Subgroups — C₆₇×#33940 #30713191

Order 17152.

520 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3396	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2243	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2226	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2225	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2210	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2206	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2177	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#559×C₆₇	—	2	1	no	—
13	D₄×Q₈×C₆₇	—	2	1	no	—
14	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
16	#500×C₆₇	C₂²	1	2	yes	—
17	#500×C₆₇	C₂²	1	1	yes	characteristic
18	#495×C₆₇	C₂²	1	2	yes	—
19	#493×C₆₇	C₂²	1	1	yes	characteristic
20	#474×C₆₇	—	2	1	no	—
21	#474×C₆₇	—	2	2	no	—
22	#463×C₆₇	—	2	2	no	—
23	#460×C₆₇	—	2	1	no	—
24	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₆₇	C₂²	1	2	yes	—
28	C₄×D₈×C₆₇	C₂²	1	2	yes	—
29	#367×C₆₇	C₂²	1	1	yes	characteristic
30	#367×C₆₇	C₂²	1	2	yes	—
31	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
33	#358×C₆₇	C₂²	1	1	yes	characteristic
34	#358×C₆₇	C₂²	1	2	yes	—
35	#357×C₆₇	C₂²	1	2	yes	—
36	#357×C₆₇	C₂²	1	2	yes	—
37	#357×C₆₇	C₂²	1	1	yes	characteristic
38	#357×C₆₇	C₂²	1	2	yes	—
39	#346×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₆₇	—	4	1	no	—
42	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×D₈×C₆₇	—	2	1	no	—
45	C₂×D₈×C₆₇	C₂³	1	2	yes	—
46	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	—	2	1	no	—
49	#127×C₆₇	—	2	1	no	—
50	#124×C₆₇	—	2	2	no	—
51	#124×C₆₇	—	4	1	no	—
52	#123×C₆₇	—	4	2	no	—
53	C₂²⋊Q₈×C₆₇	—	4	1	no	—
54	C₂²⋊Q₈×C₆₇	—	4	2	no	—
55	C₄×Q₈×C₆₇	—	2	1	no	—
56	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	C₄×D₄×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	—	4	1	no	—
61	SD₃₂×C₆₇	—	2	2	no	—
62	D₁₆×C₆₇	—	2	4	no	—
63	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
64	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
66	#107×C₆₇	—	2	2	no	—
67	#107×C₆₇	—	2	1	no	—
68	#107×C₆₇	C₂³	1	1	yes	characteristic
69	#107×C₆₇	C₂³	1	2	yes	—
70	#107×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
73	#103×C₆₇	—	2	1	no	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	—	2	1	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂³	1	2	yes	—
79	C₂×Q₈×C₆₇	—	2	1	no	—
80	C₂×Q₈×C₆₇	—	4	2	no	—
81	C₂×Q₈×C₆₇	—	4	1	no	—
82	C₂×Q₈×C₆₇	—	8	4	no	—
83	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
84	C₂×D₄×C₆₇	—	2	1	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	4	1	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂²×C₄×C₆₇	—	4	2	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	—	4	2	no	—
91	Q₁₆×C₆₇	—	2	1	no	—
92	Q₁₆×C₆₇	—	2	1	no	—
93	SD₁₆×C₆₇	—	8	2	no	—
94	D₈×C₆₇	—	2	2	no	—
95	D₈×C₆₇	—	2	1	no	—
96	D₈×C₆₇	—	2	2	no	—
97	D₈×C₆₇	—	8	1	no	—
98	D₈×C₆₇	#41	1	2	yes	—
99	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
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	—	4	1	no	—
106	C₄⋊C₄×C₆₇	—	4	2	no	—
107	C₄⋊C₄×C₆₇	—	4	1	no	—
108	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
109	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
110	C₂²⋊C₄×C₆₇	—	4	2	no	—
111	C₂²⋊C₄×C₆₇	—	4	1	no	—
112	C₂²⋊C₄×C₆₇	—	4	2	no	—
113	C₂²⋊C₄×C₆₇	—	4	2	no	—
114	C₂²⋊C₄×C₆₇	—	4	2	no	—
115	C₄²×C₆₇	—	4	1	no	—
116	C₄²×C₆₇	C₂×D₄	1	2	yes	—
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₆₇	—	2	2	no	—
119	C₁₆×C₆₇	—	2	1	no	—
120	C₁₆×C₆₇	—	2	1	no	—
121	C₂³×C₆₇	—	4	1	no	—
122	C₂³×C₆₇	—	4	2	no	—
123	Q₈×C₆₇	—	2	2	no	—
124	Q₈×C₆₇	—	4	1	no	—
125	Q₈×C₆₇	—	4	1	no	—
126	Q₈×C₆₇	—	8	4	no	—
127	Q₈×C₆₇	—	8	2	no	—
128	D₄×C₆₇	—	2	2	no	—
129	D₄×C₆₇	—	4	2	no	—
130	D₄×C₆₇	—	4	1	no	—
131	D₄×C₆₇	—	4	2	no	—
132	D₄×C₆₇	—	8	2	no	—
133	D₄×C₆₇	—	16	1	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	8	4	no	—
139	C₂×C₄×C₆₇	—	8	2	no	—
140	C₂×C₄×C₆₇	—	16	2	no	—
141	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₈×C₆₇	—	2	1	no	—
147	C₈×C₆₇	—	2	2	no	—
148	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂²×C₆₇	—	4	2	no	—
151	C₂²×C₆₇	—	8	1	no	—
152	C₂²×C₆₇	—	8	1	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	2	no	—
155	C₂²×C₆₇	—	8	2	no	—
156	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	8	1	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	—	8	1	no	—
163	C₄×C₆₇	—	8	1	no	—
164	C₄×C₆₇	#509	1	2	yes	—
165	C₄×C₆₇	#548	1	1	yes	characteristic
166	C₄×C₆₇	#577	1	1	yes	characteristic
167	#33940	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₆₇	—	8	2	no	—
169	C₂×C₆₇	—	16	2	no	—
170	C₂×C₆₇	#3358	1	1	yes	characteristic
171	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
172	C₂×C₆₇	#3416	1	1	yes	characteristic
173	#3396	C₂×C₆₇	1	1	yes	characteristic, complemented
174	Q₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#2243	C₂×C₆₇	1	2	yes	complemented
176	#2240	C₂×C₆₇	1	1	yes	characteristic, complemented
177	#2226	C₂×C₆₇	1	2	yes	complemented
178	#2225	C₂×C₆₇	1	1	yes	characteristic, complemented
179	#2210	C₂×C₆₇	1	2	yes	complemented
180	#2206	C₂×C₆₇	1	1	yes	characteristic
181	#2183	C₂×C₆₇	1	1	yes	characteristic, complemented
182	#2177	C₂×C₆₇	1	2	yes	complemented
183	C₄×SD₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
184	C₆₇	#33940	1	1	yes	characteristic, direct product factor, Sylow
185	#559	—	2	1	no	—
186	D₄×Q₈	—	2	1	no	—
187	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic
188	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
189	#500	C₂²×C₆₇	1	1	yes	characteristic
190	#500	C₂²×C₆₇	1	2	yes	—
191	#495	C₂²×C₆₇	1	2	yes	—
192	#493	C₂²×C₆₇	1	1	yes	characteristic
193	#474	—	2	1	no	—
194	#474	—	2	2	no	—
195	#463	—	2	2	no	—
196	#460	—	2	1	no	—
197	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
198	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
199	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
200	C₄×D₈	C₂²×C₆₇	1	2	yes	—
201	C₄×D₈	C₂²×C₆₇	1	2	yes	—
202	#367	C₂²×C₆₇	1	1	yes	characteristic
203	#367	C₂²×C₆₇	1	2	yes	—
204	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
205	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
206	#358	C₂²×C₆₇	1	1	yes	characteristic
207	#358	C₂²×C₆₇	1	2	yes	—
208	#357	C₂²×C₆₇	1	1	yes	characteristic
209	#357	C₂²×C₆₇	1	2	yes	—
210	#357	C₂²×C₆₇	1	2	yes	—
211	#357	C₂²×C₆₇	1	2	yes	—
212	#346	C₂²×C₆₇	1	2	yes	—
213	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
214	C₂²×Q₈	—	4	1	no	—
215	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	C₂³×C₆₇	1	2	yes	—
219	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
220	C₄⋊Q₈	—	2	2	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	#127	—	2	1	no	—
223	#124	—	2	2	no	—
224	#124	—	4	1	no	—
225	#123	—	4	2	no	—
226	C₂²⋊Q₈	—	4	1	no	—
227	C₂²⋊Q₈	—	4	2	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	—	2	1	no	—
232	C₄×D₄	—	2	2	no	—
233	C₄×D₄	—	4	1	no	—
234	SD₃₂	—	2	2	no	—
235	D₁₆	—	2	4	no	—
236	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
237	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
238	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
239	#107	—	2	2	no	—
240	#107	—	2	1	no	—
241	#107	C₂³×C₆₇	1	2	yes	—
242	#107	C₂³×C₆₇	1	1	yes	characteristic
243	#107	C₂³×C₆₇	1	1	yes	characteristic
244	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
245	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
246	#103	—	2	1	no	—
247	#102	—	2	2	no	—
248	#102	—	2	2	no	—
249	#102	—	2	1	no	—
250	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
251	C₄×C₈	C₂³×C₆₇	1	2	yes	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	8	4	no	—
256	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	2	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	—	2	1	no	—
266	SD₁₆	—	8	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	2	1	no	—
269	D₈	—	2	2	no	—
270	D₈	—	8	1	no	—
271	D₈	#41×C₆₇	1	2	yes	—
272	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
274	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
282	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	C₂×D₄×C₆₇	1	2	yes	—
290	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
291	C₁₆	—	2	2	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	4	2	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	8	4	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	16	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	16	2	no	—
314	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
315	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
316	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
317	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	2	no	—
321	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
322	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
323	C₂²	—	4	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	#509×C₆₇	1	2	yes	—
338	C₄	#548×C₆₇	1	1	yes	characteristic
339	C₄	#577×C₆₇	1	1	yes	characteristic
340	C₂	—	8	2	no	—
341	C₂	—	16	2	no	—
342	C₂	C₆₇×#3358	1	1	yes	characteristic
343	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
344	C₂	C₆₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#2206	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3396	C₂	semidirect
2. Q₈×D₈×C₆₇	C₂	semidirect
3. C₆₇×#2243	C₂	semidirect
4. C₆₇×#2240	C₂	semidirect
5. C₆₇×#2226	C₂	semidirect
6. C₆₇×#2225	C₂	semidirect
7. C₆₇×#2210	C₂	semidirect
9. C₆₇×#2183	C₂	semidirect
10. C₆₇×#2177	C₂	semidirect
11. C₄×SD₃₂×C₆₇	C₂	semidirect
167. #33940	C₆₇	direct (N × H)
173. #3396	C₂×C₆₇	semidirect
174. Q₈×D₈	C₂×C₆₇	semidirect
175. #2243	C₂×C₆₇	semidirect
176. #2240	C₂×C₆₇	semidirect
177. #2226	C₂×C₆₇	semidirect
178. #2225	C₂×C₆₇	semidirect
179. #2210	C₂×C₆₇	semidirect
181. #2183	C₂×C₆₇	semidirect
182. #2177	C₂×C₆₇	semidirect
183. C₄×SD₃₂	C₂×C₆₇	semidirect
184. C₆₇	#33940	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×#3396	2
Q₈×D₈×C₆₇	2
C₆₇×#2243	2
C₆₇×#2240	2
C₆₇×#2226	2
C₆₇×#2225	2
C₆₇×#2210	2
C₆₇×#2206	2
C₆₇×#2183	2
C₆₇×#2177	2
C₄×SD₃₂×C₆₇	2
#33940	67

Subgroups — C67×#33940 #30713191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33940 #30713191