Subgroups — C₆₇×#33921 #30713172

Order 17152.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 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	1	yes	maximal, characteristic, complemented
2	C₆₇×#3334	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2234	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2233	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2218	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2159	C₂	1	2	yes	maximal, complemented
12	#555×C₆₇	—	2	1	no	—
13	#547×C₆₇	—	2	1	no	—
14	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
16	#500×C₆₇	C₂²	1	1	yes	characteristic
17	#500×C₆₇	C₂²	1	1	yes	characteristic
18	#482×C₆₇	—	2	2	no	—
19	#480×C₆₇	—	2	2	no	—
20	#476×C₆₇	—	2	1	no	—
21	#474×C₆₇	—	2	1	no	—
22	#466×C₆₇	C₂²	1	2	yes	—
23	#466×C₆₇	C₂²	1	2	yes	—
24	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	#427×C₆₇	C₂²	1	2	yes	—
28	C₂×#107×C₆₇	C₂²	1	2	yes	—
29	#367×C₆₇	C₂²	1	2	yes	—
30	#367×C₆₇	C₂²	1	2	yes	—
31	#367×C₆₇	C₂²	1	2	yes	—
32	#366×C₆₇	C₂²	1	1	yes	characteristic
33	#366×C₆₇	C₂²	1	2	yes	—
34	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	#357×C₆₇	C₂²	1	1	yes	characteristic
36	#357×C₆₇	C₂²	1	1	yes	characteristic
37	#357×C₆₇	C₂²	1	2	yes	—
38	#357×C₆₇	C₂²	1	2	yes	—
39	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#41×C₆₇	—	4	1	no	—
42	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₆₇	—	2	1	no	—
46	C₄⋊Q₈×C₆₇	—	2	1	no	—
47	#124×C₆₇	—	4	2	no	—
48	#123×C₆₇	—	4	2	no	—
49	C₂²⋊Q₈×C₆₇	—	4	1	no	—
50	#121×C₆₇	—	2	2	no	—
51	#121×C₆₇	—	2	2	no	—
52	#121×C₆₇	—	4	1	no	—
53	C₄×D₄×C₆₇	—	2	1	no	—
54	C₄×D₄×C₆₇	—	2	1	no	—
55	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
56	#117×C₆₇	—	2	2	no	—
57	#117×C₆₇	—	4	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	D₁₆×C₆₇	—	2	2	no	—
60	M₃₂×C₆₇	—	2	4	no	—
61	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
62	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
64	#107×C₆₇	—	2	2	no	—
65	#107×C₆₇	—	2	1	no	—
66	#107×C₆₇	—	2	1	no	—
67	#107×C₆₇	C₂³	1	2	yes	—
68	#107×C₆₇	C₂³	1	2	yes	—
69	#107×C₆₇	C₂³	1	1	yes	characteristic
70	#107×C₆₇	C₂³	1	1	yes	characteristic
71	#106×C₆₇	—	2	2	no	—
72	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
73	#102×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	—	2	1	no	—
76	#102×C₆₇	—	2	1	no	—
77	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	#41×C₆₇	—	8	4	no	—
80	C₂×Q₈×C₆₇	—	4	1	no	—
81	C₂×Q₈×C₆₇	—	4	1	no	—
82	C₂×D₄×C₆₇	—	2	1	no	—
83	C₂×D₄×C₆₇	—	2	1	no	—
84	C₂×D₄×C₆₇	—	4	2	no	—
85	C₂×D₄×C₆₇	—	4	2	no	—
86	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	—	4	2	no	—
88	C₂²×C₄×C₆₇	—	4	1	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
91	D₈×C₆₇	—	2	1	no	—
92	D₈×C₆₇	—	2	1	no	—
93	D₈×C₆₇	—	2	1	no	—
94	D₈×C₆₇	—	2	1	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₆₇	—	2	1	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	2	1	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄⋊C₄×C₆₇	—	4	1	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₆₇	—	4	2	no	—
109	C₂²⋊C₄×C₆₇	—	4	2	no	—
110	C₂²⋊C₄×C₆₇	—	4	1	no	—
111	C₂²⋊C₄×C₆₇	—	4	1	no	—
112	C₂²⋊C₄×C₆₇	—	4	2	no	—
113	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
114	C₄²×C₆₇	—	4	2	no	—
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₆₇	—	2	2	no	—
117	C₁₆×C₆₇	—	2	1	no	—
118	C₁₆×C₆₇	#41	1	2	yes	complemented
119	C₂³×C₆₇	—	4	1	no	—
120	C₂³×C₆₇	—	4	1	no	—
121	C₂³×C₆₇	C₂×D₈	1	2	yes	—
122	Q₈×C₆₇	—	8	2	no	—
123	Q₈×C₆₇	—	16	1	no	—
124	D₄×C₆₇	—	2	2	no	—
125	D₄×C₆₇	—	4	1	no	—
126	D₄×C₆₇	—	4	1	no	—
127	D₄×C₆₇	—	4	1	no	—
128	D₄×C₆₇	—	4	1	no	—
129	D₄×C₆₇	—	8	4	no	—
130	D₄×C₆₇	—	16	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	8	4	no	—
138	C₂×C₄×C₆₇	—	8	2	no	—
139	C₂×C₄×C₆₇	—	8	2	no	—
140	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₆₇	—	2	2	no	—
145	C₈×C₆₇	—	2	1	no	—
146	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	8	1	no	—
152	C₂²×C₆₇	—	8	1	no	—
153	C₂²×C₆₇	—	8	1	no	—
154	C₂²×C₆₇	—	8	1	no	—
155	C₂²×C₆₇	—	8	1	no	—
156	C₂²×C₆₇	—	8	1	no	—
157	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	8	1	no	—
161	C₄×C₆₇	—	8	1	no	—
162	C₄×C₆₇	—	8	2	no	—
163	C₄×C₆₇	—	8	2	no	—
164	C₄×C₆₇	#509	1	2	yes	—
165	C₄×C₆₇	#547	1	1	yes	characteristic
166	C₄×C₆₇	#578	1	1	yes	characteristic
167	#33921	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₆₇	—	4	2	no	—
169	C₂×C₆₇	—	16	1	no	—
170	C₂×C₆₇	—	16	1	no	—
171	C₂×C₆₇	#3334	1	1	yes	characteristic
172	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
173	C₂×C₆₇	#3416	1	1	yes	characteristic
174	#3341	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#3334	C₂×C₆₇	1	1	yes	characteristic, complemented
176	#2254	C₂×C₆₇	1	1	yes	characteristic, complemented
177	#2234	C₂×C₆₇	1	2	yes	complemented
178	#2233	C₂×C₆₇	1	2	yes	complemented
179	#2226	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#2224	C₂×C₆₇	1	1	yes	characteristic, complemented
181	#2218	C₂×C₆₇	1	2	yes	complemented
182	C₄×D₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
183	#2169	C₂×C₆₇	1	1	yes	characteristic, complemented
184	#2159	C₂×C₆₇	1	2	yes	complemented
185	C₆₇	#33921	1	1	yes	characteristic, direct product factor, Sylow
186	#555	—	2	1	no	—
187	#547	—	2	1	no	—
188	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
189	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
190	#500	C₂²×C₆₇	1	1	yes	characteristic
191	#500	C₂²×C₆₇	1	1	yes	characteristic
192	#482	—	2	2	no	—
193	#480	—	2	2	no	—
194	#476	—	2	1	no	—
195	#474	—	2	1	no	—
196	#466	C₂²×C₆₇	1	2	yes	—
197	#466	C₂²×C₆₇	1	2	yes	—
198	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
199	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
200	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
201	#427	C₂²×C₆₇	1	2	yes	—
202	C₂×#107	C₂²×C₆₇	1	2	yes	—
203	#367	C₂²×C₆₇	1	2	yes	—
204	#367	C₂²×C₆₇	1	2	yes	—
205	#367	C₂²×C₆₇	1	2	yes	—
206	#366	C₂²×C₆₇	1	1	yes	characteristic
207	#366	C₂²×C₆₇	1	2	yes	—
208	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
209	#357	C₂²×C₆₇	1	1	yes	characteristic
210	#357	C₂²×C₆₇	1	1	yes	characteristic
211	#357	C₂²×C₆₇	1	2	yes	—
212	#357	C₂²×C₆₇	1	2	yes	—
213	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
214	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×#41	—	4	1	no	—
216	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
217	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	#124	—	4	2	no	—
222	#123	—	4	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	#121	—	2	2	no	—
225	#121	—	2	2	no	—
226	#121	—	4	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
230	#117	—	2	2	no	—
231	#117	—	4	1	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	D₁₆	—	2	2	no	—
234	M₃₂	—	2	4	no	—
235	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
236	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
237	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
238	#107	—	2	2	no	—
239	#107	—	2	1	no	—
240	#107	—	2	1	no	—
241	#107	C₂³×C₆₇	1	1	yes	characteristic
242	#107	C₂³×C₆₇	1	1	yes	characteristic
243	#107	C₂³×C₆₇	1	2	yes	—
244	#107	C₂³×C₆₇	1	2	yes	—
245	#106	—	2	2	no	—
246	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
247	#102	—	2	1	no	—
248	#102	—	2	1	no	—
249	#102	—	2	2	no	—
250	#102	—	2	2	no	—
251	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
252	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
253	#41	—	8	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	2	1	no	—
268	D₈	—	2	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
273	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
288	C₄²	—	4	2	no	—
289	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	#41×C₆₇	1	2	yes	complemented
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	C₂×D₈×C₆₇	1	2	yes	—
296	Q₈	—	8	2	no	—
297	Q₈	—	16	1	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	4	1	no	—
303	D₄	—	8	4	no	—
304	D₄	—	16	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
315	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
316	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	1	no	—
320	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
321	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	2	no	—
338	C₄	#509×C₆₇	1	2	yes	—
339	C₄	#547×C₆₇	1	1	yes	characteristic
340	C₄	#578×C₆₇	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	16	1	no	—
343	C₂	—	16	1	no	—
344	C₂	C₆₇×#3334	1	1	yes	characteristic
345	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
346	C₂	C₆₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×D₄×C₆₇ ◁ C₈×D₄×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×#3334	C₂	semidirect
3. C₆₇×#2254	C₂	semidirect
4. C₆₇×#2234	C₂	semidirect
5. C₆₇×#2233	C₂	semidirect
6. C₆₇×#2226	C₂	semidirect
7. C₆₇×#2224	C₂	semidirect
8. C₆₇×#2218	C₂	semidirect
9. C₄×D₁₆×C₆₇	C₂	semidirect
10. C₆₇×#2169	C₂	semidirect
11. C₆₇×#2159	C₂	semidirect
118. C₁₆×C₆₇	#41	semidirect
167. #33921	C₆₇	direct (N × H)
174. #3341	C₂×C₆₇	semidirect
175. #3334	C₂×C₆₇	semidirect
176. #2254	C₂×C₆₇	semidirect
177. #2234	C₂×C₆₇	semidirect
178. #2233	C₂×C₆₇	semidirect
179. #2226	C₂×C₆₇	semidirect
180. #2224	C₂×C₆₇	semidirect
181. #2218	C₂×C₆₇	semidirect
182. C₄×D₁₆	C₂×C₆₇	semidirect
183. #2169	C₂×C₆₇	semidirect
184. #2159	C₂×C₆₇	semidirect
185. C₆₇	#33921	direct (N × H)
292. C₁₆	#41×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×#3334	2
C₆₇×#2254	2
C₆₇×#2234	2
C₆₇×#2233	2
C₆₇×#2226	2
C₆₇×#2224	2
C₆₇×#2218	2
C₄×D₁₆×C₆₇	2
C₆₇×#2169	2
C₆₇×#2159	2
#33921	67

Subgroups — C67×#33921 #30713172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33921 #30713172