Subgroups — C₆₇×#20097 #30699348

Order 17152.

550 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3387	C₂	1	2	yes	maximal
2	C₆₇×#3363	C₂	1	2	yes	maximal
3	C₆₇×#1716	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1704	C₂	1	2	yes	maximal
5	C₆₇×#1672	C₂	1	2	yes	maximal
6	C₆₇×#1653	C₂	1	2	yes	maximal
7	C₆₇×#1633	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1633	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1587	C₂	1	2	yes	maximal
10	Q₈²×C₆₇	C₂²	1	1	yes	characteristic
11	Q₈²×C₆₇	C₂²	1	1	yes	characteristic
12	#557×C₆₇	C₂²	1	2	yes	—
13	#501×C₆₇	—	2	2	no	—
14	#498×C₆₇	C₂²	1	2	yes	—
15	#498×C₆₇	C₂²	1	1	yes	characteristic
16	#494×C₆₇	C₂²	1	1	yes	characteristic
17	#487×C₆₇	—	2	2	no	—
18	#487×C₆₇	C₂²	1	2	yes	—
19	#479×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	#475×C₆₇	—	2	2	no	—
24	#475×C₆₇	C₂²	1	2	yes	—
25	#462×C₆₇	—	2	2	no	—
26	#462×C₆₇	C₂²	1	2	yes	—
27	#446×C₆₇	C₂²	1	2	yes	—
28	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
29	#441×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	1	yes	characteristic
35	#332×C₆₇	C₂²	1	2	yes	—
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	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	—	4	1	no	—
41	C₂×Q₁₆×C₆₇	D₄	1	2	yes	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	C₄⋊Q₈×C₆₇	—	2	2	no	—
44	C₄⋊Q₈×C₆₇	—	2	2	no	—
45	C₄⋊Q₈×C₆₇	—	4	1	no	—
46	C₄⋊Q₈×C₆₇	—	4	1	no	—
47	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
50	#125×C₆₇	—	2	2	no	—
51	#125×C₆₇	—	4	2	no	—
52	C₄×Q₈×C₆₇	—	2	2	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	#107×C₆₇	—	2	2	no	—
59	#107×C₆₇	—	2	2	no	—
60	#107×C₆₇	D₄	1	2	yes	—
61	#106×C₆₇	—	2	2	no	—
62	#106×C₆₇	—	2	2	no	—
63	#106×C₆₇	—	2	2	no	—
64	#106×C₆₇	—	4	1	no	—
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	—	2	2	no	—
73	#103×C₆₇	—	2	2	no	—
74	#103×C₆₇	—	2	2	no	—
75	#103×C₆₇	D₄	1	2	yes	—
76	#103×C₆₇	D₄	1	2	yes	—
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	2	no	—
82	C₂×Q₈×C₆₇	—	2	1	no	—
83	C₂×Q₈×C₆₇	—	2	1	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×Q₈×C₆₇	—	4	1	no	—
86	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
87	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
88	Q₁₆×C₆₇	—	2	4	no	—
89	Q₁₆×C₆₇	—	4	4	no	—
90	Q₁₆×C₆₇	—	8	2	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
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	—	4	2	no	—
103	C₄⋊C₄×C₆₇	—	4	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	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	4	2	no	—
112	C₄²×C₆₇	—	4	1	no	—
113	C₄²×C₆₇	—	4	1	no	—
114	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
115	Q₈×C₆₇	—	2	2	no	—
116	Q₈×C₆₇	—	2	2	no	—
117	Q₈×C₆₇	—	2	2	no	—
118	Q₈×C₆₇	—	4	2	no	—
119	Q₈×C₆₇	—	4	2	no	—
120	Q₈×C₆₇	—	8	1	no	—
121	Q₈×C₆₇	—	8	1	no	—
122	Q₈×C₆₇	—	16	1	no	—
123	Q₈×C₆₇	—	16	1	no	—
124	Q₈×C₆₇	#137	1	4	yes	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	1	no	—
131	C₂×C₄×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₆₇	—	2	2	no	—
135	C₈×C₆₇	—	2	2	no	—
136	C₈×C₆₇	—	2	2	no	—
137	C₈×C₆₇	—	4	2	no	—
138	C₈×C₆₇	—	4	2	no	—
139	C₂²×C₆₇	D₄²	1	1	yes	characteristic
140	C₄×C₆₇	—	2	4	no	—
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	4	2	no	—
143	C₄×C₆₇	—	4	2	no	—
144	C₄×C₆₇	—	4	2	no	—
145	C₄×C₆₇	—	8	2	no	—
146	C₄×C₆₇	—	8	1	no	—
147	C₄×C₆₇	—	8	1	no	—
148	C₄×C₆₇	C₂×#137	1	2	yes	—
149	C₄×C₆₇	#578	1	2	yes	—
150	#20097	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₆₇	#3270	1	2	yes	—
152	C₂×C₆₇	#3291	1	1	yes	characteristic
153	#3387	C₂×C₆₇	1	2	yes	—
154	#3363	C₂×C₆₇	1	2	yes	—
155	#1716	C₂×C₆₇	1	1	yes	characteristic
156	#1704	C₂×C₆₇	1	2	yes	—
157	#1672	C₂×C₆₇	1	2	yes	—
158	#1653	C₂×C₆₇	1	2	yes	—
159	#1633	C₂×C₆₇	1	1	yes	characteristic
160	#1633	C₂×C₆₇	1	1	yes	characteristic
161	#1587	C₂×C₆₇	1	2	yes	—
162	C₆₇	#20097	1	1	yes	characteristic, direct product factor, Sylow
163	Q₈²	C₂²×C₆₇	1	1	yes	characteristic
164	Q₈²	C₂²×C₆₇	1	1	yes	characteristic
165	#557	C₂²×C₆₇	1	2	yes	—
166	#501	—	2	2	no	—
167	#498	C₂²×C₆₇	1	2	yes	—
168	#498	C₂²×C₆₇	1	1	yes	characteristic
169	#494	C₂²×C₆₇	1	1	yes	characteristic
170	#487	—	2	2	no	—
171	#487	C₂²×C₆₇	1	2	yes	—
172	#479	—	2	2	no	—
173	#477	—	2	2	no	—
174	#477	C₂²×C₆₇	1	2	yes	—
175	#477	C₂²×C₆₇	1	2	yes	—
176	#475	—	2	2	no	—
177	#475	C₂²×C₆₇	1	2	yes	—
178	#462	—	2	2	no	—
179	#462	C₂²×C₆₇	1	2	yes	—
180	#446	C₂²×C₆₇	1	2	yes	—
181	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
182	#441	—	2	2	no	—
183	C₄×Q₁₆	—	2	2	no	—
184	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
185	#334	C₂²×C₆₇	1	2	yes	—
186	#333	C₂²×C₆₇	1	1	yes	characteristic
187	#333	C₂²×C₆₇	1	1	yes	characteristic
188	#332	C₂²×C₆₇	1	2	yes	—
189	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
190	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
191	#322	C₂²×C₆₇	1	1	yes	characteristic
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	—	4	1	no	—
194	C₂×Q₁₆	D₄×C₆₇	1	2	yes	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	4	1	no	—
199	C₄⋊Q₈	—	4	1	no	—
200	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
201	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
202	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
203	#125	—	2	2	no	—
204	#125	—	4	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
210	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
211	#107	—	2	2	no	—
212	#107	—	2	2	no	—
213	#107	D₄×C₆₇	1	2	yes	—
214	#106	—	2	2	no	—
215	#106	—	2	2	no	—
216	#106	—	2	2	no	—
217	#106	—	4	1	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
221	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
222	#103	—	2	2	no	—
223	#103	—	2	2	no	—
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#103	D₄×C₆₇	1	2	yes	—
229	#103	D₄×C₆₇	1	2	yes	—
230	#97	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₆₇	1	2	yes	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
240	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	4	4	no	—
243	Q₁₆	—	8	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
261	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	1	no	—
266	C₄²	—	4	1	no	—
267	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	Q₈	—	2	2	no	—
270	Q₈	—	2	2	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	8	1	no	—
274	Q₈	—	8	1	no	—
275	Q₈	—	16	1	no	—
276	Q₈	—	16	1	no	—
277	Q₈	#137×C₆₇	1	4	yes	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
286	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₂²	D₄²×C₆₇	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	2	1	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	1	no	—
300	C₄	—	8	1	no	—
301	C₄	C₂×#137×C₆₇	1	2	yes	—
302	C₄	#578×C₆₇	1	2	yes	—
303	C₂	C₆₇×#3270	1	2	yes	—
304	C₂	C₆₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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
150. #20097	C₆₇	direct (N × H)
162. C₆₇	#20097	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×#3387	2
C₆₇×#3363	2
C₆₇×#1716	2
C₆₇×#1704	2
C₆₇×#1672	2
C₆₇×#1653	2
C₆₇×#1633	2
C₆₇×#1587	2
#20097	67

Subgroups — C67×#20097 #30699348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20097 #30699348