Subgroups — C₆₇×#21547 #30700798

Order 17152.

906 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×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×#444×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2930	C₂	1	2	yes	maximal, complemented
4	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1937	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1897	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1773	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1764	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#131×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#525×C₆₇	C₂²	1	2	yes	—
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	#467×C₆₇	C₄	1	2	yes	complemented
19	#466×C₆₇	C₄	1	2	yes	complemented
20	#465×C₆₇	C₄	1	4	yes	complemented
21	#444×C₆₇	—	2	4	no	—
22	#436×C₆₇	—	2	4	no	—
23	#435×C₆₇	—	2	4	no	—
24	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
26	#427×C₆₇	C₂²	1	1	yes	characteristic
27	#424×C₆₇	C₂²	1	2	yes	—
28	C₂×#104×C₆₇	C₂²	1	2	yes	—
29	C₂×#104×C₆₇	C₂²	1	2	yes	—
30	#416×C₆₇	C₂²	1	2	yes	—
31	#408×C₆₇	C₂²	1	2	yes	—
32	#407×C₆₇	C₂²	1	2	yes	—
33	#405×C₆₇	—	2	4	no	—
34	#405×C₆₇	C₂²	1	2	yes	—
35	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	#341×C₆₇	C₂²	1	2	yes	—
40	#339×C₆₇	C₂²	1	2	yes	—
41	#339×C₆₇	C₂²	1	2	yes	—
42	C₂×#41×C₆₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	#135×C₆₇	—	2	2	no	—
45	#135×C₆₇	—	2	2	no	—
46	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	#131×C₆₇	—	2	4	no	—
50	#131×C₆₇	—	2	4	no	—
51	C₂×M₁₆×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	—	2	2	no	—
53	C₂×M₁₆×C₆₇	—	2	2	no	—
54	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
55	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
56	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	—	2	2	no	—
60	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	#123×C₆₇	—	2	2	no	—
67	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
68	#121×C₆₇	C₂×C₄	1	4	yes	—
69	#120×C₆₇	—	2	2	no	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	#117×C₆₇	—	2	2	no	—
72	#117×C₆₇	—	2	2	no	—
73	#117×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	#108×C₆₇	—	2	2	no	—
79	#108×C₆₇	—	2	2	no	—
80	#107×C₆₇	C₂×C₄	1	2	yes	—
81	#106×C₆₇	C₂×C₄	1	2	yes	—
82	C₄⋊C₈×C₆₇	—	2	4	no	—
83	#104×C₆₇	—	2	4	no	—
84	#104×C₆₇	—	2	4	no	—
85	#103×C₆₇	C₂×C₄	1	4	yes	—
86	#102×C₆₇	C₂×C₄	1	4	yes	—
87	C₂²⋊C₈×C₆₇	—	2	4	no	—
88	C₂²⋊C₈×C₆₇	—	2	4	no	—
89	#97×C₆₇	—	2	2	no	—
90	#97×C₆₇	—	2	2	no	—
91	#97×C₆₇	D₄	1	4	yes	complemented
92	C₄×C₈×C₆₇	—	2	2	no	—
93	C₄×C₈×C₆₇	—	2	2	no	—
94	C₄×C₈×C₆₇	—	2	2	no	—
95	C₂⁴×C₆₇	#41	1	1	yes	characteristic
96	#41×C₆₇	—	2	2	no	—
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	—	2	2	no	—
99	#41×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	Q₁₆×C₆₇	—	2	2	no	—
116	SD₁₆×C₆₇	—	2	4	no	—
117	D₈×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	2	8	no	—
119	M₁₆×C₆₇	—	2	2	no	—
120	M₁₆×C₆₇	—	2	2	no	—
121	M₁₆×C₆₇	—	2	2	no	—
122	M₁₆×C₆₇	—	2	2	no	—
123	M₁₆×C₆₇	—	4	4	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	4	no	—
129	C₂×C₈×C₆₇	—	2	4	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	4	4	no	—
133	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
138	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₆₇	#41	1	2	yes	—
141	C₂×C₈×C₆₇	#41	1	2	yes	—
142	C₄⋊C₄×C₆₇	—	2	4	no	—
143	C₄⋊C₄×C₆₇	—	2	4	no	—
144	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
145	C₂²⋊C₄×C₆₇	—	2	4	no	—
146	C₂²⋊C₄×C₆₇	—	2	4	no	—
147	C₂²⋊C₄×C₆₇	—	2	2	no	—
148	C₂²⋊C₄×C₆₇	—	2	4	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	C₂×D₄	1	2	yes	—
153	C₄²×C₆₇	#41	1	2	yes	—
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	—	2	1	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	C₄×D₄	1	2	yes	—
159	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
160	Q₈×C₆₇	—	2	2	no	—
161	Q₈×C₆₇	—	2	2	no	—
162	D₄×C₆₇	—	2	2	no	—
163	D₄×C₆₇	—	2	2	no	—
164	D₄×C₆₇	—	4	2	no	—
165	D₄×C₆₇	—	4	2	no	—
166	D₄×C₆₇	—	8	4	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₈×C₆₇	—	2	2	no	—
189	C₈×C₆₇	—	2	2	no	—
190	C₈×C₆₇	—	4	2	no	—
191	C₈×C₆₇	—	4	2	no	—
192	C₈×C₆₇	C₄×D₄	1	2	yes	—
193	C₈×C₆₇	C₄×D₄	1	2	yes	—
194	C₈×C₆₇	#121	1	4	yes	—
195	C₂²×C₆₇	—	2	2	no	—
196	C₂²×C₆₇	—	2	2	no	—
197	C₂²×C₆₇	—	2	2	no	—
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	4	2	no	—
200	C₂²×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂²×C₆₇	#525	1	1	yes	characteristic
205	C₂²×C₆₇	#529	1	1	yes	characteristic
206	C₄×C₆₇	—	2	2	no	—
207	C₄×C₆₇	—	4	2	no	—
208	C₄×C₆₇	—	4	2	no	—
209	C₄×C₆₇	—	4	4	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
212	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
213	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
214	C₄×C₆₇	#532	1	1	yes	characteristic
215	#21547	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂×C₆₇	—	2	1	no	—
217	C₂×C₆₇	—	2	1	no	—
218	C₂×C₆₇	—	4	2	no	—
219	C₂×C₆₇	—	8	2	no	—
220	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
221	C₂×C₆₇	#2957	1	2	yes	—
222	#3050	C₂×C₆₇	1	1	yes	characteristic
223	C₂×#444	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#2930	C₂×C₆₇	1	2	yes	complemented
225	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#1975	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#1946	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#1937	C₂×C₆₇	1	2	yes	complemented
229	#1897	C₂×C₆₇	1	2	yes	complemented
230	#1810	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1773	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1764	C₂×C₆₇	1	2	yes	complemented
233	C₆₇	#21547	1	1	yes	characteristic, direct product factor, Sylow
234	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#131	C₂²×C₆₇	1	2	yes	—
236	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
237	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
238	#525	C₂²×C₆₇	1	2	yes	—
239	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
240	#467	C₄×C₆₇	1	2	yes	complemented
241	#466	C₄×C₆₇	1	2	yes	complemented
242	#465	C₄×C₆₇	1	4	yes	complemented
243	#444	—	2	4	no	—
244	#436	—	2	4	no	—
245	#435	—	2	4	no	—
246	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
248	#427	C₂²×C₆₇	1	1	yes	characteristic
249	#424	C₂²×C₆₇	1	2	yes	—
250	C₂×#104	C₂²×C₆₇	1	2	yes	—
251	C₂×#104	C₂²×C₆₇	1	2	yes	—
252	#416	C₂²×C₆₇	1	2	yes	—
253	#408	C₂²×C₆₇	1	2	yes	—
254	#407	C₂²×C₆₇	1	2	yes	—
255	#405	—	2	4	no	—
256	#405	C₂²×C₆₇	1	2	yes	—
257	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
261	#341	C₂²×C₆₇	1	2	yes	—
262	#339	C₂²×C₆₇	1	2	yes	—
263	#339	C₂²×C₆₇	1	2	yes	—
264	C₂×#41	C₂³×C₆₇	1	2	yes	—
265	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
266	#135	—	2	2	no	—
267	#135	—	2	2	no	—
268	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
269	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
270	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
271	#131	—	2	4	no	—
272	#131	—	2	4	no	—
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	—	2	2	no	—
275	C₂×M₁₆	—	2	2	no	—
276	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
277	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
278	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
279	C₂²×C₈	—	2	2	no	—
280	C₂²×C₈	—	2	2	no	—
281	C₂²×C₈	—	2	2	no	—
282	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
283	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
284	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
288	#123	—	2	2	no	—
289	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
290	#121	C₂×C₄×C₆₇	1	4	yes	—
291	#120	—	2	2	no	—
292	C₄×D₄	—	2	4	no	—
293	#117	—	2	2	no	—
294	#117	—	2	2	no	—
295	#117	C₂³×C₆₇	1	2	yes	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₂²⋊C₄	—	2	1	no	—
298	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
299	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
300	#108	—	2	2	no	—
301	#108	—	2	2	no	—
302	#107	C₂×C₄×C₆₇	1	2	yes	—
303	#106	C₂×C₄×C₆₇	1	2	yes	—
304	C₄⋊C₈	—	2	4	no	—
305	#104	—	2	4	no	—
306	#104	—	2	4	no	—
307	#103	C₂×C₄×C₆₇	1	4	yes	—
308	#102	C₂×C₄×C₆₇	1	4	yes	—
309	C₂²⋊C₈	—	2	4	no	—
310	C₂²⋊C₈	—	2	4	no	—
311	#97	—	2	2	no	—
312	#97	—	2	2	no	—
313	#97	D₄×C₆₇	1	4	yes	complemented
314	C₄×C₈	—	2	2	no	—
315	C₄×C₈	—	2	2	no	—
316	C₄×C₈	—	2	2	no	—
317	C₂⁴	#41×C₆₇	1	1	yes	characteristic
318	#41	—	2	2	no	—
319	#41	—	2	2	no	—
320	#41	—	2	2	no	—
321	#41	—	2	2	no	—
322	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	4	no	—
325	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
334	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
335	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
336	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
337	Q₁₆	—	2	2	no	—
338	SD₁₆	—	2	4	no	—
339	D₈	—	2	2	no	—
340	M₁₆	—	2	8	no	—
341	M₁₆	—	2	2	no	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	2	2	no	—
345	M₁₆	—	4	4	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	4	no	—
351	C₂×C₈	—	2	4	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	4	4	no	—
355	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
356	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
357	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
358	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
359	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
360	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
361	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
362	C₂×C₈	#41×C₆₇	1	2	yes	—
363	C₂×C₈	#41×C₆₇	1	2	yes	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
367	C₂²⋊C₄	—	2	4	no	—
368	C₂²⋊C₄	—	2	4	no	—
369	C₂²⋊C₄	—	2	2	no	—
370	C₂²⋊C₄	—	2	4	no	—
371	C₄²	—	2	1	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	1	no	—
374	C₄²	C₂×D₄×C₆₇	1	2	yes	—
375	C₄²	#41×C₆₇	1	2	yes	—
376	C₂³	—	2	1	no	—
377	C₂³	—	2	1	no	—
378	C₂³	—	2	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	C₄×D₄×C₆₇	1	2	yes	—
381	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
382	Q₈	—	2	2	no	—
383	Q₈	—	2	2	no	—
384	D₄	—	2	2	no	—
385	D₄	—	2	2	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	2	no	—
388	D₄	—	8	4	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
404	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
405	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
406	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
407	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
409	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
410	C₈	—	2	2	no	—
411	C₈	—	2	2	no	—
412	C₈	—	4	2	no	—
413	C₈	—	4	2	no	—
414	C₈	C₄×D₄×C₆₇	1	2	yes	—
415	C₈	C₄×D₄×C₆₇	1	2	yes	—
416	C₈	#121×C₆₇	1	4	yes	—
417	C₂²	—	2	2	no	—
418	C₂²	—	2	2	no	—
419	C₂²	—	2	2	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	4	4	no	—
425	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
426	C₂²	#525×C₆₇	1	1	yes	characteristic
427	C₂²	#529×C₆₇	1	1	yes	characteristic
428	C₄	—	2	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	4	no	—
433	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
434	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
435	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
436	C₄	#532×C₆₇	1	1	yes	characteristic
437	C₂	—	2	1	no	—
438	C₂	—	2	1	no	—
439	C₂	—	4	2	no	—
440	C₂	—	8	2	no	—
441	C₂	C₄×#121×C₆₇	1	1	yes	characteristic
442	C₂	C₆₇×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₆₇×#3050	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#444×C₆₇	C₂	semidirect
3. C₆₇×#2930	C₂	semidirect
4. C₂×#405×C₆₇	C₂	semidirect
5. C₆₇×#1975	C₂	semidirect
6. C₆₇×#1946	C₂	semidirect
7. C₆₇×#1937	C₂	semidirect
8. C₆₇×#1897	C₂	semidirect
9. C₆₇×#1810	C₂	semidirect
10. C₆₇×#1773	C₂	semidirect
11. C₆₇×#1764	C₂	semidirect
18. #467×C₆₇	C₄	semidirect
19. #466×C₆₇	C₄	semidirect
20. #465×C₆₇	C₄	semidirect
91. #97×C₆₇	D₄	semidirect
215. #21547	C₆₇	direct (N × H)
223. C₂×#444	C₂×C₆₇	semidirect
224. #2930	C₂×C₆₇	semidirect
225. C₂×#405	C₂×C₆₇	semidirect
226. #1975	C₂×C₆₇	semidirect
227. #1946	C₂×C₆₇	semidirect
228. #1937	C₂×C₆₇	semidirect
229. #1897	C₂×C₆₇	semidirect
230. #1810	C₂×C₆₇	semidirect
231. #1773	C₂×C₆₇	semidirect
232. #1764	C₂×C₆₇	semidirect
233. C₆₇	#21547	direct (N × H)
240. #467	C₄×C₆₇	semidirect
241. #466	C₄×C₆₇	semidirect
242. #465	C₄×C₆₇	semidirect
313. #97	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3050	2
C₂×#444×C₆₇	2
C₆₇×#2930	2
C₂×#405×C₆₇	2
C₆₇×#1975	2
C₆₇×#1946	2
C₆₇×#1937	2
C₆₇×#1897	2
C₆₇×#1810	2
C₆₇×#1773	2
C₆₇×#1764	2
#21547	67

Subgroups — C67×#21547 #30700798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21547 #30700798