Subgroups — C₂×C₆₇×#2214 #30712844

Order 17152.

686 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₆₇	C₂	1	2	yes	maximal
2	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2214	C₂	1	8	yes	maximal, direct product factor
5	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#362×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×Q₁₆×C₆₇	—	2	1	no	—
9	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
10	C₂×#131×C₆₇	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
12	#510×C₆₇	—	2	4	no	—
13	#510×C₆₇	C₂²	1	8	yes	—
14	C₂×Q₃₂×C₆₇	—	2	2	no	—
15	C₂×SD₃₂×C₆₇	—	2	2	no	—
16	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
17	#494×C₆₇	—	2	2	no	—
18	#494×C₆₇	C₂²	1	4	yes	complemented
19	#443×C₆₇	—	2	2	no	—
20	#443×C₆₇	C₂²	1	4	yes	—
21	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#104×C₆₇	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#364×C₆₇	C₂²	1	4	yes	complemented
25	#362×C₆₇	C₂²	1	8	yes	complemented
26	C₂×#41×C₆₇	—	2	1	no	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	#135×C₆₇	—	2	1	no	—
29	#135×C₆₇	—	2	1	no	—
30	#135×C₆₇	D₄	1	2	yes	—
31	#135×C₆₇	C₂³	1	2	yes	—
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	—	2	4	no	—
36	C₂×Q₁₆×C₆₇	—	2	2	no	—
37	C₂×Q₁₆×C₆₇	—	2	2	no	—
38	C₂×Q₁₆×C₆₇	—	4	1	no	—
39	C₂×Q₁₆×C₆₇	—	4	1	no	—
40	C₂×Q₁₆×C₆₇	—	4	2	no	—
41	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
42	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
43	C₂×SD₁₆×C₆₇	—	2	1	no	—
44	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
45	#131×C₆₇	—	2	4	no	—
46	#131×C₆₇	—	4	2	no	—
47	C₂×M₁₆×C₆₇	—	2	1	no	—
48	C₂×M₁₆×C₆₇	—	4	1	no	—
49	C₂²×C₈×C₆₇	—	4	1	no	—
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	—	2	4	no	—
53	C₄⋊Q₈×C₆₇	—	4	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	Q₃₂×C₆₇	—	2	8	no	—
57	SD₃₂×C₆₇	—	2	8	no	—
58	M₃₂×C₆₇	—	2	2	no	—
59	M₃₂×C₆₇	C₂³	1	4	yes	—
60	C₂×C₁₆×C₆₇	—	2	2	no	—
61	#108×C₆₇	—	2	1	no	—
62	#108×C₆₇	C₂³	1	2	yes	—
63	#104×C₆₇	—	2	4	no	—
64	C₄×C₈×C₆₇	D₄	1	2	yes	—
65	C₄×C₈×C₆₇	C₂³	1	2	yes	—
66	#41×C₆₇	—	2	2	no	—
67	#41×C₆₇	—	2	2	no	—
68	#41×C₆₇	—	4	1	no	—
69	#41×C₆₇	—	4	1	no	—
70	C₂×Q₈×C₆₇	—	2	1	no	—
71	C₂×Q₈×C₆₇	—	2	4	no	—
72	C₂×Q₈×C₆₇	—	4	2	no	—
73	C₂×Q₈×C₆₇	—	4	2	no	—
74	C₂×Q₈×C₆₇	—	4	2	no	—
75	C₂×Q₈×C₆₇	—	4	2	no	—
76	C₂×Q₈×C₆₇	—	4	2	no	—
77	C₂×D₄×C₆₇	—	2	1	no	—
78	C₂×D₄×C₆₇	—	4	1	no	—
79	C₂²×C₄×C₆₇	—	2	1	no	—
80	C₂²×C₄×C₆₇	—	4	1	no	—
81	C₂²×C₄×C₆₇	—	4	2	no	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₆₇	—	2	1	no	—
84	Q₁₆×C₆₇	—	2	4	no	—
85	Q₁₆×C₆₇	—	2	4	no	—
86	Q₁₆×C₆₇	—	4	2	no	—
87	Q₁₆×C₆₇	—	4	2	no	—
88	Q₁₆×C₆₇	—	4	4	no	—
89	Q₁₆×C₆₇	—	4	4	no	—
90	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
91	SD₁₆×C₆₇	—	2	2	no	—
92	SD₁₆×C₆₇	—	2	2	no	—
93	D₈×C₆₇	—	2	1	no	—
94	D₈×C₆₇	C₂×D₄	1	2	yes	—
95	M₁₆×C₆₇	—	2	2	no	—
96	M₁₆×C₆₇	—	4	1	no	—
97	M₁₆×C₆₇	—	4	4	no	—
98	C₂×C₈×C₆₇	—	4	1	no	—
99	C₂×C₈×C₆₇	—	4	4	no	—
100	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
101	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
102	C₂×C₈×C₆₇	D₈	1	2	yes	—
103	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
108	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₆₇	—	4	8	no	—
111	C₄²×C₆₇	—	2	1	no	—
112	C₄²×C₆₇	C₂×D₄	1	2	yes	—
113	C₁₆×C₆₇	—	2	4	no	—
114	C₂³×C₆₇	—	4	1	no	—
115	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
116	Q₈×C₆₇	—	2	2	no	—
117	Q₈×C₆₇	—	4	1	no	—
118	Q₈×C₆₇	—	4	4	no	—
119	Q₈×C₆₇	—	4	4	no	—
120	Q₈×C₆₇	—	8	2	no	—
121	Q₈×C₆₇	—	8	2	no	—
122	D₄×C₆₇	—	2	2	no	—
123	D₄×C₆₇	—	4	2	no	—
124	D₄×C₆₇	—	4	2	no	—
125	D₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	4	no	—
132	C₂×C₄×C₆₇	—	8	2	no	—
133	C₂×C₄×C₆₇	—	8	2	no	—
134	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₆₇	—	4	2	no	—
141	C₈×C₆₇	#121	1	1	yes	characteristic
142	C₈×C₆₇	#121	1	1	yes	characteristic
143	C₈×C₆₇	C₂×D₈	1	2	yes	—
144	C₈×C₆₇	C₂×D₈	1	2	yes	—
145	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₂²×C₆₇	—	2	2	no	—
148	C₂²×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	#459	1	1	yes	characteristic
151	C₂²×C₆₇	#459	1	1	yes	characteristic
152	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄×C₆₇	—	2	4	no	—
154	C₄×C₆₇	—	4	2	no	—
155	C₄×C₆₇	—	8	4	no	—
156	C₄×C₆₇	#459	1	1	yes	characteristic
157	C₄×C₆₇	#459	1	1	yes	characteristic
158	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
159	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
160	C₂×#2214	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₆₇	—	2	1	no	—
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	—	8	2	no	—
164	C₂×C₆₇	#2214	1	2	yes	direct product factor
165	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
166	C₂×#510	C₂×C₆₇	1	2	yes	—
167	C₂×#494	C₂×C₆₇	1	1	yes	characteristic, complemented
168	C₂×#443	C₂×C₆₇	1	1	yes	characteristic
169	#2214	C₂×C₆₇	1	8	yes	direct product factor
170	C₂×#364	C₂×C₆₇	1	1	yes	characteristic, complemented
171	C₂×#362	C₂×C₆₇	1	2	yes	complemented
172	C₆₇	C₂×#2214	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
174	C₂²×Q₁₆	—	2	1	no	—
175	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
176	C₂×#131	—	2	1	no	—
177	C₂×C₄⋊Q₈	—	2	1	no	—
178	#510	—	2	4	no	—
179	#510	C₂²×C₆₇	1	8	yes	—
180	C₂×Q₃₂	—	2	2	no	—
181	C₂×SD₃₂	—	2	2	no	—
182	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
183	#494	—	2	2	no	—
184	#494	C₂²×C₆₇	1	4	yes	complemented
185	#443	—	2	2	no	—
186	#443	C₂²×C₆₇	1	4	yes	—
187	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
188	C₂×#104	—	2	1	no	—
189	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
190	#364	C₂²×C₆₇	1	4	yes	complemented
191	#362	C₂²×C₆₇	1	8	yes	complemented
192	C₂×#41	—	2	1	no	—
193	C₂²×Q₈	—	2	2	no	—
194	#135	—	2	1	no	—
195	#135	—	2	1	no	—
196	#135	D₄×C₆₇	1	2	yes	—
197	#135	C₂³×C₆₇	1	2	yes	—
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×Q₁₆	—	2	2	no	—
200	C₂×Q₁₆	—	2	4	no	—
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	4	1	no	—
205	C₂×Q₁₆	—	4	1	no	—
206	C₂×Q₁₆	—	4	2	no	—
207	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
208	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
209	C₂×SD₁₆	—	2	1	no	—
210	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
211	#131	—	2	4	no	—
212	#131	—	4	2	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	4	1	no	—
215	C₂²×C₈	—	4	1	no	—
216	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	4	no	—
219	C₄⋊Q₈	—	4	2	no	—
220	C₂×C₄⋊C₄	—	4	2	no	—
221	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
222	Q₃₂	—	2	8	no	—
223	SD₃₂	—	2	8	no	—
224	M₃₂	—	2	2	no	—
225	M₃₂	C₂³×C₆₇	1	4	yes	—
226	C₂×C₁₆	—	2	2	no	—
227	#108	—	2	1	no	—
228	#108	C₂³×C₆₇	1	2	yes	—
229	#104	—	2	4	no	—
230	C₄×C₈	D₄×C₆₇	1	2	yes	—
231	C₄×C₈	C₂³×C₆₇	1	2	yes	—
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	4	no	—
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	4	2	no	—
254	Q₁₆	—	4	4	no	—
255	Q₁₆	—	4	4	no	—
256	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
257	SD₁₆	—	2	2	no	—
258	SD₁₆	—	2	2	no	—
259	D₈	—	2	1	no	—
260	D₈	C₂×D₄×C₆₇	1	2	yes	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
267	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
268	C₂×C₈	D₈×C₆₇	1	2	yes	—
269	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
273	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
274	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
275	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
276	C₄⋊C₄	—	4	8	no	—
277	C₄²	—	2	1	no	—
278	C₄²	C₂×D₄×C₆₇	1	2	yes	—
279	C₁₆	—	2	4	no	—
280	C₂³	—	4	1	no	—
281	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
301	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
304	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	#121×C₆₇	1	1	yes	characteristic
308	C₈	#121×C₆₇	1	1	yes	characteristic
309	C₈	C₂×D₈×C₆₇	1	2	yes	—
310	C₈	C₂×D₈×C₆₇	1	2	yes	—
311	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
312	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	#459×C₆₇	1	1	yes	characteristic
317	C₂²	#459×C₆₇	1	1	yes	characteristic
318	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	4	no	—
322	C₄	#459×C₆₇	1	1	yes	characteristic
323	C₄	#459×C₆₇	1	1	yes	characteristic
324	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
325	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	8	2	no	—
329	C₂	C₆₇×#2214	1	2	yes	direct product factor
330	C₂	C₂×#459×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#494×C₆₇	C₂	semidirect
4. C₆₇×#2214	C₂	direct (N × H)
5. C₂×#364×C₆₇	C₂	semidirect
6. C₂×#362×C₆₇	C₂	semidirect
18. #494×C₆₇	C₂²	semidirect
24. #364×C₆₇	C₂²	semidirect
25. #362×C₆₇	C₂²	semidirect
160. C₂×#2214	C₆₇	direct (N × H)
164. C₂×C₆₇	#2214	direct (N × H)
167. C₂×#494	C₂×C₆₇	semidirect
169. #2214	C₂×C₆₇	direct (N × H)
170. C₂×#364	C₂×C₆₇	semidirect
171. C₂×#362	C₂×C₆₇	semidirect
172. C₆₇	C₂×#2214	direct (N × H)
184. #494	C₂²×C₆₇	semidirect
190. #364	C₂²×C₆₇	semidirect
191. #362	C₂²×C₆₇	semidirect
329. C₂	C₆₇×#2214	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×#510×C₆₇	2
C₂×#494×C₆₇	2
C₂×#443×C₆₇	2
C₆₇×#2214	2
C₂×#364×C₆₇	2
C₂×#362×C₆₇	2
C₂×#2214	67

Subgroups — C2×C67×#2214 #30712844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2214 #30712844