Subgroups — C₆₇×#20117 #30699368

Order 17152.

568 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3375	C₂	1	2	yes	maximal
3	C₆₇×#1722	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1696	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1678	C₂	1	2	yes	maximal
6	C₆₇×#1633	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1630	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1623	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1587	C₂	1	2	yes	maximal, complemented
10	#559×C₆₇	C₂²	1	1	yes	characteristic
11	Q₈²×C₆₇	C₂²	1	2	yes	—
12	Q₈²×C₆₇	C₂²	1	1	yes	characteristic
13	#500×C₆₇	—	2	2	no	—
14	#500×C₆₇	C₂²	1	2	yes	—
15	#494×C₆₇	C₂²	1	1	yes	characteristic
16	#492×C₆₇	C₂²	1	1	yes	characteristic
17	#489×C₆₇	—	2	2	no	—
18	#486×C₆₇	C₂²	1	2	yes	—
19	#477×C₆₇	—	2	2	no	—
20	#477×C₆₇	—	2	2	no	—
21	#477×C₆₇	C₂²	1	2	yes	—
22	#477×C₆₇	C₂²	1	2	yes	—
23	#477×C₆₇	C₂²	1	2	yes	—
24	#474×C₆₇	—	2	2	no	—
25	#464×C₆₇	—	2	2	no	—
26	#460×C₆₇	C₂²	1	2	yes	—
27	#446×C₆₇	C₂²	1	2	yes	—
28	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
29	#440×C₆₇	—	2	2	no	—
30	C₄×Q₁₆×C₆₇	—	2	2	no	—
31	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
32	#334×C₆₇	C₂²	1	2	yes	—
33	#333×C₆₇	C₂²	1	1	yes	characteristic
34	#333×C₆₇	C₂²	1	2	yes	—
35	#331×C₆₇	C₂²	1	1	yes	characteristic
36	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
37	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#322×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₆₇	—	4	1	no	—
40	C₂×Q₁₆×C₆₇	D₄	1	2	yes	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×SD₁₆×C₆₇	—	4	1	no	—
43	C₄⋊Q₈×C₆₇	—	2	2	no	—
44	C₄⋊Q₈×C₆₇	—	2	2	no	—
45	C₄⋊Q₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	—	4	2	no	—
47	C₄⋊Q₈×C₆₇	—	4	1	no	—
48	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
50	#127×C₆₇	C₂³	1	1	yes	characteristic
51	#124×C₆₇	—	2	2	no	—
52	#123×C₆₇	—	4	1	no	—
53	C₄×Q₈×C₆₇	—	2	2	no	—
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
57	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	#107×C₆₇	—	2	2	no	—
60	#107×C₆₇	—	2	2	no	—
61	#107×C₆₇	—	2	2	no	—
62	#107×C₆₇	—	2	2	no	—
63	#107×C₆₇	—	2	2	no	—
64	#107×C₆₇	D₄	1	2	yes	—
65	C₄⋊C₈×C₆₇	—	2	2	no	—
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	—	2	2	no	—
71	#103×C₆₇	—	2	2	no	—
72	#103×C₆₇	D₄	1	2	yes	—
73	#103×C₆₇	D₄	1	2	yes	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	—	2	2	no	—
77	#97×C₆₇	—	2	2	no	—
78	C₄×C₈×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂×Q₈×C₆₇	—	2	1	no	—
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	4	2	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	1	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	Q₁₆×C₆₇	—	2	4	no	—
91	Q₁₆×C₆₇	—	8	2	no	—
92	SD₁₆×C₆₇	—	4	4	no	—
93	SD₁₆×C₆₇	—	8	2	no	—
94	C₂×C₈×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	2	yes	—
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄⋊C₄×C₆₇	—	4	2	no	—
106	C₄⋊C₄×C₆₇	—	4	2	no	—
107	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄×C₆₇	—	4	2	no	—
110	C₂²⋊C₄×C₆₇	—	4	1	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	2	2	no	—
113	C₄²×C₆₇	—	4	1	no	—
114	C₄²×C₆₇	—	4	2	no	—
115	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₆₇	—	4	1	no	—
117	Q₈×C₆₇	—	2	2	no	—
118	Q₈×C₆₇	—	2	2	no	—
119	Q₈×C₆₇	—	2	2	no	—
120	Q₈×C₆₇	—	8	1	no	—
121	Q₈×C₆₇	—	8	2	no	—
122	Q₈×C₆₇	—	16	2	no	—
123	Q₈×C₆₇	—	16	1	no	—
124	Q₈×C₆₇	#137	1	4	yes	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	4	2	no	—
127	D₄×C₆₇	—	8	1	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	16	1	no	—
135	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₆₇	—	2	2	no	—
138	C₈×C₆₇	—	2	2	no	—
139	C₈×C₆₇	—	2	2	no	—
140	C₈×C₆₇	—	4	2	no	—
141	C₈×C₆₇	—	4	2	no	—
142	C₂²×C₆₇	—	8	2	no	—
143	C₂²×C₆₇	—	8	1	no	—
144	C₂²×C₆₇	D₄²	1	1	yes	characteristic
145	C₄×C₆₇	—	2	4	no	—
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	4	2	no	—
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	—	4	2	no	—
150	C₄×C₆₇	—	8	1	no	—
151	C₄×C₆₇	—	8	2	no	—
152	C₄×C₆₇	C₂×#137	1	2	yes	—
153	C₄×C₆₇	#578	1	2	yes	—
154	#20117	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₆₇	—	16	1	no	—
156	C₂×C₆₇	#3270	1	2	yes	—
157	C₂×C₆₇	#3292	1	1	yes	characteristic
158	Q₈×Q₁₆	C₂×C₆₇	1	2	yes	complemented
159	#3375	C₂×C₆₇	1	2	yes	—
160	#1722	C₂×C₆₇	1	1	yes	characteristic
161	#1696	C₂×C₆₇	1	2	yes	complemented
162	#1678	C₂×C₆₇	1	2	yes	—
163	#1633	C₂×C₆₇	1	2	yes	complemented
164	#1630	C₂×C₆₇	1	1	yes	characteristic
165	#1623	C₂×C₆₇	1	1	yes	characteristic
166	#1587	C₂×C₆₇	1	2	yes	complemented
167	C₆₇	#20117	1	1	yes	characteristic, direct product factor, Sylow
168	#559	C₂²×C₆₇	1	1	yes	characteristic
169	Q₈²	C₂²×C₆₇	1	2	yes	—
170	Q₈²	C₂²×C₆₇	1	1	yes	characteristic
171	#500	—	2	2	no	—
172	#500	C₂²×C₆₇	1	2	yes	—
173	#494	C₂²×C₆₇	1	1	yes	characteristic
174	#492	C₂²×C₆₇	1	1	yes	characteristic
175	#489	—	2	2	no	—
176	#486	C₂²×C₆₇	1	2	yes	—
177	#477	—	2	2	no	—
178	#477	—	2	2	no	—
179	#477	C₂²×C₆₇	1	2	yes	—
180	#477	C₂²×C₆₇	1	2	yes	—
181	#477	C₂²×C₆₇	1	2	yes	—
182	#474	—	2	2	no	—
183	#464	—	2	2	no	—
184	#460	C₂²×C₆₇	1	2	yes	—
185	#446	C₂²×C₆₇	1	2	yes	—
186	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
187	#440	—	2	2	no	—
188	C₄×Q₁₆	—	2	2	no	—
189	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
190	#334	C₂²×C₆₇	1	2	yes	—
191	#333	C₂²×C₆₇	1	1	yes	characteristic
192	#333	C₂²×C₆₇	1	2	yes	—
193	#331	C₂²×C₆₇	1	1	yes	characteristic
194	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
195	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
196	#322	C₂²×C₆₇	1	1	yes	characteristic
197	C₂×Q₁₆	—	4	1	no	—
198	C₂×Q₁₆	D₄×C₆₇	1	2	yes	—
199	C₂×SD₁₆	—	2	2	no	—
200	C₂×SD₁₆	—	4	1	no	—
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	4	2	no	—
205	C₄⋊Q₈	—	4	1	no	—
206	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
208	#127	C₂³×C₆₇	1	1	yes	characteristic
209	#124	—	2	2	no	—
210	#123	—	4	1	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
215	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
216	C₄×D₄	—	2	2	no	—
217	#107	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	2	2	no	—
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#107	D₄×C₆₇	1	2	yes	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
226	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
227	#103	—	2	2	no	—
228	#103	—	2	2	no	—
229	#103	—	2	2	no	—
230	#103	D₄×C₆₇	1	2	yes	—
231	#103	D₄×C₆₇	1	2	yes	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#97	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₆₇	1	2	yes	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
244	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	Q₁₆	—	2	4	no	—
249	Q₁₆	—	8	2	no	—
250	SD₁₆	—	4	4	no	—
251	SD₁₆	—	8	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
256	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
266	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
274	C₂³	—	4	1	no	—
275	Q₈	—	2	2	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	8	1	no	—
279	Q₈	—	8	2	no	—
280	Q₈	—	16	2	no	—
281	Q₈	—	16	1	no	—
282	Q₈	#137×C₆₇	1	4	yes	—
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	16	1	no	—
293	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
294	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	1	no	—
302	C₂²	D₄²×C₆₇	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	2	no	—
310	C₄	C₂×#137×C₆₇	1	2	yes	—
311	C₄	#578×C₆₇	1	2	yes	—
312	C₂	—	16	1	no	—
313	C₂	C₆₇×#3270	1	2	yes	—
314	C₂	C₆₇×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	#127×C₆₇	C₂³
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. Q₈×Q₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1696	C₂	semidirect
6. C₆₇×#1633	C₂	semidirect
9. C₆₇×#1587	C₂	semidirect
154. #20117	C₆₇	direct (N × H)
158. Q₈×Q₁₆	C₂×C₆₇	semidirect
161. #1696	C₂×C₆₇	semidirect
163. #1633	C₂×C₆₇	semidirect
166. #1587	C₂×C₆₇	semidirect
167. C₆₇	#20117	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×Q₁₆×C₆₇	2
C₆₇×#3375	2
C₆₇×#1722	2
C₆₇×#1696	2
C₆₇×#1678	2
C₆₇×#1633	2
C₆₇×#1630	2
C₆₇×#1623	2
C₆₇×#1587	2
#20117	67

Subgroups — C67×#20117 #30699368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20117 #30699368