Subgroups — C₂×C₆₇×#2222 #30712866

Order 17152.

638 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#473×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#2222	C₂	1	8	yes	maximal, direct product factor
5	C₂×#368×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#362×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#137×C₆₇	—	2	2	no	—
8	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×Q₃₂×C₆₇	—	2	1	no	—
11	C₂×Q₃₂×C₆₇	—	2	1	no	—
12	C₂×Q₃₂×C₆₇	—	2	2	no	—
13	C₂×Q₃₂×C₆₇	C₂²	1	4	yes	complemented
14	#504×C₆₇	—	2	2	no	—
15	#504×C₆₇	C₂²	1	4	yes	—
16	C₂×M₃₂×C₆₇	—	2	1	no	—
17	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₆₇	—	2	1	no	—
19	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	#473×C₆₇	—	2	4	no	—
21	#473×C₆₇	C₂²	1	8	yes	—
22	C₂×#108×C₆₇	C₂²	1	2	yes	—
23	C₂×#101×C₆₇	—	2	2	no	—
24	#368×C₆₇	C₂²	1	4	yes	complemented
25	#362×C₆₇	C₂²	1	8	yes	complemented
26	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	#137×C₆₇	—	2	8	no	—
29	#137×C₆₇	—	4	4	no	—
30	C₂×Q₁₆×C₆₇	—	2	2	no	—
31	C₂×Q₁₆×C₆₇	—	2	2	no	—
32	C₂×Q₁₆×C₆₇	—	2	2	no	—
33	C₂×Q₁₆×C₆₇	—	2	2	no	—
34	C₂×Q₁₆×C₆₇	—	2	2	no	—
35	C₂×Q₁₆×C₆₇	—	4	2	no	—
36	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
37	C₂×SD₁₆×C₆₇	—	4	2	no	—
38	#131×C₆₇	—	2	1	no	—
39	#131×C₆₇	—	2	1	no	—
40	#131×C₆₇	D₄	1	2	yes	—
41	#131×C₆₇	C₂³	1	2	yes	—
42	C₂×M₁₆×C₆₇	—	2	1	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	Q₃₂×C₆₇	—	2	4	no	—
48	Q₃₂×C₆₇	—	2	4	no	—
49	M₃₂×C₆₇	—	2	1	no	—
50	M₃₂×C₆₇	—	2	4	no	—
51	M₃₂×C₆₇	C₂³	1	2	yes	—
52	C₂×C₁₆×C₆₇	—	2	1	no	—
53	C₂×C₁₆×C₆₇	—	2	4	no	—
54	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
57	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
58	#108×C₆₇	—	2	2	no	—
59	#108×C₆₇	C₂³	1	4	yes	—
60	#101×C₆₇	—	2	8	no	—
61	#41×C₆₇	—	2	1	no	—
62	#41×C₆₇	—	2	1	no	—
63	#41×C₆₇	D₈	1	2	yes	—
64	#41×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×Q₈×C₆₇	—	2	4	no	—
66	C₂×Q₈×C₆₇	—	4	2	no	—
67	C₂×Q₈×C₆₇	—	4	2	no	—
68	C₂×Q₈×C₆₇	—	4	2	no	—
69	C₂×Q₈×C₆₇	—	4	2	no	—
70	C₂×Q₈×C₆₇	—	4	2	no	—
71	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
72	C₂×D₄×C₆₇	—	2	1	no	—
73	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	4	2	no	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₆₇	—	2	4	no	—
78	Q₁₆×C₆₇	—	2	4	no	—
79	Q₁₆×C₆₇	—	4	8	no	—
80	Q₁₆×C₆₇	—	4	2	no	—
81	Q₁₆×C₆₇	—	4	2	no	—
82	SD₁₆×C₆₇	—	4	8	no	—
83	M₁₆×C₆₇	—	2	1	no	—
84	M₁₆×C₆₇	—	2	2	no	—
85	M₁₆×C₆₇	—	2	2	no	—
86	M₁₆×C₆₇	—	2	4	no	—
87	M₁₆×C₆₇	—	4	2	no	—
88	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	4	2	no	—
93	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
97	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₁₆×C₆₇	—	2	2	no	—
100	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
101	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
102	C₂³×C₆₇	—	2	1	no	—
103	C₂³×C₆₇	#135	1	1	yes	characteristic
104	Q₈×C₆₇	—	2	1	no	—
105	Q₈×C₆₇	—	4	4	no	—
106	Q₈×C₆₇	—	4	4	no	—
107	Q₈×C₆₇	—	8	2	no	—
108	Q₈×C₆₇	—	8	2	no	—
109	Q₈×C₆₇	C₂×D₈	1	2	yes	—
110	D₄×C₆₇	—	2	2	no	—
111	D₄×C₆₇	—	2	2	no	—
112	D₄×C₆₇	—	2	1	no	—
113	D₄×C₆₇	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	8	2	no	—
119	C₂×C₄×C₆₇	—	8	2	no	—
120	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₆₇	—	2	2	no	—
127	C₈×C₆₇	—	4	4	no	—
128	C₈×C₆₇	#121	1	1	yes	characteristic
129	C₈×C₆₇	#121	1	1	yes	characteristic
130	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	#466	1	1	yes	characteristic
136	C₂²×C₆₇	#466	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	8	4	no	—
140	C₄×C₆₇	#466	1	1	yes	characteristic
141	C₄×C₆₇	#466	1	1	yes	characteristic
142	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
143	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
144	C₂×#2222	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₆₇	—	2	1	no	—
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	4	2	no	—
148	C₂×C₆₇	#2222	1	2	yes	direct product factor
149	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
150	C₂²×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
151	C₂×#504	C₂×C₆₇	1	1	yes	characteristic
152	C₂×#473	C₂×C₆₇	1	2	yes	—
153	#2222	C₂×C₆₇	1	8	yes	direct product factor
154	C₂×#368	C₂×C₆₇	1	1	yes	characteristic, complemented
155	C₂×#362	C₂×C₆₇	1	2	yes	complemented
156	C₆₇	C₂×#2222	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×#137	—	2	2	no	—
158	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
159	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
160	C₂×Q₃₂	—	2	1	no	—
161	C₂×Q₃₂	—	2	1	no	—
162	C₂×Q₃₂	—	2	2	no	—
163	C₂×Q₃₂	C₂²×C₆₇	1	4	yes	complemented
164	#504	—	2	2	no	—
165	#504	C₂²×C₆₇	1	4	yes	—
166	C₂×M₃₂	—	2	1	no	—
167	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
168	C₂²×C₁₆	—	2	1	no	—
169	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
170	#473	—	2	4	no	—
171	#473	C₂²×C₆₇	1	8	yes	—
172	C₂×#108	C₂²×C₆₇	1	2	yes	—
173	C₂×#101	—	2	2	no	—
174	#368	C₂²×C₆₇	1	4	yes	complemented
175	#362	C₂²×C₆₇	1	8	yes	complemented
176	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
177	C₂²×Q₈	—	2	2	no	—
178	#137	—	2	8	no	—
179	#137	—	4	4	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	4	2	no	—
186	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
187	C₂×SD₁₆	—	4	2	no	—
188	#131	—	2	1	no	—
189	#131	—	2	1	no	—
190	#131	D₄×C₆₇	1	2	yes	—
191	#131	C₂³×C₆₇	1	2	yes	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
197	Q₃₂	—	2	4	no	—
198	Q₃₂	—	2	4	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	4	no	—
201	M₃₂	C₂³×C₆₇	1	2	yes	—
202	C₂×C₁₆	—	2	1	no	—
203	C₂×C₁₆	—	2	4	no	—
204	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
205	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
206	C₂×C₁₆	D₄×C₆₇	1	2	yes	—
207	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
208	#108	—	2	2	no	—
209	#108	C₂³×C₆₇	1	4	yes	—
210	#101	—	2	8	no	—
211	#41	—	2	1	no	—
212	#41	—	2	1	no	—
213	#41	D₈×C₆₇	1	2	yes	—
214	#41	C₂×D₄×C₆₇	1	2	yes	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	2	4	no	—
229	Q₁₆	—	4	8	no	—
230	Q₁₆	—	4	2	no	—
231	Q₁₆	—	4	2	no	—
232	SD₁₆	—	4	8	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	4	2	no	—
238	M₁₆	C₂×D₄×C₆₇	1	2	yes	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
246	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
247	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
249	C₁₆	—	2	2	no	—
250	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
251	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
252	C₂³	—	2	1	no	—
253	C₂³	#135×C₆₇	1	1	yes	characteristic
254	Q₈	—	2	1	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	8	2	no	—
258	Q₈	—	8	2	no	—
259	Q₈	C₂×D₈×C₆₇	1	2	yes	—
260	D₄	—	2	2	no	—
261	D₄	—	2	2	no	—
262	D₄	—	2	1	no	—
263	D₄	C₂×D₈×C₆₇	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
271	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
274	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	4	4	no	—
278	C₈	#121×C₆₇	1	1	yes	characteristic
279	C₈	#121×C₆₇	1	1	yes	characteristic
280	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
281	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	#466×C₆₇	1	1	yes	characteristic
286	C₂²	#466×C₆₇	1	1	yes	characteristic
287	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	8	4	no	—
290	C₄	#466×C₆₇	1	1	yes	characteristic
291	C₄	#466×C₆₇	1	1	yes	characteristic
292	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
293	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	C₆₇×#2222	1	2	yes	direct product factor
298	C₂	C₂×#466×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#41×C₆₇	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₃₂×C₆₇	C₂	semidirect
4. C₆₇×#2222	C₂	direct (N × H)
5. C₂×#368×C₆₇	C₂	semidirect
6. C₂×#362×C₆₇	C₂	semidirect
13. C₂×Q₃₂×C₆₇	C₂²	semidirect
24. #368×C₆₇	C₂²	semidirect
25. #362×C₆₇	C₂²	semidirect
144. C₂×#2222	C₆₇	direct (N × H)
148. C₂×C₆₇	#2222	direct (N × H)
150. C₂²×Q₃₂	C₂×C₆₇	semidirect
153. #2222	C₂×C₆₇	direct (N × H)
154. C₂×#368	C₂×C₆₇	semidirect
155. C₂×#362	C₂×C₆₇	semidirect
156. C₆₇	C₂×#2222	direct (N × H)
163. C₂×Q₃₂	C₂²×C₆₇	semidirect
174. #368	C₂²×C₆₇	semidirect
175. #362	C₂²×C₆₇	semidirect
297. C₂	C₆₇×#2222	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂×C₆₇	2
C₂×#504×C₆₇	2
C₂×#473×C₆₇	2
C₆₇×#2222	2
C₂×#368×C₆₇	2
C₂×#362×C₆₇	2
C₂×#2222	67

Subgroups — C2×C67×#2222 #30712866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2222 #30712866