Subgroups — C₆₇×#20138 #30699389

Order 17152.

652 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3399	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3305	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1721	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1698	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1667	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1630	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1624	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1621	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
10	#559×C₆₇	C₂²	1	1	yes	characteristic
11	#555×C₆₇	C₂²	1	1	yes	characteristic
12	#550×C₆₇	C₂²	1	2	yes	—
13	#496×C₆₇	—	2	2	no	—
14	#493×C₆₇	C₂²	1	2	yes	—
15	#493×C₆₇	C₂²	1	1	yes	characteristic
16	#492×C₆₇	C₂²	1	1	yes	characteristic
17	#486×C₆₇	C₂²	1	2	yes	—
18	#482×C₆₇	—	2	2	no	—
19	#477×C₆₇	C₂²	1	2	yes	—
20	#466×C₆₇	—	2	2	no	—
21	#463×C₆₇	—	2	2	no	—
22	#460×C₆₇	C₂²	1	2	yes	—
23	#459×C₆₇	—	2	2	no	—
24	#459×C₆₇	C₂²	1	2	yes	—
25	#459×C₆₇	C₂²	1	2	yes	—
26	#449×C₆₇	—	2	2	no	—
27	#446×C₆₇	C₂²	1	2	yes	—
28	#442×C₆₇	—	2	2	no	—
29	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
30	C₈×D₄×C₆₇	C₂²	1	2	yes	—
31	#416×C₆₇	—	2	2	no	—
32	#335×C₆₇	C₂²	1	2	yes	—
33	#333×C₆₇	C₂²	1	1	yes	characteristic
34	#331×C₆₇	C₂²	1	2	yes	—
35	#331×C₆₇	C₂²	1	1	yes	characteristic
36	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
37	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#322×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×SD₁₆×C₆₇	—	2	2	no	—
41	C₂×SD₁₆×C₆₇	—	4	1	no	—
42	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
43	C₂×D₈×C₆₇	—	2	2	no	—
44	C₂×D₈×C₆₇	—	2	2	no	—
45	C₂×D₈×C₆₇	—	2	2	no	—
46	C₂×D₈×C₆₇	—	2	2	no	—
47	C₂×D₈×C₆₇	—	4	1	no	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
50	#127×C₆₇	—	2	2	no	—
51	#127×C₆₇	C₂³	1	1	yes	characteristic
52	#127×C₆₇	C₂³	1	2	yes	—
53	#124×C₆₇	—	2	2	no	—
54	#124×C₆₇	—	4	1	no	—
55	#123×C₆₇	—	4	1	no	—
56	#121×C₆₇	—	2	2	no	—
57	#121×C₆₇	—	2	2	no	—
58	#121×C₆₇	—	4	2	no	—
59	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	C₂³	1	2	yes	—
63	#117×C₆₇	—	2	2	no	—
64	#107×C₆₇	—	2	2	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	#97×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	#41×C₆₇	—	4	4	no	—
82	#41×C₆₇	—	8	2	no	—
83	C₂×Q₈×C₆₇	—	2	1	no	—
84	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	2	no	—
89	C₂×D₄×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	4	2	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	4	1	no	—
93	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	4	2	no	—
96	C₂²×C₄×C₆₇	—	4	1	no	—
97	SD₁₆×C₆₇	—	4	2	no	—
98	SD₁₆×C₆₇	—	8	2	no	—
99	SD₁₆×C₆₇	—	8	2	no	—
100	D₈×C₆₇	—	4	4	no	—
101	D₈×C₆₇	—	8	2	no	—
102	D₈×C₆₇	—	8	2	no	—
103	D₈×C₆₇	—	8	2	no	—
104	D₈×C₆₇	—	8	2	no	—
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	8	2	no	—
109	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²⋊C₄×C₆₇	—	2	2	no	—
116	C₂²⋊C₄×C₆₇	—	4	2	no	—
117	C₂²⋊C₄×C₆₇	—	4	2	no	—
118	C₂²⋊C₄×C₆₇	—	4	1	no	—
119	C₂²⋊C₄×C₆₇	—	4	2	no	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₂³×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	—	4	1	no	—
125	C₂³×C₆₇	—	4	2	no	—
126	Q₈×C₆₇	—	2	2	no	—
127	Q₈×C₆₇	—	2	2	no	—
128	Q₈×C₆₇	—	8	1	no	—
129	D₄×C₆₇	—	2	2	no	—
130	D₄×C₆₇	—	2	2	no	—
131	D₄×C₆₇	—	4	2	no	—
132	D₄×C₆₇	—	4	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	2	no	—
138	D₄×C₆₇	—	8	1	no	—
139	D₄×C₆₇	—	8	4	no	—
140	D₄×C₆₇	—	16	2	no	—
141	D₄×C₆₇	—	16	1	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	1	no	—
148	C₂×C₄×C₆₇	—	8	4	no	—
149	C₂×C₄×C₆₇	—	16	1	no	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
151	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₈×C₆₇	—	2	2	no	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	—	2	2	no	—
155	C₈×C₆₇	—	4	2	no	—
156	C₈×C₆₇	—	4	2	no	—
157	C₂²×C₆₇	—	4	2	no	—
158	C₂²×C₆₇	—	4	2	no	—
159	C₂²×C₆₇	—	4	2	no	—
160	C₂²×C₆₇	—	8	2	no	—
161	C₂²×C₆₇	—	8	2	no	—
162	C₂²×C₆₇	—	8	2	no	—
163	C₂²×C₆₇	—	8	1	no	—
164	C₂²×C₆₇	—	8	2	no	—
165	C₂²×C₆₇	D₄²	1	1	yes	characteristic
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	4	2	no	—
168	C₄×C₆₇	—	4	2	no	—
169	C₄×C₆₇	—	4	2	no	—
170	C₄×C₆₇	—	8	1	no	—
171	C₄×C₆₇	#575	1	2	yes	—
172	C₄×C₆₇	#576	1	2	yes	—
173	#20138	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂×C₆₇	—	8	2	no	—
175	C₂×C₆₇	—	16	2	no	—
176	C₂×C₆₇	—	16	1	no	—
177	C₂×C₆₇	#3268	1	2	yes	—
178	C₂×C₆₇	#3292	1	1	yes	characteristic
179	#3399	C₂×C₆₇	1	2	yes	complemented
180	#3305	C₂×C₆₇	1	2	yes	complemented
181	#1721	C₂×C₆₇	1	1	yes	characteristic, complemented
182	#1698	C₂×C₆₇	1	2	yes	complemented
183	#1667	C₂×C₆₇	1	2	yes	complemented
184	#1630	C₂×C₆₇	1	1	yes	characteristic, complemented
185	#1624	C₂×C₆₇	1	1	yes	characteristic, complemented
186	#1621	C₂×C₆₇	1	2	yes	complemented
187	#1585	C₂×C₆₇	1	2	yes	complemented
188	C₆₇	#20138	1	1	yes	characteristic, direct product factor, Sylow
189	#559	C₂²×C₆₇	1	1	yes	characteristic
190	#555	C₂²×C₆₇	1	1	yes	characteristic
191	#550	C₂²×C₆₇	1	2	yes	—
192	#496	—	2	2	no	—
193	#493	C₂²×C₆₇	1	2	yes	—
194	#493	C₂²×C₆₇	1	1	yes	characteristic
195	#492	C₂²×C₆₇	1	1	yes	characteristic
196	#486	C₂²×C₆₇	1	2	yes	—
197	#482	—	2	2	no	—
198	#477	C₂²×C₆₇	1	2	yes	—
199	#466	—	2	2	no	—
200	#463	—	2	2	no	—
201	#460	C₂²×C₆₇	1	2	yes	—
202	#459	—	2	2	no	—
203	#459	C₂²×C₆₇	1	2	yes	—
204	#459	C₂²×C₆₇	1	2	yes	—
205	#449	—	2	2	no	—
206	#446	C₂²×C₆₇	1	2	yes	—
207	#442	—	2	2	no	—
208	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
209	C₈×D₄	C₂²×C₆₇	1	2	yes	—
210	#416	—	2	2	no	—
211	#335	C₂²×C₆₇	1	2	yes	—
212	#333	C₂²×C₆₇	1	1	yes	characteristic
213	#331	C₂²×C₆₇	1	2	yes	—
214	#331	C₂²×C₆₇	1	1	yes	characteristic
215	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
216	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
217	#322	C₂²×C₆₇	1	1	yes	characteristic
218	C₂×#41	—	2	2	no	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×SD₁₆	—	4	1	no	—
221	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
222	C₂×D₈	—	2	2	no	—
223	C₂×D₈	—	2	2	no	—
224	C₂×D₈	—	2	2	no	—
225	C₂×D₈	—	2	2	no	—
226	C₂×D₈	—	4	1	no	—
227	C₂²×C₈	—	2	2	no	—
228	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
229	#127	—	2	2	no	—
230	#127	C₂³×C₆₇	1	2	yes	—
231	#127	C₂³×C₆₇	1	1	yes	characteristic
232	#124	—	2	2	no	—
233	#124	—	4	1	no	—
234	#123	—	4	1	no	—
235	#121	—	2	2	no	—
236	#121	—	2	2	no	—
237	#121	—	4	2	no	—
238	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	—	2	2	no	—
241	C₄×D₄	C₂³×C₆₇	1	2	yes	—
242	#117	—	2	2	no	—
243	#107	—	2	2	no	—
244	#106	D₄×C₆₇	1	2	yes	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
247	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
248	#103	—	2	2	no	—
249	#103	D₄×C₆₇	1	2	yes	—
250	#102	—	2	2	no	—
251	#102	—	2	2	no	—
252	#102	—	2	2	no	—
253	#102	—	2	2	no	—
254	#102	—	2	2	no	—
255	#102	D₄×C₆₇	1	2	yes	—
256	C₂²⋊C₈	—	2	2	no	—
257	#97	—	2	2	no	—
258	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
259	C₄×C₈	C₂³×C₆₇	1	2	yes	—
260	#41	—	4	4	no	—
261	#41	—	8	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
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	1	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	1	no	—
276	SD₁₆	—	4	2	no	—
277	SD₁₆	—	8	2	no	—
278	SD₁₆	—	8	2	no	—
279	D₈	—	4	4	no	—
280	D₈	—	8	2	no	—
281	D₈	—	8	2	no	—
282	D₈	—	8	2	no	—
283	D₈	—	8	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	8	2	no	—
288	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
293	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
302	C₂³	—	2	2	no	—
303	C₂³	—	4	1	no	—
304	C₂³	—	4	2	no	—
305	Q₈	—	2	2	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	8	1	no	—
308	D₄	—	2	2	no	—
309	D₄	—	2	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	8	2	no	—
317	D₄	—	8	1	no	—
318	D₄	—	8	4	no	—
319	D₄	—	16	2	no	—
320	D₄	—	16	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	8	4	no	—
328	C₂×C₄	—	16	1	no	—
329	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
330	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₂²	—	4	2	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	—	8	1	no	—
343	C₂²	—	8	2	no	—
344	C₂²	D₄²×C₆₇	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	4	2	no	—
347	C₄	—	4	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	8	1	no	—
350	C₄	#575×C₆₇	1	2	yes	—
351	C₄	#576×C₆₇	1	2	yes	—
352	C₂	—	8	2	no	—
353	C₂	—	16	2	no	—
354	C₂	—	16	1	no	—
355	C₂	C₆₇×#3268	1	2	yes	—
356	C₂	C₆₇×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
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×#3399	C₂	semidirect
2. C₆₇×#3305	C₂	semidirect
3. C₆₇×#1721	C₂	semidirect
4. C₆₇×#1698	C₂	semidirect
5. C₆₇×#1667	C₂	semidirect
6. C₆₇×#1630	C₂	semidirect
7. C₆₇×#1624	C₂	semidirect
8. C₆₇×#1621	C₂	semidirect
9. C₆₇×#1585	C₂	semidirect
173. #20138	C₆₇	direct (N × H)
179. #3399	C₂×C₆₇	semidirect
180. #3305	C₂×C₆₇	semidirect
181. #1721	C₂×C₆₇	semidirect
182. #1698	C₂×C₆₇	semidirect
183. #1667	C₂×C₆₇	semidirect
184. #1630	C₂×C₆₇	semidirect
185. #1624	C₂×C₆₇	semidirect
186. #1621	C₂×C₆₇	semidirect
187. #1585	C₂×C₆₇	semidirect
188. C₆₇	#20138	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×#3399	2
C₆₇×#3305	2
C₆₇×#1721	2
C₆₇×#1698	2
C₆₇×#1667	2
C₆₇×#1630	2
C₆₇×#1624	2
C₆₇×#1621	2
C₆₇×#1585	2
#20138	67

Subgroups — C67×#20138 #30699389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20138 #30699389