Subgroups — C₆₇×#10145 #30689396

Order 17152.

672 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3208	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₆₇	—	2	2	no	—
9	#533×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
12	#493×C₆₇	—	2	2	no	—
13	#493×C₆₇	—	2	2	no	—
14	#492×C₆₇	—	2	1	no	—
15	#490×C₆₇	—	2	2	no	—
16	#486×C₆₇	—	2	1	no	—
17	#486×C₆₇	—	2	1	no	—
18	#486×C₆₇	—	2	1	no	—
19	#460×C₆₇	—	2	2	no	—
20	#450×C₆₇	—	2	2	no	—
21	#431×C₆₇	C₂²	1	1	yes	characteristic
22	#423×C₆₇	C₂²	1	1	yes	characteristic
23	#418×C₆₇	—	2	2	no	—
24	C₂×#100×C₆₇	—	2	2	no	—
25	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#331×C₆₇	—	2	2	no	—
28	Q₈⋊C₈×C₆₇	—	2	2	no	—
29	C₂²×D₄×C₆₇	—	2	1	no	—
30	C₂²×D₄×C₆₇	—	2	1	no	—
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂²×D₄×C₆₇	—	2	1	no	—
34	C₂²×D₄×C₆₇	D₄	1	2	yes	—
35	C₂×SD₁₆×C₆₇	—	2	4	no	—
36	C₂×D₈×C₆₇	—	2	8	no	—
37	C₂×D₈×C₆₇	—	4	8	no	—
38	C₂×M₁₆×C₆₇	—	2	1	no	—
39	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	—	2	1	no	—
41	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
42	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
43	#127×C₆₇	—	2	1	no	—
44	#127×C₆₇	—	2	1	no	—
45	#127×C₆₇	—	2	1	no	—
46	#127×C₆₇	—	2	1	no	—
47	#127×C₆₇	—	2	1	no	—
48	#127×C₆₇	—	2	1	no	—
49	#127×C₆₇	—	2	1	no	—
50	#127×C₆₇	D₄	1	2	yes	—
51	#125×C₆₇	—	2	1	no	—
52	C₂²⋊Q₈×C₆₇	—	2	1	no	—
53	C₂²⋊Q₈×C₆₇	—	2	1	no	—
54	C₄×Q₈×C₆₇	—	2	1	no	—
55	C₄×Q₈×C₆₇	—	2	1	no	—
56	#117×C₆₇	—	2	1	no	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₆₇	—	2	1	no	—
59	C₄⋊C₈×C₆₇	—	2	1	no	—
60	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
61	#102×C₆₇	—	2	4	no	—
62	#102×C₆₇	—	2	4	no	—
63	#102×C₆₇	—	2	4	no	—
64	#100×C₆₇	—	4	4	no	—
65	C₂²⋊C₈×C₆₇	—	2	1	no	—
66	#97×C₆₇	—	2	1	no	—
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₄×C₈×C₆₇	—	2	1	no	—
69	C₄×C₈×C₆₇	—	2	1	no	—
70	C₄×C₈×C₆₇	D₄	1	2	yes	—
71	C₂⁴×C₆₇	—	2	2	no	—
72	C₂⁴×C₆₇	—	4	1	no	—
73	C₂×Q₈×C₆₇	—	2	1	no	—
74	C₂×Q₈×C₆₇	—	2	1	no	—
75	C₂×D₄×C₆₇	—	2	2	no	—
76	C₂×D₄×C₆₇	—	2	2	no	—
77	C₂×D₄×C₆₇	—	2	2	no	—
78	C₂×D₄×C₆₇	—	2	2	no	—
79	C₂×D₄×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	—	4	4	no	—
81	C₂×D₄×C₆₇	—	4	4	no	—
82	C₂×D₄×C₆₇	—	4	1	no	—
83	C₂×D₄×C₆₇	—	4	1	no	—
84	C₂×D₄×C₆₇	—	4	1	no	—
85	C₂×D₄×C₆₇	—	4	1	no	—
86	C₂×D₄×C₆₇	—	4	1	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂×D₄×C₆₇	—	4	2	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	4	2	no	—
93	C₂×D₄×C₆₇	—	8	2	no	—
94	C₂×D₄×C₆₇	—	8	2	no	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
98	SD₁₆×C₆₇	—	8	4	no	—
99	D₈×C₆₇	—	4	16	no	—
100	M₁₆×C₆₇	—	2	2	no	—
101	M₁₆×C₆₇	—	2	2	no	—
102	M₁₆×C₆₇	—	4	2	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	4	1	no	—
108	C₄⋊C₄×C₆₇	—	2	1	no	—
109	C₄⋊C₄×C₆₇	—	2	1	no	—
110	C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₄⋊C₄×C₆₇	—	2	1	no	—
112	C₄⋊C₄×C₆₇	—	4	1	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₂²⋊C₄×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	—	4	1	no	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₆₇	—	4	2	no	—
121	C₂³×C₆₇	—	4	2	no	—
122	C₂³×C₆₇	—	4	4	no	—
123	C₂³×C₆₇	—	8	2	no	—
124	C₂³×C₆₇	—	8	2	no	—
125	C₂³×C₆₇	—	8	2	no	—
126	C₂³×C₆₇	—	8	2	no	—
127	C₂³×C₆₇	—	8	2	no	—
128	C₂³×C₆₇	#121	1	1	yes	characteristic
129	Q₈×C₆₇	—	4	2	no	—
130	Q₈×C₆₇	—	4	2	no	—
131	D₄×C₆₇	—	4	4	no	—
132	D₄×C₆₇	—	4	4	no	—
133	D₄×C₆₇	—	4	4	no	—
134	D₄×C₆₇	—	4	4	no	—
135	D₄×C₆₇	—	8	2	no	—
136	D₄×C₆₇	—	8	2	no	—
137	D₄×C₆₇	—	8	2	no	—
138	D₄×C₆₇	—	8	2	no	—
139	D₄×C₆₇	—	8	2	no	—
140	D₄×C₆₇	—	8	2	no	—
141	D₄×C₆₇	—	8	1	no	—
142	D₄×C₆₇	—	8	1	no	—
143	D₄×C₆₇	—	8	1	no	—
144	D₄×C₆₇	—	8	1	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	4	1	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
151	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
152	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
154	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
155	C₂×C₄×C₆₇	#135	1	2	yes	—
156	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
157	C₈×C₆₇	—	2	4	no	—
158	C₈×C₆₇	—	4	2	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	—	8	1	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	8	2	no	—
164	C₂²×C₆₇	—	8	2	no	—
165	C₂²×C₆₇	—	8	2	no	—
166	C₂²×C₆₇	—	8	2	no	—
167	C₂²×C₆₇	—	8	2	no	—
168	C₂²×C₆₇	—	8	2	no	—
169	C₂²×C₆₇	—	16	4	no	—
170	C₂²×C₆₇	—	16	4	no	—
171	C₂²×C₆₇	#392	1	1	yes	characteristic
172	C₂²×C₆₇	#463	1	2	yes	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	8	1	no	—
176	C₄×C₆₇	—	8	1	no	—
177	C₄×C₆₇	#449	1	2	yes	—
178	C₄×C₆₇	#453	1	2	yes	—
179	C₄×C₆₇	#472	1	2	yes	—
180	C₄×C₆₇	#495	1	1	yes	characteristic
181	C₄×C₆₇	#496	1	1	yes	characteristic
182	#10145	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂×C₆₇	—	2	2	no	—
184	C₂×C₆₇	—	16	2	no	—
185	C₂×C₆₇	—	16	2	no	—
186	C₂×C₆₇	#2002	1	1	yes	characteristic
187	C₂×C₆₇	#2012	1	1	yes	characteristic
188	C₂×C₆₇	#2018	1	1	yes	characteristic
189	#3208	C₂×C₆₇	1	1	yes	characteristic, complemented
190	C₂×#493	C₂×C₆₇	1	1	yes	characteristic
191	#3138	C₂×C₆₇	1	1	yes	characteristic, complemented
192	#1965	C₂×C₆₇	1	1	yes	characteristic, complemented
193	#1545	C₂×C₆₇	1	1	yes	characteristic, complemented
194	#1493	C₂×C₆₇	1	1	yes	characteristic, complemented
195	#1278	C₂×C₆₇	1	1	yes	characteristic, complemented
196	C₆₇	#10145	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×D₈	—	2	2	no	—
198	#533	C₂²×C₆₇	1	1	yes	characteristic
199	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
200	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
201	#493	—	2	2	no	—
202	#493	—	2	2	no	—
203	#492	—	2	1	no	—
204	#490	—	2	2	no	—
205	#486	—	2	1	no	—
206	#486	—	2	1	no	—
207	#486	—	2	1	no	—
208	#460	—	2	2	no	—
209	#450	—	2	2	no	—
210	#431	C₂²×C₆₇	1	1	yes	characteristic
211	#423	C₂²×C₆₇	1	1	yes	characteristic
212	#418	—	2	2	no	—
213	C₂×#100	—	2	2	no	—
214	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
216	#331	—	2	2	no	—
217	Q₈⋊C₈	—	2	2	no	—
218	C₂²×D₄	—	2	1	no	—
219	C₂²×D₄	—	2	1	no	—
220	C₂²×D₄	—	2	1	no	—
221	C₂²×D₄	—	2	1	no	—
222	C₂²×D₄	—	2	1	no	—
223	C₂²×D₄	D₄×C₆₇	1	2	yes	—
224	C₂×SD₁₆	—	2	4	no	—
225	C₂×D₈	—	2	8	no	—
226	C₂×D₈	—	4	8	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
229	C₂²×C₈	—	2	1	no	—
230	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
231	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
232	#127	—	2	1	no	—
233	#127	—	2	1	no	—
234	#127	—	2	1	no	—
235	#127	—	2	1	no	—
236	#127	—	2	1	no	—
237	#127	—	2	1	no	—
238	#127	—	2	1	no	—
239	#127	D₄×C₆₇	1	2	yes	—
240	#125	—	2	1	no	—
241	C₂²⋊Q₈	—	2	1	no	—
242	C₂²⋊Q₈	—	2	1	no	—
243	C₄×Q₈	—	2	1	no	—
244	C₄×Q₈	—	2	1	no	—
245	#117	—	2	1	no	—
246	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
250	#102	—	2	4	no	—
251	#102	—	2	4	no	—
252	#102	—	2	4	no	—
253	#100	—	4	4	no	—
254	C₂²⋊C₈	—	2	1	no	—
255	#97	—	2	1	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	D₄×C₆₇	1	2	yes	—
260	C₂⁴	—	2	2	no	—
261	C₂⁴	—	4	1	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	4	no	—
270	C₂×D₄	—	4	4	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	—	8	2	no	—
283	C₂×D₄	—	8	2	no	—
284	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
286	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
287	SD₁₆	—	8	4	no	—
288	D₈	—	4	16	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	4	2	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	4	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	4	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂³	—	4	2	no	—
310	C₂³	—	4	2	no	—
311	C₂³	—	4	4	no	—
312	C₂³	—	8	2	no	—
313	C₂³	—	8	2	no	—
314	C₂³	—	8	2	no	—
315	C₂³	—	8	2	no	—
316	C₂³	—	8	2	no	—
317	C₂³	#121×C₆₇	1	1	yes	characteristic
318	Q₈	—	4	2	no	—
319	Q₈	—	4	2	no	—
320	D₄	—	4	4	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	4	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	2	no	—
327	D₄	—	8	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	2	no	—
330	D₄	—	8	1	no	—
331	D₄	—	8	1	no	—
332	D₄	—	8	1	no	—
333	D₄	—	8	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
339	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
340	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
341	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
342	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
343	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
344	C₂×C₄	#135×C₆₇	1	2	yes	—
345	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
346	C₈	—	2	4	no	—
347	C₈	—	4	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	8	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	8	2	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	16	4	no	—
359	C₂²	—	16	4	no	—
360	C₂²	#392×C₆₇	1	1	yes	characteristic
361	C₂²	#463×C₆₇	1	2	yes	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	#449×C₆₇	1	2	yes	—
367	C₄	#453×C₆₇	1	2	yes	—
368	C₄	#472×C₆₇	1	2	yes	—
369	C₄	#495×C₆₇	1	1	yes	characteristic
370	C₄	#496×C₆₇	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	16	2	no	—
373	C₂	—	16	2	no	—
374	C₂	C₆₇×#2002	1	1	yes	characteristic
375	C₂	C₆₇×#2012	1	1	yes	characteristic
376	C₂	C₆₇×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂×#493×C₆₇	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3208	C₂	semidirect
3. C₆₇×#3138	C₂	semidirect
4. C₆₇×#1965	C₂	semidirect
5. C₆₇×#1545	C₂	semidirect
6. C₆₇×#1493	C₂	semidirect
7. C₆₇×#1278	C₂	semidirect
182. #10145	C₆₇	direct (N × H)
189. #3208	C₂×C₆₇	semidirect
191. #3138	C₂×C₆₇	semidirect
192. #1965	C₂×C₆₇	semidirect
193. #1545	C₂×C₆₇	semidirect
194. #1493	C₂×C₆₇	semidirect
195. #1278	C₂×C₆₇	semidirect
196. C₆₇	#10145	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3208	2
C₂×#493×C₆₇	2
C₆₇×#3138	2
C₆₇×#1965	2
C₆₇×#1545	2
C₆₇×#1493	2
C₆₇×#1278	2
#10145	67

Subgroups — C67×#10145 #30689396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10145 #30689396