Subgroups — C₂×C₆₇×#2213 #30712843

Order 17152.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#509×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2213	C₂	1	8	yes	maximal, direct product factor
5	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#361×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×D₈×C₆₇	—	2	1	no	—
9	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#131×C₆₇	—	2	1	no	—
11	C₂×#127×C₆₇	—	2	1	no	—
12	#509×C₆₇	—	2	4	no	—
13	#509×C₆₇	C₂²	1	8	yes	complemented
14	C₂×SD₃₂×C₆₇	—	2	2	no	—
15	C₂×D₁₆×C₆₇	—	2	2	no	—
16	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
17	#493×C₆₇	—	2	2	no	—
18	#493×C₆₇	C₂²	1	4	yes	complemented
19	#443×C₆₇	—	2	2	no	—
20	#443×C₆₇	C₂²	1	4	yes	complemented
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	#361×C₆₇	C₂²	1	8	yes	complemented
26	C₂×#41×C₆₇	—	2	1	no	—
27	C₂²×D₄×C₆₇	—	2	2	no	—
28	C₂²×D₄×C₆₇	—	4	2	no	—
29	#135×C₆₇	—	2	1	no	—
30	#135×C₆₇	—	2	1	no	—
31	#135×C₆₇	D₄	1	2	yes	complemented
32	#135×C₆₇	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic, complemented
34	C₂×SD₁₆×C₆₇	—	2	1	no	—
35	C₂×D₈×C₆₇	—	2	2	no	—
36	C₂×D₈×C₆₇	—	2	2	no	—
37	C₂×D₈×C₆₇	—	2	2	no	—
38	C₂×D₈×C₆₇	—	2	4	no	—
39	C₂×D₈×C₆₇	—	2	2	no	—
40	C₂×D₈×C₆₇	—	2	2	no	—
41	C₂×D₈×C₆₇	—	4	1	no	—
42	C₂×D₈×C₆₇	—	4	1	no	—
43	C₂×D₈×C₆₇	—	4	2	no	—
44	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic, complemented
45	C₂×D₈×C₆₇	C₂³	1	4	yes	—
46	#131×C₆₇	—	2	4	no	—
47	#131×C₆₇	—	4	2	no	—
48	C₂×M₁₆×C₆₇	—	2	1	no	—
49	C₂×M₁₆×C₆₇	—	4	1	no	—
50	C₂²×C₈×C₆₇	—	4	1	no	—
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	#127×C₆₇	—	2	4	no	—
54	#127×C₆₇	—	4	2	no	—
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	SD₃₂×C₆₇	—	2	8	no	—
57	D₁₆×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	C₂⁴×C₆₇	—	4	2	no	—
67	#41×C₆₇	—	2	2	no	—
68	#41×C₆₇	—	2	2	no	—
69	#41×C₆₇	—	4	1	no	—
70	#41×C₆₇	—	4	1	no	—
71	C₂×Q₈×C₆₇	—	2	1	no	—
72	C₂×D₄×C₆₇	—	2	1	no	—
73	C₂×D₄×C₆₇	—	2	4	no	—
74	C₂×D₄×C₆₇	—	4	8	no	—
75	C₂×D₄×C₆₇	—	4	1	no	—
76	C₂×D₄×C₆₇	—	4	2	no	—
77	C₂×D₄×C₆₇	—	4	2	no	—
78	C₂×D₄×C₆₇	—	4	2	no	—
79	C₂×D₄×C₆₇	—	4	2	no	—
80	C₂×D₄×C₆₇	—	4	2	no	—
81	C₂×D₄×C₆₇	—	8	8	no	—
82	C₂²×C₄×C₆₇	—	2	1	no	—
83	C₂²×C₄×C₆₇	—	4	1	no	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	Q₁₆×C₆₇	—	2	1	no	—
86	Q₁₆×C₆₇	C₂×D₄	1	2	yes	complemented
87	SD₁₆×C₆₇	—	2	2	no	—
88	SD₁₆×C₆₇	—	2	2	no	—
89	D₈×C₆₇	—	2	1	no	—
90	D₈×C₆₇	—	2	4	no	—
91	D₈×C₆₇	—	2	4	no	—
92	D₈×C₆₇	—	4	2	no	—
93	D₈×C₆₇	—	4	2	no	—
94	D₈×C₆₇	—	4	4	no	—
95	D₈×C₆₇	—	4	4	no	—
96	D₈×C₆₇	C₂×D₄	1	2	yes	complemented
97	M₁₆×C₆₇	—	2	2	no	—
98	M₁₆×C₆₇	—	4	1	no	—
99	M₁₆×C₆₇	—	4	4	no	—
100	C₂×C₈×C₆₇	—	4	1	no	—
101	C₂×C₈×C₆₇	—	4	4	no	—
102	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
103	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
104	C₂×C₈×C₆₇	D₈	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
110	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
112	C₄²×C₆₇	—	2	1	no	—
113	C₄²×C₆₇	C₂×D₄	1	2	yes	—
114	C₁₆×C₆₇	—	2	4	no	—
115	C₂³×C₆₇	—	4	1	no	—
116	C₂³×C₆₇	—	4	4	no	—
117	C₂³×C₆₇	—	8	8	no	—
118	C₂³×C₆₇	—	8	2	no	—
119	C₂³×C₆₇	—	8	2	no	—
120	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
121	Q₈×C₆₇	—	2	2	no	—
122	Q₈×C₆₇	—	4	1	no	—
123	D₄×C₆₇	—	2	2	no	—
124	D₄×C₆₇	—	4	2	no	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	4	4	no	—
127	D₄×C₆₇	—	4	4	no	—
128	D₄×C₆₇	—	4	1	no	—
129	D₄×C₆₇	—	8	16	no	—
130	D₄×C₆₇	—	8	2	no	—
131	D₄×C₆₇	—	8	2	no	—
132	C₂×C₄×C₆₇	—	2	4	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₆₇	—	4	2	no	—
144	C₈×C₆₇	#121	1	1	yes	characteristic
145	C₈×C₆₇	#121	1	1	yes	characteristic
146	C₈×C₆₇	C₂×D₈	1	2	yes	—
147	C₈×C₆₇	C₂×D₈	1	2	yes	—
148	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	—	8	8	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	4	no	—
155	C₂²×C₆₇	—	16	4	no	—
156	C₂²×C₆₇	—	16	4	no	—
157	C₂²×C₆₇	#459	1	1	yes	characteristic
158	C₂²×C₆₇	#459	1	1	yes	characteristic
159	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₄×C₆₇	—	2	4	no	—
161	C₄×C₆₇	—	4	2	no	—
162	C₄×C₆₇	#459	1	1	yes	characteristic
163	C₄×C₆₇	#459	1	1	yes	characteristic
164	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
165	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
166	C₂×#2213	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂×C₆₇	—	2	1	no	—
168	C₂×C₆₇	—	2	1	no	—
169	C₂×C₆₇	—	8	2	no	—
170	C₂×C₆₇	—	16	4	no	—
171	C₂×C₆₇	#2213	1	2	yes	direct product factor
172	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
173	C₂×#509	C₂×C₆₇	1	2	yes	complemented
174	C₂×#493	C₂×C₆₇	1	1	yes	characteristic, complemented
175	C₂×#443	C₂×C₆₇	1	1	yes	characteristic, complemented
176	#2213	C₂×C₆₇	1	8	yes	direct product factor
177	C₂×#364	C₂×C₆₇	1	1	yes	characteristic, complemented
178	C₂×#361	C₂×C₆₇	1	2	yes	complemented
179	C₆₇	C₂×#2213	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
181	C₂²×D₈	—	2	1	no	—
182	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
183	C₂×#131	—	2	1	no	—
184	C₂×#127	—	2	1	no	—
185	#509	—	2	4	no	—
186	#509	C₂²×C₆₇	1	8	yes	complemented
187	C₂×SD₃₂	—	2	2	no	—
188	C₂×D₁₆	—	2	2	no	—
189	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
190	#493	—	2	2	no	—
191	#493	C₂²×C₆₇	1	4	yes	complemented
192	#443	—	2	2	no	—
193	#443	C₂²×C₆₇	1	4	yes	complemented
194	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
195	C₂×#104	—	2	1	no	—
196	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
197	#364	C₂²×C₆₇	1	4	yes	complemented
198	#361	C₂²×C₆₇	1	8	yes	complemented
199	C₂×#41	—	2	1	no	—
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	—	4	2	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	D₄×C₆₇	1	2	yes	complemented
205	#135	C₂³×C₆₇	1	2	yes	—
206	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	—	2	4	no	—
211	C₂×D₈	—	2	2	no	—
212	C₂×D₈	—	2	2	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	—	4	1	no	—
215	C₂×D₈	—	4	1	no	—
216	C₂×D₈	—	4	2	no	—
217	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic, complemented
218	C₂×D₈	C₂³×C₆₇	1	4	yes	—
219	#131	—	2	4	no	—
220	#131	—	4	2	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	4	1	no	—
223	C₂²×C₈	—	4	1	no	—
224	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
226	#127	—	2	4	no	—
227	#127	—	4	2	no	—
228	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
229	SD₃₂	—	2	8	no	—
230	D₁₆	—	2	8	no	—
231	M₃₂	—	2	2	no	—
232	M₃₂	C₂³×C₆₇	1	4	yes	—
233	C₂×C₁₆	—	2	2	no	—
234	#108	—	2	1	no	—
235	#108	C₂³×C₆₇	1	2	yes	—
236	#104	—	2	4	no	—
237	C₄×C₈	D₄×C₆₇	1	2	yes	—
238	C₄×C₈	C₂³×C₆₇	1	2	yes	—
239	C₂⁴	—	4	2	no	—
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	4	8	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	8	8	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	C₂×D₄×C₆₇	1	2	yes	complemented
260	SD₁₆	—	2	2	no	—
261	SD₁₆	—	2	2	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	4	no	—
264	D₈	—	2	4	no	—
265	D₈	—	4	2	no	—
266	D₈	—	4	2	no	—
267	D₈	—	4	4	no	—
268	D₈	—	4	4	no	—
269	D₈	C₂×D₄×C₆₇	1	2	yes	complemented
270	M₁₆	—	2	2	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	4	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
276	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
277	C₂×C₈	D₈×C₆₇	1	2	yes	—
278	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
282	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
283	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
284	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	C₂×D₄×C₆₇	1	2	yes	—
287	C₁₆	—	2	4	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	4	no	—
290	C₂³	—	8	8	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	4	no	—
301	D₄	—	4	4	no	—
302	D₄	—	8	16	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
311	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
312	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
314	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	#121×C₆₇	1	1	yes	characteristic
318	C₈	#121×C₆₇	1	1	yes	characteristic
319	C₈	C₂×D₈×C₆₇	1	2	yes	—
320	C₈	C₂×D₈×C₆₇	1	2	yes	—
321	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
322	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	8	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	16	4	no	—
329	C₂²	—	16	4	no	—
330	C₂²	#459×C₆₇	1	1	yes	characteristic
331	C₂²	#459×C₆₇	1	1	yes	characteristic
332	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	#459×C₆₇	1	1	yes	characteristic
336	C₄	#459×C₆₇	1	1	yes	characteristic
337	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
338	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	16	4	no	—
343	C₂	C₆₇×#2213	1	2	yes	direct product factor
344	C₂	C₂×#459×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
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
1. C₂×#509×C₆₇	C₂	semidirect
2. C₂×#493×C₆₇	C₂	semidirect
3. C₂×#443×C₆₇	C₂	semidirect
4. C₆₇×#2213	C₂	direct (N × H)
5. C₂×#364×C₆₇	C₂	semidirect
6. C₂×#361×C₆₇	C₂	semidirect
13. #509×C₆₇	C₂²	semidirect
18. #493×C₆₇	C₂²	semidirect
20. #443×C₆₇	C₂²	semidirect
24. #364×C₆₇	C₂²	semidirect
25. #361×C₆₇	C₂²	semidirect
31. #135×C₆₇	D₄	semidirect
33. C₂×Q₁₆×C₆₇	D₄	semidirect
44. C₂×D₈×C₆₇	D₄	semidirect
86. Q₁₆×C₆₇	C₂×D₄	semidirect
96. D₈×C₆₇	C₂×D₄	semidirect
166. C₂×#2213	C₆₇	direct (N × H)
171. C₂×C₆₇	#2213	direct (N × H)
173. C₂×#509	C₂×C₆₇	semidirect
174. C₂×#493	C₂×C₆₇	semidirect
175. C₂×#443	C₂×C₆₇	semidirect
176. #2213	C₂×C₆₇	direct (N × H)
177. C₂×#364	C₂×C₆₇	semidirect
178. C₂×#361	C₂×C₆₇	semidirect
179. C₆₇	C₂×#2213	direct (N × H)
186. #509	C₂²×C₆₇	semidirect
191. #493	C₂²×C₆₇	semidirect
193. #443	C₂²×C₆₇	semidirect
197. #364	C₂²×C₆₇	semidirect
198. #361	C₂²×C₆₇	semidirect
204. #135	D₄×C₆₇	semidirect
206. C₂×Q₁₆	D₄×C₆₇	semidirect
217. C₂×D₈	D₄×C₆₇	semidirect
259. Q₁₆	C₂×D₄×C₆₇	semidirect
269. D₈	C₂×D₄×C₆₇	semidirect
343. C₂	C₆₇×#2213	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×#509×C₆₇	2
C₂×#493×C₆₇	2
C₂×#443×C₆₇	2
C₆₇×#2213	2
C₂×#364×C₆₇	2
C₂×#361×C₆₇	2
C₂×#2213	67

Subgroups — C2×C67×#2213 #30712843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2213 #30712843