Subgroups — C₆₇×#17863 #30697114

Order 17152.

688 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2965	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1584	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1583	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1582	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1581	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1476	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₄	1	1	yes	characteristic
13	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	C₂²	1	2	yes	—
16	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
18	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
20	C₄×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
21	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
24	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
25	#436×C₆₇	C₂²	1	2	yes	—
26	#435×C₆₇	—	2	2	no	—
27	C₈×D₄×C₆₇	—	2	2	no	—
28	C₈×D₄×C₆₇	C₂²	1	2	yes	—
29	#431×C₆₇	—	2	1	no	—
30	#431×C₆₇	—	2	1	no	—
31	#431×C₆₇	C₂²	1	1	yes	characteristic
32	#431×C₆₇	C₂²	1	1	yes	characteristic
33	#427×C₆₇	C₄	1	1	yes	characteristic
34	#416×C₆₇	C₄	1	1	yes	characteristic
35	#416×C₆₇	C₄	1	1	yes	characteristic
36	C₄×M₁₆×C₆₇	—	2	1	no	—
37	C₂×C₄×C₈×C₆₇	—	2	1	no	—
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#335×C₆₇	C₂²	1	2	yes	—
42	#334×C₆₇	C₂²	1	2	yes	—
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
45	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
46	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	#322×C₆₇	C₂²	1	2	yes	—
48	#322×C₆₇	C₂²	1	2	yes	—
49	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
51	#135×C₆₇	—	2	4	no	—
52	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂²×C₈×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	D₄	1	2	yes	—
62	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	—	2	1	no	—
68	C₄×D₄×C₆₇	—	2	1	no	—
69	C₄×D₄×C₆₇	—	2	1	no	—
70	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
74	#117×C₆₇	—	2	1	no	—
75	#117×C₆₇	—	2	1	no	—
76	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
77	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	—	2	1	no	—
79	C₂×C₄²×C₆₇	—	2	1	no	—
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
82	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
83	#106×C₆₇	C₂×C₄	1	1	yes	characteristic
84	#106×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₆₇	—	2	2	no	—
86	C₄⋊C₈×C₆₇	—	2	2	no	—
87	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
88	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
89	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
90	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
91	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
92	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
93	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
94	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
95	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
96	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂²⋊C₈×C₆₇	—	2	1	no	—
98	C₂²⋊C₈×C₆₇	—	2	1	no	—
99	C₂²⋊C₈×C₆₇	—	2	2	no	—
100	C₂²⋊C₈×C₆₇	—	2	2	no	—
101	#97×C₆₇	—	2	2	no	—
102	C₄×C₈×C₆₇	—	2	2	no	—
103	C₄×C₈×C₆₇	—	2	1	no	—
104	C₄×C₈×C₆₇	—	2	1	no	—
105	C₄×C₈×C₆₇	—	2	1	no	—
106	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
108	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
109	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
110	C₄×C₈×C₆₇	C₂³	1	2	yes	—
111	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
112	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₆₇	C₂³	1	2	yes	—
114	#41×C₆₇	—	2	2	no	—
115	#41×C₆₇	—	2	2	no	—
116	#41×C₆₇	—	2	2	no	—
117	#41×C₆₇	M₁₆	1	4	yes	complemented
118	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₄×C₆₇	—	2	1	no	—
121	C₂×D₄×C₆₇	—	2	1	no	—
122	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	1	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
133	Q₁₆×C₆₇	—	2	2	no	—
134	SD₁₆×C₆₇	—	2	2	no	—
135	SD₁₆×C₆₇	—	2	2	no	—
136	D₈×C₆₇	—	2	2	no	—
137	M₁₆×C₆₇	—	4	4	no	—
138	C₂×C₈×C₆₇	—	2	2	no	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	4	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	4	2	no	—
145	C₂×C₈×C₆₇	—	4	4	no	—
146	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
151	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
152	C₄⋊C₄×C₆₇	—	2	1	no	—
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂²⋊C₄×C₆₇	—	2	1	no	—
161	C₂²⋊C₄×C₆₇	—	2	1	no	—
162	C₂²⋊C₄×C₆₇	—	2	1	no	—
163	C₂²⋊C₄×C₆₇	—	2	1	no	—
164	C₂²⋊C₄×C₆₇	—	2	1	no	—
165	C₂²⋊C₄×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	—	2	1	no	—
167	C₄²×C₆₇	—	2	1	no	—
168	C₄²×C₆₇	—	2	1	no	—
169	C₄²×C₆₇	—	2	1	no	—
170	C₄²×C₆₇	—	2	1	no	—
171	C₄²×C₆₇	—	2	1	no	—
172	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₆₇	#41	1	1	yes	characteristic
175	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂³×C₆₇	—	2	1	no	—
177	C₂³×C₆₇	—	2	1	no	—
178	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
179	Q₈×C₆₇	—	2	1	no	—
180	Q₈×C₆₇	—	2	1	no	—
181	Q₈×C₆₇	—	2	1	no	—
182	Q₈×C₆₇	C₂×M₁₆	1	2	yes	—
183	D₄×C₆₇	—	2	4	no	—
184	D₄×C₆₇	—	2	1	no	—
185	D₄×C₆₇	—	2	1	no	—
186	D₄×C₆₇	—	2	1	no	—
187	D₄×C₆₇	—	4	2	no	—
188	D₄×C₆₇	C₂×M₁₆	1	2	yes	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	4	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	2	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
207	C₂×C₄×C₆₇	#131	1	2	yes	—
208	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
211	C₈×C₆₇	—	2	1	no	—
212	C₈×C₆₇	—	2	1	no	—
213	C₈×C₆₇	—	2	1	no	—
214	C₈×C₆₇	—	2	1	no	—
215	C₈×C₆₇	—	2	2	no	—
216	C₈×C₆₇	—	4	2	no	—
217	C₈×C₆₇	—	4	2	no	—
218	C₈×C₆₇	—	4	2	no	—
219	C₈×C₆₇	#135	1	4	yes	complemented
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	—	2	1	no	—
222	C₂²×C₆₇	—	2	2	no	—
223	C₂²×C₆₇	—	4	1	no	—
224	C₂²×C₆₇	—	4	1	no	—
225	C₂²×C₆₇	—	4	1	no	—
226	C₂²×C₆₇	—	4	1	no	—
227	C₂²×C₆₇	—	4	1	no	—
228	C₂²×C₆₇	#435	1	2	yes	—
229	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	1	no	—
232	C₄×C₆₇	—	2	1	no	—
233	C₄×C₆₇	—	4	1	no	—
234	C₄×C₆₇	—	4	1	no	—
235	C₄×C₆₇	—	4	1	no	—
236	C₄×C₆₇	—	4	1	no	—
237	C₄×C₆₇	—	4	1	no	—
238	C₄×C₆₇	#435	1	2	yes	—
239	C₄×C₆₇	#444	1	2	yes	—
240	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
241	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
242	C₄×C₆₇	C₂×#135	1	2	yes	—
243	#17863	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₂×C₆₇	—	2	2	no	—
245	C₂×C₆₇	—	4	2	no	—
246	C₂×C₆₇	—	8	1	no	—
247	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
248	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
249	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
250	#2965	C₂×C₆₇	1	1	yes	characteristic, complemented
251	C₈×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
252	C₄×#135	C₂×C₆₇	1	1	yes	characteristic
253	#1584	C₂×C₆₇	1	2	yes	complemented
254	#1583	C₂×C₆₇	1	2	yes	complemented
255	#1582	C₂×C₆₇	1	2	yes	complemented
256	#1581	C₂×C₆₇	1	2	yes	complemented
257	#1564	C₂×C₆₇	1	1	yes	characteristic, complemented
258	#1476	C₂×C₆₇	1	1	yes	characteristic, complemented
259	#1476	C₂×C₆₇	1	1	yes	characteristic, complemented
260	C₂×#322	C₂×C₆₇	1	1	yes	characteristic, complemented
261	C₆₇	#17863	1	1	yes	characteristic, direct product factor, Sylow
262	C₂×#135	C₄×C₆₇	1	1	yes	characteristic
263	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
264	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
265	#446	C₂²×C₆₇	1	2	yes	—
266	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
267	C₄×Q₁₆	C₄×C₆₇	1	1	yes	characteristic
268	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
269	C₄×SD₁₆	C₄×C₆₇	1	1	yes	characteristic
270	C₄×SD₁₆	C₄×C₆₇	1	1	yes	characteristic
271	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
272	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
273	C₄×D₈	C₄×C₆₇	1	1	yes	characteristic
274	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
275	#436	C₂²×C₆₇	1	2	yes	—
276	#435	—	2	2	no	—
277	C₈×D₄	—	2	2	no	—
278	C₈×D₄	C₂²×C₆₇	1	2	yes	—
279	#431	—	2	1	no	—
280	#431	—	2	1	no	—
281	#431	C₂²×C₆₇	1	1	yes	characteristic
282	#431	C₂²×C₆₇	1	1	yes	characteristic
283	#427	C₄×C₆₇	1	1	yes	characteristic
284	#416	C₄×C₆₇	1	1	yes	characteristic
285	#416	C₄×C₆₇	1	1	yes	characteristic
286	C₄×M₁₆	—	2	1	no	—
287	C₂×C₄×C₈	—	2	1	no	—
288	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
289	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
290	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
291	#335	C₂²×C₆₇	1	2	yes	—
292	#334	C₂²×C₆₇	1	2	yes	—
293	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
294	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
295	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
296	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
297	#322	C₂²×C₆₇	1	2	yes	—
298	#322	C₂²×C₆₇	1	2	yes	—
299	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
300	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
301	#135	—	2	4	no	—
302	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
303	C₂×SD₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
304	C₂×SD₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
305	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
306	C₂×M₁₆	—	2	2	no	—
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	—	2	2	no	—
309	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₈	D₄×C₆₇	1	2	yes	—
312	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
313	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
314	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
315	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
321	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
322	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
323	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
324	#117	—	2	1	no	—
325	#117	—	2	1	no	—
326	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
327	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
331	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
332	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
333	#106	C₂×C₄×C₆₇	1	1	yes	characteristic
334	#106	C₂×C₄×C₆₇	1	1	yes	characteristic
335	C₄⋊C₈	—	2	2	no	—
336	C₄⋊C₈	—	2	2	no	—
337	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
338	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
339	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
340	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
341	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
342	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
343	#102	C₂×C₄×C₆₇	1	1	yes	characteristic
344	#102	C₂×C₄×C₆₇	1	1	yes	characteristic
345	#102	C₂×C₄×C₆₇	1	1	yes	characteristic
346	#102	C₂×C₄×C₆₇	1	1	yes	characteristic
347	C₂²⋊C₈	—	2	1	no	—
348	C₂²⋊C₈	—	2	1	no	—
349	C₂²⋊C₈	—	2	2	no	—
350	C₂²⋊C₈	—	2	2	no	—
351	#97	—	2	2	no	—
352	C₄×C₈	—	2	2	no	—
353	C₄×C₈	—	2	1	no	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	—	2	1	no	—
356	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
357	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
358	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
359	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
360	C₄×C₈	C₂³×C₆₇	1	2	yes	—
361	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
362	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
363	C₄×C₈	C₂³×C₆₇	1	2	yes	—
364	#41	—	2	2	no	—
365	#41	—	2	2	no	—
366	#41	—	2	2	no	—
367	#41	M₁₆×C₆₇	1	4	yes	complemented
368	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
369	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
373	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
382	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
383	Q₁₆	—	2	2	no	—
384	SD₁₆	—	2	2	no	—
385	SD₁₆	—	2	2	no	—
386	D₈	—	2	2	no	—
387	M₁₆	—	4	4	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	4	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	4	2	no	—
395	C₂×C₈	—	4	4	no	—
396	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
397	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
398	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
399	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
400	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
401	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
407	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
408	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
409	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
410	C₂²⋊C₄	—	2	1	no	—
411	C₂²⋊C₄	—	2	1	no	—
412	C₂²⋊C₄	—	2	1	no	—
413	C₂²⋊C₄	—	2	1	no	—
414	C₂²⋊C₄	—	2	1	no	—
415	C₂²⋊C₄	—	2	1	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	—	2	1	no	—
419	C₄²	—	2	1	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	1	no	—
422	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
423	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
424	C₄²	#41×C₆₇	1	1	yes	characteristic
425	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
426	C₂³	—	2	1	no	—
427	C₂³	—	2	1	no	—
428	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
429	Q₈	—	2	1	no	—
430	Q₈	—	2	1	no	—
431	Q₈	—	2	1	no	—
432	Q₈	C₂×M₁₆×C₆₇	1	2	yes	—
433	D₄	—	2	4	no	—
434	D₄	—	2	1	no	—
435	D₄	—	2	1	no	—
436	D₄	—	2	1	no	—
437	D₄	—	4	2	no	—
438	D₄	C₂×M₁₆×C₆₇	1	2	yes	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
454	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
455	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
456	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
457	C₂×C₄	#131×C₆₇	1	2	yes	—
458	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	2	1	no	—
464	C₈	—	2	1	no	—
465	C₈	—	2	2	no	—
466	C₈	—	4	2	no	—
467	C₈	—	4	2	no	—
468	C₈	—	4	2	no	—
469	C₈	#135×C₆₇	1	4	yes	complemented
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	1	no	—
477	C₂²	—	4	1	no	—
478	C₂²	#435×C₆₇	1	2	yes	—
479	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	1	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	1	no	—
487	C₄	—	4	1	no	—
488	C₄	#435×C₆₇	1	2	yes	—
489	C₄	#444×C₆₇	1	2	yes	—
490	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
491	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
492	C₄	C₂×#135×C₆₇	1	2	yes	—
493	C₂	—	2	2	no	—
494	C₂	—	4	2	no	—
495	C₂	—	8	1	no	—
496	C₂	C₂×#435×C₆₇	1	1	yes	characteristic
497	C₂	C₄×#135×C₆₇	1	1	yes	characteristic
498	C₂	C₂×#444×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2965	C₂	semidirect
2. C₈×#41×C₆₇	C₂	semidirect
4. C₆₇×#1584	C₂	semidirect
5. C₆₇×#1583	C₂	semidirect
6. C₆₇×#1582	C₂	semidirect
7. C₆₇×#1581	C₂	semidirect
8. C₆₇×#1564	C₂	semidirect
9. C₆₇×#1476	C₂	semidirect
10. C₆₇×#1476	C₂	semidirect
11. C₂×#322×C₆₇	C₂	semidirect
117. #41×C₆₇	M₁₆	semidirect
219. C₈×C₆₇	#135	semidirect
243. #17863	C₆₇	direct (N × H)
250. #2965	C₂×C₆₇	semidirect
251. C₈×#41	C₂×C₆₇	semidirect
253. #1584	C₂×C₆₇	semidirect
254. #1583	C₂×C₆₇	semidirect
255. #1582	C₂×C₆₇	semidirect
256. #1581	C₂×C₆₇	semidirect
257. #1564	C₂×C₆₇	semidirect
258. #1476	C₂×C₆₇	semidirect
259. #1476	C₂×C₆₇	semidirect
260. C₂×#322	C₂×C₆₇	semidirect
261. C₆₇	#17863	direct (N × H)
367. #41	M₁₆×C₆₇	semidirect
469. C₈	#135×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2965	2
C₈×#41×C₆₇	2
C₄×#135×C₆₇	2
C₆₇×#1584	2
C₆₇×#1583	2
C₆₇×#1582	2
C₆₇×#1581	2
C₆₇×#1564	2
C₆₇×#1476	2
C₂×#322×C₆₇	2
#17863	67

Subgroups — C67×#17863 #30697114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17863 #30697114