Subgroups — D₄×#108×C₆₇ #30701918

Order 17152.

894 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×M₁₆×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#108×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1950	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1855	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1810	C₂	1	2	yes	maximal, complemented
8	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
10	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	#436×C₆₇	—	2	2	no	—
13	#435×C₆₇	—	2	4	no	—
14	C₈×D₄×C₆₇	—	2	2	no	—
15	C₈×D₄×C₆₇	C₄	1	4	yes	—
16	C₈×D₄×C₆₇	C₄	1	2	yes	—
17	C₈×D₄×C₆₇	C₄	1	2	yes	—
18	C₂×#108×C₆₇	—	2	8	no	—
19	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#108×C₆₇	C₂²	1	2	yes	—
21	C₂×#108×C₆₇	C₂²	1	4	yes	—
22	C₂×#108×C₆₇	C₂²	1	2	yes	—
23	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
24	#423×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#407×C₆₇	C₂²	1	4	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#341×C₆₇	C₂²	1	2	yes	—
31	#341×C₆₇	C₂²	1	4	yes	—
32	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
34	C₂×M₁₆×C₆₇	—	2	4	no	—
35	C₂×M₁₆×C₆₇	—	2	8	no	—
36	C₂×M₁₆×C₆₇	—	4	4	no	—
37	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
38	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
39	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	—	2	4	no	—
41	C₂²×C₈×C₆₇	—	2	4	no	—
42	C₂²×C₈×C₆₇	—	2	4	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
52	C₄×D₄×C₆₇	D₄	1	2	yes	—
53	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
55	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	#108×C₆₇	—	2	8	no	—
58	#108×C₆₇	—	2	4	no	—
59	#108×C₆₇	—	2	2	no	—
60	#108×C₆₇	—	4	2	no	—
61	#108×C₆₇	D₄	1	4	yes	direct product factor
62	C₄⋊C₈×C₆₇	—	2	2	no	—
63	C₄⋊C₈×C₆₇	—	2	2	no	—
64	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
65	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₆₇	—	2	4	no	—
67	C₂²⋊C₈×C₆₇	—	2	4	no	—
68	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
70	#97×C₆₇	—	2	2	no	—
71	C₄×C₈×C₆₇	—	2	2	no	—
72	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
74	C₂⁴×C₆₇	C₂×Q₈	1	2	yes	—
75	C₂×D₄×C₆₇	—	2	4	no	—
76	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
91	M₁₆×C₆₇	—	2	2	no	—
92	M₁₆×C₆₇	—	2	4	no	—
93	M₁₆×C₆₇	—	2	2	no	—
94	M₁₆×C₆₇	—	2	4	no	—
95	M₁₆×C₆₇	—	2	2	no	—
96	M₁₆×C₆₇	—	4	8	no	—
97	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	2	1	no	—
102	C₂×C₈×C₆₇	—	2	1	no	—
103	C₂×C₈×C₆₇	—	2	8	no	—
104	C₂×C₈×C₆₇	—	2	8	no	—
105	C₂×C₈×C₆₇	—	2	4	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	4	8	no	—
108	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₆₇	#41	1	2	yes	—
113	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
115	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
116	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
117	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
119	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
120	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂³×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	2	4	no	—
127	C₂³×C₆₇	—	4	2	no	—
128	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
131	D₄×C₆₇	—	2	4	no	—
132	D₄×C₆₇	—	2	2	no	—
133	D₄×C₆₇	#108	1	4	yes	direct product factor
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	4	1	no	—
146	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
158	C₈×C₆₇	—	2	4	no	—
159	C₈×C₆₇	—	2	2	no	—
160	C₈×C₆₇	—	2	4	no	—
161	C₈×C₆₇	—	4	4	no	—
162	C₈×C₆₇	—	4	2	no	—
163	C₈×C₆₇	C₄×D₄	1	4	yes	—
164	C₂²×C₆₇	—	2	2	no	—
165	C₂²×C₆₇	—	2	1	no	—
166	C₂²×C₆₇	—	2	1	no	—
167	C₂²×C₆₇	—	2	2	no	—
168	C₂²×C₆₇	—	2	2	no	—
169	C₂²×C₆₇	—	2	2	no	—
170	C₂²×C₆₇	—	2	2	no	—
171	C₂²×C₆₇	—	4	4	no	—
172	C₂²×C₆₇	—	4	4	no	—
173	C₂²×C₆₇	C₂×#108	1	4	yes	complemented
174	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
175	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
176	C₂²×C₆₇	#554	1	1	yes	characteristic
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	—	2	2	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	C₂×#108	1	2	yes	complemented
183	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₆₇	#520	1	1	yes	characteristic
185	C₄×C₆₇	D₄²	1	1	yes	characteristic
186	C₄×C₆₇	#547	1	1	yes	characteristic
187	D₄×#108	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂×C₆₇	—	2	1	no	—
189	C₂×C₆₇	—	2	1	no	—
190	C₂×C₆₇	—	2	4	no	—
191	C₂×C₆₇	—	4	2	no	—
192	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
194	C₂×C₆₇	#2916	1	1	yes	characteristic
195	D₄×M₁₆	C₂×C₆₇	1	2	yes	—
196	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic
197	C₂²×#108	C₂×C₆₇	1	2	yes	complemented
198	#1950	C₂×C₆₇	1	2	yes	complemented
199	#1855	C₂×C₆₇	1	4	yes	complemented
200	#1848	C₂×C₆₇	1	1	yes	characteristic, complemented
201	#1810	C₂×C₆₇	1	2	yes	complemented
202	C₄×#108	C₂×C₆₇	1	1	yes	characteristic, complemented
203	C₆₇	D₄×#108	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
205	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
206	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
207	#436	—	2	2	no	—
208	#435	—	2	4	no	—
209	C₈×D₄	—	2	2	no	—
210	C₈×D₄	C₄×C₆₇	1	4	yes	—
211	C₈×D₄	C₄×C₆₇	1	2	yes	—
212	C₈×D₄	C₄×C₆₇	1	2	yes	—
213	C₂×#108	—	2	8	no	—
214	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×#108	C₂²×C₆₇	1	2	yes	—
216	C₂×#108	C₂²×C₆₇	1	4	yes	—
217	C₂×#108	C₂²×C₆₇	1	2	yes	—
218	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
219	#423	C₂²×C₆₇	1	2	yes	—
220	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
221	#407	C₂²×C₆₇	1	4	yes	—
222	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
223	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
224	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
225	#341	C₂²×C₆₇	1	4	yes	—
226	#341	C₂²×C₆₇	1	2	yes	—
227	C₂²×D₄	Q₈×C₆₇	1	1	yes	characteristic
228	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
229	C₂×M₁₆	—	2	4	no	—
230	C₂×M₁₆	—	2	8	no	—
231	C₂×M₁₆	—	4	4	no	—
232	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
233	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
234	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
235	C₂²×C₈	—	2	4	no	—
236	C₂²×C₈	—	2	4	no	—
237	C₂²×C₈	—	2	4	no	—
238	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
239	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
240	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
241	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
242	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
243	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
244	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
245	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
246	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
247	C₄×D₄	D₄×C₆₇	1	2	yes	—
248	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
249	C₂×C₄⋊C₄	Q₈×C₆₇	1	1	yes	characteristic
250	C₂×C₂²⋊C₄	Q₈×C₆₇	1	2	yes	—
251	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
252	#108	—	2	8	no	—
253	#108	—	2	4	no	—
254	#108	—	2	2	no	—
255	#108	—	4	2	no	—
256	#108	D₄×C₆₇	1	4	yes	direct product factor
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
260	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
261	C₂²⋊C₈	—	2	4	no	—
262	C₂²⋊C₈	—	2	4	no	—
263	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
264	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
265	#97	—	2	2	no	—
266	C₄×C₈	—	2	2	no	—
267	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
268	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
269	C₂⁴	C₂×Q₈×C₆₇	1	2	yes	—
270	C₂×D₄	—	2	4	no	—
271	C₂×D₄	C₄⋊C₄×C₆₇	1	2	yes	—
272	C₂×D₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
273	C₂×D₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
279	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
282	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
283	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
284	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	4	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	4	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	4	8	no	—
292	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	8	no	—
299	C₂×C₈	—	2	8	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	4	no	—
302	C₂×C₈	—	4	8	no	—
303	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
304	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
305	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
306	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
307	C₂×C₈	#41×C₆₇	1	2	yes	—
308	C₄⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
309	C₄⋊C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
310	C₄⋊C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
311	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
312	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
313	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
314	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
315	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
316	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
317	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	—	2	2	no	—
321	C₂³	—	2	4	no	—
322	C₂³	—	4	2	no	—
323	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
324	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
325	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
326	D₄	—	2	4	no	—
327	D₄	—	2	2	no	—
328	D₄	#108×C₆₇	1	4	yes	direct product factor
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
342	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
343	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
345	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
346	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
347	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
351	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
352	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
353	C₈	—	2	4	no	—
354	C₈	—	2	2	no	—
355	C₈	—	2	4	no	—
356	C₈	—	4	4	no	—
357	C₈	—	4	2	no	—
358	C₈	C₄×D₄×C₆₇	1	4	yes	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	2	no	—
363	C₂²	—	2	2	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	4	no	—
367	C₂²	—	4	4	no	—
368	C₂²	C₂×#108×C₆₇	1	4	yes	complemented
369	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
370	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
371	C₂²	#554×C₆₇	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	C₂×#108×C₆₇	1	2	yes	complemented
378	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
379	C₄	#520×C₆₇	1	1	yes	characteristic
380	C₄	D₄²×C₆₇	1	1	yes	characteristic
381	C₄	#547×C₆₇	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	2	4	no	—
385	C₂	—	4	2	no	—
386	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
387	C₂	C₂²×#108×C₆₇	1	1	yes	characteristic
388	C₂	C₆₇×#2916	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	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₆₇	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×#108×C₆₇	C₂	semidirect
4. C₆₇×#1950	C₂	semidirect
5. C₆₇×#1855	C₂	semidirect
6. C₆₇×#1848	C₂	semidirect
7. C₆₇×#1810	C₂	semidirect
8. C₄×#108×C₆₇	C₂	semidirect
61. #108×C₆₇	D₄	direct (N × H)
133. D₄×C₆₇	#108	direct (N × H)
173. C₂²×C₆₇	C₂×#108	semidirect
182. C₄×C₆₇	C₂×#108	semidirect
187. D₄×#108	C₆₇	direct (N × H)
197. C₂²×#108	C₂×C₆₇	semidirect
198. #1950	C₂×C₆₇	semidirect
199. #1855	C₂×C₆₇	semidirect
200. #1848	C₂×C₆₇	semidirect
201. #1810	C₂×C₆₇	semidirect
202. C₄×#108	C₂×C₆₇	semidirect
203. C₆₇	D₄×#108	direct (N × H)
256. #108	D₄×C₆₇	direct (N × H)
328. D₄	#108×C₆₇	direct (N × H)
368. C₂²	C₂×#108×C₆₇	semidirect
377. C₄	C₂×#108×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
D₄×M₁₆×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂²×#108×C₆₇	2
C₆₇×#1950	2
C₆₇×#1855	2
C₆₇×#1848	2
C₆₇×#1810	2
C₄×#108×C₆₇	2
D₄×#108	67

Subgroups — D4×#108×C67 #30701918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#108×C₆₇ #30701918