Subgroups — C₆₇×#22130 #30701381

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3076	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1986	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1945	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1937	C₂	1	2	yes	maximal
8	C₆₇×#1929	C₂	1	2	yes	maximal
9	C₆₇×#1922	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1806	C₂	1	1	yes	maximal, characteristic
11	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#533×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
14	#479×C₆₇	C₄	1	2	yes	complemented
15	#478×C₆₇	C₄	1	2	yes	complemented
16	#477×C₆₇	C₄	1	1	yes	characteristic, complemented
17	#476×C₆₇	C₄	1	1	yes	characteristic, complemented
18	#475×C₆₇	C₄	1	1	yes	characteristic, complemented
19	#474×C₆₇	C₄	1	1	yes	characteristic, complemented
20	#446×C₆₇	—	2	1	no	—
21	#446×C₆₇	—	2	1	no	—
22	C₈×Q₈×C₆₇	—	2	1	no	—
23	C₈×Q₈×C₆₇	—	2	1	no	—
24	#433×C₆₇	—	2	1	no	—
25	#433×C₆₇	—	2	1	no	—
26	#431×C₆₇	—	2	1	no	—
27	#431×C₆₇	—	2	1	no	—
28	C₂×#108×C₆₇	C₂²	1	2	yes	—
29	#427×C₆₇	C₂²	1	2	yes	—
30	#424×C₆₇	C₂²	1	1	yes	characteristic
31	#424×C₆₇	C₂²	1	1	yes	characteristic
32	#424×C₆₇	C₂²	1	2	yes	—
33	#423×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#104×C₆₇	C₂²	1	2	yes	—
36	#416×C₆₇	C₂²	1	2	yes	—
37	#408×C₆₇	C₂²	1	1	yes	characteristic
38	#408×C₆₇	C₂²	1	1	yes	characteristic
39	#405×C₆₇	C₂²	1	2	yes	—
40	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
44	#341×C₆₇	C₂²	1	2	yes	—
45	#339×C₆₇	C₂²	1	2	yes	—
46	#339×C₆₇	C₂²	1	2	yes	—
47	#339×C₆₇	C₂²	1	2	yes	—
48	#336×C₆₇	C₂²	1	2	yes	—
49	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	#125×C₆₇	C₂×C₄	1	2	yes	—
59	C₂²⋊Q₈×C₆₇	—	2	1	no	—
60	C₂²⋊Q₈×C₆₇	—	2	1	no	—
61	C₄×Q₈×C₆₇	—	2	1	no	—
62	C₄×Q₈×C₆₇	—	2	1	no	—
63	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄×D₄×C₆₇	—	2	1	no	—
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	#117×C₆₇	—	2	1	no	—
70	#117×C₆₇	C₂³	1	1	yes	characteristic
71	#117×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄²×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#108×C₆₇	—	4	2	no	—
77	#107×C₆₇	C₂×C₄	1	2	yes	—
78	#107×C₆₇	C₂×C₄	1	2	yes	—
79	#106×C₆₇	C₂×C₄	1	2	yes	—
80	#106×C₆₇	C₂×C₄	1	2	yes	—
81	C₄⋊C₈×C₆₇	—	2	1	no	—
82	C₄⋊C₈×C₆₇	—	2	1	no	—
83	C₄⋊C₈×C₆₇	—	2	1	no	—
84	C₄⋊C₈×C₆₇	—	2	1	no	—
85	C₄⋊C₈×C₆₇	—	2	2	no	—
86	C₄⋊C₈×C₆₇	—	2	2	no	—
87	C₄⋊C₈×C₆₇	—	2	1	no	—
88	C₄⋊C₈×C₆₇	—	2	1	no	—
89	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
92	#104×C₆₇	—	4	2	no	—
93	#103×C₆₇	C₂×C₄	1	2	yes	—
94	#103×C₆₇	C₂×C₄	1	2	yes	—
95	#102×C₆₇	C₂×C₄	1	2	yes	—
96	#102×C₆₇	C₂×C₄	1	2	yes	—
97	C₂²⋊C₈×C₆₇	—	2	1	no	—
98	C₂²⋊C₈×C₆₇	—	2	1	no	—
99	C₂²⋊C₈×C₆₇	—	2	1	no	—
100	C₂²⋊C₈×C₆₇	—	2	1	no	—
101	#97×C₆₇	—	2	2	no	—
102	#97×C₆₇	—	2	1	no	—
103	C₄×C₈×C₆₇	—	2	2	no	—
104	C₄×C₈×C₆₇	—	2	1	no	—
105	C₄×C₈×C₆₇	—	4	2	no	—
106	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
107	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
108	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
109	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
110	#95×C₆₇	—	2	1	no	—
111	#41×C₆₇	—	2	1	no	—
112	#41×C₆₇	—	2	1	no	—
113	#41×C₆₇	—	2	1	no	—
114	#41×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	—	2	1	no	—
117	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄×C₆₇	—	2	1	no	—
119	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
127	M₁₆×C₆₇	—	2	2	no	—
128	M₁₆×C₆₇	—	2	1	no	—
129	M₁₆×C₆₇	—	2	1	no	—
130	M₁₆×C₆₇	—	2	1	no	—
131	M₁₆×C₆₇	—	2	1	no	—
132	M₁₆×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	1	no	—
135	C₂×C₈×C₆₇	—	2	2	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	—	2	1	no	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	1	no	—
141	C₂×C₈×C₆₇	—	2	1	no	—
142	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
143	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₄⋊C₄×C₆₇	—	2	4	no	—
149	C₄⋊C₄×C₆₇	—	2	1	no	—
150	C₄⋊C₄×C₆₇	—	2	1	no	—
151	C₄⋊C₄×C₆₇	—	2	2	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
156	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
157	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂²⋊C₄×C₆₇	—	2	1	no	—
160	C₂²⋊C₄×C₆₇	—	2	2	no	—
161	C₂²⋊C₄×C₆₇	—	2	1	no	—
162	C₂²⋊C₄×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	—	2	1	no	—
164	C₄²×C₆₇	—	2	1	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	—	2	2	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂³×C₆₇	—	2	1	no	—
174	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
175	Q₈×C₆₇	—	2	1	no	—
176	Q₈×C₆₇	—	2	1	no	—
177	Q₈×C₆₇	—	4	2	no	—
178	Q₈×C₆₇	—	4	2	no	—
179	D₄×C₆₇	—	2	1	no	—
180	D₄×C₆₇	—	2	1	no	—
181	D₄×C₆₇	—	4	1	no	—
182	D₄×C₆₇	—	4	1	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	1	no	—
195	C₂×C₄×C₆₇	—	4	1	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
205	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
206	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
207	C₈×C₆₇	—	2	2	no	—
208	C₈×C₆₇	—	2	2	no	—
209	C₈×C₆₇	—	2	1	no	—
210	C₈×C₆₇	—	2	1	no	—
211	C₈×C₆₇	—	2	1	no	—
212	C₈×C₆₇	—	2	1	no	—
213	C₈×C₆₇	—	4	1	no	—
214	C₈×C₆₇	—	4	2	no	—
215	C₈×C₆₇	—	4	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	—	4	1	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	—	4	1	no	—
221	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₆₇	#541	1	1	yes	characteristic
223	C₂²×C₆₇	#542	1	1	yes	characteristic
224	C₄×C₆₇	—	2	1	no	—
225	C₄×C₆₇	—	2	1	no	—
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	4	4	no	—
228	C₄×C₆₇	—	4	1	no	—
229	C₄×C₆₇	—	4	1	no	—
230	C₄×C₆₇	—	4	2	no	—
231	C₄×C₆₇	—	4	1	no	—
232	C₄×C₆₇	—	4	2	no	—
233	C₄×C₆₇	#443	1	2	yes	—
234	C₄×C₆₇	#519	1	1	yes	characteristic
235	C₄×C₆₇	#520	1	1	yes	characteristic
236	C₄×C₆₇	#536	1	1	yes	characteristic
237	C₄×C₆₇	#537	1	1	yes	characteristic
238	#22130	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂×C₆₇	—	2	1	no	—
240	C₂×C₆₇	—	2	1	no	—
241	C₂×C₆₇	—	8	1	no	—
242	C₂×C₆₇	#2323	1	1	yes	characteristic
243	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
244	C₂×C₆₇	#2955	1	1	yes	characteristic
245	#3076	C₂×C₆₇	1	1	yes	characteristic
246	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#2922	C₂×C₆₇	1	1	yes	characteristic
248	#1986	C₂×C₆₇	1	1	yes	characteristic, complemented
249	#1950	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#1945	C₂×C₆₇	1	2	yes	complemented
251	#1937	C₂×C₆₇	1	2	yes	—
252	#1929	C₂×C₆₇	1	2	yes	—
253	#1922	C₂×C₆₇	1	2	yes	complemented
254	#1806	C₂×C₆₇	1	1	yes	characteristic
255	C₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
256	C₆₇	#22130	1	1	yes	characteristic, direct product factor, Sylow
257	#533	C₂²×C₆₇	1	1	yes	characteristic
258	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
259	#479	C₄×C₆₇	1	2	yes	complemented
260	#478	C₄×C₆₇	1	2	yes	complemented
261	#477	C₄×C₆₇	1	1	yes	characteristic, complemented
262	#476	C₄×C₆₇	1	1	yes	characteristic, complemented
263	#475	C₄×C₆₇	1	1	yes	characteristic, complemented
264	#474	C₄×C₆₇	1	1	yes	characteristic, complemented
265	#446	—	2	1	no	—
266	#446	—	2	1	no	—
267	C₈×Q₈	—	2	1	no	—
268	C₈×Q₈	—	2	1	no	—
269	#433	—	2	1	no	—
270	#433	—	2	1	no	—
271	#431	—	2	1	no	—
272	#431	—	2	1	no	—
273	C₂×#108	C₂²×C₆₇	1	2	yes	—
274	#427	C₂²×C₆₇	1	2	yes	—
275	#424	C₂²×C₆₇	1	1	yes	characteristic
276	#424	C₂²×C₆₇	1	1	yes	characteristic
277	#424	C₂²×C₆₇	1	2	yes	—
278	#423	C₂²×C₆₇	1	1	yes	characteristic
279	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×#104	C₂²×C₆₇	1	2	yes	—
281	#416	C₂²×C₆₇	1	2	yes	—
282	#408	C₂²×C₆₇	1	1	yes	characteristic
283	#408	C₂²×C₆₇	1	1	yes	characteristic
284	#405	C₂²×C₆₇	1	2	yes	—
285	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
287	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
288	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
289	#341	C₂²×C₆₇	1	2	yes	—
290	#339	C₂²×C₆₇	1	2	yes	—
291	#339	C₂²×C₆₇	1	2	yes	—
292	#339	C₂²×C₆₇	1	2	yes	—
293	#336	C₂²×C₆₇	1	2	yes	—
294	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
295	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
296	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
297	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
298	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
299	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
301	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
302	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
303	#125	C₂×C₄×C₆₇	1	2	yes	—
304	C₂²⋊Q₈	—	2	1	no	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
309	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
313	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
314	#117	—	2	1	no	—
315	#117	C₂³×C₆₇	1	1	yes	characteristic
316	#117	C₂³×C₆₇	1	2	yes	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
320	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
321	#108	—	4	2	no	—
322	#107	C₂×C₄×C₆₇	1	2	yes	—
323	#107	C₂×C₄×C₆₇	1	2	yes	—
324	#106	C₂×C₄×C₆₇	1	2	yes	—
325	#106	C₂×C₄×C₆₇	1	2	yes	—
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	—	2	1	no	—
328	C₄⋊C₈	—	2	1	no	—
329	C₄⋊C₈	—	2	1	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	2	2	no	—
332	C₄⋊C₈	—	2	1	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
335	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
336	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
337	#104	—	4	2	no	—
338	#103	C₂×C₄×C₆₇	1	2	yes	—
339	#103	C₂×C₄×C₆₇	1	2	yes	—
340	#102	C₂×C₄×C₆₇	1	2	yes	—
341	#102	C₂×C₄×C₆₇	1	2	yes	—
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	2	1	no	—
344	C₂²⋊C₈	—	2	1	no	—
345	C₂²⋊C₈	—	2	1	no	—
346	#97	—	2	2	no	—
347	#97	—	2	1	no	—
348	C₄×C₈	—	2	2	no	—
349	C₄×C₈	—	2	1	no	—
350	C₄×C₈	—	4	2	no	—
351	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
352	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
353	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
354	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
355	#95	—	2	1	no	—
356	#41	—	2	1	no	—
357	#41	—	2	1	no	—
358	#41	—	2	1	no	—
359	#41	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
370	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
371	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
372	M₁₆	—	2	2	no	—
373	M₁₆	—	2	1	no	—
374	M₁₆	—	2	1	no	—
375	M₁₆	—	2	1	no	—
376	M₁₆	—	2	1	no	—
377	M₁₆	—	2	2	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
388	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
389	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
390	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
391	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
392	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
401	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
402	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
403	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
404	C₂²⋊C₄	—	2	1	no	—
405	C₂²⋊C₄	—	2	2	no	—
406	C₂²⋊C₄	—	2	1	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	2	1	no	—
410	C₄²	—	2	2	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
415	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
416	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
417	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
418	C₂³	—	2	1	no	—
419	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
420	Q₈	—	2	1	no	—
421	Q₈	—	2	1	no	—
422	Q₈	—	4	2	no	—
423	Q₈	—	4	2	no	—
424	D₄	—	2	1	no	—
425	D₄	—	2	1	no	—
426	D₄	—	4	1	no	—
427	D₄	—	4	1	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
443	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
444	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
445	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
446	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
448	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
449	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
450	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
451	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
452	C₈	—	2	2	no	—
453	C₈	—	2	2	no	—
454	C₈	—	2	1	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	1	no	—
458	C₈	—	4	1	no	—
459	C₈	—	4	1	no	—
460	C₈	—	4	2	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	1	no	—
466	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
467	C₂²	#541×C₆₇	1	1	yes	characteristic
468	C₂²	#542×C₆₇	1	1	yes	characteristic
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	4	no	—
473	C₄	—	4	1	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	4	2	no	—
478	C₄	#443×C₆₇	1	2	yes	—
479	C₄	#519×C₆₇	1	1	yes	characteristic
480	C₄	#520×C₆₇	1	1	yes	characteristic
481	C₄	#536×C₆₇	1	1	yes	characteristic
482	C₄	#537×C₆₇	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	8	1	no	—
486	C₂	C₆₇×#2323	1	1	yes	characteristic
487	C₂	C₂×#443×C₆₇	1	1	yes	characteristic
488	C₂	C₆₇×#2955	1	1	yes	characteristic

	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	C₂×#41×C₆₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2960	C₂	semidirect
4. C₆₇×#1986	C₂	semidirect
5. C₆₇×#1950	C₂	semidirect
6. C₆₇×#1945	C₂	semidirect
9. C₆₇×#1922	C₂	semidirect
11. C₈×C₄⋊C₄×C₆₇	C₂	semidirect
14. #479×C₆₇	C₄	semidirect
15. #478×C₆₇	C₄	semidirect
16. #477×C₆₇	C₄	semidirect
17. #476×C₆₇	C₄	semidirect
18. #475×C₆₇	C₄	semidirect
19. #474×C₆₇	C₄	semidirect
238. #22130	C₆₇	direct (N × H)
246. #2960	C₂×C₆₇	semidirect
248. #1986	C₂×C₆₇	semidirect
249. #1950	C₂×C₆₇	semidirect
250. #1945	C₂×C₆₇	semidirect
253. #1922	C₂×C₆₇	semidirect
255. C₈×C₄⋊C₄	C₂×C₆₇	semidirect
256. C₆₇	#22130	direct (N × H)
259. #479	C₄×C₆₇	semidirect
260. #478	C₄×C₆₇	semidirect
261. #477	C₄×C₆₇	semidirect
262. #476	C₄×C₆₇	semidirect
263. #475	C₄×C₆₇	semidirect
264. #474	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3076	2
C₆₇×#2960	2
C₆₇×#2922	2
C₆₇×#1986	2
C₆₇×#1950	2
C₆₇×#1945	2
C₆₇×#1937	2
C₆₇×#1929	2
C₆₇×#1922	2
C₆₇×#1806	2
C₈×C₄⋊C₄×C₆₇	2
#22130	67

Subgroups — C67×#22130 #30701381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22130 #30701381