Subgroups — C₆₇×#33210 #30712461

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2857	C₂	1	2	yes	maximal
4	C₆₇×#1990	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1943	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1809	C₂	1	4	yes	maximal
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	#498×C₆₇	—	2	4	no	—
13	C₂×#107×C₆₇	C₂²	1	4	yes	—
14	C₂×#106×C₆₇	C₂²	1	4	yes	—
15	C₄×M₁₆×C₆₇	—	2	4	no	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#391×C₆₇	—	2	2	no	—
20	#389×C₆₇	—	2	2	no	—
21	#389×C₆₇	C₂²	1	4	yes	—
22	#384×C₆₇	C₂²	1	2	yes	—
23	#380×C₆₇	—	2	2	no	—
24	#380×C₆₇	C₂²	1	4	yes	—
25	#340×C₆₇	C₂²	1	8	yes	—
26	C₂²×Q₈×C₆₇	—	2	2	no	—
27	C₂³×C₄×C₆₇	D₄	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₆₇	—	2	4	no	—
30	C₂×M₁₆×C₆₇	—	2	4	no	—
31	C₂×M₁₆×C₆₇	Q₈	1	8	yes	complemented
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₆₇	—	2	4	no	—
35	C₄⋊Q₈×C₆₇	—	2	4	no	—
36	C₂²⋊Q₈×C₆₇	—	4	8	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	D₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	#107×C₆₇	—	2	8	no	—
50	#106×C₆₇	—	2	8	no	—
51	#97×C₆₇	—	2	4	no	—
52	C₄×C₈×C₆₇	—	2	2	no	—
53	C₄×C₈×C₆₇	D₄	1	4	yes	—
54	#95×C₆₇	—	2	4	no	—
55	#95×C₆₇	—	2	4	no	—
56	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×Q₈×C₆₇	—	4	4	no	—
58	C₂×Q₈×C₆₇	—	4	4	no	—
59	C₂×Q₈×C₆₇	—	4	4	no	—
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	—	2	1	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₆₇	—	2	16	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
79	C₂×C₈×C₆₇	#41	1	4	yes	—
80	C₄⋊C₄×C₆₇	—	2	4	no	—
81	C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	—	4	8	no	—
84	C₄⋊C₄×C₆₇	—	4	4	no	—
85	C₄⋊C₄×C₆₇	—	4	4	no	—
86	C₄⋊C₄×C₆₇	—	4	4	no	—
87	C₂²⋊C₄×C₆₇	—	4	8	no	—
88	C₄²×C₆₇	—	2	4	no	—
89	C₄²×C₆₇	—	2	1	no	—
90	C₄²×C₆₇	—	2	1	no	—
91	C₄²×C₆₇	C₂×D₄	1	2	yes	—
92	C₄²×C₆₇	#41	1	2	yes	—
93	C₂³×C₆₇	—	2	1	no	—
94	C₂³×C₆₇	—	2	1	no	—
95	C₂³×C₆₇	—	2	2	no	—
96	C₂³×C₆₇	—	2	2	no	—
97	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
98	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₆₇	—	8	8	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	4	8	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	#121	1	4	yes	—
115	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
117	C₂×C₄×C₆₇	#137	1	4	yes	—
118	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
122	C₈×C₆₇	—	2	4	no	—
123	C₈×C₆₇	C₂²⋊Q₈	1	8	yes	complemented
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	4	1	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	—	4	2	no	—
129	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
130	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₂²×C₆₇	#532	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
133	C₂²×C₆₇	C₂×#137	1	2	yes	—
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	4	2	no	—
136	C₄×C₆₇	—	8	4	no	—
137	C₄×C₆₇	—	8	4	no	—
138	C₄×C₆₇	#468	1	8	yes	—
139	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
140	C₄×C₆₇	#533	1	2	yes	—
141	#33210	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₆₇	—	4	2	no	—
143	C₂×C₆₇	#2857	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#468	1	2	yes	—
145	C₂×C₆₇	#3165	1	2	yes	—
146	C₂×C₆₇	#3184	1	1	yes	characteristic
147	C₂×C₆₇	#3185	1	1	yes	characteristic
148	C₂×#498	C₂×C₆₇	1	1	yes	characteristic, complemented
149	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
150	#2857	C₂×C₆₇	1	2	yes	—
151	#1990	C₂×C₆₇	1	2	yes	complemented
152	#1943	C₂×C₆₇	1	4	yes	complemented
153	#1845	C₂×C₆₇	1	1	yes	characteristic, complemented
154	#1809	C₂×C₆₇	1	4	yes	—
155	C₆₇	#33210	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
157	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
158	C₂×C₂²⋊Q₈	—	2	2	no	—
159	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
160	#498	—	2	4	no	—
161	C₂×#107	C₂²×C₆₇	1	4	yes	—
162	C₂×#106	C₂²×C₆₇	1	4	yes	—
163	C₄×M₁₆	—	2	4	no	—
164	C₂×#97	C₂²×C₆₇	1	2	yes	—
165	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
167	#391	—	2	2	no	—
168	#389	—	2	2	no	—
169	#389	C₂²×C₆₇	1	4	yes	—
170	#384	C₂²×C₆₇	1	2	yes	—
171	#380	—	2	2	no	—
172	#380	C₂²×C₆₇	1	4	yes	—
173	#340	C₂²×C₆₇	1	8	yes	—
174	C₂²×Q₈	—	2	2	no	—
175	C₂³×C₄	D₄×C₆₇	1	2	yes	—
176	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	Q₈×C₆₇	1	8	yes	complemented
180	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
181	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
182	C₄⋊Q₈	—	2	4	no	—
183	C₄⋊Q₈	—	2	4	no	—
184	C₂²⋊Q₈	—	4	8	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	4	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	D₄×C₆₇	1	2	yes	—
195	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
196	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
197	#107	—	2	8	no	—
198	#106	—	2	8	no	—
199	#97	—	2	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	D₄×C₆₇	1	4	yes	—
202	#95	—	2	4	no	—
203	#95	—	2	4	no	—
204	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
205	C₂×Q₈	—	4	4	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	4	4	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
219	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
221	M₁₆	—	2	16	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
225	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
226	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
227	C₂×C₈	#41×C₆₇	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	8	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₂²⋊C₄	—	4	8	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	C₂×D₄×C₆₇	1	2	yes	—
240	C₄²	#41×C₆₇	1	2	yes	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
246	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
247	Q₈	—	8	8	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	#121×C₆₇	1	4	yes	—
263	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
264	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
265	C₂×C₄	#137×C₆₇	1	4	yes	—
266	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
268	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
269	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	C₂²⋊Q₈×C₆₇	1	8	yes	complemented
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	1	no	—
277	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
278	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
279	C₂²	#532×C₆₇	1	1	yes	characteristic
280	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
281	C₂²	C₂×#137×C₆₇	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	8	4	no	—
285	C₄	—	8	4	no	—
286	C₄	#468×C₆₇	1	8	yes	—
287	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
288	C₄	#533×C₆₇	1	2	yes	—
289	C₂	—	4	2	no	—
290	C₂	C₆₇×#2857	1	1	yes	characteristic
291	C₂	C₂×#468×C₆₇	1	2	yes	—
292	C₂	C₆₇×#3165	1	2	yes	—
293	C₂	C₆₇×#3184	1	1	yes	characteristic
294	C₂	C₆₇×#3185	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#498×C₆₇	C₂	semidirect
4. C₆₇×#1990	C₂	semidirect
5. C₆₇×#1943	C₂	semidirect
6. C₆₇×#1845	C₂	semidirect
31. C₂×M₁₆×C₆₇	Q₈	semidirect
123. C₈×C₆₇	C₂²⋊Q₈	semidirect
141. #33210	C₆₇	direct (N × H)
148. C₂×#498	C₂×C₆₇	semidirect
151. #1990	C₂×C₆₇	semidirect
152. #1943	C₂×C₆₇	semidirect
153. #1845	C₂×C₆₇	semidirect
155. C₆₇	#33210	direct (N × H)
179. C₂×M₁₆	Q₈×C₆₇	semidirect
271. C₈	C₂²⋊Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#498×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2857	2
C₆₇×#1990	2
C₆₇×#1943	2
C₆₇×#1845	2
C₆₇×#1809	2
#33210	67

Subgroups — C67×#33210 #30712461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33210 #30712461