Subgroups — C₆₇×#13004 #30692255

Order 17152.

560 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₇	—	2	1	no	—
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#121×C₆₇	—	2	1	no	—
10	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
11	#466×C₆₇	—	2	1	no	—
12	#466×C₆₇	—	2	1	no	—
13	#466×C₆₇	C₄	1	2	yes	—
14	#466×C₆₇	C₄	1	2	yes	—
15	C₂×#108×C₆₇	—	2	1	no	—
16	C₂×#108×C₆₇	—	2	4	no	—
17	C₂×#108×C₆₇	C₂²	1	2	yes	—
18	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#102×C₆₇	—	2	1	no	—
20	#361×C₆₇	—	2	4	no	—
21	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
22	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
23	C₂²×D₄×C₆₇	—	2	1	no	—
24	C₂²×D₄×C₆₇	—	4	1	no	—
25	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂×D₈×C₆₇	—	2	2	no	—
27	C₂×D₈×C₆₇	—	2	1	no	—
28	C₂×D₈×C₆₇	—	2	2	no	—
29	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₆₇	—	2	4	no	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	—	4	2	no	—
33	C₂×M₁₆×C₆₇	—	4	1	no	—
34	C₂²×C₈×C₆₇	—	2	2	no	—
35	C₂²×C₈×C₆₇	—	2	2	no	—
36	C₂²×C₈×C₆₇	—	4	1	no	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#121×C₆₇	—	2	2	no	—
44	#121×C₆₇	—	2	2	no	—
45	#121×C₆₇	—	4	1	no	—
46	#121×C₆₇	—	4	1	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
49	M₃₂×C₆₇	—	2	2	no	—
50	M₃₂×C₆₇	—	2	2	no	—
51	C₂×C₁₆×C₆₇	—	2	2	no	—
52	#108×C₆₇	—	2	4	no	—
53	#108×C₆₇	—	2	2	no	—
54	#108×C₆₇	D₄	1	4	yes	complemented
55	#107×C₆₇	—	2	1	no	—
56	#107×C₆₇	C₂×C₄	1	2	yes	—
57	#102×C₆₇	—	2	2	no	—
58	#102×C₆₇	—	2	2	no	—
59	C₂⁴×C₆₇	—	4	1	no	—
60	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
61	C₂×D₄×C₆₇	—	2	2	no	—
62	C₂×D₄×C₆₇	—	4	2	no	—
63	C₂×D₄×C₆₇	—	4	2	no	—
64	C₂×D₄×C₆₇	—	4	1	no	—
65	C₂×D₄×C₆₇	—	4	1	no	—
66	C₂×D₄×C₆₇	—	4	1	no	—
67	C₂×D₄×C₆₇	—	4	1	no	—
68	C₂×D₄×C₆₇	—	4	1	no	—
69	C₂×D₄×C₆₇	—	8	4	no	—
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	—	4	1	no	—
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	D₈×C₆₇	—	2	4	no	—
82	D₈×C₆₇	—	4	2	no	—
83	M₁₆×C₆₇	—	2	4	no	—
84	M₁₆×C₆₇	—	4	4	no	—
85	M₁₆×C₆₇	—	4	2	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	4	4	no	—
93	C₂×C₈×C₆₇	—	4	2	no	—
94	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	#41	1	2	yes	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	4	1	no	—
102	C₂²⋊C₄×C₆₇	—	4	2	no	—
103	C₂²⋊C₄×C₆₇	—	4	2	no	—
104	C₁₆×C₆₇	—	4	2	no	—
105	C₂³×C₆₇	—	2	2	no	—
106	C₂³×C₆₇	—	2	1	no	—
107	C₂³×C₆₇	—	2	1	no	—
108	C₂³×C₆₇	—	4	1	no	—
109	C₂³×C₆₇	—	4	2	no	—
110	C₂³×C₆₇	—	8	2	no	—
111	C₂³×C₆₇	—	8	2	no	—
112	C₂³×C₆₇	—	8	1	no	—
113	C₂³×C₆₇	—	8	1	no	—
114	C₂³×C₆₇	#102	1	1	yes	characteristic
115	C₂³×C₆₇	#102	1	1	yes	characteristic
116	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
117	D₄×C₆₇	—	4	2	no	—
118	D₄×C₆₇	—	4	2	no	—
119	D₄×C₆₇	—	8	4	no	—
120	D₄×C₆₇	—	8	4	no	—
121	D₄×C₆₇	—	8	1	no	—
122	D₄×C₆₇	—	8	1	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	8	2	no	—
135	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
139	C₂×C₄×C₆₇	#123	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	#137	1	1	yes	characteristic
143	C₈×C₆₇	—	2	1	no	—
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	4	4	no	—
146	C₈×C₆₇	C₄×D₄	1	2	yes	—
147	C₈×C₆₇	#121	1	2	yes	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	—	2	1	no	—
153	C₂²×C₆₇	—	2	1	no	—
154	C₂²×C₆₇	—	2	1	no	—
155	C₂²×C₆₇	—	4	2	no	—
156	C₂²×C₆₇	—	4	2	no	—
157	C₂²×C₆₇	—	8	2	no	—
158	C₂²×C₆₇	—	8	2	no	—
159	C₂²×C₆₇	—	8	2	no	—
160	C₂²×C₆₇	—	16	2	no	—
161	C₂²×C₆₇	—	16	2	no	—
162	C₂²×C₆₇	#361	1	2	yes	complemented
163	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
164	C₂²×C₆₇	#448	1	1	yes	characteristic
165	C₂²×C₆₇	#481	1	1	yes	characteristic
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	8	2	no	—
169	C₄×C₆₇	#386	1	1	yes	characteristic
170	C₄×C₆₇	#417	1	1	yes	characteristic
171	C₄×C₆₇	#447	1	1	yes	characteristic
172	C₄×C₆₇	#480	1	1	yes	characteristic
173	#13004	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂×C₆₇	—	2	1	no	—
175	C₂×C₆₇	—	2	1	no	—
176	C₂×C₆₇	—	2	2	no	—
177	C₂×C₆₇	—	4	1	no	—
178	C₂×C₆₇	—	16	2	no	—
179	C₂×C₆₇	#1874	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#361	1	1	yes	characteristic
181	C₂×C₆₇	#2148	1	1	yes	characteristic
182	C₂×#466	C₂×C₆₇	1	1	yes	characteristic
183	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic, complemented
184	C₂×#361	C₂×C₆₇	1	2	yes	complemented
185	#2112	C₂×C₆₇	1	1	yes	characteristic, complemented
186	#1387	C₂×C₆₇	1	2	yes	complemented
187	C₆₇	#13004	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
189	C₂²×M₁₆	—	2	1	no	—
190	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
191	C₂×#121	—	2	1	no	—
192	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
193	#466	—	2	1	no	—
194	#466	—	2	1	no	—
195	#466	C₄×C₆₇	1	2	yes	—
196	#466	C₄×C₆₇	1	2	yes	—
197	C₂×#108	—	2	1	no	—
198	C₂×#108	—	2	4	no	—
199	C₂×#108	C₂²×C₆₇	1	2	yes	—
200	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
201	C₂×#102	—	2	1	no	—
202	#361	—	2	4	no	—
203	C₂²⋊C₁₆	—	2	1	no	—
204	C₂²⋊C₁₆	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	—	4	1	no	—
207	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	2	no	—
211	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
212	C₂×M₁₆	—	2	4	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	4	2	no	—
215	C₂×M₁₆	—	4	1	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	2	2	no	—
218	C₂²×C₈	—	4	1	no	—
219	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
220	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
221	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
222	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
223	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
225	#121	—	2	2	no	—
226	#121	—	2	2	no	—
227	#121	—	4	1	no	—
228	#121	—	4	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₂²⋊C₄	—	4	1	no	—
231	M₃₂	—	2	2	no	—
232	M₃₂	—	2	2	no	—
233	C₂×C₁₆	—	2	2	no	—
234	#108	—	2	4	no	—
235	#108	—	2	2	no	—
236	#108	D₄×C₆₇	1	4	yes	complemented
237	#107	—	2	1	no	—
238	#107	C₂×C₄×C₆₇	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	C₂⁴	—	4	1	no	—
242	C₂⁴	SD₁₆×C₆₇	1	1	yes	characteristic
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	8	4	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
259	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
260	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
261	C₂²×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	M₁₆	—	2	4	no	—
266	M₁₆	—	4	4	no	—
267	M₁₆	—	4	2	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
277	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
278	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₂×C₈	#41×C₆₇	1	2	yes	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₁₆	—	4	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	1	no	—
295	C₂³	—	8	1	no	—
296	C₂³	#102×C₆₇	1	1	yes	characteristic
297	C₂³	#102×C₆₇	1	1	yes	characteristic
298	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	8	1	no	—
304	D₄	—	8	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
318	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
320	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
321	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
323	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
324	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	4	4	no	—
328	C₈	C₄×D₄×C₆₇	1	2	yes	—
329	C₈	#121×C₆₇	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	16	2	no	—
343	C₂²	—	16	2	no	—
344	C₂²	#361×C₆₇	1	2	yes	complemented
345	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
346	C₂²	#448×C₆₇	1	1	yes	characteristic
347	C₂²	#481×C₆₇	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	8	2	no	—
351	C₄	#386×C₆₇	1	1	yes	characteristic
352	C₄	#417×C₆₇	1	1	yes	characteristic
353	C₄	#447×C₆₇	1	1	yes	characteristic
354	C₄	#480×C₆₇	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	2	no	—
358	C₂	—	4	1	no	—
359	C₂	—	16	2	no	—
360	C₂	C₆₇×#1874	1	1	yes	characteristic
361	C₂	C₂×#361×C₆₇	1	1	yes	characteristic
362	C₂	C₆₇×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
prime radical	C₂×#466×C₆₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#108×C₆₇	C₂	semidirect
3. C₂×#361×C₆₇	C₂	semidirect
4. C₆₇×#2112	C₂	semidirect
5. C₆₇×#1387	C₂	semidirect
54. #108×C₆₇	D₄	semidirect
162. C₂²×C₆₇	#361	semidirect
173. #13004	C₆₇	direct (N × H)
183. C₂²×#108	C₂×C₆₇	semidirect
184. C₂×#361	C₂×C₆₇	semidirect
185. #2112	C₂×C₆₇	semidirect
186. #1387	C₂×C₆₇	semidirect
187. C₆₇	#13004	direct (N × H)
236. #108	D₄×C₆₇	semidirect
344. C₂²	#361×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#466×C₆₇	2
C₂²×#108×C₆₇	2
C₂×#361×C₆₇	2
C₆₇×#2112	2
C₆₇×#1387	2
#13004	67

Subgroups — C67×#13004 #30692255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13004 #30692255