Subgroups — C₆₇×#10153 #30689404

Order 17152.

620 conjugacy classes of subgroups.

The subgroup lattice has 350 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
2	C₆₇×#3165	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1965	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1549	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×SD₁₆×C₆₇	—	2	2	no	—
9	#533×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
12	#501×C₆₇	—	2	2	no	—
13	#500×C₆₇	—	2	1	no	—
14	#498×C₆₇	—	2	1	no	—
15	#492×C₆₇	—	2	2	no	—
16	#492×C₆₇	—	2	2	no	—
17	#492×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	#428×C₆₇	—	2	2	no	—
23	#423×C₆₇	C₂²	1	1	yes	characteristic
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	#332×C₆₇	—	2	2	no	—
28	Q₈⋊C₈×C₆₇	—	2	2	no	—
29	C₂²×Q₈×C₆₇	—	2	1	no	—
30	C₂²×D₄×C₆₇	—	2	1	no	—
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	D₄	1	2	yes	—
33	C₂×SD₁₆×C₆₇	—	2	8	no	—
34	C₂×SD₁₆×C₆₇	—	2	4	no	—
35	C₂×SD₁₆×C₆₇	—	4	8	no	—
36	C₂×M₁₆×C₆₇	—	2	1	no	—
37	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
38	C₂²×C₈×C₆₇	—	2	1	no	—
39	C₄⋊Q₈×C₆₇	—	2	1	no	—
40	C₄⋊Q₈×C₆₇	—	2	1	no	—
41	C₄⋊Q₈×C₆₇	—	2	1	no	—
42	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
43	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
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	#125×C₆₇	—	2	1	no	—
50	C₂²⋊Q₈×C₆₇	—	2	1	no	—
51	C₂²⋊Q₈×C₆₇	—	2	1	no	—
52	C₄×Q₈×C₆₇	—	2	1	no	—
53	C₄×Q₈×C₆₇	—	2	1	no	—
54	#117×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	#107×C₆₇	—	2	4	no	—
59	#106×C₆₇	—	2	4	no	—
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
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	C₂×Q₈×C₆₇	—	2	1	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×Q₈×C₆₇	—	2	1	no	—
75	C₂×Q₈×C₆₇	—	4	4	no	—
76	C₂×Q₈×C₆₇	—	4	1	no	—
77	C₂×D₄×C₆₇	—	2	2	no	—
78	C₂×D₄×C₆₇	—	2	2	no	—
79	C₂×D₄×C₆₇	—	4	4	no	—
80	C₂×D₄×C₆₇	—	4	1	no	—
81	C₂×D₄×C₆₇	—	4	1	no	—
82	C₂×D₄×C₆₇	—	4	1	no	—
83	C₂×D₄×C₆₇	—	4	1	no	—
84	C₂×D₄×C₆₇	—	4	2	no	—
85	C₂×D₄×C₆₇	—	4	2	no	—
86	C₂×D₄×C₆₇	—	8	2	no	—
87	C₂²×C₄×C₆₇	—	4	1	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	SD₁₆×C₆₇	—	4	16	no	—
92	SD₁₆×C₆₇	—	8	4	no	—
93	M₁₆×C₆₇	—	2	2	no	—
94	M₁₆×C₆₇	—	4	2	no	—
95	M₁₆×C₆₇	SD₁₆	1	4	yes	complemented
96	C₂×C₈×C₆₇	—	2	4	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	4	1	no	—
101	C₄⋊C₄×C₆₇	—	2	1	no	—
102	C₄⋊C₄×C₆₇	—	2	1	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	1	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₄⋊C₄×C₆₇	—	4	1	no	—
108	C₄⋊C₄×C₆₇	—	4	1	no	—
109	C₄⋊C₄×C₆₇	—	4	1	no	—
110	C₄⋊C₄×C₆₇	—	4	1	no	—
111	C₂²⋊C₄×C₆₇	—	4	1	no	—
112	C₄²×C₆₇	—	4	1	no	—
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂³×C₆₇	—	4	2	no	—
118	C₂³×C₆₇	—	4	4	no	—
119	C₂³×C₆₇	—	8	2	no	—
120	C₂³×C₆₇	—	8	2	no	—
121	C₂³×C₆₇	#121	1	1	yes	characteristic
122	Q₈×C₆₇	—	4	4	no	—
123	Q₈×C₆₇	—	4	2	no	—
124	Q₈×C₆₇	—	4	2	no	—
125	Q₈×C₆₇	—	8	2	no	—
126	D₄×C₆₇	—	4	4	no	—
127	D₄×C₆₇	—	4	4	no	—
128	D₄×C₆₇	—	8	2	no	—
129	D₄×C₆₇	—	8	2	no	—
130	D₄×C₆₇	—	8	2	no	—
131	D₄×C₆₇	—	8	1	no	—
132	D₄×C₆₇	—	8	1	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	8	2	no	—
139	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
142	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₆₇	#137	1	2	yes	—
147	C₈×C₆₇	—	2	4	no	—
148	C₈×C₆₇	—	2	4	no	—
149	C₈×C₆₇	—	4	2	no	—
150	C₈×C₆₇	—	8	1	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	—	2	1	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	2	no	—
155	C₂²×C₆₇	—	8	2	no	—
156	C₂²×C₆₇	—	16	4	no	—
157	C₂²×C₆₇	#392	1	1	yes	characteristic
158	C₂²×C₆₇	#461	1	2	yes	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	2	1	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	—	8	1	no	—
163	C₄×C₆₇	—	8	1	no	—
164	C₄×C₆₇	#448	1	2	yes	—
165	C₄×C₆₇	#453	1	2	yes	—
166	C₄×C₆₇	#471	1	2	yes	—
167	C₄×C₆₇	#492	1	1	yes	characteristic
168	C₄×C₆₇	#497	1	1	yes	characteristic
169	#10153	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂×C₆₇	—	2	2	no	—
171	C₂×C₆₇	—	16	2	no	—
172	C₂×C₆₇	#2002	1	1	yes	characteristic
173	C₂×C₆₇	#2013	1	1	yes	characteristic
174	C₂×C₆₇	#2019	1	1	yes	characteristic
175	#3208	C₂×C₆₇	1	1	yes	characteristic
176	#3165	C₂×C₆₇	1	1	yes	characteristic, complemented
177	C₂×#492	C₂×C₆₇	1	1	yes	characteristic
178	#1965	C₂×C₆₇	1	1	yes	characteristic
179	#1549	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#1493	C₂×C₆₇	1	1	yes	characteristic, complemented
181	#1278	C₂×C₆₇	1	1	yes	characteristic, complemented
182	C₆₇	#10153	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×SD₁₆	—	2	2	no	—
184	#533	C₂²×C₆₇	1	1	yes	characteristic
185	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
186	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
187	#501	—	2	2	no	—
188	#500	—	2	1	no	—
189	#498	—	2	1	no	—
190	#492	—	2	2	no	—
191	#492	—	2	1	no	—
192	#492	—	2	2	no	—
193	#486	—	2	1	no	—
194	#460	—	2	2	no	—
195	#450	—	2	2	no	—
196	#431	C₂²×C₆₇	1	1	yes	characteristic
197	#428	—	2	2	no	—
198	#423	C₂²×C₆₇	1	1	yes	characteristic
199	C₂×#100	—	2	2	no	—
200	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
202	#332	—	2	2	no	—
203	Q₈⋊C₈	—	2	2	no	—
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	—	2	1	no	—
207	C₂²×D₄	D₄×C₆₇	1	2	yes	—
208	C₂×SD₁₆	—	2	8	no	—
209	C₂×SD₁₆	—	2	4	no	—
210	C₂×SD₁₆	—	4	8	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
213	C₂²×C₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
218	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
219	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
220	#127	—	2	1	no	—
221	#127	—	2	1	no	—
222	#127	—	2	1	no	—
223	#127	—	2	1	no	—
224	#125	—	2	1	no	—
225	C₂²⋊Q₈	—	2	1	no	—
226	C₂²⋊Q₈	—	2	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	#117	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
233	#107	—	2	4	no	—
234	#106	—	2	4	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
238	#102	—	2	4	no	—
239	#100	—	4	4	no	—
240	C₂²⋊C₈	—	2	1	no	—
241	#97	—	2	1	no	—
242	C₄×C₈	—	2	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	D₄×C₆₇	1	2	yes	—
246	C₂⁴	—	2	2	no	—
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	4	4	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	8	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
265	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
266	SD₁₆	—	4	16	no	—
267	SD₁₆	—	8	4	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	SD₁₆×C₆₇	1	4	yes	complemented
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₄²	—	4	1	no	—
288	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
292	C₂³	—	4	2	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	8	2	no	—
295	C₂³	—	8	2	no	—
296	C₂³	#121×C₆₇	1	1	yes	characteristic
297	Q₈	—	4	4	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	4	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	1	no	—
307	D₄	—	8	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
315	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
316	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
317	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
318	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
319	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
321	C₂×C₄	#137×C₆₇	1	2	yes	—
322	C₈	—	2	4	no	—
323	C₈	—	2	4	no	—
324	C₈	—	4	2	no	—
325	C₈	—	8	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	16	4	no	—
332	C₂²	#392×C₆₇	1	1	yes	characteristic
333	C₂²	#461×C₆₇	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	#448×C₆₇	1	2	yes	—
340	C₄	#453×C₆₇	1	2	yes	—
341	C₄	#471×C₆₇	1	2	yes	—
342	C₄	#492×C₆₇	1	1	yes	characteristic
343	C₄	#497×C₆₇	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	16	2	no	—
346	C₂	C₆₇×#2002	1	1	yes	characteristic
347	C₂	C₆₇×#2013	1	1	yes	characteristic
348	C₂	C₆₇×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
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
2. C₆₇×#3165	C₂	semidirect
5. C₆₇×#1549	C₂	semidirect
6. C₆₇×#1493	C₂	semidirect
7. C₆₇×#1278	C₂	semidirect
95. M₁₆×C₆₇	SD₁₆	semidirect
169. #10153	C₆₇	direct (N × H)
176. #3165	C₂×C₆₇	semidirect
179. #1549	C₂×C₆₇	semidirect
180. #1493	C₂×C₆₇	semidirect
181. #1278	C₂×C₆₇	semidirect
182. C₆₇	#10153	direct (N × H)
270. M₁₆	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3208	2
C₆₇×#3165	2
C₂×#492×C₆₇	2
C₆₇×#1965	2
C₆₇×#1549	2
C₆₇×#1493	2
C₆₇×#1278	2
#10153	67

Subgroups — C67×#10153 #30689404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10153 #30689404