Subgroups — C₆₇×#20152 #30699403

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3394	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3324	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1715	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1685	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1682	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1652	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1620	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1619	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1588	C₂	1	2	yes	maximal, complemented
10	#559×C₆₇	C₂²	1	1	yes	characteristic, complemented
11	#556×C₆₇	C₂²	1	2	yes	—
12	D₄²×C₆₇	C₂²	1	1	yes	characteristic
13	#498×C₆₇	C₂²	1	1	yes	characteristic, complemented
14	#495×C₆₇	—	2	2	no	—
15	#493×C₆₇	C₂²	1	1	yes	characteristic
16	#492×C₆₇	C₂²	1	2	yes	complemented
17	#486×C₆₇	C₂²	1	2	yes	—
18	#482×C₆₇	—	2	2	no	—
19	#474×C₆₇	C₂²	1	2	yes	—
20	#468×C₆₇	—	2	2	no	—
21	#464×C₆₇	—	2	2	no	—
22	#463×C₆₇	—	2	2	no	—
23	#462×C₆₇	C₂²	1	2	yes	complemented
24	#460×C₆₇	C₂²	1	2	yes	—
25	#459×C₆₇	C₂²	1	2	yes	—
26	#450×C₆₇	—	2	2	no	—
27	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
28	C₄×SD₁₆×C₆₇	—	2	2	no	—
29	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
30	#436×C₆₇	C₂²	1	2	yes	—
31	#418×C₆₇	—	2	2	no	—
32	#335×C₆₇	C₂²	1	2	yes	complemented
33	#332×C₆₇	C₂²	1	2	yes	complemented
34	#331×C₆₇	C₂²	1	1	yes	characteristic
35	#331×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	complemented
37	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#322×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₆₇	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	—	2	2	no	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×SD₁₆×C₆₇	—	2	2	no	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
45	C₂×D₈×C₆₇	—	2	2	no	—
46	C₂×D₈×C₆₇	—	4	1	no	—
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
49	#127×C₆₇	C₂³	1	1	yes	characteristic
50	#127×C₆₇	C₂³	1	1	yes	characteristic
51	#126×C₆₇	—	4	2	no	—
52	#124×C₆₇	—	2	2	no	—
53	#124×C₆₇	—	2	2	no	—
54	#123×C₆₇	—	4	1	no	—
55	C₂²⋊Q₈×C₆₇	—	2	2	no	—
56	#121×C₆₇	—	2	2	no	—
57	#120×C₆₇	—	4	1	no	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	C₂³	1	2	yes	—
62	#117×C₆₇	—	2	2	no	—
63	#107×C₆₇	—	2	2	no	—
64	#106×C₆₇	—	4	1	no	—
65	#106×C₆₇	D₄	1	2	yes	—
66	C₄⋊C₈×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
69	#103×C₆₇	—	2	2	no	—
70	#103×C₆₇	D₄	1	2	yes	—
71	#102×C₆₇	—	2	2	no	—
72	#102×C₆₇	—	2	2	no	—
73	#102×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	D₄	1	2	yes	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₄×C₈×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂⁴×C₆₇	—	4	1	no	—
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	1	no	—
87	C₂×D₄×C₆₇	—	2	1	no	—
88	C₂×D₄×C₆₇	—	4	2	no	—
89	C₂×D₄×C₆₇	—	4	4	no	—
90	C₂×D₄×C₆₇	—	8	2	no	—
91	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	4	1	no	—
94	Q₁₆×C₆₇	—	8	2	no	—
95	SD₁₆×C₆₇	—	4	2	no	—
96	SD₁₆×C₆₇	—	4	4	no	—
97	SD₁₆×C₆₇	—	8	2	no	—
98	SD₁₆×C₆₇	—	8	2	no	—
99	D₈×C₆₇	—	8	2	no	—
100	D₈×C₆₇	—	8	2	no	—
101	M₁₆×C₆₇	—	8	2	no	—
102	C₂×C₈×C₆₇	—	2	1	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₆₇	—	2	2	no	—
114	C₂²⋊C₄×C₆₇	—	4	2	no	—
115	C₂²⋊C₄×C₆₇	—	4	1	no	—
116	C₂²⋊C₄×C₆₇	—	4	1	no	—
117	C₂²⋊C₄×C₆₇	—	4	2	no	—
118	C₂²⋊C₄×C₆₇	—	4	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	4	2	no	—
121	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
122	C₂³×C₆₇	—	2	2	no	—
123	C₂³×C₆₇	—	4	1	no	—
124	C₂³×C₆₇	—	4	1	no	—
125	C₂³×C₆₇	—	8	4	no	—
126	C₂³×C₆₇	—	16	1	no	—
127	Q₈×C₆₇	—	2	2	no	—
128	Q₈×C₆₇	—	2	2	no	—
129	Q₈×C₆₇	—	4	2	no	—
130	Q₈×C₆₇	—	4	2	no	—
131	D₄×C₆₇	—	2	2	no	—
132	D₄×C₆₇	—	2	2	no	—
133	D₄×C₆₇	—	4	2	no	—
134	D₄×C₆₇	—	4	2	no	—
135	D₄×C₆₇	—	4	2	no	—
136	D₄×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	8	1	no	—
138	D₄×C₆₇	—	8	1	no	—
139	D₄×C₆₇	—	8	4	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	16	1	no	—
147	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
148	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₈×C₆₇	—	2	2	no	—
150	C₈×C₆₇	—	2	2	no	—
151	C₈×C₆₇	—	2	2	no	—
152	C₈×C₆₇	—	4	2	no	—
153	C₈×C₆₇	—	4	2	no	—
154	C₂²×C₆₇	—	4	2	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	1	no	—
160	C₂²×C₆₇	—	8	1	no	—
161	C₂²×C₆₇	—	16	4	no	—
162	C₂²×C₆₇	D₄²	1	1	yes	characteristic
163	C₄×C₆₇	—	2	1	no	—
164	C₄×C₆₇	—	4	2	no	—
165	C₄×C₆₇	—	4	2	no	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	#575	1	2	yes	—
169	C₄×C₆₇	#576	1	2	yes	—
170	#20152	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂×C₆₇	—	8	2	no	—
172	C₂×C₆₇	—	16	1	no	—
173	C₂×C₆₇	—	16	1	no	—
174	C₂×C₆₇	#3268	1	2	yes	—
175	C₂×C₆₇	#3291	1	1	yes	characteristic
176	#3394	C₂×C₆₇	1	2	yes	complemented
177	#3324	C₂×C₆₇	1	2	yes	complemented
178	#1715	C₂×C₆₇	1	1	yes	characteristic, complemented
179	#1685	C₂×C₆₇	1	2	yes	complemented
180	#1682	C₂×C₆₇	1	2	yes	complemented
181	#1652	C₂×C₆₇	1	2	yes	complemented
182	#1620	C₂×C₆₇	1	1	yes	characteristic, complemented
183	#1619	C₂×C₆₇	1	1	yes	characteristic
184	#1588	C₂×C₆₇	1	2	yes	complemented
185	C₆₇	#20152	1	1	yes	characteristic, direct product factor, Sylow
186	#559	C₂²×C₆₇	1	1	yes	characteristic, complemented
187	#556	C₂²×C₆₇	1	2	yes	—
188	D₄²	C₂²×C₆₇	1	1	yes	characteristic
189	#498	C₂²×C₆₇	1	1	yes	characteristic, complemented
190	#495	—	2	2	no	—
191	#493	C₂²×C₆₇	1	1	yes	characteristic
192	#492	C₂²×C₆₇	1	2	yes	complemented
193	#486	C₂²×C₆₇	1	2	yes	—
194	#482	—	2	2	no	—
195	#474	C₂²×C₆₇	1	2	yes	—
196	#468	—	2	2	no	—
197	#464	—	2	2	no	—
198	#463	—	2	2	no	—
199	#462	C₂²×C₆₇	1	2	yes	complemented
200	#460	C₂²×C₆₇	1	2	yes	—
201	#459	C₂²×C₆₇	1	2	yes	—
202	#450	—	2	2	no	—
203	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
204	C₄×SD₁₆	—	2	2	no	—
205	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
206	#436	C₂²×C₆₇	1	2	yes	—
207	#418	—	2	2	no	—
208	#335	C₂²×C₆₇	1	2	yes	complemented
209	#332	C₂²×C₆₇	1	2	yes	complemented
210	#331	C₂²×C₆₇	1	1	yes	characteristic
211	#331	C₂²×C₆₇	1	1	yes	characteristic, complemented
212	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	complemented
213	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
214	#322	C₂²×C₆₇	1	1	yes	characteristic, complemented
215	C₂²×D₄	—	2	2	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	—	4	1	no	—
223	C₂×M₁₆	—	2	2	no	—
224	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
225	#127	C₂³×C₆₇	1	1	yes	characteristic
226	#127	C₂³×C₆₇	1	1	yes	characteristic
227	#126	—	4	2	no	—
228	#124	—	2	2	no	—
229	#124	—	2	2	no	—
230	#123	—	4	1	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	#121	—	2	2	no	—
233	#120	—	4	1	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	C₂³×C₆₇	1	2	yes	—
238	#117	—	2	2	no	—
239	#107	—	2	2	no	—
240	#106	—	4	1	no	—
241	#106	D₄×C₆₇	1	2	yes	—
242	C₄⋊C₈	—	2	2	no	—
243	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
244	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
245	#103	—	2	2	no	—
246	#103	D₄×C₆₇	1	2	yes	—
247	#102	—	2	2	no	—
248	#102	—	2	2	no	—
249	#102	—	2	2	no	—
250	#102	—	2	2	no	—
251	#102	—	2	2	no	—
252	#102	D₄×C₆₇	1	2	yes	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₄×C₈	—	2	2	no	—
255	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₆₇	1	2	yes	—
257	C₂⁴	—	4	1	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	8	2	no	—
267	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	4	1	no	—
270	Q₁₆	—	8	2	no	—
271	SD₁₆	—	4	2	no	—
272	SD₁₆	—	4	4	no	—
273	SD₁₆	—	8	2	no	—
274	SD₁₆	—	8	2	no	—
275	D₈	—	8	2	no	—
276	D₈	—	8	2	no	—
277	M₁₆	—	8	2	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
282	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
288	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	1	no	—
292	C₂²⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	4	2	no	—
297	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
298	C₂³	—	2	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	1	no	—
301	C₂³	—	8	4	no	—
302	C₂³	—	16	1	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	2	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	8	1	no	—
314	D₄	—	8	1	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	16	1	no	—
323	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
324	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	—	4	2	no	—
329	C₈	—	4	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	16	4	no	—
338	C₂²	D₄²×C₆₇	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	8	2	no	—
344	C₄	#575×C₆₇	1	2	yes	—
345	C₄	#576×C₆₇	1	2	yes	—
346	C₂	—	8	2	no	—
347	C₂	—	16	1	no	—
348	C₂	—	16	1	no	—
349	C₂	C₆₇×#3268	1	2	yes	—
350	C₂	C₆₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#1619	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3394	C₂	semidirect
2. C₆₇×#3324	C₂	semidirect
3. C₆₇×#1715	C₂	semidirect
4. C₆₇×#1685	C₂	semidirect
5. C₆₇×#1682	C₂	semidirect
6. C₆₇×#1652	C₂	semidirect
7. C₆₇×#1620	C₂	semidirect
9. C₆₇×#1588	C₂	semidirect
10. #559×C₆₇	C₂²	semidirect
13. #498×C₆₇	C₂²	semidirect
16. #492×C₆₇	C₂²	semidirect
23. #462×C₆₇	C₂²	semidirect
27. C₈×Q₈×C₆₇	C₂²	semidirect
32. #335×C₆₇	C₂²	semidirect
33. #332×C₆₇	C₂²	semidirect
35. #331×C₆₇	C₂²	semidirect
36. Q₈⋊C₈×C₆₇	C₂²	semidirect
38. #322×C₆₇	C₂²	semidirect
170. #20152	C₆₇	direct (N × H)
176. #3394	C₂×C₆₇	semidirect
177. #3324	C₂×C₆₇	semidirect
178. #1715	C₂×C₆₇	semidirect
179. #1685	C₂×C₆₇	semidirect
180. #1682	C₂×C₆₇	semidirect
181. #1652	C₂×C₆₇	semidirect
182. #1620	C₂×C₆₇	semidirect
184. #1588	C₂×C₆₇	semidirect
185. C₆₇	#20152	direct (N × H)
186. #559	C₂²×C₆₇	semidirect
189. #498	C₂²×C₆₇	semidirect
192. #492	C₂²×C₆₇	semidirect
199. #462	C₂²×C₆₇	semidirect
203. C₈×Q₈	C₂²×C₆₇	semidirect
208. #335	C₂²×C₆₇	semidirect
209. #332	C₂²×C₆₇	semidirect
211. #331	C₂²×C₆₇	semidirect
212. Q₈⋊C₈	C₂²×C₆₇	semidirect
214. #322	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3394	2
C₆₇×#3324	2
C₆₇×#1715	2
C₆₇×#1685	2
C₆₇×#1682	2
C₆₇×#1652	2
C₆₇×#1620	2
C₆₇×#1619	2
C₆₇×#1588	2
#20152	67

Subgroups — C67×#20152 #30699403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20152 #30699403