Subgroups — C₆₇×#17874 #30697125

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2989	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2982	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1588	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1587	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1584	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1583	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1476	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	C₄	1	1	yes	characteristic
13	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	C₂²	1	2	yes	—
16	C₈×Q₈×C₆₇	—	2	1	no	—
17	C₈×Q₈×C₆₇	—	2	1	no	—
18	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
20	#441×C₆₇	C₄	1	2	yes	—
21	#440×C₆₇	C₄	1	2	yes	—
22	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
23	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
24	#436×C₆₇	C₂²	1	2	yes	—
25	#435×C₆₇	—	2	2	no	—
26	#432×C₆₇	—	2	1	no	—
27	#431×C₆₇	—	2	2	no	—
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	#431×C₆₇	C₂²	1	1	yes	characteristic
30	#428×C₆₇	C₄	1	1	yes	characteristic
31	#423×C₆₇	—	2	1	no	—
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#419×C₆₇	C₄	1	1	yes	characteristic
34	#417×C₆₇	C₄	1	1	yes	characteristic
35	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#335×C₆₇	C₂²	1	2	yes	—
38	#334×C₆₇	C₂²	1	2	yes	—
39	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
40	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
45	#322×C₆₇	C₂²	1	2	yes	—
46	#322×C₆₇	C₂²	1	2	yes	—
47	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	#137×C₆₇	—	2	4	no	—
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₆₇	—	2	2	no	—
53	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₄×Q₈×C₆₇	—	2	1	no	—
58	C₄×Q₈×C₆₇	—	2	1	no	—
59	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
62	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₆₇	—	2	1	no	—
64	C₄×D₄×C₆₇	—	2	1	no	—
65	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	#117×C₆₇	—	2	1	no	—
68	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	—	2	1	no	—
72	C₂×C₄²×C₆₇	—	2	1	no	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	C₂×C₄	1	2	yes	—
75	#106×C₆₇	C₂×C₄	1	2	yes	—
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	1	no	—
80	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
85	#103×C₆₇	C₂×C₄	1	2	yes	—
86	#103×C₆₇	C₂×C₄	1	2	yes	—
87	#103×C₆₇	C₂×C₄	1	2	yes	—
88	#102×C₆₇	C₂×C₄	1	2	yes	—
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₂²⋊C₈×C₆₇	—	2	2	no	—
91	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
92	#97×C₆₇	—	2	2	no	—
93	#97×C₆₇	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₆₇	—	2	2	no	—
95	C₄×C₈×C₆₇	—	2	2	no	—
96	C₄×C₈×C₆₇	—	2	1	no	—
97	C₄×C₈×C₆₇	—	2	1	no	—
98	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	2	yes	—
101	C₄×C₈×C₆₇	C₂³	1	2	yes	—
102	#41×C₆₇	—	2	2	no	—
103	#41×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×Q₈×C₆₇	—	2	2	no	—
106	C₂×Q₈×C₆₇	M₁₆	1	4	yes	complemented
107	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
119	Q₁₆×C₆₇	—	2	4	no	—
120	SD₁₆×C₆₇	—	2	4	no	—
121	M₁₆×C₆₇	—	2	2	no	—
122	M₁₆×C₆₇	—	2	2	no	—
123	M₁₆×C₆₇	—	8	2	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	4	2	no	—
130	C₂×C₈×C₆₇	—	8	1	no	—
131	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
133	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	2	1	no	—
140	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
141	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₆₇	—	2	1	no	—
144	C₂²⋊C₄×C₆₇	—	2	1	no	—
145	C₂²⋊C₄×C₆₇	—	2	1	no	—
146	C₄²×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₆₇	#41	1	1	yes	characteristic
154	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
155	C₂³×C₆₇	—	2	1	no	—
156	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
157	Q₈×C₆₇	—	2	2	no	—
158	Q₈×C₆₇	—	2	4	no	—
159	Q₈×C₆₇	—	2	1	no	—
160	Q₈×C₆₇	—	2	1	no	—
161	Q₈×C₆₇	C₂×M₁₆	1	4	yes	—
162	D₄×C₆₇	—	2	1	no	—
163	D₄×C₆₇	—	2	1	no	—
164	D₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	8	1	no	—
178	C₂×C₄×C₆₇	—	8	1	no	—
179	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
183	C₂×C₄×C₆₇	#131	1	2	yes	—
184	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₈×C₆₇	—	2	2	no	—
188	C₈×C₆₇	—	2	2	no	—
189	C₈×C₆₇	—	2	2	no	—
190	C₈×C₆₇	—	2	2	no	—
191	C₈×C₆₇	—	2	2	no	—
192	C₈×C₆₇	—	4	1	no	—
193	C₈×C₆₇	—	4	1	no	—
194	C₈×C₆₇	—	4	2	no	—
195	C₈×C₆₇	—	4	1	no	—
196	C₂²×C₆₇	—	2	1	no	—
197	C₂²×C₆₇	—	2	1	no	—
198	C₂²×C₆₇	—	4	1	no	—
199	C₂²×C₆₇	—	4	1	no	—
200	C₂²×C₆₇	—	4	1	no	—
201	C₂²×C₆₇	#435	1	2	yes	—
202	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
203	C₄×C₆₇	—	2	4	no	—
204	C₄×C₆₇	—	2	1	no	—
205	C₄×C₆₇	—	2	1	no	—
206	C₄×C₆₇	—	2	1	no	—
207	C₄×C₆₇	—	4	1	no	—
208	C₄×C₆₇	—	4	1	no	—
209	C₄×C₆₇	—	4	1	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	#435	1	2	yes	—
212	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
213	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
214	C₄×C₆₇	#577	1	1	yes	characteristic
215	C₄×C₆₇	#578	1	1	yes	characteristic
216	#17874	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₆₇	—	2	2	no	—
218	C₂×C₆₇	—	8	1	no	—
219	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
220	C₂×C₆₇	#2947	1	1	yes	characteristic
221	C₂×C₆₇	#2956	1	1	yes	characteristic
222	#2989	C₂×C₆₇	1	1	yes	characteristic
223	#2982	C₂×C₆₇	1	1	yes	characteristic, complemented
224	C₄×#137	C₂×C₆₇	1	1	yes	characteristic
225	#1588	C₂×C₆₇	1	2	yes	complemented
226	#1587	C₂×C₆₇	1	2	yes	complemented
227	#1584	C₂×C₆₇	1	2	yes	complemented
228	#1583	C₂×C₆₇	1	2	yes	complemented
229	#1565	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#1476	C₂×C₆₇	1	1	yes	characteristic
231	C₂×Q₈⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1451	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₆₇	#17874	1	1	yes	characteristic, direct product factor, Sylow
234	C₂×#137	C₄×C₆₇	1	1	yes	characteristic
235	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
237	#446	C₂²×C₆₇	1	2	yes	—
238	C₈×Q₈	—	2	1	no	—
239	C₈×Q₈	—	2	1	no	—
240	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
241	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
242	#441	C₄×C₆₇	1	2	yes	—
243	#440	C₄×C₆₇	1	2	yes	—
244	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
245	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
246	#436	C₂²×C₆₇	1	2	yes	—
247	#435	—	2	2	no	—
248	#432	—	2	1	no	—
249	#431	—	2	2	no	—
250	#431	C₂²×C₆₇	1	1	yes	characteristic
251	#431	C₂²×C₆₇	1	1	yes	characteristic
252	#428	C₄×C₆₇	1	1	yes	characteristic
253	#423	—	2	1	no	—
254	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
255	#419	C₄×C₆₇	1	1	yes	characteristic
256	#417	C₄×C₆₇	1	1	yes	characteristic
257	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
259	#335	C₂²×C₆₇	1	2	yes	—
260	#334	C₂²×C₆₇	1	2	yes	—
261	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
262	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
263	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
264	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
265	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
266	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
267	#322	C₂²×C₆₇	1	2	yes	—
268	#322	C₂²×C₆₇	1	2	yes	—
269	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
270	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
271	#137	—	2	4	no	—
272	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
273	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
274	C₂×M₁₆	—	2	2	no	—
275	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
276	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
277	C₂²×C₈	—	2	1	no	—
278	C₂²×C₈	—	2	1	no	—
279	C₄×Q₈	—	2	1	no	—
280	C₄×Q₈	—	2	1	no	—
281	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
282	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
283	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
284	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
285	C₄×D₄	—	2	1	no	—
286	C₄×D₄	—	2	1	no	—
287	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
288	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
289	#117	—	2	1	no	—
290	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
296	#107	C₂×C₄×C₆₇	1	2	yes	—
297	#106	C₂×C₄×C₆₇	1	2	yes	—
298	C₄⋊C₈	—	2	2	no	—
299	C₄⋊C₈	—	2	2	no	—
300	C₄⋊C₈	—	2	2	no	—
301	C₄⋊C₈	—	2	1	no	—
302	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
303	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
304	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
305	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
306	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
307	#103	C₂×C₄×C₆₇	1	2	yes	—
308	#103	C₂×C₄×C₆₇	1	2	yes	—
309	#103	C₂×C₄×C₆₇	1	2	yes	—
310	#102	C₂×C₄×C₆₇	1	2	yes	—
311	C₂²⋊C₈	—	2	1	no	—
312	C₂²⋊C₈	—	2	2	no	—
313	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
314	#97	—	2	2	no	—
315	#97	C₂×C₄×C₆₇	1	2	yes	—
316	C₄×C₈	—	2	2	no	—
317	C₄×C₈	—	2	2	no	—
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	—	2	1	no	—
320	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
321	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
322	C₄×C₈	C₂³×C₆₇	1	2	yes	—
323	C₄×C₈	C₂³×C₆₇	1	2	yes	—
324	#41	—	2	2	no	—
325	#41	—	2	2	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	M₁₆×C₆₇	1	4	yes	complemented
329	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
330	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
331	C₂×D₄	—	2	1	no	—
332	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
339	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
340	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
341	Q₁₆	—	2	4	no	—
342	SD₁₆	—	2	4	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	2	2	no	—
345	M₁₆	—	8	2	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	4	2	no	—
352	C₂×C₈	—	8	1	no	—
353	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
354	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
355	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
356	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
363	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
364	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
365	C₂²⋊C₄	—	2	1	no	—
366	C₂²⋊C₄	—	2	1	no	—
367	C₂²⋊C₄	—	2	1	no	—
368	C₄²	—	2	1	no	—
369	C₄²	—	2	1	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	1	no	—
373	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
374	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
375	C₄²	#41×C₆₇	1	1	yes	characteristic
376	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
377	C₂³	—	2	1	no	—
378	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
379	Q₈	—	2	2	no	—
380	Q₈	—	2	4	no	—
381	Q₈	—	2	1	no	—
382	Q₈	—	2	1	no	—
383	Q₈	C₂×M₁₆×C₆₇	1	4	yes	—
384	D₄	—	2	1	no	—
385	D₄	—	2	1	no	—
386	D₄	—	4	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	8	1	no	—
400	C₂×C₄	—	8	1	no	—
401	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
403	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
404	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
405	C₂×C₄	#131×C₆₇	1	2	yes	—
406	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
407	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	—	2	2	no	—
411	C₈	—	2	2	no	—
412	C₈	—	2	2	no	—
413	C₈	—	2	2	no	—
414	C₈	—	4	1	no	—
415	C₈	—	4	2	no	—
416	C₈	—	4	1	no	—
417	C₈	—	4	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	4	1	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	4	1	no	—
423	C₂²	#435×C₆₇	1	2	yes	—
424	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
425	C₄	—	2	4	no	—
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	4	1	no	—
430	C₄	—	4	1	no	—
431	C₄	—	4	1	no	—
432	C₄	—	4	2	no	—
433	C₄	#435×C₆₇	1	2	yes	—
434	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
435	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
436	C₄	#577×C₆₇	1	1	yes	characteristic
437	C₄	#578×C₆₇	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	8	1	no	—
440	C₂	C₂×#435×C₆₇	1	1	yes	characteristic
441	C₂	C₆₇×#2947	1	1	yes	characteristic
442	C₂	C₆₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#41×C₆₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ #431×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2982	C₂	semidirect
4. C₆₇×#1588	C₂	semidirect
5. C₆₇×#1587	C₂	semidirect
6. C₆₇×#1584	C₂	semidirect
7. C₆₇×#1583	C₂	semidirect
8. C₆₇×#1565	C₂	semidirect
10. C₂×Q₈⋊C₈×C₆₇	C₂	semidirect
11. C₆₇×#1451	C₂	semidirect
106. C₂×Q₈×C₆₇	M₁₆	semidirect
216. #17874	C₆₇	direct (N × H)
223. #2982	C₂×C₆₇	semidirect
225. #1588	C₂×C₆₇	semidirect
226. #1587	C₂×C₆₇	semidirect
227. #1584	C₂×C₆₇	semidirect
228. #1583	C₂×C₆₇	semidirect
229. #1565	C₂×C₆₇	semidirect
231. C₂×Q₈⋊C₈	C₂×C₆₇	semidirect
232. #1451	C₂×C₆₇	semidirect
233. C₆₇	#17874	direct (N × H)
328. C₂×Q₈	M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2989	2
C₆₇×#2982	2
C₄×#137×C₆₇	2
C₆₇×#1588	2
C₆₇×#1587	2
C₆₇×#1584	2
C₆₇×#1583	2
C₆₇×#1565	2
C₆₇×#1476	2
C₂×Q₈⋊C₈×C₆₇	2
C₆₇×#1451	2
#17874	67

Subgroups — C67×#17874 #30697125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17874 #30697125