—	4	3	no	—
114	C₂×C₄×C₆₇	#121	1	6	yes	—
115	C₂×C₄×C₆₇	#121	1	6	yes	—
116	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
117	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
118	C₈×C₆₇	—	4	3	no	—
119	C₈×C₆₇	—	4	1	no	—
120	C₂²×C₆₇	—	8	3	no	—
121	C₂²×C₆₇	—	8	6	no	—
122	C₂²×C₆₇	—	8	3	no	—
123	C₂²×C₆₇	—	8	6	no	—
124	C₂²×C₆₇	—	8	3	no	—
125	C₂²×C₆₇	C₂×#121	1	3	yes	—
126	C₂²×C₆₇	#576	1	1	yes	characteristic
127	C₂²×C₆₇	#577	1	3	yes	—
128	C₄×C₆₇	—	2	4	no	—
129	C₄×C₆₇	—	2	6	no	—
130	C₄×C₆₇	—	2	6	no	—
131	C₄×C₆₇	—	4	6	no	—
132	C₄×C₆₇	—	8	1	no	—
133	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
134	C₄×C₆₇	#532	1	3	yes	—
135	#33167	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₆₇	—	16	3	no	—
137	C₂×C₆₇	#2856	1	1	yes	characteristic
138	C₂×C₆₇	#3043	1	3	yes	—
139	C₂×C₆₇	#3045	1	3	yes	—
140	C₂×#464	C₂×C₆₇	1	3	yes	complemented
141	C₂×#460	C₂×C₆₇	1	3	yes	complemented
142	#2657	C₂×C₆₇	1	1	yes	characteristic
143	#2276	C₂×C₆₇	1	1	yes	characteristic, complemented
144	#1983	C₂×C₆₇	1	3	yes	complemented
145	#1982	C₂×C₆₇	1	3	yes	complemented
146	#1843	C₂×C₆₇	1	1	yes	characteristic, complemented
147	C₆₇	#33167	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
149	C₂²×SD₁₆	C₂²×C₆₇	1	3	yes	—
150	C₂×#127	C₂²×C₆₇	1	3	yes	—
151	C₂×#124	C₂²×C₆₇	1	3	yes	—
152	C₂×C₄×Q₈	C₂²×C₆₇	1	3	yes	—
153	#464	—	2	12	no	—
154	#460	—	2	12	no	—
155	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
156	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
157	C₂×#103	C₂²×C₆₇	1	3	yes	—
158	C₂×#102	C₂²×C₆₇	1	3	yes	—
159	C₂×#102	C₂²×C₆₇	1	6	yes	—
160	#392	—	2	1	no	—
161	#392	—	2	3	no	—
162	C₄×C₄⋊C₄	—	2	3	no	—
163	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
164	#376	—	2	1	no	—
165	#376	C₂²×C₆₇	1	1	yes	characteristic
166	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
167	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
168	C₂²×D₄	—	2	3	no	—
169	C₂²×D₄	—	2	3	no	—
170	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
171	C₂×Q₁₆	—	2	6	no	—
172	C₂×SD₁₆	—	2	12	no	—
173	C₂×SD₁₆	—	2	6	no	—
174	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
175	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
176	#127	—	2	6	no	—
177	#127	—	2	6	no	—
178	#124	—	2	6	no	—
179	#124	—	2	6	no	—
180	C₄×Q₈	—	2	6	no	—
181	C₄×Q₈	D₄×C₆₇	1	12	yes	complemented
182	C₂×C₄⋊C₄	—	2	3	no	—
183	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
184	C₂×C₂²⋊C₄	—	2	3	no	—
185	C₂×C₂²⋊C₄	—	2	3	no	—
186	C₂×C₄²	—	2	3	no	—
187	C₂×C₄²	—	2	3	no	—
188	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
189	C₄⋊C₈	—	2	6	no	—
190	C₄⋊C₈	—	2	6	no	—
191	#103	—	2	6	no	—
192	#102	—	2	12	no	—
193	#102	—	2	6	no	—
194	#95	—	2	3	no	—
195	#95	D₄×C₆₇	1	1	yes	characteristic
196	#95	D₄×C₆₇	1	3	yes	—
197	C₂⁴	—	2	3	no	—
198	C₂×Q₈	—	2	3	no	—
199	C₂×Q₈	—	2	3	no	—
200	C₂×Q₈	—	2	3	no	—
201	C₂×Q₈	C₂×D₄×C₆₇	1	6	yes	—
202	C₂×D₄	—	2	3	no	—
203	C₂×D₄	—	2	6	no	—
204	C₂×D₄	—	2	3	no	—
205	C₂×D₄	—	2	6	no	—
206	C₂×D₄	—	4	6	no	—
207	C₂×D₄	—	4	6	no	—
208	C₂×D₄	—	8	3	no	—
209	C₂×D₄	—	8	6	no	—
210	C₂×D₄	—	8	3	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	3	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
215	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
216	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
217	Q₁₆	—	4	4	no	—
218	SD₁₆	—	4	12	no	—
219	C₂×C₈	—	2	6	no	—
220	C₂×C₈	—	2	3	no	—
221	C₂×C₈	—	2	3	no	—
222	C₄⋊C₄	—	2	12	no	—
223	C₄⋊C₄	—	2	3	no	—
224	C₄⋊C₄	C₂×D₄×C₆₇	1	6	yes	—
225	C₂²⋊C₄	—	4	6	no	—
226	C₂²⋊C₄	—	4	6	no	—
227	C₄²	—	2	12	no	—
228	C₄²	—	8	3	no	—
229	C₄²	C₂×D₄×C₆₇	1	6	yes	—
230	C₄²	#41×C₆₇	1	6	yes	—
231	C₂³	—	4	3	no	—
232	C₂³	—	4	6	no	—
233	C₂³	—	4	3	no	—
234	C₂³	—	4	3	no	—
235	C₂³	—	4	6	no	—
236	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
237	Q₈	—	2	6	no	—
238	Q₈	—	4	1	no	—
239	Q₈	—	4	3	no	—
240	Q₈	#127×C₆₇	1	4	yes	—
241	D₄	—	4	3	no	—
242	D₄	—	4	3	no	—
243	D₄	—	4	6	no	—
244	D₄	—	8	12	no	—
245	D₄	—	8	12	no	—
246	C₂×C₄	—	2	6	no	—
247	C₂×C₄	—	2	3	no	—
248	C₂×C₄	—	2	6	no	—
249	C₂×C₄	—	2	3	no	—
250	C₂×C₄	—	2	6	no	—
251	C₂×C₄	—	4	3	no	—
252	C₂×C₄	—	4	3	no	—
253	C₂×C₄	—	4	3	no	—
254	C₂×C₄	#121×C₆₇	1	6	yes	—
255	C₂×C₄	#121×C₆₇	1	6	yes	—
256	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
257	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
258	C₈	—	4	3	no	—
259	C₈	—	4	1	no	—
260	C₂²	—	8	6	no	—
261	C₂²	—	8	3	no	—
262	C₂²	—	8	6	no	—
263	C₂²	—	8	3	no	—
264	C₂²	—	8	3	no	—
265	C₂²	C₂×#121×C₆₇	1	3	yes	—
266	C₂²	#576×C₆₇	1	1	yes	characteristic
267	C₂²	#577×C₆₇	1	3	yes	—
268	C₄	—	2	6	no	—
269	C₄	—	2	6	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	6	no	—
272	C₄	—	8	1	no	—
273	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
274	C₄	#532×C₆₇	1	3	yes	—
275	C₂	—	16	3	no	—
276	C₂	C₆₇×#2856	1	1	yes	characteristic
277	C₂	C₆₇×#3043	1	3	yes	—
278	C₂	C₆₇×#3045	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2657	C₂
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
1. C₂×#464×C₆₇	C₂	semidirect
2. C₂×#460×C₆₇	C₂	semidirect
4. C₆₇×#2276	C₂	semidirect
5. C₆₇×#1983	C₂	semidirect
6. C₆₇×#1982	C₂	semidirect
7. C₆₇×#1843	C₂	semidirect
41. C₄×Q₈×C₆₇	D₄	semidirect
135. #33167	C₆₇	direct (N × H)
140. C₂×#464	C₂×C₆₇	semidirect
141. C₂×#460	C₂×C₆₇	semidirect
143. #2276	C₂×C₆₇	semidirect
144. #1983	C₂×C₆₇	semidirect
145. #1982	C₂×C₆₇	semidirect
146. #1843	C₂×C₆₇	semidirect
147. C₆₇	#33167	direct (N × H)
181. C₄×Q₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#464×C₆₇	2
C₂×#460×C₆₇	2
C₆₇×#2657	2
C₆₇×#2276	2
C₆₇×#1983	2
C₆₇×#1982	2
C₆₇×#1843	2
#33167	67

Subgroups — C67×#33167 #30712418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33167 #30712418