Subgroups — C₆₇×#33942 #30713193

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 334 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
2	Q₈×Q₁₆×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
5	C₆₇×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2224	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2211	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2207	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2175	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#559×C₆₇	—	2	1	no	—
13	Q₈²×C₆₇	—	2	1	no	—
14	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×SD₃₂×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	#477×C₆₇	—	2	1	no	—
21	#477×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×Q₁₆×C₆₇	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₆₇	C₂²	1	2	yes	—
29	#366×C₆₇	C₂²	1	1	yes	characteristic
30	#366×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	#358×C₆₇	C₂²	1	1	yes	characteristic
36	#357×C₆₇	C₂²	1	2	yes	—
37	#357×C₆₇	C₂²	1	2	yes	—
38	#357×C₆₇	C₂²	1	2	yes	—
39	#346×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
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	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	—	2	1	no	—
48	C₄⋊Q₈×C₆₇	—	4	1	no	—
49	C₄⋊Q₈×C₆₇	—	4	2	no	—
50	#127×C₆₇	—	2	1	no	—
51	#124×C₆₇	—	2	2	no	—
52	#124×C₆₇	—	4	1	no	—
53	#123×C₆₇	—	4	2	no	—
54	C₄×Q₈×C₆₇	—	2	1	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	1	no	—
57	C₄×Q₈×C₆₇	—	4	1	no	—
58	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
59	C₄×D₄×C₆₇	—	2	2	no	—
60	C₄×D₄×C₆₇	—	4	1	no	—
61	Q₃₂×C₆₇	—	2	2	no	—
62	SD₃₂×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	#103×C₆₇	—	2	2	no	—
75	#103×C₆₇	—	2	1	no	—
76	#102×C₆₇	—	2	2	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	—	2	1	no	—
81	C₂×Q₈×C₆₇	—	4	2	no	—
82	C₂×Q₈×C₆₇	—	4	1	no	—
83	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	4	1	no	—
86	C₂×D₄×C₆₇	—	4	1	no	—
87	C₂²×C₄×C₆₇	—	4	2	no	—
88	Q₁₆×C₆₇	—	2	1	no	—
89	Q₁₆×C₆₇	—	2	1	no	—
90	Q₁₆×C₆₇	—	2	1	no	—
91	Q₁₆×C₆₇	#41	1	2	yes	—
92	SD₁₆×C₆₇	—	8	2	no	—
93	D₈×C₆₇	—	2	2	no	—
94	D₈×C₆₇	—	2	2	no	—
95	D₈×C₆₇	—	8	1	no	—
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	1	no	—
102	C₄⋊C₄×C₆₇	—	4	1	no	—
103	C₄⋊C₄×C₆₇	—	4	2	no	—
104	C₄⋊C₄×C₆₇	—	4	1	no	—
105	C₄⋊C₄×C₆₇	—	4	2	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
108	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
109	C₂²⋊C₄×C₆₇	—	4	2	no	—
110	C₂²⋊C₄×C₆₇	—	4	2	no	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₄²×C₆₇	—	4	1	no	—
113	C₄²×C₆₇	—	4	2	no	—
114	C₄²×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	C₂×D₄	1	2	yes	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₇	—	2	2	no	—
118	C₁₆×C₆₇	—	2	1	no	—
119	C₁₆×C₆₇	—	2	1	no	—
120	C₂³×C₆₇	—	4	2	no	—
121	Q₈×C₆₇	—	2	2	no	—
122	Q₈×C₆₇	—	2	2	no	—
123	Q₈×C₆₇	—	4	1	no	—
124	Q₈×C₆₇	—	4	1	no	—
125	Q₈×C₆₇	—	4	1	no	—
126	Q₈×C₆₇	—	16	2	no	—
127	Q₈×C₆₇	—	16	1	no	—
128	D₄×C₆₇	—	4	2	no	—
129	D₄×C₆₇	—	4	2	no	—
130	D₄×C₆₇	—	8	2	no	—
131	D₄×C₆₇	—	16	1	no	—
132	C₂×C₄×C₆₇	—	4	1	no	—
133	C₂×C₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	16	2	no	—
138	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₆₇	—	2	1	no	—
144	C₈×C₆₇	—	2	2	no	—
145	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂²×C₆₇	—	8	2	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	4	2	no	—
155	C₄×C₆₇	—	8	1	no	—
156	C₄×C₆₇	—	8	1	no	—
157	C₄×C₆₇	—	8	1	no	—
158	C₄×C₆₇	—	8	2	no	—
159	C₄×C₆₇	#510	1	2	yes	—
160	C₄×C₆₇	#548	1	1	yes	characteristic
161	C₄×C₆₇	#577	1	1	yes	characteristic
162	#33942	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₆₇	—	16	2	no	—
164	C₂×C₆₇	#3358	1	1	yes	characteristic
165	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
166	C₂×C₆₇	#3415	1	1	yes	characteristic
167	#3396	C₂×C₆₇	1	1	yes	characteristic
168	Q₈×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#2243	C₂×C₆₇	1	2	yes	complemented
170	#2240	C₂×C₆₇	1	1	yes	characteristic
171	#2227	C₂×C₆₇	1	1	yes	characteristic, complemented
172	#2224	C₂×C₆₇	1	2	yes	complemented
173	#2211	C₂×C₆₇	1	2	yes	complemented
174	#2207	C₂×C₆₇	1	1	yes	characteristic
175	#2183	C₂×C₆₇	1	1	yes	characteristic, complemented
176	#2175	C₂×C₆₇	1	2	yes	complemented
177	C₄×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
178	C₆₇	#33942	1	1	yes	characteristic, direct product factor, Sylow
179	#559	—	2	1	no	—
180	Q₈²	—	2	1	no	—
181	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
182	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
183	#500	C₂²×C₆₇	1	2	yes	—
184	#500	C₂²×C₆₇	1	1	yes	characteristic
185	#495	C₂²×C₆₇	1	2	yes	—
186	#493	C₂²×C₆₇	1	1	yes	characteristic
187	#477	—	2	1	no	—
188	#477	—	2	2	no	—
189	#463	—	2	2	no	—
190	#460	—	2	1	no	—
191	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
192	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
193	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
194	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
195	C₄×D₈	C₂²×C₆₇	1	2	yes	—
196	#366	C₂²×C₆₇	1	1	yes	characteristic
197	#366	C₂²×C₆₇	1	2	yes	—
198	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
199	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
200	#358	C₂²×C₆₇	1	1	yes	characteristic
201	#358	C₂²×C₆₇	1	1	yes	characteristic
202	#358	C₂²×C₆₇	1	2	yes	—
203	#357	C₂²×C₆₇	1	2	yes	—
204	#357	C₂²×C₆₇	1	2	yes	—
205	#357	C₂²×C₆₇	1	2	yes	—
206	#346	C₂²×C₆₇	1	2	yes	—
207	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
208	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
209	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	C₂³×C₆₇	1	2	yes	—
213	C₄⋊Q₈	—	2	2	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	4	1	no	—
216	C₄⋊Q₈	—	4	2	no	—
217	#127	—	2	1	no	—
218	#124	—	2	2	no	—
219	#124	—	4	1	no	—
220	#123	—	4	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	—	4	1	no	—
225	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	4	1	no	—
228	Q₃₂	—	2	2	no	—
229	SD₃₂	—	2	4	no	—
230	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
232	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
233	#107	—	2	2	no	—
234	#107	—	2	1	no	—
235	#107	C₂³×C₆₇	1	2	yes	—
236	#107	C₂³×C₆₇	1	1	yes	characteristic
237	#107	C₂³×C₆₇	1	1	yes	characteristic
238	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
239	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
240	#103	—	2	1	no	—
241	#103	—	2	2	no	—
242	#103	—	2	1	no	—
243	#102	—	2	2	no	—
244	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₆₇	1	2	yes	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂²×C₄	—	4	2	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	1	no	—
258	Q₁₆	#41×C₆₇	1	2	yes	—
259	SD₁₆	—	8	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	8	1	no	—
263	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
275	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	4	1	no	—
282	C₄²	C₂×D₄×C₆₇	1	2	yes	—
283	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₂³	—	4	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	16	2	no	—
294	Q₈	—	16	1	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	16	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
306	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
307	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
313	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	2	no	—
326	C₄	#510×C₆₇	1	2	yes	—
327	C₄	#548×C₆₇	1	1	yes	characteristic
328	C₄	#577×C₆₇	1	1	yes	characteristic
329	C₂	—	16	2	no	—
330	C₂	C₆₇×#3358	1	1	yes	characteristic
331	C₂	C₂×#510×C₆₇	1	1	yes	characteristic
332	C₂	C₆₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	#493×C₆₇	C₂²
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
2. Q₈×Q₁₆×C₆₇	C₂	semidirect
3. C₆₇×#2243	C₂	semidirect
5. C₆₇×#2227	C₂	semidirect
6. C₆₇×#2224	C₂	semidirect
7. C₆₇×#2211	C₂	semidirect
9. C₆₇×#2183	C₂	semidirect
10. C₆₇×#2175	C₂	semidirect
11. C₄×Q₃₂×C₆₇	C₂	semidirect
162. #33942	C₆₇	direct (N × H)
168. Q₈×Q₁₆	C₂×C₆₇	semidirect
169. #2243	C₂×C₆₇	semidirect
171. #2227	C₂×C₆₇	semidirect
172. #2224	C₂×C₆₇	semidirect
173. #2211	C₂×C₆₇	semidirect
175. #2183	C₂×C₆₇	semidirect
176. #2175	C₂×C₆₇	semidirect
177. C₄×Q₃₂	C₂×C₆₇	semidirect
178. C₆₇	#33942	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×Q₁₆×C₆₇	2
C₆₇×#2243	2
C₆₇×#2240	2
C₆₇×#2227	2
C₆₇×#2224	2
C₆₇×#2211	2
C₆₇×#2207	2
C₆₇×#2183	2
C₆₇×#2175	2
C₄×Q₃₂×C₆₇	2
#33942	67

Subgroups — C67×#33942 #30713193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33942 #30713193