Subgroups — C₆₇×#20131 #30699382

Order 17152.

640 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×#3341	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3325	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1722	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1695	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1677	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1634	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1629	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1624	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1586	C₂	1	2	yes	maximal, complemented
10	#555×C₆₇	C₂²	1	2	yes	—
11	#555×C₆₇	C₂²	1	1	yes	characteristic, complemented
12	D₄²×C₆₇	C₂²	1	1	yes	characteristic
13	#500×C₆₇	C₂²	1	2	yes	complemented
14	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	#492×C₆₇	C₂²	1	1	yes	characteristic
16	#486×C₆₇	C₂²	1	2	yes	—
17	#482×C₆₇	—	2	2	no	—
18	#482×C₆₇	—	2	2	no	—
19	#482×C₆₇	—	2	2	no	—
20	#476×C₆₇	C₂²	1	2	yes	—
21	#476×C₆₇	C₂²	1	2	yes	—
22	#476×C₆₇	C₂²	1	2	yes	complemented
23	#469×C₆₇	—	2	2	no	—
24	#466×C₆₇	—	2	2	no	—
25	#459×C₆₇	C₂²	1	2	yes	—
26	#447×C₆₇	—	2	2	no	—
27	C₄×D₈×C₆₇	C₂²	1	2	yes	—
28	#436×C₆₇	C₂²	1	2	yes	—
29	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
30	#427×C₆₇	—	2	2	no	—
31	#418×C₆₇	—	2	2	no	—
32	#334×C₆₇	C₂²	1	2	yes	complemented
33	#333×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	#333×C₆₇	C₂²	1	2	yes	complemented
35	#331×C₆₇	C₂²	1	1	yes	characteristic
36	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
37	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
38	#322×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₆₇	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	—	4	1	no	—
41	C₂×SD₁₆×C₆₇	—	4	1	no	—
42	C₂×D₈×C₆₇	—	2	2	no	—
43	C₂×D₈×C₆₇	D₄	1	2	yes	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
48	#127×C₆₇	C₂³	1	1	yes	characteristic
49	#124×C₆₇	—	4	2	no	—
50	#124×C₆₇	—	4	1	no	—
51	#121×C₆₇	—	2	2	no	—
52	#121×C₆₇	—	2	2	no	—
53	#121×C₆₇	—	2	2	no	—
54	#121×C₆₇	—	2	2	no	—
55	#120×C₆₇	—	4	1	no	—
56	C₄×D₄×C₆₇	C₂³	1	2	yes	—
57	C₄×D₄×C₆₇	C₂³	1	2	yes	—
58	#117×C₆₇	—	2	2	no	—
59	#117×C₆₇	—	2	2	no	—
60	#117×C₆₇	—	2	2	no	—
61	#107×C₆₇	—	2	2	no	—
62	#107×C₆₇	—	2	2	no	—
63	#107×C₆₇	D₄	1	2	yes	—
64	#106×C₆₇	—	2	2	no	—
65	#106×C₆₇	—	2	2	no	—
66	#106×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
69	#102×C₆₇	—	2	2	no	—
70	#102×C₆₇	—	2	2	no	—
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	D₄	1	2	yes	—
76	#102×C₆₇	D₄	1	2	yes	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₂²⋊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	1	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	1	no	—
86	C₂×D₄×C₆₇	—	4	2	no	—
87	C₂×D₄×C₆₇	—	4	4	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂×D₄×C₆₇	—	8	2	no	—
91	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	Q₁₆×C₆₇	—	8	2	no	—
96	SD₁₆×C₆₇	—	8	2	no	—
97	D₈×C₆₇	—	4	2	no	—
98	D₈×C₆₇	—	8	2	no	—
99	M₁₆×C₆₇	—	8	2	no	—
100	C₂×C₈×C₆₇	—	2	1	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	4	4	no	—
104	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂²⋊C₄×C₆₇	—	4	1	no	—
114	C₂²⋊C₄×C₆₇	—	4	2	no	—
115	C₂²⋊C₄×C₆₇	—	4	2	no	—
116	C₂²⋊C₄×C₆₇	—	4	2	no	—
117	C₄²×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	—	4	2	no	—
119	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂³×C₆₇	—	2	2	no	—
121	C₂³×C₆₇	—	2	2	no	—
122	C₂³×C₆₇	—	4	1	no	—
123	C₂³×C₆₇	—	8	4	no	—
124	C₂³×C₆₇	—	16	1	no	—
125	Q₈×C₆₇	—	8	1	no	—
126	Q₈×C₆₇	—	8	2	no	—
127	D₄×C₆₇	—	2	2	no	—
128	D₄×C₆₇	—	2	2	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	—	8	1	no	—
134	D₄×C₆₇	—	8	4	no	—
135	D₄×C₆₇	—	16	1	no	—
136	D₄×C₆₇	—	16	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	1	no	—
145	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
147	C₈×C₆₇	—	2	2	no	—
148	C₈×C₆₇	—	2	2	no	—
149	C₈×C₆₇	—	2	2	no	—
150	C₈×C₆₇	—	4	2	no	—
151	C₈×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	—	4	2	no	—
153	C₂²×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	—	4	2	no	—
158	C₂²×C₆₇	—	8	2	no	—
159	C₂²×C₆₇	—	8	1	no	—
160	C₂²×C₆₇	—	16	4	no	—
161	C₂²×C₆₇	D₄²	1	1	yes	characteristic
162	C₄×C₆₇	—	2	1	no	—
163	C₄×C₆₇	—	4	2	no	—
164	C₄×C₆₇	—	4	2	no	—
165	C₄×C₆₇	—	8	1	no	—
166	C₄×C₆₇	—	8	2	no	—
167	C₄×C₆₇	#575	1	2	yes	—
168	C₄×C₆₇	#577	1	2	yes	—
169	#20131	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂×C₆₇	—	8	2	no	—
171	C₂×C₆₇	—	8	2	no	—
172	C₂×C₆₇	—	16	1	no	—
173	C₂×C₆₇	#3267	1	2	yes	—
174	C₂×C₆₇	#3292	1	1	yes	characteristic
175	#3341	C₂×C₆₇	1	2	yes	complemented
176	#3325	C₂×C₆₇	1	2	yes	complemented
177	#1722	C₂×C₆₇	1	1	yes	characteristic, complemented
178	#1695	C₂×C₆₇	1	2	yes	complemented
179	#1677	C₂×C₆₇	1	2	yes	complemented
180	#1634	C₂×C₆₇	1	2	yes	complemented
181	#1629	C₂×C₆₇	1	1	yes	characteristic, complemented
182	#1624	C₂×C₆₇	1	1	yes	characteristic, complemented
183	#1586	C₂×C₆₇	1	2	yes	complemented
184	C₆₇	#20131	1	1	yes	characteristic, direct product factor, Sylow
185	#555	C₂²×C₆₇	1	2	yes	—
186	#555	C₂²×C₆₇	1	1	yes	characteristic, complemented
187	D₄²	C₂²×C₆₇	1	1	yes	characteristic
188	#500	C₂²×C₆₇	1	2	yes	complemented
189	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
190	#492	C₂²×C₆₇	1	1	yes	characteristic
191	#486	C₂²×C₆₇	1	2	yes	—
192	#482	—	2	2	no	—
193	#482	—	2	2	no	—
194	#482	—	2	2	no	—
195	#476	C₂²×C₆₇	1	2	yes	—
196	#476	C₂²×C₆₇	1	2	yes	—
197	#476	C₂²×C₆₇	1	2	yes	complemented
198	#469	—	2	2	no	—
199	#466	—	2	2	no	—
200	#459	C₂²×C₆₇	1	2	yes	—
201	#447	—	2	2	no	—
202	C₄×D₈	C₂²×C₆₇	1	2	yes	—
203	#436	C₂²×C₆₇	1	2	yes	—
204	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
205	#427	—	2	2	no	—
206	#418	—	2	2	no	—
207	#334	C₂²×C₆₇	1	2	yes	complemented
208	#333	C₂²×C₆₇	1	1	yes	characteristic, complemented
209	#333	C₂²×C₆₇	1	2	yes	complemented
210	#331	C₂²×C₆₇	1	1	yes	characteristic
211	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
212	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
213	#322	C₂²×C₆₇	1	1	yes	characteristic, complemented
214	C₂²×D₄	—	2	2	no	—
215	C₂×Q₁₆	—	4	1	no	—
216	C₂×SD₁₆	—	4	1	no	—
217	C₂×D₈	—	2	2	no	—
218	C₂×D₈	D₄×C₆₇	1	2	yes	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂²×C₈	—	2	2	no	—
221	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
222	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
223	#127	C₂³×C₆₇	1	1	yes	characteristic
224	#124	—	4	2	no	—
225	#124	—	4	1	no	—
226	#121	—	2	2	no	—
227	#121	—	2	2	no	—
228	#121	—	2	2	no	—
229	#121	—	2	2	no	—
230	#120	—	4	1	no	—
231	C₄×D₄	C₂³×C₆₇	1	2	yes	—
232	C₄×D₄	C₂³×C₆₇	1	2	yes	—
233	#117	—	2	2	no	—
234	#117	—	2	2	no	—
235	#117	—	2	2	no	—
236	#107	—	2	2	no	—
237	#107	—	2	2	no	—
238	#107	D₄×C₆₇	1	2	yes	—
239	#106	—	2	2	no	—
240	#106	—	2	2	no	—
241	#106	—	2	2	no	—
242	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
243	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	#102	—	2	2	no	—
248	#102	—	2	2	no	—
249	#102	—	2	2	no	—
250	#102	D₄×C₆₇	1	2	yes	—
251	#102	D₄×C₆₇	1	2	yes	—
252	C₂²⋊C₈	—	2	2	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₆₇	1	2	yes	—
256	C₂⁴	—	4	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	4	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	8	2	no	—
266	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
267	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	Q₁₆	—	8	2	no	—
271	SD₁₆	—	8	2	no	—
272	D₈	—	4	2	no	—
273	D₈	—	8	2	no	—
274	M₁₆	—	8	2	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	4	no	—
279	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
285	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	2	no	—
294	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
295	C₂³	—	2	2	no	—
296	C₂³	—	2	2	no	—
297	C₂³	—	4	1	no	—
298	C₂³	—	8	4	no	—
299	C₂³	—	16	1	no	—
300	Q₈	—	8	1	no	—
301	Q₈	—	8	2	no	—
302	D₄	—	2	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	1	no	—
309	D₄	—	8	4	no	—
310	D₄	—	16	1	no	—
311	D₄	—	16	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
321	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	4	2	no	—
326	C₈	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	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	1	no	—
335	C₂²	—	16	4	no	—
336	C₂²	D₄²×C₆₇	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	4	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	8	1	no	—
341	C₄	—	8	2	no	—
342	C₄	#575×C₆₇	1	2	yes	—
343	C₄	#577×C₆₇	1	2	yes	—
344	C₂	—	8	2	no	—
345	C₂	—	8	2	no	—
346	C₂	—	16	1	no	—
347	C₂	C₆₇×#3267	1	2	yes	—
348	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×#3341	C₂	semidirect
2. C₆₇×#3325	C₂	semidirect
3. C₆₇×#1722	C₂	semidirect
4. C₆₇×#1695	C₂	semidirect
5. C₆₇×#1677	C₂	semidirect
6. C₆₇×#1634	C₂	semidirect
7. C₆₇×#1629	C₂	semidirect
8. C₆₇×#1624	C₂	semidirect
9. C₆₇×#1586	C₂	semidirect
11. #555×C₆₇	C₂²	semidirect
13. #500×C₆₇	C₂²	semidirect
14. #494×C₆₇	C₂²	semidirect
22. #476×C₆₇	C₂²	semidirect
29. C₈×D₄×C₆₇	C₂²	semidirect
32. #334×C₆₇	C₂²	semidirect
33. #333×C₆₇	C₂²	semidirect
34. #333×C₆₇	C₂²	semidirect
37. D₄⋊C₈×C₆₇	C₂²	semidirect
38. #322×C₆₇	C₂²	semidirect
169. #20131	C₆₇	direct (N × H)
175. #3341	C₂×C₆₇	semidirect
176. #3325	C₂×C₆₇	semidirect
177. #1722	C₂×C₆₇	semidirect
178. #1695	C₂×C₆₇	semidirect
179. #1677	C₂×C₆₇	semidirect
180. #1634	C₂×C₆₇	semidirect
181. #1629	C₂×C₆₇	semidirect
182. #1624	C₂×C₆₇	semidirect
183. #1586	C₂×C₆₇	semidirect
184. C₆₇	#20131	direct (N × H)
186. #555	C₂²×C₆₇	semidirect
188. #500	C₂²×C₆₇	semidirect
189. #494	C₂²×C₆₇	semidirect
197. #476	C₂²×C₆₇	semidirect
204. C₈×D₄	C₂²×C₆₇	semidirect
207. #334	C₂²×C₆₇	semidirect
208. #333	C₂²×C₆₇	semidirect
209. #333	C₂²×C₆₇	semidirect
212. D₄⋊C₈	C₂²×C₆₇	semidirect
213. #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×#3341	2
C₆₇×#3325	2
C₆₇×#1722	2
C₆₇×#1695	2
C₆₇×#1677	2
C₆₇×#1634	2
C₆₇×#1629	2
C₆₇×#1624	2
C₆₇×#1586	2
#20131	67

Subgroups — C67×#20131 #30699382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20131 #30699382